Project_606

# # custAG[order(custAG$seq),] %>%spread(key = Qt,value = order_unit, fill=0)
#  custAG[order(custAG$seq),] %>%spread(key = Qt,value = order_unit, fill=0)
# Creating QUarter data over each column
# Q1 Q2 Q3 ...
# 1  2  3  ...
# 1  2  3  ...
# 1  2  3  ...

#Conbined Data
Dt_ANOVA <- custAG[,c(3,2)]
 Dt_ANOVA2017 <- Dt_ANOVA[which(Dt_ANOVA$Qt %in% c("Q1_17", "Q2_17", "Q3_17" ,"Q4_17")),] 
  boxplot(Dt_ANOVA2017$order_unit~Dt_ANOVA2017$Qt, 
        main="Boxplot comparing Qty of Quarter", 
        col= rainbow(4), 
        horizontal = TRUE)

library(Hmisc)  
  ggplot(Dt_ANOVA2017, aes(reorder(Qt,order_unit),order_unit,fill=Qt))+
# ggplot(tyre, aes(Brands,Mileage,fill=Brands))+ # if you want to leave them alphabetic
  geom_jitter(colour = "dark gray",width=.1) +
  stat_boxplot(geom ='errorbar',width = 0.4) +
  geom_boxplot()+
  labs(title="Boxplot, dotplot and SEM plot of mileage for four Quarters of Sales", 
       x = "Quarter (sorted)",
       y = "Sales",
       subtitle ="Gray dots=sample data points, Black dot=outlier, Blue dot=mean, Red=99% confidence interval",
caption = "No Major obvious difference in mean is noted") + 
    guides(fill=FALSE) +
  stat_summary(fun.data = "mean_cl_normal", colour = "red", size = 1.5, fun.args = list(conf.int=.99)) +
  stat_summary(geom="point", fun.y=mean, color="blue") +
  theme_bw()

Statistical data analysis

All_Qt_row <- custA[,-c(1)]
head(All_Qt_row)

summary(All_Qt_row)

##      Q1_17            Q1_18            Q2_17             Q2_18        
##  Min.   : 1.000   Min.   :  1.00   Min.   :  1.000   Min.   :  1.000  
##  1st Qu.: 1.000   1st Qu.:  1.00   1st Qu.:  1.000   1st Qu.:  1.000  
##  Median : 1.000   Median :  1.00   Median :  1.000   Median :  1.000  
##  Mean   : 3.494   Mean   :  4.12   Mean   :  3.331   Mean   :  3.193  
##  3rd Qu.: 1.000   3rd Qu.:  1.00   3rd Qu.:  1.000   3rd Qu.:  1.000  
##  Max.   :85.000   Max.   :156.00   Max.   :260.000   Max.   :101.000  
##      Q3_17             Q3_18            Q4_17            Q4_18       
##  Min.   :  1.000   Min.   : 1.000   Min.   : 1.000   Min.   : 1.000  
##  1st Qu.:  1.000   1st Qu.: 1.000   1st Qu.: 1.000   1st Qu.: 1.000  
##  Median :  1.000   Median : 1.000   Median : 1.000   Median : 1.000  
##  Mean   :  3.176   Mean   : 2.928   Mean   : 2.522   Mean   : 2.719  
##  3rd Qu.:  1.000   3rd Qu.: 1.000   3rd Qu.: 1.000   3rd Qu.: 1.000  
##  Max.   :160.000   Max.   :81.000   Max.   :87.000   Max.   :64.000

# Creating stacked quarter data.
# Q1
# Q2
# Q3 
All_Qt_stack <- custAG[,c(3,2)]
head(All_Qt_stack,n=10)

print(t(describe.by(All_Qt_stack)))

##            order_unit           Qt*
## vars     1.000000e+00    2.00000000
## n        8.000000e+03 8000.00000000
## mean     3.185375e+00    4.50000000
## sd       8.371232e+00    2.29143107
## median   1.000000e+00    4.50000000
## trimmed  1.222656e+00    4.50000000
## mad      0.000000e+00    2.96520000
## min      1.000000e+00    1.00000000
## max      2.600000e+02    8.00000000
## range    2.590000e+02    7.00000000
## skew     9.792949e+00    0.00000000
## kurtosis 1.801081e+02   -1.23853569
## se       9.359322e-02    0.02561898

AOV test

head(All_Qt_stack)

AOV_RESULT <- aov(order_unit~Qt,All_Qt_stack) # Qt is predictor 

class(AOV_RESULT)

## [1] "aov" "lm"

 # The names command will give you some sense of all the information contained in the list object.
names(AOV_RESULT)

##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "contrasts"     "xlevels"       "call"          "terms"        
## [13] "model"

# The summary command gives us the key ANOVA data we need and produces a classic ANOVA table
summary(AOV_RESULT)

##               Df Sum Sq Mean Sq F value   Pr(>F)    
## Qt             7   1714  244.85   3.502 0.000937 ***
## Residuals   7992 558836   69.92                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# One-way ANOVA showed a no significant effect of Quarter on Order Quantity gain (F(7, 7992) = 1.504 , , p a .001).


print( AOV_RESULT )

## Call:
##    aov(formula = order_unit ~ Qt, data = All_Qt_stack)
## 
## Terms:
##                       Qt Residuals
## Sum of Squares    1713.9  558836.1
## Deg. of Freedom        7      7992
## 
## Residual standard error: 8.362084
## Estimated effects may be unbalanced

F-test is always one sided And that we only reject the null hypothesis for very large F-values That means we’re only interested in the upper tail of the F-distribution

pf( 1.504, 7, 7992,lower.tail=FALSE)

## [1] 0.1607537

# R doesn't use the names "between-group"
# and "within-group". Instead, it tries to assign more meaningful names: in our case # "between groups" variance corresponds to the effect that the "Qt" has on the outcome variable; "within groups" variance is corresponds to the "leftover" variability, so it calls that the residuals
# There's a few different ways you could measure the effect size in an ANOVA, but the most commonly
# used measures are ??2 (eta squared) and partial ??2. For a one way analysis of variance they're identical
# to each other, so for the moment I'll just explain ??2. The definition of ??2 is actually really simple

SSb <- 3473
SSt <-   (3473+2636684)  # total sums of squares

# eta squared
# The interpretation of ??2 (eta squared)  is equally straightforward: it refers to the proportion of the variability in the
# outcome variable (Order_Quantity) that can be explained in terms of the predictor (Quarter). 
# A value of ??2 ((eta squared) = 0 
# means that there is no relationship at all between the two, 
# whereas a value of ??2 (eta squared) = 1 means that the relationship is perfect.
# eta_Sqrd <- sqrt(SSb / SSt) 
eta_Sqrd <- sqrt(SSb / SSt)  # eta-squared value
eta_Sqrd 

## [1] 0.03626916

### 

Here we can conclude that there is no strong relation between the order Qty over Quarter.

Check Condition

We have assumed 3 things; independence, homogeneity of variance (homoscedasticity) and normality for considering ANOVA result for our probelm. Lets see if our result meets these conditions.
Independence : Eacu customer is indendent and buying pattern is too.
our errors or residuals are normally distributed : Our Residual is Right Skewed as its just the promotion sales data.
The final is homogeneity of variance also known as (homoscedasticity). : Due to some big outlier the sample is not showing the plot , but we can asumet this based on small sample size .

# Plot each one by one
par(mfrow=c(2,2))
plot(AOV_RESULT)

#PLotting Residual only 
hist(AOV_RESULT$residuals)

# homoscedasticity
plot(AOV_RESULT$residuals)

print(t(psych::describe(AOV_RESULT$residuals) ))# skew kurtosis are very high , which shows that data is not fully normalised , given the sample size we will 

##                     X1
## vars      1.000000e+00
## n         8.000000e+03
## mean      5.771812e-16
## sd        8.358424e+00
## median   -2.176000e+00
## trimmed  -1.838967e+00
## mad       4.714668e-01
## min      -3.120000e+00
## max       2.566690e+02
## range     2.597890e+02
## skew      9.769740e+00
## kurtosis  1.799567e+02
## se        9.345002e-02

#using sample of size 500 to see data.



par(mfrow=c(2,2))

for (i in 1:10)
{
  AOV_RESULT_Sam <- aov(order_unit~Qt,(Dt_ANOVA[sample(nrow(Dt_ANOVA),500),]))
 plot(AOV_RESULT_Sam$residuals)
}

Running “pairwise” t-tests

It’s tempting to conclude that Q1 is better than the other Quarters. Afetr running multiple T test , it was clear that the pvalue 0.0178 < 0.05 , so we can reject null hypothesis. Suport atlernate hypothesis that there is evidence of a difference in sleas of Quarter 1 and Quarter 2.

Please Note : I have done pairewise t test using function but not sure how to iterpret the result . Including it for future study.

# How we can say which quarter is more or less in terms of data 

library(gplots)
plotmeans( formula = order_unit~Qt, # plot Order Qty by Quarter
 data = All_Qt_stack, # the data frame
 xlab = "Quarter", # x-axis label
 ylab = "Order Qty", # y-axis label
 n.label = FALSE # don't display sample size
 )

plotmeans( formula = order_unit~seq, # plot Order Qty by Quarter
 data = custAG[,c(3,4,2)], # the data frame
 xlab = "Quarter", # x-axis label
 ylab = "Order Qty", # y-axis label
 n.label = FALSE # don't display sample size,
 
 )

# it's tempting to conclude that Q1 is better than the other Quarters excpet Q2 and better than Anxifree,
# but there's no real difference between each quarter. However, if we want to get a clearer
# answer about this, it might help to run some tests.


#Running "pairwise" t-tests
# set p.adjust.method = "none" since we're not doing any adjustments. 
pairwise.t.test(x = Dt_ANOVA$order_unit,
                g = Dt_ANOVA$Qt,
                p.adjust.method = "none")

## 
##  Pairwise comparisons using t tests with pooled SD 
## 
## data:  Dt_ANOVA$order_unit and Dt_ANOVA$Qt 
## 
##       Q1_17   Q1_18   Q2_17   Q2_18   Q3_17   Q3_18   Q4_17  
## Q1_18 0.09418 -       -       -       -       -       -      
## Q2_17 0.66294 0.03490 -       -       -       -       -      
## Q2_18 0.42091 0.01320 0.71212 -       -       -       -      
## Q3_17 0.39516 0.01161 0.67853 0.96374 -       -       -      
## Q3_18 0.13019 0.00144 0.28122 0.47858 0.50724 -       -      
## Q4_17 0.00936 1.9e-05 0.03055 0.07280 0.08036 0.27766 -      
## Q4_18 0.03826 0.00018 0.10177 0.20501 0.22173 0.57626 0.59835
## 
## P value adjustment method: none

library(DATA606)

## 
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics 
## This package is designed to support this course. The text book used 
## is OpenIntro Statistics, 3rd Edition. You can read this by typing 
## vignette('os3') or visit www.OpenIntro.org. 
##  
## The getLabs() function will return a list of the labs available. 
##  
## The demo(package='DATA606') will list the demos that are available.

Dt_ANOVA2017_q12 <- Dt_ANOVA2017[which(Dt_ANOVA2017$Qt %in% c("Q1_17","Q2_17")),] 
Dt_ANOVA2017_q12$Qt <- factor(Dt_ANOVA2017_q12$Qt)
inference(y = as.numeric(Dt_ANOVA2017_q12$order_unit), 
          x = Dt_ANOVA2017_q12$Qt, est = "mean", type = "ht", null = 0, 
          alternative = "greater", method = "theoretical")

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_Q1_17 = 1000, mean_Q1_17 = 3.494, sd_Q1_17 = 7.6124
## n_Q2_17 = 1000, mean_Q2_17 = 3.331, sd_Q2_17 = 11.4136

## Observed difference between means (Q1_17-Q2_17) = 0.163
## 
## H0: mu_Q1_17 - mu_Q2_17 = 0 
## HA: mu_Q1_17 - mu_Q2_17 > 0 
## Standard error = 0.434 
## Test statistic: Z =  0.376 
## p-value =  0.3536

Dt_ANOVA2017_q23 <- Dt_ANOVA2017[which(Dt_ANOVA2017$Qt %in% c("Q2_17","Q3_17")),] 
Dt_ANOVA2017_q23$Qt <- factor(Dt_ANOVA2017_q23$Qt)
inference(y = as.numeric(Dt_ANOVA2017_q23$order_unit), 
          x = Dt_ANOVA2017_q23$Qt, est = "mean", type = "ht", null = 0, 
          alternative = "greater", method = "theoretical")

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_Q2_17 = 1000, mean_Q2_17 = 3.331, sd_Q2_17 = 11.4136
## n_Q3_17 = 1000, mean_Q3_17 = 3.176, sd_Q3_17 = 8.692

## Observed difference between means (Q2_17-Q3_17) = 0.155
## 
## H0: mu_Q2_17 - mu_Q3_17 = 0 
## HA: mu_Q2_17 - mu_Q3_17 > 0 
## Standard error = 0.454 
## Test statistic: Z =  0.342 
## p-value =  0.3663

Dt_ANOVA2017_q34 <- Dt_ANOVA2017[which(Dt_ANOVA2017$Qt %in% c("Q3_17","Q4_17")),] 
Dt_ANOVA2017_q34$Qt <- factor(Dt_ANOVA2017_q34$Qt)
inference(y = as.numeric(Dt_ANOVA2017_q34$order_unit), 
          x = Dt_ANOVA2017_q34$Qt, est = "mean", type = "ht", null = 0, 
          alternative = "greater", method = "theoretical")

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_Q3_17 = 1000, mean_Q3_17 = 3.176, sd_Q3_17 = 8.692
## n_Q4_17 = 1000, mean_Q4_17 = 2.522, sd_Q4_17 = 5.7909

## Observed difference between means (Q3_17-Q4_17) = 0.654
## 
## H0: mu_Q3_17 - mu_Q4_17 = 0 
## HA: mu_Q3_17 - mu_Q4_17 > 0 
## Standard error = 0.33 
## Test statistic: Z =  1.98 
## p-value =  0.0238

# (t.test( formula = Dt_ANOVA2017_q12$order_unit ~ Dt_ANOVA2017_q12$Qt,
#  data = Dt_ANOVA2017,
#  subset = Qt %in% c("Q1_17","Q2_17"),
#  var.equal = TRUE
#  ))

# Suppose that my post hoc analysis consists of m separate tests, and I want to ensure
# that the total probability of making any Type I errors at all is at most ??.9 If so, then the Bonferroni
# correction just says "multiply all your raw p-values by m". If we let p denote the original p-value, and
# let p1
# j be the corrected value, then the Bonferroni correction tells that:
# p1 " m ^ p
# And therefore, if you're using the Bonferroni correction, you would reject the null hypothesis if p1 a ??.
# The logic behind this correction is very straightforward. We're doing m different tests; so if we arrange
# it so that each test has a Type I error rate of at most ??{m, then the total Type I error rate across these
# tests cannot be larger than ??.
# here we have 8 Quaters data so comb(8,2) = 28 , with this method all the results were multiplied by 28 

pairwise.t.test(x = Dt_ANOVA$order_unit,
                g = Dt_ANOVA$Qt,
                p.adjust.method = "bonferroni")

## 
##  Pairwise comparisons using t tests with pooled SD 
## 
## data:  Dt_ANOVA$order_unit and Dt_ANOVA$Qt 
## 
##       Q1_17   Q1_18   Q2_17   Q2_18   Q3_17   Q3_18   Q4_17  
## Q1_18 1.00000 -       -       -       -       -       -      
## Q2_17 1.00000 0.97731 -       -       -       -       -      
## Q2_18 1.00000 0.36963 1.00000 -       -       -       -      
## Q3_17 1.00000 0.32513 1.00000 1.00000 -       -       -      
## Q3_18 1.00000 0.04034 1.00000 1.00000 1.00000 -       -      
## Q4_17 0.26213 0.00055 0.85531 1.00000 1.00000 1.00000 -      
## Q4_18 1.00000 0.00506 1.00000 1.00000 1.00000 1.00000 1.00000
## 
## P value adjustment method: bonferroni

(2) formulate an analysis plan,

We will create Sample of 1000 datapoints where records are spread in column for each quarter. we would be doing regular chi square test and then we would switch to Multiple simulation

Significance level. significance levels 0.05, or 0.10; but any value between 0 and 1 can be used.

(3) analyze sample data

custA$KUNNR_NEW <- as.character(custA$KUNNR_NEW )
summary(custA)

##   KUNNR_NEW             Q1_17            Q1_18            Q2_17        
##  Length:1000        Min.   : 1.000   Min.   :  1.00   Min.   :  1.000  
##  Class :character   1st Qu.: 1.000   1st Qu.:  1.00   1st Qu.:  1.000  
##  Mode  :character   Median : 1.000   Median :  1.00   Median :  1.000  
##                     Mean   : 3.494   Mean   :  4.12   Mean   :  3.331  
##                     3rd Qu.: 1.000   3rd Qu.:  1.00   3rd Qu.:  1.000  
##                     Max.   :85.000   Max.   :156.00   Max.   :260.000  
##      Q2_18             Q3_17             Q3_18            Q4_17       
##  Min.   :  1.000   Min.   :  1.000   Min.   : 1.000   Min.   : 1.000  
##  1st Qu.:  1.000   1st Qu.:  1.000   1st Qu.: 1.000   1st Qu.: 1.000  
##  Median :  1.000   Median :  1.000   Median : 1.000   Median : 1.000  
##  Mean   :  3.193   Mean   :  3.176   Mean   : 2.928   Mean   : 2.522  
##  3rd Qu.:  1.000   3rd Qu.:  1.000   3rd Qu.: 1.000   3rd Qu.: 1.000  
##  Max.   :101.000   Max.   :160.000   Max.   :81.000   Max.   :87.000  
##      Q4_18       
##  Min.   : 1.000  
##  1st Qu.: 1.000  
##  Median : 1.000  
##  Mean   : 2.719  
##  3rd Qu.: 1.000  
##  Max.   :64.000

head(custA)

# List test condtion for CHi-Square
# 1.  Are the value independent for each Quarter : 
# 2. 

# Ho : Customer are not baised on speding over Quarter . pq1 = pq2 = pq3 ...
# Ha : Customer are spending less or more compared to last quarter.




# 1. convert the data as a table with only Quater data .

dt <- as.table(as.matrix(custA[,]))


# Test 1 
chi_custA <-  custA[1:50,c(2,4,6,8,3,5,7,9)]  %>% 
            chisq.test(test_custA)
chi_custA

## 
##  Pearson's Chi-squared test
## 
## data:  .
## X-squared = 5112.2, df = 343, p-value < 2.2e-16

head(chi_custA$observed)

##      Q1_17 Q2_17 Q3_17 Q4_17 Q1_18 Q2_18 Q3_18 Q4_18
## [1,]     3     1     1     1     1     1     1     1
## [2,]     1     1     1     1     1     1     2     1
## [3,]     1     1     6     1     1     5     1     1
## [4,]     1    19     1     1     1     1     1     1
## [5,]     1     1     1    15     1     1     1     1
## [6,]     1     1     1     6     1     1     1     1

head(chi_custA$expected)

##          Q1_17    Q2_17    Q3_17     Q4_17    Q1_18     Q2_18    Q3_18
## [1,] 0.6643151 1.787184 1.904762 0.8877131 1.528513 0.8406820 1.069959
## [2,] 0.5978836 1.608466 1.714286 0.7989418 1.375661 0.7566138 0.962963
## [3,] 1.1293357 3.038213 3.238095 1.5091123 2.598471 1.4291593 1.818930
## [4,] 1.7272193 4.646678 4.952381 2.3080541 3.974133 2.1857731 2.781893
## [5,] 1.4614932 3.931805 4.190476 1.9529688 3.362728 1.8495003 2.353909
## [6,] 0.8636096 2.323339 2.476190 1.1540270 1.987066 1.0928865 1.390947
##         Q4_18
## [1,] 1.316872
## [2,] 1.185185
## [3,] 2.238683
## [4,] 3.423868
## [5,] 2.897119
## [6,] 1.711934

plot(chi_custA$residuals)

chi_custA <-  custA[1:50,c(2,4,6,8,3)]  %>% 
            chisq.test()
chi_custA

## 
##  Pearson's Chi-squared test
## 
## data:  .
## X-squared = 2496.8, df = 196, p-value < 2.2e-16

head(chi_custA$observed)

##      Q1_17 Q2_17 Q3_17 Q4_17 Q1_18
## [1,]     3     1     1     1     1
## [2,]     1     1     1     1     1
## [3,]     1     1     6     1     1
## [4,]     1    19     1     1     1
## [5,]     1     1     1    15     1
## [6,]     1     1     1     6     1

head(chi_custA$expected)

##          Q1_17    Q2_17   Q3_17     Q4_17    Q1_18
## [1,] 0.6866319 1.847222 1.96875 0.9175347 1.579861
## [2,] 0.4904514 1.319444 1.40625 0.6553819 1.128472
## [3,] 0.9809028 2.638889 2.81250 1.3107639 2.256944
## [4,] 2.2560764 6.069444 6.46875 3.0147569 5.190972
## [5,] 1.8637153 5.013889 5.34375 2.4904514 4.288194
## [6,] 0.9809028 2.638889 2.81250 1.3107639 2.256944

plot(chi_custA$residuals)

(gather((as.data.frame(chi_custA$residuals)),"Qt","Val")) %>% ggplot(mapping = aes(x=Qt,y= Val))+
  geom_col()

# Test 2 
chi_custA <- custA[sample(1:nrow(custA),500),c(2,4,6,8,3,5,7,9)]   %>% 
             chisq.test(simulate.p.value = TRUE)
chi_custA

## 
##  Pearson's Chi-squared test with simulated p-value (based on 2000
##  replicates)
## 
## data:  .
## X-squared = 37353, df = NA, p-value = 0.0004998

head(chi_custA$observed)

##      Q1_17 Q2_17 Q3_17 Q4_17 Q1_18 Q2_18 Q3_18 Q4_18
## [1,]     1     1     1     1     4     1     1     1
## [2,]     1     1     1     1     3     1     1     1
## [3,]     1     1     1     1     1     2     1     1
## [4,]     1     1     1     1     1     1     1     6
## [5,]     1     1    10     1     1     5     1     1
## [6,]     1     1     1     1    10     1     1     1

head(chi_custA$expected)

##         Q1_17    Q2_17    Q3_17     Q4_17    Q1_18    Q2_18    Q3_18
## [1,] 1.625146 1.396785 1.473191 1.0636853 1.801998 1.292047 1.327246
## [2,] 1.477406 1.269804 1.339265 0.9669867 1.638180 1.174588 1.206587
## [3,] 1.329665 1.142824 1.205338 0.8702880 1.474362 1.057129 1.085928
## [4,] 1.920627 1.650745 1.741044 1.2570827 2.129634 1.526965 1.568563
## [5,] 3.102552 2.666589 2.812456 2.0306720 3.440178 2.466635 2.533833
## [6,] 2.511590 2.158667 2.276750 1.6438773 2.784906 1.996800 2.051198
##          Q4_18
## [1,] 1.0199017
## [2,] 0.9271833
## [3,] 0.8344650
## [4,] 1.2053383
## [5,] 1.9470850
## [6,] 1.5762117

round(chi_custA$residuals, 3)

##         Q1_17  Q2_17  Q3_17  Q4_17  Q1_18  Q2_18  Q3_18  Q4_18
##   [1,] -0.490 -0.336 -0.390 -0.062  1.637 -0.257 -0.284 -0.020
##   [2,] -0.393 -0.239 -0.293  0.034  1.064 -0.161 -0.188  0.076
##   [3,] -0.286 -0.134 -0.187  0.139 -0.391  0.917 -0.082  0.181
##   [4,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426 -0.454  4.367
##   [5,] -1.194 -1.021  4.286 -0.723 -1.316  1.613 -0.964 -0.679
##   [6,] -0.954 -0.789 -0.846 -0.502  4.324 -0.705 -0.734 -0.459
##   [7,] -0.286 -0.134 -0.187  0.139 -0.391  0.917 -0.082  0.181
##   [8,] -0.393 -0.239 -0.293  0.034 -0.499 -0.161 -0.188  2.153
##   [9,] -2.055 -1.843 -1.917 -1.489 10.528 -1.739 -1.775 -1.437
##  [10,] -2.124 -1.909 12.602 -1.549 -2.277 -1.802 -1.839 -1.496
##  [11,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
##  [12,] -2.500 -2.264 -2.346 -1.873 -2.669 -2.148 -2.188 18.998
##  [13,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
##  [14,] -1.714 -1.520 -1.587 -1.191 -1.853  4.767 -1.456  5.825
##  [15,] -1.947 -1.741 -1.813  7.995 -2.094 -1.639 -1.674  5.049
##  [16,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
##  [17,]  5.844 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910 -0.627
##  [18,] -1.450 -1.267 -1.330 -0.955  7.143 -1.175 -1.207 -0.909
##  [19,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
##  [20,] -2.641 -2.398 -2.482 -1.994 13.770 -2.278 -2.319 -1.935
##  [21,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
##  [22,] -0.817 -0.656 -0.712  5.438 -0.930 -0.574 -0.602 -0.331
##  [23,] -0.817 -0.656 -0.712 -0.374 -0.930  4.700 -0.602 -0.331
##  [24,] -0.887 -0.724 -0.781 -0.440 -1.001 -0.641  5.088 -0.397
##  [25,] -1.079 -0.909  5.927 -0.618 -1.197 -0.825 -0.854 -0.574
##  [26,] -0.580 -0.424 -0.479  3.564 -0.689 -0.345 -0.372 -0.107
##  [27,] -1.018 -0.850 -0.909  7.018 -1.135 -0.766 -0.795 -0.518
##  [28,] -2.020  4.402  3.734 -1.458 -2.169 -1.706 -1.742  1.190
##  [29,] -0.393 -0.239 -0.293  0.034 -0.499 -0.161 -0.188  2.153
##  [30,] -1.402 -1.220 -1.283 -0.912 -1.530 -1.130  8.627 -0.866
##  [31,] -1.402 -1.220 -1.283 -0.912 -1.530  8.791 -1.161 -0.866
##  [32,] -0.954 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734  6.710
##  [33,] -0.664 -0.506  3.228 -0.229 -0.774 -0.426 -0.454 -0.187
##  [34,] -3.147  7.480 -2.968 -2.422 -3.344 -2.739 -2.785 11.963
##  [35,]  1.965 -1.520 -1.587 -1.191  5.135 -1.423 -1.456 -1.142
##  [36,]  7.145 -1.173 -1.235 -0.867 -1.479 -1.083 -1.114 -0.821
##  [37,]  5.955 -2.345  6.542 -1.946 -2.758 -2.227 -2.267 -1.888
##  [38,] -1.018 -0.850 -0.909 -0.561  4.689 -0.766 -0.795 -0.518
##  [39,] -0.743 -0.583 -0.639 -0.304 -0.854 -0.503  4.086 -0.262
##  [40,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
##  [41,] -1.910 -1.706 -1.777 -1.363 -2.056 12.306 -1.640 -1.312
##  [42,] -0.664 -0.506  2.470 -0.229 -0.089 -0.426 -0.454 -0.187
##  [43,] -1.714 -1.520 -1.587 -1.191  8.629 -1.423 -1.456 -1.142
##  [44,]  6.682  1.291 -1.665 -1.262 -1.936 -1.498 -1.038 -1.212
##  [45,]  7.592 -1.398 -1.463 -1.077 -1.721 -1.304 -0.267 -1.030
##  [46,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
##  [47,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574  4.601 -0.331
##  [48,] -1.873 -1.670 -1.740 -1.330 -2.017 -1.570 11.830 -1.280
##  [49,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
##  [50,] -1.018 -0.850 -0.909 -0.561  4.689 -0.766 -0.795 -0.518
##  [51,] -1.079 -0.909 -0.968 -0.618  5.038 -0.825 -0.854 -0.574
##  [52,] -0.393 -0.239  1.435  0.034 -0.499 -0.161 -0.188  0.076
##  [53,] -0.743 -0.583 -0.639 -0.304 -0.854  4.176 -0.530 -0.262
##  [54,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825  6.411 -0.574
##  [55,] -1.587 -1.398 -1.463 -1.077 -1.721 10.075 -1.336 -1.030
##  [56,]  7.125 -2.550 -2.637 -2.130  5.227 -2.425 -2.468 -2.069
##  [57,] -0.286 -0.134 -0.187  1.211 -0.391 -0.056 -0.082  0.181
##  [58,]  7.444 -1.220 -1.283 -0.912 -1.530 -1.130 -1.161 -0.866
##  [59,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
##  [60,] 10.493 -1.706 -1.777 -1.363 -2.056 -1.605 -1.640 -1.312
##  [61,] -0.490 -0.336 -0.390 -0.062  1.637 -0.257 -0.284 -0.020
##  [62,] -0.887  4.889 -0.781 -0.440 -1.001 -0.641 -0.670 -0.397
##  [63,] -1.137 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910  8.185
##  [64,] -1.947 -1.741 11.486 -1.395 -2.094 -1.639 -1.674 -1.344
##  [65,] -3.306 -3.023 -3.120 -2.556 17.416 -2.884 -2.931 -2.488
##  [66,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574  4.601 -0.331
##  [67,] -0.954 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734  6.710
##  [68,] -2.090 -1.876 -1.950 15.352 -2.242 -1.771 -1.807 -1.467
##  [69,]  8.820 -1.439 -1.506 -1.116 -1.766 -1.344 -1.377 -1.068
##  [70,] -0.887 -0.724 -0.781 -0.440  3.940 -0.641 -0.670 -0.397
##  [71,] -0.490 -0.336 -0.390 -0.062 -0.597 -0.257  2.320 -0.020
##  [72,] -2.055 -1.843 -1.490 -1.489 -2.206  4.185  6.768 -1.437
##  [73,] -0.817 -0.656 -0.712  5.438 -0.930 -0.574 -0.602 -0.331
##  [74,]  5.115 -0.850 -0.909 -0.561 -1.135 -0.766 -0.795 -0.518
##  [75,] -0.393 -0.239  1.435  0.034 -0.499 -0.161 -0.188  0.076
##  [76,] -1.194 -1.021 -1.081  8.399 -1.316 -0.934 -0.964 -0.679
##  [77,] -1.497 -1.312 -1.376 -0.997 -1.628 -1.219 -1.251 11.058
##  [78,] -1.873 -1.670 -1.740 -1.330 -2.017 -1.570  7.512  3.647
##  [79,] -1.497 -1.312 -1.376 -0.997 -1.628 -1.219  9.276 -0.950
##  [80,] -1.587 -1.398  6.148 -1.077 -1.721  1.947 -1.336 -1.030
##  [81,]  5.844 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910 -0.627
##  [82,] -1.714 -1.520 10.006 -1.191 -1.853 -1.423 -1.456 -1.142
##  [83,] -0.490 -0.336 -0.390 -0.062  1.637 -0.257 -0.284 -0.020
##  [84,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
##  [85,]  0.581 -0.134 -0.187  0.139 -0.391 -0.056 -0.082  0.181
##  [86,] -1.673 -1.480 -1.547  1.734 -1.810 -0.336  6.856 -1.106
##  [87,] -1.714 10.386 -1.587 -1.191 -1.853 -1.423 -1.456 -1.142
##  [88,] -1.194 -1.021 -1.081  2.084 -1.316 -0.934  4.690 -0.679
##  [89,] -1.137 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910  8.185
##  [90,] -1.137  4.681  0.807 -0.672 -1.258 -0.880 -0.910 -0.627
##  [91,] -1.301 -1.124 -1.185 -0.821 -1.426  3.224  3.737 -0.776
##  [92,] -0.664 -0.506 -0.562 -0.229 -0.774  3.620 -0.454 -0.187
##  [93,] -1.910 11.673 -1.777 -1.363 -2.056 -1.605 -1.640 -1.312
##  [94,] -1.248  7.303 -1.134 -0.773 -1.372 -0.985 -1.015 -0.728
##  [95,] -1.795 -1.597 -1.665 13.077 -1.936 -1.498 -1.532 -1.212
##  [96,] -2.256 -2.034 -2.110 16.631 -2.414 -1.924 -1.961 -1.608
##  [97,] -1.630 -1.439 -1.506 -1.116 -1.766 -1.344 10.186 -1.068
##  [98,]  8.432  1.867 -2.016 -1.578 -2.312 -1.833 -1.870 -1.525
##  [99,] -1.497 -1.312 -1.376 10.762 -1.628 -1.219 -1.251 -0.950
## [100,] -3.052 -2.785 -2.877 19.422 -3.245 -2.653  0.274 -2.278
## [101,] -1.630 -1.439 -1.506 -1.116  8.158 -1.344 -1.377 -1.068
## [102,]  3.015 -0.724  0.586 -0.440 -1.001 -0.641 -0.670 -0.397
## [103,]  4.725 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734 -0.459
## [104,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
## [105,] -1.301 -1.124  1.664 -0.821 -1.426  5.049 -1.066 -0.776
## [106,]  5.491 -1.559  2.654 -1.227 -1.895 -1.461 -1.494 -1.178
## [107,] -0.664 -0.506  3.228 -0.229 -0.774 -0.426 -0.454 -0.187
## [108,] -1.587 -1.398 -1.463 -1.077  7.914 -1.304 -1.336 -1.030
## [109,]  5.513 -1.312 -1.376  2.097 -1.628 -1.219 -1.251 -0.950
## [110,] -1.673 -1.480  9.741 -1.154 -1.810 -1.384 -1.417 -1.106
## [111,] -1.450 -1.267 -1.330 -0.955  7.143 -1.175 -1.207 -0.909
## [112,] -0.954 -0.789 -0.846 -0.502  4.324 -0.705 -0.734 -0.459
## [113,]  9.806 -1.597 -1.665 -1.262 -1.936 -1.498 -1.532 -1.212
## [114,] -1.137  6.564 -1.026 -0.672 -1.258 -0.880 -0.910 -0.627
## [115,] -1.587  7.982 -1.463 -1.077 -1.721 -1.304 -1.336  0.800
## [116,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [117,] -2.441 -2.209 14.602 -1.822 -2.607 -2.094 -2.133 -1.766
## [118,] -0.954  5.337 -0.846 -0.502 -1.070 -0.705 -0.734 -0.459
## [119,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [120,] -1.137 -0.966 -1.026 -0.672 -1.258  6.949 -0.910 -0.627
## [121,] -1.673  8.096 -1.547 -1.154 -0.035 -1.384 -1.417 -1.106
## [122,] -1.542 -1.355 -1.420 -1.038 -1.675 -1.262  9.587 -0.991
## [123,] -1.137 -0.966 -1.026 -0.672 -1.258  6.949 -0.910 -0.627
## [124,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602  5.604
## [125,] -1.910 -1.706 11.251 -1.363 -2.056 -1.605 -1.640 -1.312
## [126,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [127,] -0.490  2.203 -0.390 -0.062 -0.597 -0.257 -0.284 -0.020
## [128,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602  5.604
## [129,] -0.817 -0.656 -0.712 -0.374 -0.930  4.700 -0.602 -0.331
## [130,]  5.487 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [131,]  6.187 -1.021 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [132,] -0.286 -0.134 -0.187  0.139  0.433 -0.056 -0.082  0.181
## [133,] -2.319 -2.094 -2.171 12.983 -2.480 -1.982  1.681 -1.662
## [134,] -1.873 -1.670 11.012 -1.330 -2.017 -1.570 -1.604 -1.280
## [135,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [136,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [137,] -1.947 -1.741 -1.813  1.735 -2.094  9.721 -1.674 -1.344
## [138,] -1.542  9.252 -1.420 -1.038 -1.675 -1.262 -1.294 -0.991
## [139,] -1.018 -0.850 -0.909 -0.561  3.524 -0.766  0.562 -0.518
## [140,] -1.587 -1.398 -1.463 -1.077 -1.721 -1.304  9.890 -1.030
## [141,] -0.490 -0.336 -0.390 -0.062  1.637 -0.257 -0.284 -0.020
## [142,] -1.542 -1.355  8.908 -1.038 -1.675 -1.262 -1.294 -0.991
## [143,] -1.947 11.916 -1.813 -1.395 -2.094 -1.639 -1.674 -1.344
## [144,] -2.191 -1.972 -2.048 -1.607 -2.347 14.229 -1.901 -1.553
## [145,] -4.065 25.634 -3.847 -3.189 -4.304 -3.571 -3.626 -3.111
## [146,] -2.191 -1.972 -2.048 -1.607 -2.347 14.229 -1.901 -1.553
## [147,]  2.157  7.367 -2.016 -0.123 -2.312 -1.833 -1.870 -1.525
## [148,] -1.497 -1.312 -1.376 10.762 -1.628 -1.219 -1.251 -0.950
## [149,]  5.487 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [150,] -1.301 -1.124 -1.185 -0.821 -1.426 -1.035 -1.066  9.496
## [151,] -0.743  3.917 -0.639 -0.304 -0.854 -0.503 -0.530 -0.262
## [152,] -0.580 -0.424 -0.479 -0.149 -0.689  3.024 -0.372 -0.107
## [153,]  6.517 -1.073 -1.134 -0.773 -1.372 -0.985 -1.015 -0.728
## [154,] -1.352 -1.173 -1.235 -0.867 -1.479  3.086  4.175 -0.821
## [155,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [156,] -0.393 -0.239 -0.293  2.067 -0.499 -0.161 -0.188  0.076
## [157,] -1.194 -1.021 -1.081 -0.723 -1.316 -0.934 -0.964  8.638
## [158,] -1.402 -1.220 -1.283 -0.912 -1.530  8.791 -1.161 -0.866
## [159,]  0.581 -0.134 -0.187  0.139 -0.391 -0.056 -0.082  0.181
## [160,]  9.074 -1.480 -1.547 -1.154 -1.810 -1.384 -1.417 -1.106
## [161,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
## [162,] -1.301  7.653 -1.185 -0.821 -1.426 -1.035 -1.066 -0.776
## [163,] -3.479 -3.185 16.618 -2.701 -3.691 -3.041 -1.023  0.402
## [164,] -1.248  7.303 -1.134 -0.773 -1.372 -0.985 -1.015 -0.728
## [165,] -2.124 -1.909  5.934 -1.549 -2.277 -1.802  5.186 -1.496
## [166,] -2.319  6.726  5.246 -1.717 -2.480 -1.982 -2.020 -1.662
## [167,] -0.817 -0.656 -0.712 -0.374 -0.930  4.700 -0.602 -0.331
## [168,]  5.844 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910 -0.627
## [169,] -1.755 -1.559 -1.627 -1.227  4.986  3.110 -1.494 -1.178
## [170,] -1.137 -0.966 -1.026 -0.672 -1.258 -0.880  6.815 -0.627
## [171,]  6.517 -1.073 -1.134 -0.773 -1.372 -0.985 -1.015 -0.728
## [172,] 11.353 -1.843 -1.917 -1.489 -2.206 -1.739 -1.775 -1.437
## [173,] -0.954 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734  6.710
## [174,] -2.223  4.203 -0.870 -1.635  4.905 -1.894 -1.931 -1.581
## [175,]  4.725 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734 -0.459
## [176,] -0.743 -0.583 -0.639 -0.304 -0.854  4.176 -0.530 -0.262
## [177,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985  7.577 -0.728
## [178,] -1.984 -1.776 -1.848 -1.427 -2.132 -1.673 12.582 -1.376
## [179,] -1.542 -1.355 -1.420 -1.038 -1.675 -1.262 -1.294 11.422
## [180,] -1.587  3.813 -1.463 -1.077 -1.721  4.656 -1.336 -1.030
## [181,] -1.248 -1.073 -1.134  8.826 -1.372 -0.985 -1.015 -0.728
## [182,] -1.194  6.940 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [183,] -0.393 -0.239  1.435  0.034 -0.499 -0.161 -0.188  0.076
## [184,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [185,] -1.079 -0.909 -0.968 -0.618 -1.197  6.539 -0.854 -0.574
## [186,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [187,] -0.887 -0.724 -0.781 -0.440 -1.001  5.194 -0.670 -0.397
## [188,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [189,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985  7.577 -0.728
## [190,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [191,] -1.834 -1.634 -1.703 -1.296 -1.977 11.782 -1.568 -1.246
## [192,] -0.580  2.816 -0.479 -0.149 -0.689 -0.345 -0.372 -0.107
## [193,] -0.393 -0.239 -0.293  0.034  1.064 -0.161 -0.188  0.076
## [194,]  8.017 -1.312 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [195,] -0.580  2.816 -0.479 -0.149 -0.689 -0.345 -0.372 -0.107
## [196,] -0.393 -0.239 -0.293  0.034 -0.499  1.684 -0.188  0.076
## [197,] -2.020 -1.810 11.943 -1.458 -2.169 -1.706 -1.742 -1.407
## [198,] -1.450 -1.267 -1.330 10.397 -1.579 -1.175 -1.207 -0.909
## [199,] -1.079 -0.909 -0.341  6.760 -1.197 -0.825 -0.854 -0.574
## [200,] -2.319 -2.094 -2.171 -1.717 -2.480 15.108 -2.020 -1.662
## [201,] -0.887 -0.724 -0.781 -0.440 -1.001 -0.641  5.088 -0.397
## [202,] -0.490 -0.336 -0.390 -0.062 -0.597 -0.257  2.320 -0.020
## [203,] -2.055 -1.843 -1.917 15.084 -2.206 -1.739 -1.775 -1.437
## [204,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985  7.577 -0.728
## [205,] -0.664  3.385 -0.562 -0.229 -0.774 -0.426 -0.454 -0.187
## [206,] -1.194 -1.021  6.671 -0.723 -1.316 -0.934 -0.964 -0.679
## [207,] -0.664 -0.506  3.228 -0.229 -0.774 -0.426 -0.454 -0.187
## [208,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985  7.577 -0.728
## [209,] -0.743 -0.583 -0.639  4.853 -0.854 -0.503 -0.530 -0.262
## [210,] -2.641 -2.398 -2.482 -1.994  9.623  1.112 -0.832 -1.935
## [211,]  9.323 -1.520 -1.587 -1.191 -1.853 -1.423 -1.456 -1.142
## [212,] 12.545 -2.034 -2.110 -1.663 -2.414 -1.924 -1.961 -1.608
## [213,] -0.817 -0.656  4.227 -0.374 -0.930 -0.574 -0.602 -0.331
## [214,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [215,] -1.352 -1.173 -1.235 -0.867 -1.479 -1.083  8.288 -0.821
## [216,] -2.124 -1.909 -1.983 12.673 -0.017 -1.802 -1.839 -1.496
## [217,] -0.817 -0.656 -0.712  0.456  2.898 -0.574 -0.602 -0.331
## [218,] -1.450 -1.267 -1.330 -0.955  7.143 -1.175 -1.207 -0.909
## [219,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [220,] -0.887 -0.724 -0.781 -0.440  3.940 -0.641 -0.670 -0.397
## [221,] -0.887 -0.724 -0.781 -0.440 -1.001 -0.641  5.088 -0.397
## [222,] -1.248 -1.073 -1.134 -0.773  3.896 -0.985  1.440 -0.728
## [223,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825  6.411 -0.574
## [224,] -2.223 -2.003 -0.870 -1.635 -2.381 -1.894  0.615 12.461
## [225,] -0.286 -0.134 -0.187  0.139  0.433 -0.056 -0.082  0.181
## [226,]  8.661 -1.559 -1.627 -1.227 -1.895 -1.461 -0.492 -1.178
## [227,] -1.497 -1.312 -1.376 -0.997 -1.628 -1.219 -1.251 11.058
## [228,] -1.194 -1.021 -1.081  8.399 -1.316 -0.934 -0.964 -0.679
## [229,]  3.057 -2.034 -2.110 -1.663  6.595 -1.924 -1.961 -1.608
## [230,] -0.580 -0.424 -0.479 -0.149 -0.689 -0.345  2.952 -0.107
## [231,] -0.743 -0.583 -0.639 -0.304 -0.854 -0.503  4.086 -0.262
## [232,] -1.079  6.172 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [233,]  2.170 -1.439 -1.506  7.104 -1.766 -1.344 -1.377 -1.068
## [234,] -0.954 -0.789  5.118 -0.502 -1.070 -0.705 -0.734 -0.459
## [235,] -2.287 -2.064 -2.141 -1.690 -2.448 14.893 -1.991 -1.636
## [236,] -1.352 -1.173 -1.235 -0.867 -1.479 -1.083 -1.114  9.904
## [237,] -3.004 -2.739 -2.830 -2.301 -3.195 14.350  2.701 -2.238
## [238,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [239,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574  4.601 -0.331
## [240,] -1.079  6.172 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [241,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [242,]  9.074 -1.480 -1.547 -1.154 -1.810 -1.384 -1.417 -1.106
## [243,] -0.580 -0.424 -0.479 -0.149 -0.689  3.024 -0.372 -0.107
## [244,] -2.256 -2.034 -2.110 -1.663 -2.414 14.675 -1.961 -1.608
## [245,] -1.587 -1.398 -1.463 -1.077 -1.721 -1.304 -1.336 11.777
## [246,] -1.497 -1.312 -1.376 -0.997  7.407 -1.219 -1.251 -0.950
## [247,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [248,]  7.145 -1.173 -1.235 -0.867 -1.479 -1.083 -1.114 -0.821
## [249,]  5.844 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910 -0.627
## [250,] -3.803 -3.488 15.078 -2.971  1.833 -1.672 -3.386 -2.896
## [251,] -1.248 -1.073  7.022 -0.773 -1.372 -0.985 -1.015 -0.728
## [252,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [253,] -0.490 -0.336 -0.390 -0.062  1.637 -0.257 -0.284 -0.020
## [254,] -1.497  8.950 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [255,] -1.402 -1.220 -1.283 10.022 -1.530 -1.130 -1.161 -0.866
## [256,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426 -0.454  4.367
## [257,] -2.500 -2.264 -2.346 -1.873  1.662 -0.181  9.069 -1.815
## [258,] -2.381 -2.152 -2.231 12.639 -0.469 -2.039 -2.078  0.585
## [259,] -1.194 -1.021 -1.081 -0.723 -1.316  7.344 -0.964 -0.679
## [260,] -1.248 -1.073 -1.134 -0.773  6.003 -0.985 -1.015 -0.728
## [261,] -1.497 -1.312  8.616 -0.997 -1.628 -1.219 -1.251 -0.950
## [262,] 10.039 -1.634 -1.703 -1.296 -1.977 -1.534 -1.568 -1.246
## [263,] 10.493 -1.706 -1.777 -1.363 -2.056 -1.605 -1.640 -1.312
## [264,] -0.954 -0.789 -0.846 -0.502 -1.070  5.664 -0.734 -0.459
## [265,] -0.887  4.889 -0.781 -0.440 -1.001 -0.641 -0.670 -0.397
## [266,] -1.947 -1.741 -1.813 -1.395  9.930 -1.639 -1.674 -1.344
## [267,] -1.630  9.832 -1.506 -1.116 -1.766 -1.344 -1.377 -1.068
## [268,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985  7.577 -0.728
## [269,] -0.490  1.357 -0.390 -0.062 -0.597 -0.257 -0.284  0.970
## [270,] -0.490 -0.336 -0.390  2.847 -0.597 -0.257 -0.284 -0.020
## [271,] -1.402 -1.220 -1.283 -0.912 -1.530 -1.130  8.627 -0.866
## [272,] -0.817 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602  5.604
## [273,] -1.018 -0.850  5.532 -0.561 -1.135 -0.766 -0.795 -0.518
## [274,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
## [275,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [276,] -0.954 -0.789 -0.846 -0.502  4.324 -0.705 -0.734 -0.459
## [277,] -0.393  1.535 -0.293  0.034 -0.499 -0.161 -0.188  0.076
## [278,]  6.836 -1.124 -1.185 -0.821 -1.426 -1.035 -1.066 -0.776
## [279,] -1.673 -1.480 -1.547 12.130 -1.810 -1.384 -1.417 -1.106
## [280,] -1.673 -1.480 -1.547 -1.154 -1.810 -1.384 10.475 -1.106
## [281,] -0.817  2.967 -0.712  1.287 -0.930 -0.574 -0.602 -0.331
## [282,] -1.834 -1.634 -1.703 -1.296  9.299 -1.534 -1.568 -1.246
## [283,] -1.630 -1.439 -1.506 11.801 -1.766 -1.344 -1.377 -1.068
## [284,]  8.404 -1.520 -0.621 -1.191 -1.853 -1.423 -1.456 -1.142
## [285,] -0.393 -0.239 -0.293  0.034 -0.499 -0.161 -0.188  2.153
## [286,]  4.316 -0.724 -0.781 -0.440 -1.001 -0.641 -0.670 -0.397
## [287,]  7.444 -1.220 -1.283 -0.912 -1.530 -1.130 -1.161 -0.866
## [288,] -1.450 -1.267 -1.330 -0.955 -1.579 -1.175  8.956 -0.909
## [289,] -0.393 -0.239 -0.293  0.034 -0.499 -0.161  1.633  0.076
## [290,] -3.328 -3.044 -3.142 -2.574 -3.533 -2.904 21.599 -2.506
## [291,]  0.430 -0.239  0.571  0.034 -0.499 -0.161 -0.188  0.076
## [292,] -0.286 -0.134 -0.187  0.139 -0.391  0.917 -0.082  0.181
## [293,]  4.334 -1.876 -1.950 -1.519 -2.242 -1.771  6.189 -1.467
## [294,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426  3.538 -0.187
## [295,] -1.194 -1.021 -1.081 -0.723  5.693 -0.934 -0.964 -0.679
## [296,]  9.074 -1.480 -1.547 -1.154 -1.810 -1.384 -1.417 -1.106
## [297,] -1.542 -1.355 -1.420 -1.038 -1.675 -1.262 -1.294 11.422
## [298,] -0.743 -0.583 -0.639 -0.304  3.108 -0.503 -0.530 -0.262
## [299,] -0.664 -0.506  3.228 -0.229 -0.774 -0.426 -0.454 -0.187
## [300,] -0.490 -0.336  2.082 -0.062 -0.597 -0.257 -0.284 -0.020
## [301,] -0.743 -0.583 -0.639 -0.304  3.108 -0.503 -0.530 -0.262
## [302,] -2.696  8.243 -2.535 -2.040  1.833 -2.328  0.921 -1.981
## [303,] -0.817 -0.656 -0.712  5.438 -0.930 -0.574 -0.602 -0.331
## [304,] -1.194 -1.021 -1.081 -0.723 -1.316 -0.934  2.806  4.338
## [305,] -0.743 -0.583 -0.639 -0.304  3.108 -0.503 -0.530 -0.262
## [306,] -1.137 -0.966 -1.026 -0.672 -1.258  6.949 -0.910 -0.627
## [307,]  3.429 -0.583 -0.639 -0.304 -0.854 -0.503 -0.530 -0.262
## [308,]  8.291 -1.355 -1.420 -1.038 -1.675 -1.262 -1.294 -0.991
## [309,] -0.817 -0.656 -0.712  5.438 -0.930 -0.574 -0.602 -0.331
## [310,] -0.954 -0.789  5.118 -0.502 -1.070 -0.705 -0.734 -0.459
## [311,] -0.286 -0.134 -0.187  0.139 -0.391  0.917 -0.082  0.181
## [312,] -1.137 -0.966  6.307 -0.672 -1.258 -0.880 -0.910 -0.627
## [313,] -2.557 -2.319 -2.401 -1.922 -2.729  9.393 -2.241  6.401
## [314,] -0.954 -0.789 -0.846  6.517 -1.070 -0.705 -0.734 -0.459
## [315,] -1.834 -1.634  0.606 -1.296  3.035 -1.534  3.298 -1.246
## [316,] -1.450  8.640 -1.330 -0.955 -1.579 -1.175 -1.207 -0.909
## [317,] -1.910 11.673 -1.777 -1.363 -2.056 -1.605 -1.640 -1.312
## [318,] -1.630 -0.415  3.483 -1.116 -1.766 -1.344  2.828  0.131
## [319,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [320,] -0.580  2.006 -0.479 -0.149  0.024 -0.345 -0.372 -0.107
## [321,] -4.766 -4.388 29.103 -3.770 -5.039 -4.203 -4.266 -3.681
## [322,] -2.381 -2.152 -2.231 -1.770 12.332 -2.039 -2.078 -1.715
## [323,] -0.393 -0.239 -0.293  0.034 -0.499  1.684 -0.188  0.076
## [324,] -0.580 -0.424 -0.479 -0.149 -0.689  3.024 -0.372 -0.107
## [325,] -0.580 -0.424  2.676 -0.149 -0.689 -0.345 -0.372 -0.107
## [326,] -1.402 -1.220  8.007 -0.912 -1.530 -1.130 -1.161 -0.866
## [327,] -2.157 -1.941 12.815 -1.578 -2.312 -1.833 -1.870 -1.525
## [328,] -1.079 -0.909 -0.968  7.498 -1.197 -0.825 -0.854 -0.574
## [329,] -1.910 -1.706 -1.777 -1.363 -2.056 -1.605 -1.640 14.345
## [330,] -0.490 -0.336 -0.390  2.847 -0.597 -0.257 -0.284 -0.020
## [331,] -2.529 -2.292 -2.373 -1.897 -2.699 -2.175 16.251 -1.840
## [332,] -2.020  0.852 -1.882 -1.458  4.472  2.446 -1.742 -1.407
## [333,] -0.286 -0.134  0.724  0.139 -0.391 -0.056 -0.082  0.181
## [334,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [335,] -1.137 -0.966  3.251 -0.672 -1.258 -0.880 -0.910  3.044
## [336,] -2.124 -1.909 -1.983 -1.549 -0.017 -1.802 10.893 -1.496
## [337,] -2.090 12.846 -1.950 -1.519 -2.242 -1.771 -1.807 -1.467
## [338,] -1.352 -1.173 -1.235 -0.867  6.591 -1.083 -1.114 -0.821
## [339,] -1.194  6.940 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [340,] -2.441 -2.209 -2.289 -1.822 12.665 -2.094 -2.133 -1.766
## [341,] -2.470  9.602 -2.317 -1.847 -2.638 -2.121  3.716 -1.791
## [342,] -0.393 -0.239 -0.293  0.034  1.064 -0.161 -0.188  0.076
## [343,] -1.947 -1.741 -1.813  2.256  7.124 -1.639 -1.674 -1.344
## [344,] -1.497 -1.312 -1.376 -0.997 -1.628 -1.219 -1.251 11.058
## [345,] -1.587  9.546 -1.463 -1.077 -1.721 -1.304 -1.336 -1.030
## [346,] -0.393 -0.239 -0.293  0.034 -0.499  1.684 -0.188  0.076
## [347,] -1.248 -1.073  7.022 -0.773 -1.372 -0.985 -1.015 -0.728
## [348,] -1.630  9.832 -1.506 -1.116 -1.766 -1.344 -1.377 -1.068
## [349,] -2.557 -2.319 -2.401  2.338 10.034 -2.201 -2.241 -1.864
## [350,] -2.614 16.261 -2.455 -1.970 -2.787 -2.253 -2.293 -1.911
## [351,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426  3.538 -0.187
## [352,] -0.286  0.802 -0.187  0.139 -0.391 -0.056 -0.082  0.181
## [353,] -0.393 -0.239 -0.293  0.034 -0.499 -0.161  1.633  0.076
## [354,] -0.887 -0.724 -0.781 -0.440  3.940 -0.641 -0.670 -0.397
## [355,] -1.450 -1.267 -1.330 10.397 -1.579 -1.175 -1.207 -0.909
## [356,] -1.352 -1.173  7.689 -0.867 -1.479 -1.083 -1.114 -0.821
## [357,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854  7.714
## [358,] -1.262 -1.819 15.616 -2.342 -3.245 -2.653 -2.698 -2.278
## [359,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
## [360,]  9.806 -1.597 -1.665 -1.262 -1.936 -1.498 -1.532 -1.212
## [361,] -1.352 -1.173 -1.235 -0.867 -1.479  8.446 -1.114 -0.821
## [362,] -1.714  6.914 -1.587 -1.191 -1.853  2.188 -1.456 -1.142
## [363,] -0.664 -0.506 -0.562  4.230 -0.774 -0.426 -0.454 -0.187
## [364,] -0.954 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734  6.710
## [365,] -0.490 -0.336 -0.390 -0.062 -0.597 -0.257  2.320 -0.020
## [366,]  6.187 -1.021 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [367,] -1.194 -1.021 -1.081 -0.723 -1.316  7.344 -0.964 -0.679
## [368,] -0.286  0.802 -0.187  0.139 -0.391 -0.056 -0.082  0.181
## [369,] -0.393 -0.239  1.435  0.034 -0.499 -0.161 -0.188  0.076
## [370,] -1.673 -1.480 -1.547 -1.154 -1.810 -1.384 10.475 -1.106
## [371,] -0.664 -0.506 -0.562 -0.229 -0.774  3.620 -0.454 -0.187
## [372,] -2.287 -2.064  3.381 10.378 -2.448 -1.953 -1.991 -1.636
## [373,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854  7.714
## [374,] -1.450 -0.166 -1.330 -0.955 -1.579  4.547  2.181 -0.909
## [375,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [376,] -1.542 -1.355 -1.420 -1.038 -1.675 -1.262  9.587 -0.991
## [377,] -0.286 -0.134  0.724  0.139 -0.391 -0.056 -0.082  0.181
## [378,] -0.743 -0.583 -0.639  4.853 -0.854 -0.503 -0.530 -0.262
## [379,] -3.004 -2.739 -2.830 -2.301  2.263 13.333 -2.653 -2.238
## [380,] -1.673 10.113 -1.547 -1.154 -1.810 -1.384 -1.417 -1.106
## [381,] -2.223 -2.003 13.231 -1.635 -2.381 -1.894 -1.931 -1.581
## [382,] -1.714 -1.520  3.726 -1.191  3.825 -1.423 -1.456 -1.142
## [383,] -2.641 -2.398 -2.482 -1.994 -2.816 -2.278 17.007 -1.935
## [384,]  0.581 -0.134 -0.187  0.139 -0.391 -0.056 -0.082  0.181
## [385,] -0.286 -0.134  0.724  0.139 -0.391 -0.056 -0.082  0.181
## [386,] -1.834 -1.634 -1.703 -1.296  9.299 -1.534 -1.568 -1.246
## [387,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [388,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
## [389,] -1.497 -1.312 -1.376  2.716 -1.628 -1.219  5.952 -0.950
## [390,] -0.664 -0.506  3.228 -0.229 -0.774 -0.426 -0.454 -0.187
## [391,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
## [392,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426  3.538 -0.187
## [393,] -0.580 -0.424 -0.479 -0.149 -0.689 -0.345 -0.372  3.685
## [394,] 10.493 -1.706 -1.777 -1.363 -2.056 -1.605 -1.640 -1.312
## [395,]  2.424 -0.424 -0.479 -0.149 -0.689 -0.345 -0.372 -0.107
## [396,] -0.817  4.417 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [397,]  8.669 -1.972  1.619 -1.607 -2.347 -1.864 -1.901 -1.553
## [398,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [399,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [400,] -1.873 -1.670 -1.740 -1.330 -2.017 -1.570 -1.604 14.046
## [401,] -4.462 -4.104 -4.227 -3.519 23.712 -3.930 -3.989 -3.434
## [402,] 10.454 -0.784  0.100 -2.108 -2.957 -2.401 -2.443 -0.826
## [403,] -1.984 -1.776 -1.848 -1.427 10.132 -1.673 -1.708 -1.376
## [404,] -1.587 -1.398 -1.463 -1.077  6.079  0.864 -1.336 -1.030
## [405,]  2.353 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910  3.779
## [406,] -2.723 -2.475 -2.560 -2.063 -2.901 17.866 -2.394 -2.003
## [407,] -1.079  6.172 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [408,] -0.010  8.991 -1.665 -1.262 -1.936 -1.498 -1.532 -1.212
## [409,] -1.301 -1.124 -1.185 -0.821  6.302 -1.035 -1.066 -0.776
## [410,] -0.286 -0.134 -0.187  1.211 -0.391 -0.056 -0.082  0.181
## [411,] -0.286 -0.134 -0.187  1.211 -0.391 -0.056 -0.082  0.181
## [412,] -1.402 -1.220 -1.283 -0.912  6.871 -1.130 -1.161 -0.866
## [413,] -1.194 -1.021  5.478 -0.723 -0.237 -0.934 -0.964 -0.679
## [414,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [415,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [416,] -1.714 -1.520 -1.587 -1.191 -1.853 -1.423 -1.456 12.791
## [417,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825  6.411 -0.574
## [418,] -1.402 -1.220 -1.283 -0.912 -1.530 -1.130  8.627 -0.866
## [419,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
## [420,]  7.303 -1.559 -1.627 -1.227  0.255 -1.461 -1.494 -1.178
## [421,]  0.581 -0.134 -0.187  0.139 -0.391 -0.056 -0.082  0.181
## [422,] -1.497  8.950 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [423,] -1.402 -1.220  8.007 -0.912 -1.530 -1.130 -1.161 -0.866
## [424,] -1.194 -1.021 -1.081 -0.723 -1.316  7.344 -0.964 -0.679
## [425,] -0.664 -0.506 -0.562  4.230 -0.774 -0.426 -0.454 -0.187
## [426,] -0.490  2.203 -0.390 -0.062 -0.597 -0.257 -0.284 -0.020
## [427,]  8.186  6.721 -2.830 -2.301 -3.195 -2.609 -2.653 -2.238
## [428,] -0.954 -0.789 -0.846 -0.502 -1.070 -0.705 -0.734  6.710
## [429,]  1.863 -0.336 -0.390 -0.062 -0.597 -0.257 -0.284 -0.020
## [430,] -0.817 -0.656 -0.712 -0.374 -0.930  4.700 -0.602 -0.331
## [431,] -0.393 -0.239 -0.293  2.067 -0.499 -0.161 -0.188  0.076
## [432,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854  7.714
## [433,] -1.137 -0.966 -1.026 -0.672 -1.258 -0.880 -0.910  8.185
## [434,] -1.587  9.546 -1.463 -1.077 -1.721 -1.304 -1.336 -1.030
## [435,] -1.450 -1.267 -1.330  7.874 -1.579 -1.175 -1.207  1.668
## [436,] -1.714 -1.520 -1.587 -1.191 -1.853 10.956 -1.456 -1.142
## [437,] -1.079 -0.909 -0.968 -0.618  5.038 -0.825 -0.854 -0.574
## [438,]  3.885 -0.656 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [439,] -0.743 -0.583 -0.639 -0.304 -0.854  4.176 -0.530 -0.262
## [440,] -0.580 -0.424 -0.479 -0.149  2.164 -0.345 -0.372 -0.107
## [441,]  6.187 -1.021 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [442,] -0.954 -0.789 -0.846 -0.502  4.324 -0.705 -0.734 -0.459
## [443,]  9.323 -1.520 -1.587 -1.191 -1.853 -1.423 -1.456 -1.142
## [444,]  6.836 -1.124 -1.185 -0.821 -1.426 -1.035 -1.066 -0.776
## [445,] -1.301 -1.124 -1.185 -0.821 -1.426  8.091 -1.066 -0.776
## [446,] -3.350 -3.064 -3.163 -2.593 17.657 -2.924 -2.972 -2.525
## [447,] -1.301 -1.124  1.664 -0.821 -1.426  5.049 -1.066 -0.776
## [448,]  4.316 -0.724 -0.781 -0.440 -1.001 -0.641 -0.670 -0.397
## [449,] -0.664 -0.506 -0.562 -0.229 -0.774  3.620 -0.454 -0.187
## [450,] -0.490  2.203 -0.390 -0.062 -0.597 -0.257 -0.284 -0.020
## [451,] -0.490 -0.336  2.082 -0.062 -0.597 -0.257 -0.284 -0.020
## [452,] 13.988 -2.264 -2.346 -1.873 -2.669 -2.148 -2.188 -1.815
## [453,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [454,] -0.457 -1.597 -1.665 -1.262  3.996  3.006 -1.532 -1.212
## [455,] -2.500  7.574 -2.346 -1.873 -2.669  6.114 -2.188 -1.815
## [456,] -1.018 -0.850 -0.909  7.018 -1.135 -0.766 -0.795 -0.518
## [457,] -0.286 -0.134 -0.187  0.139 -0.391 -0.056  0.877  0.181
## [458,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [459,] -0.887 -0.724 -0.781 -0.440 -1.001 -0.641  5.088 -0.397
## [460,] -1.248 -1.073 -1.134  8.826 -1.372 -0.985 -1.015 -0.728
## [461,]  0.008 -2.237 -2.317 -1.847 10.475 -2.121 -2.161 -1.791
## [462,] -1.079 -0.909 -0.968 -0.618 -1.197 -0.825 -0.854  7.714
## [463,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [464,] -1.018 -0.850 -0.909 -0.561 -1.135  6.111 -0.795 -0.518
## [465,]  8.017 -1.312 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [466,] -0.490 -0.336  1.258 -0.062 -0.597 -0.257  0.584 -0.020
## [467,] -1.079  6.172 -0.968 -0.618 -1.197 -0.825 -0.854 -0.574
## [468,] -1.587 -1.398 -1.463 11.463 -1.721 -1.304 -1.336 -1.030
## [469,] -1.497  8.950 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [470,] -1.714 -1.520 -1.587 -1.191  8.629 -1.423 -1.456 -1.142
## [471,]  7.444 -1.220 -1.283 -0.912 -1.530 -1.130 -1.161 -0.866
## [472,] -2.191 -1.972 -2.048 -1.607 -2.347 14.229 -1.901 -1.553
## [473,] -1.497  8.950 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [474,] -0.286 -0.134 -0.187  0.139  0.433 -0.056 -0.082  0.181
## [475,]  6.836 -1.124 -1.185 -0.821 -1.426 -1.035 -1.066 -0.776
## [476,] -2.669 -2.424 -2.508 -2.017 -2.845 -2.303 17.191 -1.958
## [477,] -0.664 -0.506 -0.562 -0.229  2.652 -0.426 -0.454 -0.187
## [478,] 11.962 -1.941 -2.016 -1.578 -2.312 -1.833 -1.870 -1.525
## [479,] -1.542  6.600  1.162 -1.038 -1.675 -1.262 -1.294 -0.991
## [480,] -0.490 -0.336 -0.390  2.847 -0.597 -0.257 -0.284 -0.020
## [481,]  1.863 -0.336 -0.390 -0.062 -0.597 -0.257 -0.284 -0.020
## [482,] -0.817  4.417 -0.712 -0.374 -0.930 -0.574 -0.602 -0.331
## [483,]  4.501 -0.850 -0.909 -0.561 -1.135 -0.766 -0.117 -0.518
## [484,] -4.154 -3.817 -3.933 -3.263 22.039 -3.652 -3.708 -3.184
## [485,] -1.301 -1.124 -1.185 -0.821 -1.426  8.091 -1.066 -0.776
## [486,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426 -0.454  4.367
## [487,]  9.567 -1.559 -1.627 -1.227 -1.895 -1.461 -1.494 -1.178
## [488,]  6.187 -1.021 -1.081 -0.723 -1.316 -0.934 -0.964 -0.679
## [489,] -1.248 -1.073 -1.134 -0.773 -1.372 -0.985 -1.015  9.074
## [490,] -1.542 -1.355 -1.420 -1.038 -1.675  9.766 -1.294 -0.991
## [491,] -0.887 -0.724  4.684 -0.440 -1.001 -0.641 -0.670 -0.397
## [492,] -0.286 -0.134 -0.187  1.211 -0.391 -0.056 -0.082  0.181
## [493,] -1.301 -1.124 -1.185 -0.821 -1.426 -1.035 -1.066  9.496
## [494,] -1.497  8.950 -1.376 -0.997 -1.628 -1.219 -1.251 -0.950
## [495,] -1.910 -1.706 11.251 -1.363 -2.056 -1.605 -1.640 -1.312
## [496,] -0.167 -0.016 -0.069  0.257 -0.271  0.062  0.035  0.300
## [497,] -0.393 -0.239 -0.293  2.067 -0.499 -0.161 -0.188  0.076
## [498,] -0.664 -0.506 -0.562 -0.229 -0.774 -0.426 -0.454  4.367
## [499,]  0.115 -0.909  4.674 -0.618 -1.197 -0.825 -0.854 -0.574
## [500,]  5.715 -3.302  5.839 -2.805 -3.822 -3.154 -3.204  4.783

plot(chi_custA$residuals)

chi_custA$p.value

## [1] 0.0004997501

 # Test 3 
custA[sample(1:nrow(custA),10),c(2,4)]   %>% 
             chisq.test()

## 
##  Pearson's Chi-squared test
## 
## data:  .
## X-squared = 70.085, df = 9, p-value = 1.465e-11

custA[sample(1:nrow(custA),10),c(4,6)]   %>% 
             chisq.test()

## 
##  Pearson's Chi-squared test
## 
## data:  .
## X-squared = 11.483, df = 9, p-value = 0.2441

Since p-value = chi_custA$p.value , which is less than .05 so we can reject the null hypothesis. So we see there is dependancy in the spending over the quarter.

Using Simulation

#Multiple simulation
#Doing multiple check 
rm(p_chi)
p_chi <- data.frame(Qt12= rep(0, 50),Qt23= rep(0, 50),Qt34= rep(0, 50))
for(i in 1:50){
  
   chi_test <- custA[sample(1:nrow(custA),10),c(2,4)]   %>% 
             chisq.test()
  p_chi$Qt12[i]  <- chi_test$p.value
  
  chi_test <- custA[sample(1:nrow(custA),10),c(4,6)]   %>% 
             chisq.test()
  p_chi$Qt23[i]  <- chi_test$p.value
  
  
  
  chi_test <- custA[sample(1:nrow(custA),10),c(6,8)]   %>% 
             chisq.test()
  
  p_chi$Qt34[i]  <- chi_test$p.value
  # samp <- sample(c("atheist", "non_atheist"), n, replace = TRUE, prob = c(p, 1-p))
  # p_hats[i] <- sum(samp == "atheist")/n
}

# "Chi-squared approximation may be incorrectChi-squared "   The warning message found in the solution above is due to the small cell values in the contingency table. 
#lets check the key
table(gather(p_chi)$key)

## 
## Qt12 Qt23 Qt34 
##   50   50   50

#Gather the data so that we can plot on it.
#using geom_jitter to add some noise as most of poits would overlap in normal plot.
gather(p_chi) %>% ggplot(mapping = aes(x= key, y= round(value,3))) + geom_point()+geom_jitter()

gather(p_chi)%>% filter(value > 0.05)

#contigency table 

 rowSums(custA[,-c(1)])

##    [1]  10   9  17  26  22  13  26  25  31  27  10  13  13  32  19 182  33
##   [18]  30  32 167  18  32  28  38  20  33  86  13  30  46  12  21  40  12
##   [35]  31  32  14  19  20  21  27  25  10  17 168  53  28  30  13  27  21
##   [52]  21   8  37  16  10  11  17  23   8  43  12  20  10  17  18  20   8
##   [69]  23  17  31  55  54  13  21  22  19  19  25  13  84  18  50  10  14
##   [86]  13  19  29  11  14  30  17   8  14  41  25  15  22  32  29   9  37
##  [103]  13  39  10   9  49  32   8  31  18  22  14  11  16   9  14  34  35
##  [120]  13  20  17  15  10  29  19  53   8  34  25  15  41  27  17  19  43
##  [137]  22  22 111  18  47  10  19  16  19  37   9  20  23  15   9  29  37
##  [154]   9  14   8  42  16  19  16  53  27  44  25  26  37  45  32  26  32
##  [171]  15  24   9   9  15  16  22  19  15  12  76  10  17  24  42  43  23
##  [188]  17  15   8  74  27  27  95  25  10  13  27  18  13  15  21  19  51
##  [205]  23  35  14  25  20  19  28  25  19  34  16  47  58  27  27   9  11
##  [222]  20  12  14  24  10  50   8  15  17  94  15  20  19  55   8   8  16
##  [239]  23  29  63   8  17  40  43  21  18  30  19  10  37  15  26  66  19
##  [256]  47  22  37   8  16  27  19  46  19  24  60  34  46  28   8  27  49
##  [273]  22  21  23  23  20   8  12   9  21  21  12  13  57  25  14  41  14
##  [290]  20  20  34  14  28  11  40  28  22  28  24  38   8  23  12  13  27
##  [307]   8  25  21  65  21  37  87  48  22  17   8  55  32  13  12  16  14
##  [324]  17  26  32  23  19  46  12  19  14  11  21  17  26  24  16  76  35
##  [341]  27  20  15  15  34  10  21  21  53  19  82  61  27  13  15  37  13
##  [358]   9  25  12  20  26  38  17 101  24  15  28  45  34  14  33  34  24
##  [375]  23  19  12  23  16  16  23  55  13  26  14  25  35  27  16  10  10
##  [392]  15  15  30  27  28  17  22  19  25  22  35  59  40  33  25  27   9
##  [409]  33  22  24  24  11  47 148  30  10  21  11  37  13  88  39  13  28
##  [426]  17  76   8  19  94   9  44  32  13  23  22  12  49  18  21  16  31
##  [443]  28  13  48  30  23  10   9  27  10  23   9  19  23  37  11  21  13
##  [460]  22  21  29  21  18  25  10  16  24  10   8  11  12  12  14  12  18
##  [477]  19  21  37  25  12  18  10  30  52  51  55  14  19  14  20  13  29
##  [494]  29   9  29  26  27  22  52  30  19  15  11  10  17  31  13  13  17
##  [511]  24  32  37  27  16  27  17  18  31  13  20  16  38  23  13  22  10
##  [528]  10  18   8  27  31  13  19  24  22  25  18  13  10  26   8  16  17
##  [545]  10  26  63  36  35  11   8  39  12  30  15  20  27  44  54  25  57
##  [562]  57   9   9  25  10  18   9  12  38  10  17   8   9  42  44  27  16
##  [579]  16  18  16  18 108  45  16  21  12  22  17  15  10  23  13  12  26
##  [596]  29  19  11  19  15  57  15  11  18  55  11  20  18  28  12 125  40
##  [613]   8  74  28  17   9  29  48  60  21  30  80  13  17  14  22  12   9
##  [630]  39  10  16  30  21  17  13  45   9   9  12  22  17   9  39  28  19
##  [647]  32  13  19   9  12  12  19  11   9 163  10  24  14  15  40  48  12
##  [664]  17  30  33  15  17  10  15  22  47  29  15  20  47  11  27  38   8
##  [681]  28  27  16  21  27  32   8  15  20  24  13  14  27  16  28   9  46
##  [698]  37  21  12  47   8  16  34  11  25  19  15  12  27  48  17  17  15
##  [715]  88  39  10  15  14  11 130  21  28  11  27   9  16   9  12  22  26
##  [732]  17  19  10  22  28  13  10  21  21  16  20  27  36  18  13  16  31
##  [749]  11  25  16  13  24  33  19   9  31  12  41  39  38  10  22  35  13
##  [766]  12  30  15  18  13  54  20  37  27  12  40  13  28  34  12  29  16
##  [783]  16  32  17  11  14  32  17  30  26  10  37  31  13  16  11  17  18
##  [800]  24  20  16  40  27  12  17  13  89  26   8  11  35  37  12  31  39
##  [817]  12  19  20  47  19   8  43  31  12  14  45  13  13  71  25  13   9
##  [834]  29  31  32  59  29  15  10  10  23  27  66  20  42  15  17  45  15
##  [851]  40  17  56  14  24  26  19  45  25  27  29  18  13  19  19  27  17
##  [868]  21  32  77  18  20   9  19  11  21  31  36  22  15  13  35  15  13
##  [885]  17  21  19  32  12  34  34  74   9  22  20   9  20  16   8  14  18
##  [902]  27  49   9  29  19  21  12  62  14   8  18  22  25   9  25  17  27
##  [919]  67  17   8  24  41  22  62  27  13  10  25  32  29   8  13  37  12
##  [936]  30  21  10  38   8  17  25  19  15  13  12  11  17  20  13  23  15
##  [953]  15  13  22  92  11 339  20  47  29  46  27  17  31  13  19  17  23
##  [970]  34  18   9  17  36  13  37  33   9  32  18  31  31  15   8  38  27
##  [987]  12  35  19  16  60  32  14  29  11   8  36  37  27  18

 colSums(custA[,-c(1)])

## Q1_17 Q1_18 Q2_17 Q2_18 Q3_17 Q3_18 Q4_17 Q4_18 
##  3494  4120  3331  3193  3176  2928  2522  2719

(4) interpret results.

Since p-value = chi_custA$p.value , which is less than .05 so we can reject the null hypothesis. So we see there is dependancy in the spending over the quarter. Multiple simulation also indicatess pvlaue < 0.05 , for Aplha = 0.5 it indicates that we are 95% confident that spending over each quater is depended on spending over last quarter.

Limitation : We can’t evalaute spending is more or less using Chi-square test , as it is goodness of fittest only.

Data

# pairs.panels(lm_cust[,c(2,3)])

# Try Predicting Quantity based on Known informaiton . 
# Identifying Best model to predict Quantity 

head(lm_cust)

result <- data.frame(var = 1:10,pval = 1:10, comment=1:10)

result$var <- "A"
result$pval <- "A"
result$comment <- "A" 

# Working With Sample 
set.seed(42672)
 lm_cust_s1 <- lm_cust[sample(nrow(lm_cust),200),]
 lm_cust_s1$month <- month(lm_cust_s1$`Order Date`)
 lm_cust_s1$month <- as.factor(lm_cust_s1$month )

LM Model 1

 names(lm_cust_s1)

##  [1] "KUNNR_NEW"  "Qt"         "Brand"      "Order Date" "promo"     
##  [6] "city"       "state"      "Order_Qty"  "Doll_Val"   "month"

lm1 <- lm(Order_Qty ~ Qt + Brand + promo + month + state,lm_cust_s1)
summary(lm1)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + month + state, 
##     data = lm_cust_s1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -16.550  -4.241   0.000   3.479  32.806 
## 
## Coefficients: (4 not defined because of singularities)
##                Estimate Std. Error t value Pr(>|t|)  
## (Intercept)     -1.9078    21.6089  -0.088   0.9298  
## QtQ1_18          0.9060    17.1804   0.053   0.9581  
## QtQ2_17        -10.6690     6.7652  -1.577   0.1181  
## QtQ2_18          4.4088    16.8263   0.262   0.7939  
## QtQ3_17          4.2890    18.8786   0.227   0.8208  
## QtQ3_18          0.2220    18.6653   0.012   0.9905  
## QtQ4_17         -2.2830    17.5834  -0.130   0.8970  
## QtQ4_18         -0.7737    17.6637  -0.044   0.9652  
## BrandAX         18.3458    22.3044   0.823   0.4128  
## BrandBB          2.3696    19.4477   0.122   0.9033  
## BrandBE          4.8108    17.0912   0.281   0.7790  
## BrandBV         12.8512    18.2565   0.704   0.4832  
## BrandCH         20.3679    18.4397   1.105   0.2721  
## BrandDG         16.9778    19.2512   0.882   0.3800  
## BrandDY          0.7952    18.4361   0.043   0.9657  
## BrandEA         17.7019    17.2015   1.029   0.3060  
## BrandHC         11.0212    15.3629   0.717   0.4749  
## BrandMK         12.2266    16.2524   0.752   0.4537  
## BrandMU          5.1632    19.3904   0.266   0.7906  
## BrandOJ          8.0469    24.5189   0.328   0.7435  
## BrandOO         12.0027    16.3069   0.736   0.4635  
## BrandOX         16.9387    16.0352   1.056   0.2935  
## BrandOY          4.6321    16.9397   0.273   0.7851  
## BrandPH          0.1814    16.1386   0.011   0.9911  
## BrandPO         -6.6710    20.5100  -0.325   0.7457  
## BrandPR         13.6841    16.1577   0.847   0.3992  
## BrandPS         -1.4118    19.6018  -0.072   0.9427  
## BrandRB         11.3283    16.1044   0.703   0.4835  
## BrandRJ         -1.4976    17.3495  -0.086   0.9314  
## BrandRL          0.2288    18.2880   0.013   0.9900  
## BrandRX         11.1098    16.0246   0.693   0.4898  
## BrandRY          2.2306    16.3775   0.136   0.8919  
## BrandTF          8.5103    16.3253   0.521   0.6034  
## BrandTY          6.2068    16.5061   0.376   0.7077  
## BrandVA         -3.8398    19.1370  -0.201   0.8414  
## BrandVE          9.0235    16.2144   0.557   0.5792  
## BrandVO         11.9972    17.4822   0.686   0.4942  
## promoH15         9.6017    13.9029   0.691   0.4915  
## promoHC18       19.2841    15.2727   1.263   0.2098  
## promoLUX40      21.9169    14.8400   1.477   0.1430  
## promoMVE150     17.2033    25.5096   0.674   0.5017  
## promoNB         21.1233    21.4209   0.986   0.3266  
## promoOther       6.9515    20.5648   0.338   0.7361  
## promoPB3             NA         NA      NA       NA  
## promoRSD         5.7801    22.6733   0.255   0.7993  
## promoSUN2       32.3281    17.4618   1.851   0.0672 .
## promoSY1         9.6893    20.4632   0.473   0.6369  
## promoSY100      14.1988    21.3098   0.666   0.5068  
## promoSY1000     19.7310    23.8682   0.827   0.4105  
## promoSY2        -6.9960    25.9335  -0.270   0.7879  
## promoSY200      38.9640    25.2710   1.542   0.1264  
## promoSY250      18.7303    21.4514   0.873   0.3848  
## promoSY5        -4.9719    23.1484  -0.215   0.8304  
## promoSYNERGY54   8.2911    22.2721   0.372   0.7105  
## promoSYVE        7.7185    22.2776   0.346   0.7297  
## promoUP1        19.0150    14.2490   1.334   0.1852  
## promoUP3        11.5593    16.8913   0.684   0.4954  
## promoVIP        49.2421    23.6627   2.081   0.0401 *
## promoVPFP200     1.1052    17.1949   0.064   0.9489  
## promoWC1        13.7854    14.3078   0.963   0.3377  
## promoWild Card   0.6250    23.6542   0.026   0.9790  
## month2          -4.0531     4.5852  -0.884   0.3789  
## month3          -4.1391     4.8446  -0.854   0.3950  
## month4           1.9585     4.8677   0.402   0.6883  
## month5          -0.3939     5.9585  -0.066   0.9474  
## month6               NA         NA      NA       NA  
## month7          -5.0892     6.4939  -0.784   0.4352  
## month8           2.4540     4.3761   0.561   0.5763  
## month9               NA         NA      NA       NA  
## month10          4.5085     4.3857   1.028   0.3065  
## month11          3.0711     4.5898   0.669   0.5050  
## month12              NA         NA      NA       NA  
## stateAR          4.9067    13.5503   0.362   0.7181  
## stateAZ        -12.6404     9.1172  -1.386   0.1688  
## stateCA         -3.1682     6.8239  -0.464   0.6435  
## stateCO         -7.3438    13.2637  -0.554   0.5811  
## stateCT        -15.9275     9.3699  -1.700   0.0924 .
## stateFL         -1.0817     7.6681  -0.141   0.8881  
## stateGA        -14.0750    10.1232  -1.390   0.1676  
## stateIA        -11.6502    11.5498  -1.009   0.3157  
## stateIL          5.5588     8.2697   0.672   0.5031  
## stateIN         -2.1564    10.5735  -0.204   0.8388  
## stateKS        -11.9797    11.1076  -1.079   0.2835  
## stateKY         -0.8319    13.1444  -0.063   0.9497  
## stateLA        -19.7563    12.3596  -1.598   0.1132  
## stateMA          3.8609     9.5001   0.406   0.6854  
## stateME          5.5262    12.3303   0.448   0.6550  
## stateMI         -2.8545     9.2395  -0.309   0.7580  
## stateMN          2.4257    10.0699   0.241   0.8102  
## stateMO        -19.8191    13.5869  -1.459   0.1479  
## stateMS        -16.2905    13.0349  -1.250   0.2144  
## stateNC         -6.0919    10.8236  -0.563   0.5749  
## stateND         -3.1519    14.4050  -0.219   0.8273  
## stateNH         -5.5647    10.7039  -0.520   0.6044  
## stateNJ         -4.4015     8.2831  -0.531   0.5964  
## stateNV          2.2659    10.5634   0.215   0.8306  
## stateNY         -3.9499     7.5107  -0.526   0.6002  
## stateOH         -3.9169     8.4383  -0.464   0.6436  
## stateOK        -11.1089     9.1041  -1.220   0.2254  
## stateOR         -3.1731     9.9453  -0.319   0.7504  
## statePA         -5.7108     8.4292  -0.677   0.4997  
## statePR        -16.4884    10.2060  -1.616   0.1095  
## stateSC        -11.6695    10.7825  -1.082   0.2818  
## stateTN         -7.6426    14.5518  -0.525   0.6007  
## stateTX         -3.9184     6.9454  -0.564   0.5740  
## stateUT        -11.2160    10.3562  -1.083   0.2815  
## stateVA         11.3389    10.1637   1.116   0.2674  
## stateWA        -13.5126    10.2216  -1.322   0.1893  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.09 on 96 degrees of freedom
## Multiple R-squared:  0.5671, Adjusted R-squared:  0.1027 
## F-statistic: 1.221 on 103 and 96 DF,  p-value: 0.1613

anova(lm1)

result$var[1] <- "lm(formula = Order_Qty ~ Qt + Brand + promo + month + state, 
    data = lm_cust_s1)"
result$pval[1] <-  "Adjusted R-squared:  0.2356 F-statistic: 1.568 on 108 and 91 DF,  p-value: 0.01388"

result$comment[1] <- "(Month , State ) Qt,Promo codes are significant,state is not " 

LM Model 2

#Dropping State

lm2 <- lm(Order_Qty ~ Qt + Brand + promo + month,lm_cust_s1)
summary(lm2)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + month, data = lm_cust_s1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -16.961  -4.673  -0.497   3.248  35.929 
## 
## Coefficients: (4 not defined because of singularities)
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)      1.76646   15.85513   0.111   0.9115  
## QtQ1_18          8.19326   15.10309   0.542   0.5884  
## QtQ2_17         -8.45412    5.56060  -1.520   0.1308  
## QtQ2_18          8.26195   14.70668   0.562   0.5752  
## QtQ3_17          9.13929   16.42153   0.557   0.5788  
## QtQ3_18          4.85130   16.32582   0.297   0.7668  
## QtQ4_17          5.16805   15.40193   0.336   0.7377  
## QtQ4_18          6.02604   15.43343   0.390   0.6968  
## BrandAX         13.47419   14.92339   0.903   0.3682  
## BrandBB         -1.85800   14.56726  -0.128   0.8987  
## BrandBE          0.84478   11.24694   0.075   0.9402  
## BrandBV          2.46199   13.09288   0.188   0.8511  
## BrandCH         10.16419   13.25922   0.767   0.4447  
## BrandDG         15.77592   15.14897   1.041   0.2996  
## BrandDY         -2.40774   13.09982  -0.184   0.8545  
## BrandEA          9.51967   11.71647   0.813   0.4180  
## BrandHC          8.00163   10.83654   0.738   0.4616  
## BrandMK          9.78688   11.22669   0.872   0.3849  
## BrandMU         -0.12788   14.74098  -0.009   0.9931  
## BrandOJ          3.66944   18.80247   0.195   0.8456  
## BrandOO          8.97486   10.80145   0.831   0.4075  
## BrandOX         13.51180   10.78819   1.252   0.2126  
## BrandOY         -1.75425   11.59509  -0.151   0.8800  
## BrandPH         -2.08049   11.16732  -0.186   0.8525  
## BrandPO        -15.82037   15.52513  -1.019   0.3101  
## BrandPR          6.24358   10.87677   0.574   0.5669  
## BrandPS         -2.75859   15.29122  -0.180   0.8571  
## BrandRB          5.17785   10.71694   0.483   0.6298  
## BrandRJ         -5.56978   12.09695  -0.460   0.6460  
## BrandRL         -0.40758   13.20894  -0.031   0.9754  
## BrandRX          6.84940   10.75139   0.637   0.5252  
## BrandRY         -3.06439   11.08068  -0.277   0.7826  
## BrandTF          5.68303   11.06348   0.514   0.6083  
## BrandTY          3.27404   11.37846   0.288   0.7740  
## BrandVA         -7.78735   15.01296  -0.519   0.6048  
## BrandVE          6.43716   11.16435   0.577   0.5652  
## BrandVO         10.41475   12.85835   0.810   0.4194  
## promoH15         7.75470   11.45441   0.677   0.4996  
## promoHC18       17.88883   12.80711   1.397   0.1648  
## promoLUX40      13.38640   12.53006   1.068   0.2873  
## promoMVE150     10.66944   18.80247   0.567   0.5714  
## promoNB         13.05855   16.03867   0.814   0.4170  
## promoOther      -1.99801   17.76144  -0.112   0.9106  
## promoPB3              NA         NA      NA       NA  
## promoRSD        -1.29378   19.37013  -0.067   0.9468  
## promoSUN2       25.00462   15.13055   1.653   0.1008  
## promoSY1         0.19567   17.86870   0.011   0.9913  
## promoSY100       3.85284   18.28232   0.211   0.8334  
## promoSY1000     10.29834   20.80866   0.495   0.6215  
## promoSY2       -13.57687   21.18255  -0.641   0.5227  
## promoSY200      22.06638   21.20608   1.041   0.3000  
## promoSY250      12.29655   18.50514   0.664   0.5075  
## promoSY5        -9.92659   20.77319  -0.478   0.6335  
## promoSYNERGY54  -1.14395   19.23230  -0.059   0.9527  
## promoSYVE        2.88566   19.02076   0.152   0.8796  
## promoUP1        11.09237   11.97327   0.926   0.3559  
## promoUP3         5.67156   14.53816   0.390   0.6971  
## promoVIP        40.69755   20.62294   1.973   0.0505 .
## promoVPFP200    -2.17996   15.41726  -0.141   0.8878  
## promoWC1         8.37933   12.06755   0.694   0.4887  
## promoWild Card  -6.76739   20.93979  -0.323   0.7471  
## month2          -4.74452    4.06230  -1.168   0.2449  
## month3          -3.43590    4.12213  -0.834   0.4061  
## month4           0.53474    4.14536   0.129   0.8976  
## month5           1.83826    4.72189   0.389   0.6977  
## month6                NA         NA      NA       NA  
## month7          -3.50092    5.28964  -0.662   0.5092  
## month8           4.22014    3.83925   1.099   0.2737  
## month9                NA         NA      NA       NA  
## month10          0.34721    3.82871   0.091   0.9279  
## month11          0.06351    3.46890   0.018   0.9854  
## month12               NA         NA      NA       NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.04 on 132 degrees of freedom
## Multiple R-squared:  0.4109, Adjusted R-squared:  0.1118 
## F-statistic: 1.374 on 67 and 132 DF,  p-value: 0.06178

anova(lm2)

result$var[2] <- "lm(formula = Order_Qty ~ Qt + Brand + promo + month, data = lm_cust_s1)"
result$pval[2] <- "Adjusted R-squared:  0.3198 F-statistic: 2.396 on 67 and 132 DF,  p-value: 9.788e-06"
result$comment[2] <- "(Month)Brand is not significant" 

#Promo promoVPFP200 turns out to more significant here. 

LM Model 3

#3  Dropping brand info 
lm3 <- lm(Order_Qty ~  Qt + promo + month,lm_cust_s1)
summary(lm3)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + promo + month, data = lm_cust_s1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.866  -5.922  -1.888   5.804  38.720 
## 
## Coefficients: (3 not defined because of singularities)
##                Estimate Std. Error t value Pr(>|t|)  
## (Intercept)      7.0142    11.8946   0.590   0.5562  
## QtQ1_18          1.0985    12.8603   0.085   0.9320  
## QtQ2_17         -6.6454     5.3531  -1.241   0.2163  
## QtQ2_18          2.2094    12.4434   0.178   0.8593  
## QtQ3_17          0.3547    13.2444   0.027   0.9787  
## QtQ3_18         -4.1745    13.4272  -0.311   0.7563  
## QtQ4_17         -1.3696    13.1970  -0.104   0.9175  
## QtQ4_18         -0.2888    13.1282  -0.022   0.9825  
## promoH15         6.6873    11.5699   0.578   0.5641  
## promoHC18       14.0000    12.7914   1.094   0.2754  
## promoLUX40      11.3344    12.4090   0.913   0.3624  
## promoMVE150      3.4745    15.8602   0.219   0.8269  
## promoNB          3.5945    15.9481   0.225   0.8220  
## promoOther       4.5661    16.0019   0.285   0.7757  
## promoPB3        -3.5255    15.8602  -0.222   0.8244  
## promoRSD         2.7159    17.5790   0.154   0.8774  
## promoSUN2       19.0000    14.7702   1.286   0.2002  
## promoSY1         8.1008    16.1166   0.503   0.6159  
## promoSY100      10.1683    16.3313   0.623   0.5344  
## promoSY1000     15.3760    19.4517   0.790   0.4304  
## promoSY2        -7.7943    19.3216  -0.403   0.6872  
## promoSY200      19.1929    19.3130   0.994   0.3218  
## promoSY250      13.7531    16.7025   0.823   0.4115  
## promoSY5        -0.7943    19.3216  -0.041   0.9673  
## promoSYNERGY54   4.2402    17.7321   0.239   0.8113  
## promoSYVE       13.7861    17.0650   0.808   0.4204  
## promoUP1         9.0155    12.0499   0.748   0.4555  
## promoUP3         4.0000    14.7702   0.271   0.7869  
## promoVIP        44.7764    18.9571   2.362   0.0194 *
## promoVPFP200    -2.5255    15.8602  -0.159   0.8737  
## promoWC1         6.7764    11.8833   0.570   0.5693  
## promoWild Card  -0.6240    19.4517  -0.032   0.9745  
## month2          -2.6087     3.9068  -0.668   0.5053  
## month3          -1.4887     3.8859  -0.383   0.7021  
## month4           0.5578     4.1556   0.134   0.8934  
## month5           1.6312     4.5462   0.359   0.7202  
## month6               NA         NA      NA       NA  
## month7          -1.4957     4.7273  -0.316   0.7521  
## month8           5.4989     3.5054   1.569   0.1187  
## month9               NA         NA      NA       NA  
## month10          2.0689     3.7185   0.556   0.5787  
## month11          1.1625     3.2653   0.356   0.7223  
## month12              NA         NA      NA       NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.44 on 160 degrees of freedom
## Multiple R-squared:  0.2265, Adjusted R-squared:  0.03798 
## F-statistic: 1.201 on 39 and 160 DF,  p-value: 0.2151

anova(lm3)

result$var[3] <- "lm(formula = Order_Qty ~ Qt + promo + month, data = lm_cust_s1)"
result$pval[3] <- "Adjusted R-squared:  0.2799 F-statistic:  3.09 on 37 and 162 DF,  p-value: 4.911e-07"
result$comment[3] <- "(Month) least significant" 

LM Model 4-5

#4  Dropping Month
lm4 <- lm(Order_Qty ~  Qt + promo ,lm_cust_s1)
summary(lm4)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + promo, data = lm_cust_s1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.500  -6.170  -1.375   5.923  40.099 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)  
## (Intercept)     6.86722   11.25286   0.610   0.5425  
## QtQ1_18         0.07737   12.43278   0.006   0.9950  
## QtQ2_17        -4.86722    4.42147  -1.101   0.2726  
## QtQ2_18         3.09959   12.03509   0.258   0.7971  
## QtQ3_17         3.24186   12.44002   0.261   0.7947  
## QtQ3_18        -1.17263   12.52218  -0.094   0.9255  
## QtQ4_17        -0.19445   12.44455  -0.016   0.9876  
## QtQ4_18         1.03282   12.36837   0.084   0.9335  
## promoH15        5.60581   10.99322   0.510   0.6108  
## promoHC18      14.00000   12.67344   1.105   0.2709  
## promoLUX40     10.30622   11.87453   0.868   0.3867  
## promoMVE150     2.13278   15.28739   0.140   0.8892  
## promoNB         1.13278   15.28739   0.074   0.9410  
## promoOther      4.43041   15.72944   0.282   0.7785  
## promoPB3       -4.86722   15.28739  -0.318   0.7506  
## promoRSD        3.68021   17.26820   0.213   0.8315  
## promoSUN2      19.00000   14.63403   1.298   0.1959  
## promoSY1        7.83324   15.87814   0.493   0.6224  
## promoSY100      9.79220   16.05434   0.610   0.5427  
## promoSY1000    15.05541   18.94609   0.795   0.4279  
## promoSY2       -6.90004   18.90388  -0.365   0.7156  
## promoSY200     19.32723   18.95381   1.020   0.3093  
## promoSY250     13.55541   16.42325   0.825   0.4103  
## promoSY5        0.09996   18.90388   0.005   0.9958  
## promoSYNERGY54  4.09996   17.43038   0.235   0.8143  
## promoSYVE      15.05541   16.75938   0.898   0.3703  
## promoUP1        8.47967   11.64176   0.728   0.4674  
## promoUP3        4.00000   14.63403   0.273   0.7849  
## promoVIP       44.03320   18.68752   2.356   0.0196 *
## promoVPFP200   -3.86722   15.28739  -0.253   0.8006  
## promoWC1        6.03320   11.62190   0.519   0.6044  
## promoWild Card -0.94459   18.94609  -0.050   0.9603  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.35 on 168 degrees of freedom
## Multiple R-squared:  0.2028, Adjusted R-squared:  0.05564 
## F-statistic: 1.378 on 31 and 168 DF,  p-value: 0.1034

anova(lm4)

result$var[4] <- "lm(formula = Order_Qty ~ Qt + promo, data = lm_cust_s1)"
result$pval[4] <- "Adjusted R-squared:  -0.003147 F-statistic: 0.9799 on 31 and 168 DF,  p-value: 0.5038"
result$comment[4] <- "not much significant"

# 5 Increase Sample Size
rm(lm_cust_s1)
# SInce I see no major significance increasing the sample size to 500
set.seed(12121)
lm_cust_s2 <- lm_cust[sample(nrow(lm_cust),500),]
 lm_cust_s2$month <- month(lm_cust_s2$`Order Date`)
 lm_cust_s2$month <- as.factor(lm_cust_s2$month )
 names(lm_cust_s2)

##  [1] "KUNNR_NEW"  "Qt"         "Brand"      "Order Date" "promo"     
##  [6] "city"       "state"      "Order_Qty"  "Doll_Val"   "month"

 lm5 <- lm(Order_Qty ~ Qt + Brand + promo + month + state,lm_cust_s2)
summary(lm5)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + month + state, 
##     data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -58.622  -6.576  -0.564   4.571  96.370 
## 
## Coefficients: (3 not defined because of singularities)
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -2.23836   24.91771  -0.090 0.928471    
## QtQ1_18            5.68596    7.62788   0.745 0.456493    
## QtQ2_17            4.90074    4.33482   1.131 0.258978    
## QtQ2_18            9.77888    7.96075   1.228 0.220085    
## QtQ3_17            9.26640    7.94036   1.167 0.243964    
## QtQ3_18            6.23512    8.12498   0.767 0.443334    
## QtQ4_17           10.98159    8.02317   1.369 0.171915    
## QtQ4_18            7.87855    8.17353   0.964 0.335722    
## BrandAX            6.28736    9.54950   0.658 0.510694    
## BrandBB            2.30396   10.36390   0.222 0.824198    
## BrandBE           25.20950    9.20688   2.738 0.006479 ** 
## BrandBV            6.23365    9.93958   0.627 0.530946    
## BrandCH           18.36577   12.41099   1.480 0.139780    
## BrandDG            9.22986    9.48157   0.973 0.330966    
## BrandDY           -2.43066    8.92135  -0.272 0.785425    
## BrandEA            4.22087    8.21652   0.514 0.607765    
## BrandHC           17.56763    7.28464   2.412 0.016370 *  
## BrandMK           15.26507    7.34155   2.079 0.038283 *  
## BrandMU           28.37507   16.41629   1.728 0.084741 .  
## BrandOJ            0.28410    9.39984   0.030 0.975905    
## BrandOO           18.05318    7.12518   2.534 0.011699 *  
## BrandOX           16.29683    7.31304   2.228 0.026452 *  
## BrandOY           13.78368    8.12854   1.696 0.090783 .  
## BrandPH           16.20134    8.73146   1.856 0.064320 .  
## BrandPO           41.37726   10.32638   4.007 7.44e-05 ***
## BrandPP            9.44181   13.11977   0.720 0.472188    
## BrandPR           17.81589    7.33210   2.430 0.015581 *  
## BrandPS            8.62006    9.21071   0.936 0.349950    
## BrandRA         -101.12802   22.07255  -4.582 6.32e-06 ***
## BrandRB           19.20374    6.87953   2.791 0.005520 ** 
## BrandRJ            7.60922   13.77483   0.552 0.581009    
## BrandRL            9.40933    9.68586   0.971 0.331961    
## BrandRX           17.98048    6.92617   2.596 0.009808 ** 
## BrandRY            3.57624    7.19166   0.497 0.619291    
## BrandSF           -2.41412   13.26180  -0.182 0.855655    
## BrandTF           13.30561    7.87569   1.689 0.091977 .  
## BrandTY           12.26509    8.09756   1.515 0.130713    
## BrandVA           -9.65624   13.05699  -0.740 0.460046    
## BrandVE           13.16022    7.84219   1.678 0.094168 .  
## BrandVO           12.33400    7.86830   1.568 0.117842    
## promoCODE F       -7.66331   21.77963  -0.352 0.725147    
## promoH15         -11.02374   18.64078  -0.591 0.554630    
## promoLUX40         2.01094   18.56484   0.108 0.913801    
## promoMVE150      -16.92342   20.92357  -0.809 0.419139    
## promoMVENB       -24.45911   23.95361  -1.021 0.307874    
## promoNASC75      -17.31631   20.35928  -0.851 0.395578    
## promoNB            2.27980   21.45877   0.106 0.915449    
## promoOOX40       -13.05094   20.60380  -0.633 0.526849    
## promoOOX60        -2.75317   24.34163  -0.113 0.910008    
## promoOther       -15.02685   19.72704  -0.762 0.446702    
## promoOY12        -12.91125   21.77023  -0.593 0.553498    
## promoPRE-SELL B  -14.21934   27.05889  -0.525 0.599554    
## promoPRE-SELL T  -15.02483   25.20373  -0.596 0.551450    
## promoRB15         -7.71640   19.39064  -0.398 0.690901    
## promoRSD         -13.01034   25.53166  -0.510 0.610653    
## promoSUN1        -14.23245   26.54096  -0.536 0.592112    
## promoSY1          -9.48118   20.23297  -0.469 0.639632    
## promoSY100       -13.89471   19.91631  -0.698 0.485833    
## promoSY1000       87.81025   24.64570   3.563 0.000415 ***
## promoSY2           6.64700   20.81949   0.319 0.749704    
## promoSY200         0.60032   19.31584   0.031 0.975223    
## promoSY250         9.70349   20.85748   0.465 0.642042    
## promoSY5         -11.05189   21.54875  -0.513 0.608343    
## promoSYNERGY54   -17.91337   24.88359  -0.720 0.472051    
## promoSYVE         -7.84596   20.86777  -0.376 0.707144    
## promoUP1          -6.74887   18.90346  -0.357 0.721283    
## promoUP3          -5.34776   19.01536  -0.281 0.778689    
## promoVIP         -16.80078   25.26275  -0.665 0.506439    
## promoVP1200      -14.77289   20.05127  -0.737 0.461739    
## promoVPFP150     -27.72705   23.43836  -1.183 0.237580    
## promoVPFP200      38.96072   19.18009   2.031 0.042940 *  
## promoVPNB          3.12551   24.90540   0.125 0.900200    
## promoWC1           3.86447   19.03659   0.203 0.839245    
## promoWC2         -26.74522   25.63692  -1.043 0.297525    
## promoWild Card    -7.32750   20.76494  -0.353 0.724381    
## month2             7.20628    3.20305   2.250 0.025049 *  
## month3             5.80081    3.42283   1.695 0.090968 .  
## month4            -4.29734    3.89805  -1.102 0.270992    
## month5            -5.83098    3.87120  -1.506 0.132859    
## month6                  NA         NA      NA       NA    
## month7            -0.90008    3.68034  -0.245 0.806930    
## month8             4.65983    3.76886   1.236 0.217096    
## month9                  NA         NA      NA       NA    
## month10           -0.15770    3.99211  -0.040 0.968510    
## month11            1.49643    3.81906   0.392 0.695408    
## month12                 NA         NA      NA       NA    
## stateAL            9.50445   18.76959   0.506 0.612896    
## stateAR            4.25873   17.50037   0.243 0.807869    
## stateAZ            8.75412   15.96340   0.548 0.583758    
## stateCA            5.96085   15.56816   0.383 0.702024    
## stateCO            6.89773   16.34269   0.422 0.673221    
## stateCT            8.87272   17.47015   0.508 0.611842    
## stateDE            2.94917   18.67069   0.158 0.874577    
## stateFL            5.07178   15.78812   0.321 0.748210    
## stateGA            5.08077   16.15192   0.315 0.753273    
## stateHI           -3.27474   21.54100  -0.152 0.879251    
## stateIA           -2.92012   21.57278  -0.135 0.892400    
## stateID           -0.50943   18.88718  -0.027 0.978497    
## stateIL           -0.10485   16.10398  -0.007 0.994809    
## stateIN           13.93032   16.31788   0.854 0.393834    
## stateKS           -0.24870   17.33186  -0.014 0.988559    
## stateKY            3.56168   16.30882   0.218 0.827246    
## stateLA            0.73917   16.61934   0.044 0.964549    
## stateMA            0.18353   16.46690   0.011 0.991113    
## stateMD           19.31106   16.35617   1.181 0.238499    
## stateME           -2.02199   21.92339  -0.092 0.926566    
## stateMI            7.34720   16.06901   0.457 0.647776    
## stateMN            3.27208   17.81806   0.184 0.854398    
## stateMO           -0.05557   16.95513  -0.003 0.997387    
## stateMS           -0.01067   17.85912  -0.001 0.999524    
## stateNC            0.75692   16.76942   0.045 0.964023    
## stateND           -8.31927   22.32342  -0.373 0.709608    
## stateNE            1.81415   17.71895   0.102 0.918507    
## stateNH          -29.26246   19.28305  -1.518 0.129991    
## stateNJ           20.06935   16.02458   1.252 0.211214    
## stateNM            3.80066   19.03693   0.200 0.841867    
## stateNV           -1.60254   16.24832  -0.099 0.921487    
## stateNY            3.99519   15.71092   0.254 0.799410    
## stateOH            9.43637   15.92247   0.593 0.553782    
## stateOK           12.39685   16.83889   0.736 0.462075    
## stateOR            0.48465   16.33196   0.030 0.976342    
## statePA            2.76725   15.76791   0.175 0.860784    
## statePR            9.11070   16.83870   0.541 0.588795    
## stateRI            4.01777   18.73917   0.214 0.830350    
## stateSC            5.09637   16.51896   0.309 0.757864    
## stateSD           -1.17365   21.59044  -0.054 0.956678    
## stateTN            0.23167   18.01229   0.013 0.989745    
## stateTX            8.62790   15.52548   0.556 0.578735    
## stateUT           -0.84388   17.26070  -0.049 0.961033    
## stateVA           11.47599   16.28521   0.705 0.481449    
## stateVT            1.88275   21.95387   0.086 0.931704    
## stateWA            3.68880   17.23079   0.214 0.830601    
## stateWI            3.55623   21.82413   0.163 0.870647    
## stateWY            0.82148   17.90189   0.046 0.963425    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.5 on 369 degrees of freedom
## Multiple R-squared:  0.4578, Adjusted R-squared:  0.2667 
## F-statistic: 2.396 on 130 and 369 DF,  p-value: 6.195e-11

anova(lm5)

result$var[5] <- "500: lm(formula = Order_Qty ~ Qt + Brand + promo + month + state,     data = lm_cust_s2)"
result$pval[5] <-   " Adjusted R-squared:  0.05218 F-statistic: 1.214 on 128 and 370 DF,  p-value: 0.08366"
result$comment[5] <- "(Sig: Promo) " 

LM Model 6-10

#6 Dropping Month, I want to keep state for now.

lm6 <- lm(Order_Qty ~ Qt + Brand + promo + state,lm_cust_s2)
summary(lm6)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + state, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.496  -6.406  -0.601   4.280  99.577 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       1.52877   24.63637   0.062 0.950553    
## QtQ1_18           5.06436    7.54958   0.671 0.502750    
## QtQ2_17          -3.00991    3.09533  -0.972 0.331474    
## QtQ2_18           1.72419    7.27105   0.237 0.812684    
## QtQ3_17           5.18133    7.47685   0.693 0.488747    
## QtQ3_18           2.86289    7.65075   0.374 0.708467    
## QtQ4_17           6.41013    7.33828   0.874 0.382936    
## QtQ4_18           3.55857    7.65293   0.465 0.642204    
## BrandAX           7.04313    9.49117   0.742 0.458505    
## BrandBB           4.53298   10.31383   0.440 0.660548    
## BrandBE          27.21911    9.14236   2.977 0.003096 ** 
## BrandBV           7.42837    9.90625   0.750 0.453802    
## BrandCH          20.06635   12.34992   1.625 0.105037    
## BrandDG          12.06067    9.33964   1.291 0.197376    
## BrandDY           0.08956    8.85609   0.010 0.991937    
## BrandEA           5.49184    8.20344   0.669 0.503615    
## BrandHC          18.99537    7.21785   2.632 0.008844 ** 
## BrandMK          16.48295    7.29505   2.259 0.024424 *  
## BrandMU          33.19880   16.32795   2.033 0.042727 *  
## BrandOJ           4.73391    9.19500   0.515 0.606970    
## BrandOO          19.56637    7.06140   2.771 0.005867 ** 
## BrandOX          18.13895    7.26779   2.496 0.012994 *  
## BrandOY          15.12124    8.06852   1.874 0.061690 .  
## BrandPH          17.48872    8.67507   2.016 0.044512 *  
## BrandPO          41.13267   10.30461   3.992 7.89e-05 ***
## BrandPP           8.15352   13.13639   0.621 0.535184    
## BrandPR          18.68787    7.30897   2.557 0.010954 *  
## BrandPS          10.79240    9.16730   1.177 0.239830    
## BrandRA         -99.18107   22.09677  -4.488 9.54e-06 ***
## BrandRB          20.36236    6.81353   2.989 0.002987 ** 
## BrandRJ           6.15807   13.71160   0.449 0.653607    
## BrandRL          10.66471    9.66073   1.104 0.270330    
## BrandRX          19.85388    6.84985   2.898 0.003969 ** 
## BrandRY           4.16747    7.16701   0.581 0.561264    
## BrandSF          -1.07966   13.21749  -0.082 0.934941    
## BrandTF          14.93447    7.77970   1.920 0.055654 .  
## BrandTY          13.68924    7.98476   1.714 0.087273 .  
## BrandVA          -5.67434   12.85949  -0.441 0.659280    
## BrandVE          15.09650    7.75827   1.946 0.052415 .  
## BrandVO          14.01905    7.82057   1.793 0.073841 .  
## promoCODE F     -12.74067   21.59668  -0.590 0.555587    
## promoH15        -13.10862   18.49679  -0.709 0.478950    
## promoLUX40       -0.08512   18.39459  -0.005 0.996310    
## promoMVE150     -17.37885   20.99411  -0.828 0.408309    
## promoMVENB      -24.83259   23.98568  -1.035 0.301188    
## promoNASC75     -19.86089   20.22305  -0.982 0.326684    
## promoNB          -2.23223   21.28790  -0.105 0.916543    
## promoOOX40      -13.04779   20.54753  -0.635 0.525810    
## promoOOX60       -4.77677   24.35194  -0.196 0.844594    
## promoOther      -16.37420   19.68875  -0.832 0.406132    
## promoOY12       -13.00063   21.71437  -0.599 0.549725    
## promoPRE-SELL B -13.45720   26.63859  -0.505 0.613730    
## promoPRE-SELL T -12.32346   24.96930  -0.494 0.621915    
## promoRB15        -8.75614   19.35548  -0.452 0.651251    
## promoRSD        -16.97116   25.29161  -0.671 0.502619    
## promoSUN1        -9.86427   26.39147  -0.374 0.708788    
## promoSY1        -10.77403   20.18025  -0.534 0.593732    
## promoSY100      -15.32144   19.90025  -0.770 0.441834    
## promoSY1000      87.36566   24.67576   3.541 0.000449 ***
## promoSY2          5.12432   20.75449   0.247 0.805119    
## promoSY200        1.95062   19.21182   0.102 0.919182    
## promoSY250        8.24511   20.76480   0.397 0.691539    
## promoSY5        -11.28542   21.53484  -0.524 0.600549    
## promoSYNERGY54  -17.96254   24.83461  -0.723 0.469952    
## promoSYVE       -10.46337   20.82454  -0.502 0.615642    
## promoUP1        -10.07883   18.70540  -0.539 0.590330    
## promoUP3         -7.48757   18.89459  -0.396 0.692122    
## promoVIP        -14.51965   25.20483  -0.576 0.564914    
## promoVP1200     -17.82344   19.92407  -0.895 0.371589    
## promoVPFP150    -28.07846   23.44161  -1.198 0.231746    
## promoVPFP200     39.59204   19.16920   2.065 0.039568 *  
## promoVPNB         3.11183   24.85811   0.125 0.900445    
## promoWC1          2.28277   18.98140   0.120 0.904339    
## promoWC2        -25.25581   25.61904  -0.986 0.324853    
## promoWild Card   -7.92439   20.77213  -0.381 0.703054    
## stateAL          12.09751   18.67516   0.648 0.517518    
## stateAR           6.06923   17.42167   0.348 0.727755    
## stateAZ           9.14826   15.91024   0.575 0.565639    
## stateCA           7.18733   15.51088   0.463 0.643364    
## stateCO           7.89525   16.30694   0.484 0.628550    
## stateCT          10.87731   17.36564   0.626 0.531452    
## stateDE           6.59727   18.49572   0.357 0.721522    
## stateFL           5.40488   15.71230   0.344 0.731045    
## stateGA           6.61596   16.05348   0.412 0.680486    
## stateHI          -0.78531   21.46796  -0.037 0.970839    
## stateIA          -1.92706   21.44819  -0.090 0.928456    
## stateID          -3.73818   18.74433  -0.199 0.842034    
## stateIL           1.01288   16.03717   0.063 0.949674    
## stateIN          14.41846   16.18538   0.891 0.373587    
## stateKS          -1.13977   17.23663  -0.066 0.947313    
## stateKY           5.03695   16.23614   0.310 0.756557    
## stateLA           0.45227   16.52306   0.027 0.978177    
## stateMA           2.22960   16.36496   0.136 0.891702    
## stateMD          19.09073   16.31033   1.170 0.242552    
## stateME          -1.49917   21.80032  -0.069 0.945211    
## stateMI           8.89246   16.02217   0.555 0.579217    
## stateMN           5.56969   17.73045   0.314 0.753595    
## stateMO           1.88272   16.83366   0.112 0.911008    
## stateMS          -0.12679   17.82427  -0.007 0.994328    
## stateNC           0.52579   16.69392   0.031 0.974891    
## stateND          -6.93225   22.25168  -0.312 0.755564    
## stateNE           3.48953   17.59910   0.198 0.842934    
## stateNH         -29.91738   19.23082  -1.556 0.120618    
## stateNJ          21.22228   15.93093   1.332 0.183618    
## stateNM           6.33967   18.84572   0.336 0.736758    
## stateNV          -0.80598   16.16271  -0.050 0.960255    
## stateNY           4.78374   15.61806   0.306 0.759549    
## stateOH          10.35219   15.84978   0.653 0.514062    
## stateOK          14.75048   16.75936   0.880 0.379348    
## stateOR           2.61633   16.21158   0.161 0.871876    
## statePA           3.19331   15.71870   0.203 0.839125    
## statePR           9.87068   16.80498   0.587 0.557309    
## stateRI           5.50006   18.63505   0.295 0.768045    
## stateSC           6.51450   16.42027   0.397 0.691787    
## stateSD          -1.37982   21.47017  -0.064 0.948792    
## stateTN          -0.67807   17.91974  -0.038 0.969836    
## stateTX           9.39156   15.48293   0.607 0.544498    
## stateUT          -1.20692   17.13130  -0.070 0.943872    
## stateVA          12.57227   16.19860   0.776 0.438157    
## stateVT           8.19761   21.67719   0.378 0.705519    
## stateWA           5.65411   17.19492   0.329 0.742471    
## stateWI           5.26781   21.73125   0.242 0.808596    
## stateWY           1.21667   17.82631   0.068 0.945622    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.55 on 377 degrees of freedom
## Multiple R-squared:  0.4421, Adjusted R-squared:  0.2616 
## F-statistic: 2.449 on 122 and 377 DF,  p-value: 3.965e-11

anova(lm6)

result$var[6] <- "lm(formula = Order_Qty ~ Qt + Brand + promo + state, data = lm_cust_s2)"
result$pval[6] <- "Adjusted R-squared:  0.04593 F-statistic:   1.2 on 120 and 378 DF,  p-value: 0.1016"

result$comment[6] <- "promo is very less significant now, nothing else is significant" 

#7 Drop Brand

set.seed(12123)

lm7 <- lm(Order_Qty ~  Qt + promo + state,lm_cust_s2)
summary(lm7)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + promo + state, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.897  -7.220  -1.751   4.574 118.831 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)  
## (Intercept)      17.82699   22.77007   0.783   0.4341  
## QtQ1_18           9.60323    7.94634   1.209   0.2275  
## QtQ2_17          -1.91410    3.24532  -0.590   0.5556  
## QtQ2_18           6.37288    7.66468   0.831   0.4062  
## QtQ3_17           9.74091    7.92989   1.228   0.2200  
## QtQ3_18           6.55771    8.06986   0.813   0.4169  
## QtQ4_17           9.28270    7.82815   1.186   0.2364  
## QtQ4_18           6.49315    8.07874   0.804   0.4220  
## promoCODE F     -21.30117   20.15356  -1.057   0.2912  
## promoH15        -16.22210   16.49983  -0.983   0.3261  
## promoLUX40       -5.98859   16.38959  -0.365   0.7150  
## promoMVE150     -16.49235   22.80517  -0.723   0.4700  
## promoMVENB      -24.58051   23.12634  -1.063   0.2885  
## promoNASC75     -18.45159   18.71766  -0.986   0.3248  
## promoNB          -6.10572   19.96944  -0.306   0.7599  
## promoOOX40      -20.10862   19.03007  -1.057   0.2913  
## promoOOX60      -11.93311   22.68317  -0.526   0.5991  
## promoOther      -22.19987   18.06058  -1.229   0.2197  
## promoOY12       -13.99418   20.50940  -0.682   0.4954  
## promoPRE-SELL B -28.25690   25.30282  -1.117   0.2648  
## promoPRE-SELL T -14.99829   24.01635  -0.625   0.5326  
## promoRB15        -8.81141   17.58271  -0.501   0.6165  
## promoRSD        -25.86523   24.25123  -1.067   0.2868  
## promoSUN1       -24.01901   22.90203  -1.049   0.2949  
## promoSY1        -18.84227   18.51470  -1.018   0.3094  
## promoSY100      -19.76133   18.31480  -1.079   0.2812  
## promoSY1000      27.34825   21.40559   1.278   0.2021  
## promoSY2         -6.17294   19.15482  -0.322   0.7474  
## promoSY200       -6.45183   17.32059  -0.372   0.7097  
## promoSY250       -7.59815   19.07136  -0.398   0.6905  
## promoSY5        -20.75396   20.20567  -1.027   0.3050  
## promoSYNERGY54  -21.42626   24.18976  -0.886   0.3763  
## promoSYVE       -15.85038   19.33258  -0.820   0.4128  
## promoUP1        -13.09966   16.80322  -0.780   0.4361  
## promoUP3        -10.10114   16.98647  -0.595   0.5524  
## promoVIP        -21.86523   24.25123  -0.902   0.3678  
## promoVP1200     -19.81277   18.28019  -1.084   0.2791  
## promoVPFP150    -26.93311   22.68317  -1.187   0.2358  
## promoVPFP200     40.45007   17.39671   2.325   0.0206 *
## promoVPNB        -1.26706   23.74306  -0.053   0.9575  
## promoWC1         -2.23708   17.04531  -0.131   0.8956  
## promoWC2        -21.86523   24.25123  -0.902   0.3678  
## promoWild Card  -17.18915   19.38265  -0.887   0.3757  
## stateAL           9.92992   19.62458   0.506   0.6131  
## stateAR           4.21711   17.92803   0.235   0.8142  
## stateAZ          11.78135   16.64629   0.708   0.4795  
## stateCA          10.10612   16.13699   0.626   0.5315  
## stateCO          13.39658   16.98687   0.789   0.4308  
## stateCT          15.71661   17.99018   0.874   0.3828  
## stateDE          11.38483   19.46232   0.585   0.5589  
## stateFL           9.17301   16.36422   0.561   0.5754  
## stateGA           8.75352   16.71646   0.524   0.6008  
## stateHI           3.76965   22.63566   0.167   0.8678  
## stateIA           5.09018   22.66199   0.225   0.8224  
## stateID           2.02190   19.76866   0.102   0.9186  
## stateIL           4.61969   16.85942   0.274   0.7842  
## stateIN          12.93396   16.92027   0.764   0.4451  
## stateKS           0.17131   17.93655   0.010   0.9924  
## stateKY           7.78882   16.92252   0.460   0.6456  
## stateLA           4.71040   17.24343   0.273   0.7849  
## stateMA           9.65741   17.03375   0.567   0.5711  
## stateMD          20.85035   16.89661   1.234   0.2179  
## stateME           0.09018   22.66199   0.004   0.9968  
## stateMI          10.16827   16.66883   0.610   0.5422  
## stateMN           1.89110   18.51469   0.102   0.9187  
## stateMO           4.23787   17.52649   0.242   0.8091  
## stateMS          -1.91202   18.45235  -0.104   0.9175  
## stateNC           5.53496   17.33964   0.319   0.7497  
## stateND           6.87973   22.65604   0.304   0.7615  
## stateNE           5.04570   18.96670   0.266   0.7904  
## stateNH         -15.25370   19.90810  -0.766   0.4440  
## stateNJ          17.33696   16.56044   1.047   0.2958  
## stateNM           5.19970   19.68206   0.264   0.7918  
## stateNV           0.28869   16.83305   0.017   0.9863  
## stateNY           7.66536   16.25692   0.472   0.6375  
## stateOH          13.30330   16.47950   0.807   0.4200  
## stateOK          15.07647   17.44027   0.864   0.3878  
## stateOR           3.20506   16.87591   0.190   0.8495  
## statePA           5.30031   16.44689   0.322   0.7474  
## statePR          15.44008   17.51342   0.882   0.3785  
## stateRI          11.42992   19.62458   0.582   0.5606  
## stateSC           7.98442   17.05499   0.468   0.6399  
## stateSD           4.81517   22.66903   0.212   0.8319  
## stateTN           1.02774   18.70072   0.055   0.9562  
## stateTX          10.95956   16.18455   0.677   0.4987  
## stateUT           5.22434   17.92671   0.291   0.7709  
## stateVA          12.53188   16.86834   0.743   0.4580  
## stateVT          14.18677   22.96257   0.618   0.5370  
## stateWA           3.36819   17.97034   0.187   0.8514  
## stateWI           5.87973   22.65604   0.260   0.7954  
## stateWY           3.15485   18.50057   0.171   0.8647  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.83 on 409 degrees of freedom
## Multiple R-squared:  0.283,  Adjusted R-squared:  0.1252 
## F-statistic: 1.794 on 90 and 409 DF,  p-value: 7.364e-05

anova(lm7)

result$var[7] <- "lm(formula = Order_Qty ~ Qt + promo + state, data = lm_cust_s2)"
result$pval[7] <- "Adjusted R-squared:  0.04285 F-statistic: 1.251 on 89 and 409 DF,  p-value: 0.07818"

result$comment[7] <- "not much. sig " 



#8 Drop state and Brand

set.seed(12126)

lm8 <- lm(Order_Qty ~    Qt + promo + Brand,lm_cust_s2)
summary(lm8)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + promo + Brand, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.996  -7.220  -1.227   4.521 112.894 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      10.91788   19.09197   0.572 0.567721    
## QtQ1_18           7.06051    7.25004   0.974 0.330681    
## QtQ2_17          -0.05754    2.94410  -0.020 0.984417    
## QtQ2_18           5.66405    6.97502   0.812 0.417219    
## QtQ3_17           6.95078    7.18832   0.967 0.334117    
## QtQ3_18           6.47498    7.27275   0.890 0.373805    
## QtQ4_17           9.15344    7.03005   1.302 0.193606    
## QtQ4_18           6.83850    7.35370   0.930 0.352930    
## promoCODE F     -19.32242   20.99467  -0.920 0.357913    
## promoH15        -14.90217   18.25338  -0.816 0.414725    
## promoLUX40       -0.38336   18.19446  -0.021 0.983200    
## promoMVE150     -18.00000   20.67632  -0.871 0.384485    
## promoMVENB      -24.18558   23.39922  -1.034 0.301908    
## promoNASC75     -18.60921   20.09430  -0.926 0.354923    
## promoNB          -0.55462   20.92204  -0.027 0.978864    
## promoOOX40       -9.38359   20.08931  -0.467 0.640672    
## promoOOX60       -4.36790   24.14594  -0.181 0.856535    
## promoOther      -19.56951   19.20950  -1.019 0.308905    
## promoOY12       -18.68670   20.86216  -0.896 0.370908    
## promoPRE-SELL B -14.45229   24.36992  -0.593 0.553471    
## promoPRE-SELL T -21.19629   23.51011  -0.902 0.367790    
## promoRB15       -10.69289   19.24500  -0.556 0.578763    
## promoRSD        -23.86034   24.48004  -0.975 0.330271    
## promoSUN1        -6.71018   25.46497  -0.264 0.792288    
## promoSY1        -15.86576   19.61723  -0.809 0.419102    
## promoSY100      -15.81382   19.40307  -0.815 0.415519    
## promoSY1000      85.36565   24.27172   3.517 0.000483 ***
## promoSY2         -0.17745   20.28992  -0.009 0.993026    
## promoSY200       -2.30075   18.99052  -0.121 0.903627    
## promoSY250        7.88248   20.26298   0.389 0.697465    
## promoSY5        -19.02459   20.74543  -0.917 0.359637    
## promoSYNERGY54  -20.02408   24.42109  -0.820 0.412704    
## promoSYVE       -14.99847   20.36204  -0.737 0.461778    
## promoUP1        -10.87043   18.42731  -0.590 0.555565    
## promoUP3         -8.68283   18.54037  -0.468 0.639796    
## promoVIP        -21.11033   24.43453  -0.864 0.388100    
## promoVP1200     -15.66941   19.51538  -0.803 0.422466    
## promoVPFP150    -27.16314   23.23322  -1.169 0.242998    
## promoVPFP200     34.83268   18.99865   1.833 0.067438 .  
## promoVPNB         7.03219   23.73477   0.296 0.767159    
## promoWC1         -0.20502   18.68815  -0.011 0.991252    
## promoWC2        -29.37264   24.73669  -1.187 0.235729    
## promoWild Card   -9.71568   20.40087  -0.476 0.634149    
## BrandAX           8.09437    9.27029   0.873 0.383073    
## BrandBB           4.40083    9.78474   0.450 0.653110    
## BrandBE          22.79071    8.51049   2.678 0.007694 ** 
## BrandBV           4.53441    9.79543   0.463 0.643665    
## BrandCH          16.08212   12.27795   1.310 0.190958    
## BrandDG           9.45003    9.09381   1.039 0.299316    
## BrandDY          -1.10752    8.64108  -0.128 0.898075    
## BrandEA           7.20300    7.90716   0.911 0.362839    
## BrandHC          17.34540    7.00349   2.477 0.013649 *  
## BrandMK          14.52840    7.09902   2.047 0.041318 *  
## BrandMU          28.17666   16.12137   1.748 0.081224 .  
## BrandOJ           3.21066    8.78855   0.365 0.715052    
## BrandOO          16.49300    6.88708   2.395 0.017063 *  
## BrandOX          16.72909    7.04730   2.374 0.018048 *  
## BrandOY          11.80856    7.64626   1.544 0.123246    
## BrandPH          14.04994    8.35522   1.682 0.093386 .  
## BrandPO          30.19333    9.87588   3.057 0.002375 ** 
## BrandPP           1.55707   12.23662   0.127 0.898805    
## BrandPR          15.26770    7.09767   2.151 0.032033 *  
## BrandPS           5.72306    8.91862   0.642 0.521415    
## BrandRA         -87.34404   21.71560  -4.022 6.82e-05 ***
## BrandRB          17.24526    6.64785   2.594 0.009811 ** 
## BrandRJ          -2.15016   12.25389  -0.175 0.860796    
## BrandRL          10.55406    9.33896   1.130 0.259067    
## BrandRX          17.65596    6.63757   2.660 0.008109 ** 
## BrandRY           2.07458    6.95340   0.298 0.765578    
## BrandSF           0.06096   12.60019   0.005 0.996142    
## BrandTF          13.27841    7.48407   1.774 0.076742 .  
## BrandTY           9.80262    7.69471   1.274 0.203380    
## BrandVA          -1.63487   12.07466  -0.135 0.892362    
## BrandVE          13.33501    7.54497   1.767 0.077878 .  
## BrandVO          12.43224    7.38455   1.684 0.093003 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.62 on 425 degrees of freedom
## Multiple R-squared:  0.3647, Adjusted R-squared:  0.2541 
## F-statistic: 3.297 on 74 and 425 DF,  p-value: 1.028e-14

anova(lm8)

result$var[8] <- "lm(formula = Order_Qty ~ Qt + promo + Brand, data = lm_cust_s2)"
result$pval[8] <- "Adjusted R-squared:  0.04555 F-statistic:  1.34 on 70 and 429 DF,  p-value: 0.04416"
result$comment[8] <- "not much sig." 


lm9 <- lm(formula = Order_Qty ~ Qt + Brand + promo + month, data = lm_cust_s2)  
summary(lm9)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + month, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.750  -7.022  -1.354   4.585 109.021 
## 
## Coefficients: (3 not defined because of singularities)
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       6.181159  19.234168   0.321 0.748096    
## QtQ1_18           7.576366   7.288913   1.039 0.299204    
## QtQ2_17           6.971625   4.061219   1.717 0.086789 .  
## QtQ2_18          13.110317   7.576606   1.730 0.084304 .  
## QtQ3_17          10.859031   7.562334   1.436 0.151770    
## QtQ3_18           9.437978   7.674453   1.230 0.219469    
## QtQ4_17          14.695165   7.648736   1.921 0.055381 .  
## QtQ4_18          11.955958   7.857497   1.522 0.128868    
## BrandAX           7.161199   9.309875   0.769 0.442207    
## BrandBB           3.163395   9.832283   0.322 0.747814    
## BrandBE          21.289024   8.559222   2.487 0.013263 *  
## BrandBV           4.306264   9.808624   0.439 0.660868    
## BrandCH          13.615238  12.280327   1.109 0.268197    
## BrandDG           6.527993   9.189764   0.710 0.477881    
## BrandDY          -3.075941   8.673155  -0.355 0.723031    
## BrandEA           6.097335   7.900319   0.772 0.440680    
## BrandHC          16.461769   7.044462   2.337 0.019920 *  
## BrandMK          13.242229   7.129100   1.857 0.063946 .  
## BrandMU          23.321242  16.118782   1.447 0.148694    
## BrandOJ           0.297105   8.930357   0.033 0.973476    
## BrandOO          15.286678   6.932210   2.205 0.027988 *  
## BrandOX          15.016663   7.082081   2.120 0.034564 *  
## BrandOY          11.164944   7.687309   1.452 0.147146    
## BrandPH          13.325576   8.408063   1.585 0.113757    
## BrandPO          31.029831   9.870574   3.144 0.001788 ** 
## BrandPP           3.293757  12.202494   0.270 0.787352    
## BrandPR          14.886553   7.105670   2.095 0.036772 *  
## BrandPS           4.155378   8.918288   0.466 0.641502    
## BrandRA         -88.924971  21.607401  -4.115 4.66e-05 ***
## BrandRB          16.552876   6.690870   2.474 0.013760 *  
## BrandRJ          -1.637048  12.292957  -0.133 0.894123    
## BrandRL           9.768585   9.307913   1.049 0.294559    
## BrandRX          16.075029   6.688829   2.403 0.016685 *  
## BrandRY           1.844805   6.966909   0.265 0.791298    
## BrandSF           0.314770  12.634630   0.025 0.980136    
## BrandTF          12.118302   7.567280   1.601 0.110044    
## BrandTY           8.725250   7.757849   1.125 0.261363    
## BrandVA          -6.085671  12.201965  -0.499 0.618222    
## BrandVE          11.609611   7.606555   1.526 0.127702    
## BrandVO          11.373074   7.401579   1.537 0.125156    
## promoCODE F     -17.884516  21.082302  -0.848 0.396747    
## promoH15        -13.772933  18.309700  -0.752 0.452343    
## promoLUX40        0.702886  18.275956   0.038 0.969340    
## promoMVE150     -18.000000  20.546030  -0.876 0.381490    
## promoMVENB      -26.271315  23.310316  -1.127 0.260380    
## promoNASC75     -16.569757  20.157356  -0.822 0.411535    
## promoNB           3.489081  20.994132   0.166 0.868085    
## promoOOX40      -10.146352  20.070234  -0.506 0.613445    
## promoOOX60       -3.912756  24.049823  -0.163 0.870838    
## promoOther      -19.060061  19.188540  -0.993 0.321138    
## promoOY12       -20.397441  20.848811  -0.978 0.328468    
## promoPRE-SELL B -17.534777  24.515590  -0.715 0.474855    
## promoPRE-SELL T -23.346814  23.681710  -0.986 0.324774    
## promoRB15       -10.723265  19.208508  -0.558 0.576969    
## promoRSD        -22.269129  24.612582  -0.905 0.366101    
## promoSUN1        -9.515735  25.530055  -0.373 0.709541    
## promoSY1        -15.373870  19.600746  -0.784 0.433279    
## promoSY100      -15.144423  19.366255  -0.782 0.434658    
## promoSY1000      83.963843  24.164782   3.475 0.000565 ***
## promoSY2          0.525059  20.289440   0.026 0.979367    
## promoSY200       -3.446806  19.037747  -0.181 0.856415    
## promoSY250        8.621645  20.290303   0.425 0.671118    
## promoSY5        -19.400331  20.703050  -0.937 0.349262    
## promoSYNERGY54  -21.224502  24.379222  -0.871 0.384475    
## promoSYVE       -11.979047  20.336611  -0.589 0.556154    
## promoUP1         -9.382130  18.506440  -0.507 0.612447    
## promoUP3         -7.537797  18.593049  -0.405 0.685385    
## promoVIP        -22.533705  24.431281  -0.922 0.356890    
## promoVP1200     -13.736997  19.548343  -0.703 0.482623    
## promoVPFP150    -28.937638  23.148551  -1.250 0.211970    
## promoVPFP200     32.811914  18.946381   1.732 0.084043 .  
## promoVPNB         5.289662  23.660668   0.224 0.823206    
## promoWC1          0.531093  18.676424   0.028 0.977328    
## promoWC2        -30.580500  24.709737  -1.238 0.216565    
## promoWild Card  -10.481975  20.330390  -0.516 0.606420    
## month2            8.047354   3.030940   2.655 0.008233 ** 
## month3            7.203604   3.156798   2.282 0.022996 *  
## month4           -3.279584   3.710154  -0.884 0.377234    
## month5           -3.140648   3.652239  -0.860 0.390325    
## month6                  NA         NA      NA       NA    
## month7            0.001712   3.539126   0.000 0.999614    
## month8            6.119683   3.638560   1.682 0.093338 .  
## month9                  NA         NA      NA       NA    
## month10          -1.271845   3.749527  -0.339 0.734629    
## month11           1.200833   3.534208   0.340 0.734198    
## month12                 NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.53 on 417 degrees of freedom
## Multiple R-squared:  0.3845, Adjusted R-squared:  0.2635 
## F-statistic: 3.177 on 82 and 417 DF,  p-value: 1.158e-14

anova(lm9)

result$var[9] <- "lm(formula = Order_Qty ~ Qt + Brand + promo + month, data = lm_cust_s2)"
result$pval[9] <- "Adjusted R-squared:  0.05115 F-statistic: 1.345 on 78 and 421 DF,  p-value: 0.03633"
result$comment[9] <- "Pvalue looks less than  to .05"
## 10 
lm10 <- lm(formula = Order_Qty ~ Qt + promo + month, data = lm_cust_s2)
summary(lm10)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + promo + month, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.500  -7.557  -1.736   4.402 122.450 
## 
## Coefficients: (3 not defined because of singularities)
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)      19.0988    15.7928   1.209   0.2272  
## QtQ1_18          10.9097     7.5123   1.452   0.1471  
## QtQ2_17           6.0304     4.1704   1.446   0.1489  
## QtQ2_18          15.5235     7.8028   1.989   0.0473 *
## QtQ3_17          12.8233     7.8312   1.637   0.1022  
## QtQ3_18          10.9335     7.9773   1.371   0.1712  
## QtQ4_17          17.9925     7.9371   2.267   0.0239 *
## QtQ4_18          14.5858     8.0892   1.803   0.0720 .
## promoCODE F     -23.0839    19.0287  -1.213   0.2257  
## promoH15        -14.2361    15.9386  -0.893   0.3722  
## promoLUX40       -2.6157    15.8592  -0.165   0.8691  
## promoMVE150     -18.0000    21.8516  -0.824   0.4105  
## promoMVENB      -25.0000    21.8516  -1.144   0.2532  
## promoNASC75     -12.6392    18.3653  -0.688   0.4917  
## promoNB           1.7976    19.1932   0.094   0.9254  
## promoOOX40      -13.7528    18.0629  -0.761   0.4468  
## promoOOX60      -11.0000    21.8516  -0.503   0.6149  
## promoOther      -21.2966    17.1376  -1.243   0.2146  
## promoOY12       -20.4495    18.9982  -1.076   0.2823  
## promoPRE-SELL B -25.8990    22.1075  -1.172   0.2420  
## promoPRE-SELL T -23.8990    22.1075  -1.081   0.2803  
## promoRB15        -7.3206    17.0616  -0.429   0.6681  
## promoRSD        -26.6610    23.1229  -1.153   0.2495  
## promoSUN1       -23.1292    22.2566  -1.039   0.2993  
## promoSY1        -18.6139    17.5124  -1.063   0.2884  
## promoSY100      -16.1685    17.3743  -0.931   0.3526  
## promoSY1000      30.5903    20.3606   1.502   0.1337  
## promoSY2         -4.3766    18.2818  -0.239   0.8109  
## promoSY200       -6.0972    16.8391  -0.362   0.7175  
## promoSY250       -4.2586    18.1881  -0.234   0.8150  
## promoSY5        -26.0564    18.9321  -1.376   0.1694  
## promoSYNERGY54  -21.4281    23.2410  -0.922   0.3570  
## promoSYVE       -14.1990    18.4580  -0.769   0.4421  
## promoUP1         -9.7744    16.1217  -0.606   0.5446  
## promoUP3         -7.5879    16.2651  -0.467   0.6411  
## promoVIP        -24.6224    23.1671  -1.063   0.2884  
## promoVP1200     -13.2742    17.4166  -0.762   0.4464  
## promoVPFP150    -26.0000    21.8516  -1.190   0.2347  
## promoVPFP200     35.5000    16.6895   2.127   0.0340 *
## promoVPNB         5.0000    21.8516   0.229   0.8191  
## promoWC1         -1.1299    16.3643  -0.069   0.9450  
## promoWC2        -24.6224    23.1671  -1.063   0.2884  
## promoWild Card  -15.5092    18.5234  -0.837   0.4029  
## month2            7.8002     3.1045   2.513   0.0123 *
## month3            7.9012     3.2656   2.420   0.0159 *
## month4           -2.8233     3.8337  -0.736   0.4618  
## month5           -1.9614     3.6842  -0.532   0.5947  
## month6                NA         NA      NA       NA  
## month7            1.7468     3.6685   0.476   0.6342  
## month8            6.7384     3.7294   1.807   0.0715 .
## month9                NA         NA      NA       NA  
## month10          -4.4498     3.7932  -1.173   0.2414  
## month11           0.7435     3.6338   0.205   0.8380  
## month12               NA         NA      NA       NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.45 on 449 degrees of freedom
## Multiple R-squared:  0.2504, Adjusted R-squared:  0.1669 
## F-statistic: 2.999 on 50 and 449 DF,  p-value: 6.715e-10

anova(lm10)

result$var[10] <- "lm(formula = Order_Qty ~ Qt + promo + month, data = lm_cust_s2)"
result$pval[10] <- "Adjusted R-squared:  0.0448 
F-statistic: 1.498 on 47 and 452 DF,  p-value: 0.02147"
result$comment[10] <- "not much sig."

Final Model

# p-value: 0.02331
lm_pbm <- lm(formula = Order_Qty ~  promo + Brand + month, data = lm_cust_s2)

summary(lm(formula = Order_Qty ~  promo + Brand + month, data = lm_cust_s2))

## 
## Call:
## lm(formula = Order_Qty ~ promo + Brand + month, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.900  -7.122  -1.347   4.404 108.734 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       6.4594    19.1440   0.337  0.73597    
## promoCODE F     -18.4376    21.0272  -0.877  0.38107    
## promoH15        -14.2338    18.2601  -0.780  0.43612    
## promoLUX40        0.2627    18.2282   0.014  0.98851    
## promoMVE150     -18.0000    20.5002  -0.878  0.38042    
## promoMVENB      -26.9237    23.2311  -1.159  0.24713    
## promoNASC75     -16.6401    20.0516  -0.830  0.40709    
## promoNB           3.1356    20.9442   0.150  0.88106    
## promoOOX40      -10.3352    20.0067  -0.517  0.60571    
## promoOOX60       -4.0954    23.9651  -0.171  0.86439    
## promoOther      -12.6564    18.0043  -0.703  0.48246    
## promoOY12       -21.0095    20.7784  -1.011  0.31254    
## promoPRE-SELL B -18.6297    24.4131  -0.763  0.44583    
## promoPRE-SELL T -24.6383    23.5948  -1.044  0.29698    
## promoRB15       -11.1523    19.1511  -0.582  0.56065    
## promoRSD        -17.0222    23.7796  -0.716  0.47449    
## promoSUN1        -9.2637    25.4574  -0.364  0.71612    
## promoSY1         -9.8998    18.4399  -0.537  0.59164    
## promoSY100       -7.8949    18.2787  -0.432  0.66602    
## promoSY1000      90.9055    23.0554   3.943 9.43e-05 ***
## promoSY2          5.3394    19.0799   0.280  0.77973    
## promoSY200       -2.7275    18.9362  -0.144  0.88554    
## promoSY250       15.5501    19.0269   0.817  0.41424    
## promoSY5        -14.3404    19.5531  -0.733  0.46372    
## promoSYNERGY54  -16.3377    23.4664  -0.696  0.48668    
## promoSYVE        -6.3770    19.1916  -0.332  0.73984    
## promoUP1         -9.8059    18.4535  -0.531  0.59543    
## promoUP3         -8.1234    18.5388  -0.438  0.66148    
## promoVIP        -16.4256    23.4458  -0.701  0.48395    
## promoVP1200     -14.5300    19.4735  -0.746  0.45600    
## promoVPFP150    -29.6386    23.0746  -1.284  0.19968    
## promoVPFP200     32.2610    18.8836   1.708  0.08830 .  
## promoVPNB         5.2297    23.5957   0.222  0.82470    
## promoWC1          3.4220    18.4043   0.186  0.85259    
## promoWC2        -24.6812    23.9193  -1.032  0.30273    
## promoWild Card   -3.6196    19.0762  -0.190  0.84960    
## BrandAX           7.3029     9.2693   0.788  0.43123    
## BrandBB           3.7440     9.7932   0.382  0.70243    
## BrandBE          21.9211     8.5025   2.578  0.01027 *  
## BrandBV           5.1027     9.7177   0.525  0.59980    
## BrandCH          13.5505    12.1828   1.112  0.26666    
## BrandDG           6.6459     9.0846   0.732  0.46485    
## BrandDY          -2.6249     8.6188  -0.305  0.76086    
## BrandEA           5.9863     7.8611   0.762  0.44678    
## BrandHC          17.0837     7.0012   2.440  0.01509 *  
## BrandMK          13.6655     7.0639   1.935  0.05372 .  
## BrandMU          23.3555    16.0212   1.458  0.14565    
## BrandOJ           0.8699     8.8906   0.098  0.92211    
## BrandOO          15.6215     6.8855   2.269  0.02379 *  
## BrandOX          15.4978     7.0464   2.199  0.02839 *  
## BrandOY          11.5447     7.6402   1.511  0.13153    
## BrandPH          13.3208     8.3141   1.602  0.10986    
## BrandPO          31.4128     9.8303   3.196  0.00150 ** 
## BrandPP           4.0117    12.1392   0.330  0.74120    
## BrandPR          15.4742     7.0633   2.191  0.02901 *  
## BrandPS           4.7080     8.8569   0.532  0.59531    
## BrandRA         -88.3550    21.5473  -4.101 4.95e-05 ***
## BrandRB          17.1891     6.6340   2.591  0.00990 ** 
## BrandRJ          -1.4965    12.2085  -0.123  0.90250    
## BrandRL          10.4400     9.2607   1.127  0.26024    
## BrandRX          16.6450     6.6352   2.509  0.01250 *  
## BrandRY           2.1357     6.9128   0.309  0.75751    
## BrandSF           0.9081    12.5728   0.072  0.94246    
## BrandTF          13.1113     7.5031   1.747  0.08129 .  
## BrandTY           9.2698     7.7089   1.202  0.22986    
## BrandVA          -5.4738    12.1610  -0.450  0.65286    
## BrandVE          11.7638     7.5700   1.554  0.12093    
## BrandVO          11.6822     7.3574   1.588  0.11308    
## month2            8.0676     3.0220   2.670  0.00789 ** 
## month3            6.9900     3.1444   2.223  0.02674 *  
## month4            2.8747     3.4278   0.839  0.40215    
## month5            3.4515     3.3683   1.025  0.30610    
## month6            6.3007     3.4371   1.833  0.06749 .  
## month7            2.8340     3.6265   0.781  0.43497    
## month8            8.8415     3.5805   2.469  0.01393 *  
## month9            3.0164     3.3874   0.890  0.37373    
## month10           5.5656     3.5247   1.579  0.11508    
## month11           7.4041     3.4243   2.162  0.03117 *  
## month12           6.1625     3.5228   1.749  0.08096 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.5 on 421 degrees of freedom
## Multiple R-squared:  0.3814, Adjusted R-squared:  0.2668 
## F-statistic: 3.328 on 78 and 421 DF,  p-value: 2.198e-15

anova(lm(formula = Order_Qty ~  promo + Brand + month, data = lm_cust_s2))

# p-value: 0.0177
lm_pd <- lm(formula = Order_Qty ~  promo  + `Order Date`, data = lm_cust_s2)
anova(lm(formula = Order_Qty ~  promo  + `Order Date`, data = lm_cust_s2))

summary(lm(formula = Order_Qty ~  promo  + `Order Date`, data = lm_cust_s2))

## 
## Call:
## lm(formula = Order_Qty ~ promo + `Order Date`, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.499  -7.737  -2.688   4.403 125.378 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)  
## (Intercept)      3.034e+01  1.039e+02   0.292   0.7703  
## promoCODE F     -2.499e+01  1.900e+01  -1.315   0.1891  
## promoH15        -1.737e+01  1.580e+01  -1.099   0.2722  
## promoLUX40      -6.105e+00  1.572e+01  -0.388   0.6979  
## promoMVE150     -1.800e+01  2.194e+01  -0.821   0.4123  
## promoMVENB      -2.500e+01  2.194e+01  -1.140   0.2550  
## promoNASC75     -1.732e+01  1.791e+01  -0.967   0.3340  
## promoNB         -4.502e+00  1.900e+01  -0.237   0.8128  
## promoOOX40      -1.498e+01  1.792e+01  -0.836   0.4034  
## promoOOX60      -1.100e+01  2.194e+01  -0.501   0.6163  
## promoOther      -1.420e+01  1.572e+01  -0.903   0.3669  
## promoOY12       -2.050e+01  1.900e+01  -1.079   0.2811  
## promoPRE-SELL B -2.600e+01  2.194e+01  -1.185   0.2365  
## promoPRE-SELL T -2.401e+01  2.194e+01  -1.094   0.2744  
## promoRB15       -9.391e+00  1.699e+01  -0.553   0.5808  
## promoRSD        -2.091e+01  2.209e+01  -0.947   0.3443  
## promoSUN1       -2.498e+01  2.195e+01  -1.138   0.2557  
## promoSY1        -1.257e+01  1.624e+01  -0.774   0.4395  
## promoSY100      -8.962e+00  1.579e+01  -0.567   0.5707  
## promoSY1000      4.157e+01  1.913e+01   2.173   0.0303 *
## promoSY2         1.264e+00  1.700e+01   0.074   0.9407  
## promoSY200      -6.602e+00  1.647e+01  -0.401   0.6887  
## promoSY250       3.691e+00  1.658e+01   0.223   0.8239  
## promoSY5        -1.888e+01  1.776e+01  -1.063   0.2883  
## promoSYNERGY54  -1.388e+01  2.226e+01  -0.623   0.5334  
## promoSYVE       -1.106e+01  1.709e+01  -0.647   0.5181  
## promoUP1        -1.334e+01  1.596e+01  -0.836   0.4036  
## promoUP3        -1.055e+01  1.620e+01  -0.651   0.5154  
## promoVIP        -1.691e+01  2.212e+01  -0.764   0.4451  
## promoVP1200     -1.726e+01  1.734e+01  -0.995   0.3203  
## promoVPFP150    -2.600e+01  2.194e+01  -1.185   0.2365  
## promoVPFP200     3.550e+01  1.675e+01   2.119   0.0346 *
## promoVPNB        5.003e+00  2.194e+01   0.228   0.8197  
## promoWC1         5.657e-01  1.600e+01   0.035   0.9718  
## promoWC2        -1.691e+01  2.211e+01  -0.765   0.4448  
## promoWild Card  -6.129e+00  1.713e+01  -0.358   0.7206  
## `Order Date`    -2.245e-09  6.901e-08  -0.033   0.9741  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.51 on 463 degrees of freedom
## Multiple R-squared:  0.221,  Adjusted R-squared:  0.1605 
## F-statistic:  3.65 on 36 and 463 DF,  p-value: 6.218e-11

#p-value: 0.0169
lm_py <- lm(formula = Order_Qty ~  promo  + year(`Order Date`), data = lm_cust_s2)
summary(lm(formula = Order_Qty ~  promo  + year(`Order Date`), data = lm_cust_s2))

## 
## Call:
## lm(formula = Order_Qty ~ promo + year(`Order Date`), data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.500  -7.731  -2.567   4.491 125.375 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)  
## (Intercept)        1177.2421  4227.6508   0.278   0.7808  
## promoCODE F         -25.0000    18.9955  -1.316   0.1888  
## promoH15            -17.3704    15.7943  -1.100   0.2720  
## promoLUX40           -6.1081    15.7179  -0.389   0.6977  
## promoMVE150         -18.0000    21.9341  -0.821   0.4123  
## promoMVENB          -25.0000    21.9341  -1.140   0.2550  
## promoNASC75         -17.3333    17.9091  -0.968   0.3336  
## promoNB              -4.5000    18.9955  -0.237   0.8128  
## promoOOX40          -15.0000    17.9091  -0.838   0.4027  
## promoOOX60          -11.0000    21.9341  -0.502   0.6163  
## promoOther          -13.8624    15.6149  -0.888   0.3751  
## promoOY12           -20.5000    18.9955  -1.079   0.2811  
## promoPRE-SELL B     -26.0000    21.9341  -1.185   0.2365  
## promoPRE-SELL T     -24.0000    21.9341  -1.094   0.2744  
## promoRB15            -9.4000    16.9900  -0.553   0.5803  
## promoRSD            -20.4297    22.0340  -0.927   0.3543  
## promoSUN1           -25.0000    21.9341  -1.140   0.2550  
## promoSY1            -12.0964    16.0720  -0.753   0.4520  
## promoSY100           -9.0000    15.7502  -0.571   0.5680  
## promoSY1000          42.0703    19.1107   2.201   0.0282 *
## promoSY2              1.7131    16.7125   0.103   0.9184  
## promoSY200           -6.6250    16.4505  -0.403   0.6873  
## promoSY250            4.1953    16.5835   0.253   0.8004  
## promoSY5            -18.4297    17.4666  -1.055   0.2919  
## promoSYNERGY54      -13.4297    22.0340  -0.610   0.5425  
## promoSYVE           -10.5964    16.8830  -0.628   0.5306  
## promoUP1            -13.3529    15.9594  -0.837   0.4032  
## promoUP3            -10.5455    16.1994  -0.651   0.5154  
## promoVIP            -16.4297    22.0340  -0.746   0.4563  
## promoVP1200         -17.2500    17.3404  -0.995   0.3204  
## promoVPFP150        -26.0000    21.9341  -1.185   0.2365  
## promoVPFP200         35.5000    16.7524   2.119   0.0346 *
## promoVPNB             5.0000    21.9341   0.228   0.8198  
## promoWC1              0.6971    15.9713   0.044   0.9652  
## promoWC2            -16.4297    22.0340  -0.746   0.4563  
## promoWild Card       -5.6297    17.1188  -0.329   0.7424  
## year(`Order Date`)   -0.5703     2.0960  -0.272   0.7857  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.51 on 463 degrees of freedom
## Multiple R-squared:  0.2212, Adjusted R-squared:  0.1606 
## F-statistic: 3.652 on 36 and 463 DF,  p-value: 6.05e-11

anova(lm(formula = Order_Qty ~  promo  + year(`Order Date`), data = lm_cust_s2))

#p-value: 0.4186  Not considering 
summary(lm(formula = Order_Qty ~  month + Brand + Qt , data = lm_cust_s2))

## 
## Call:
## lm(formula = Order_Qty ~ month + Brand + Qt, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.096  -7.671  -2.384   4.219 135.632 
## 
## Coefficients: (3 not defined because of singularities)
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.55179    7.20768   1.325 0.185766    
## month2        8.26770    3.34414   2.472 0.013792 *  
## month3       11.47808    3.24462   3.538 0.000446 ***
## month4       -1.24341    4.20529  -0.296 0.767612    
## month5       -0.02701    4.34391  -0.006 0.995041    
## month6        3.33782    4.11187   0.812 0.417362    
## month7       -2.21626    4.50009  -0.492 0.622611    
## month8        3.78240    4.25074   0.890 0.374035    
## month9       -0.96426    4.00482  -0.241 0.809839    
## month10       0.44848    4.59902   0.098 0.922359    
## month11       2.24251    4.09957   0.547 0.584641    
## month12       3.22126    4.13088   0.780 0.435917    
## BrandAX      -0.55422   10.09759  -0.055 0.956253    
## BrandBB      -9.70195   10.72204  -0.905 0.366021    
## BrandBE      11.51707    9.03659   1.274 0.203144    
## BrandBV      -8.64616   10.09153  -0.857 0.392024    
## BrandCH      -0.47967   10.81382  -0.044 0.964639    
## BrandDG       0.36055    9.64851   0.037 0.970208    
## BrandDY      -0.82685    9.60699  -0.086 0.931451    
## BrandEA      -0.42653    8.45039  -0.050 0.959766    
## BrandHC       6.21782    7.49232   0.830 0.407038    
## BrandMK       1.44245    7.53605   0.191 0.848293    
## BrandMU      12.66581   18.00761   0.703 0.482194    
## BrandOJ     -14.21209    9.33267  -1.523 0.128500    
## BrandOO       6.06612    7.37329   0.823 0.411104    
## BrandOX       1.19909    7.49510   0.160 0.872965    
## BrandOY      -1.48787    8.16462  -0.182 0.855481    
## BrandPH       3.08764    8.85312   0.349 0.727430    
## BrandPO      28.71730   10.70881   2.682 0.007594 ** 
## BrandPP      -4.42259   13.46163  -0.329 0.742661    
## BrandPR       5.79856    7.56799   0.766 0.443960    
## BrandPS      -5.60724    9.61817  -0.583 0.560194    
## BrandRA      -2.85636   17.94420  -0.159 0.873598    
## BrandRB       6.33812    7.10467   0.892 0.372811    
## BrandRJ     -10.96205   11.74599  -0.933 0.351185    
## BrandRL      -3.57699   10.01874  -0.357 0.721236    
## BrandRX       5.86175    7.06205   0.830 0.406957    
## BrandRY      -6.46912    7.42940  -0.871 0.384355    
## BrandSF      -2.59908   13.57246  -0.191 0.848222    
## BrandTF       2.68708    7.95910   0.338 0.735813    
## BrandTY      -1.50482    8.23192  -0.183 0.855035    
## BrandVA      -5.14599   13.69580  -0.376 0.707291    
## BrandVE      -0.37324    8.18324  -0.046 0.963641    
## BrandVO       0.65988    7.88283   0.084 0.933323    
## QtQ1_18      -2.17350    2.71350  -0.801 0.423555    
## QtQ2_17       3.94211    3.17030   1.243 0.214345    
## QtQ2_18            NA         NA      NA       NA    
## QtQ3_17       2.61036    3.28104   0.796 0.426687    
## QtQ3_18            NA         NA      NA       NA    
## QtQ4_17       2.19058    3.38054   0.648 0.517316    
## QtQ4_18            NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.4 on 452 degrees of freedom
## Multiple R-squared:  0.1504, Adjusted R-squared:  0.06202 
## F-statistic: 1.702 on 47 and 452 DF,  p-value: 0.003565

#p-value: 0.0569
lm_pbms <- lm(formula = Order_Qty ~  Brand + promo + month + state,   data = lm_cust_s2)

summary(lm(formula = Order_Qty ~  Brand + promo + month + state,   data = lm_cust_s2))

## 
## Call:
## lm(formula = Order_Qty ~ Brand + promo + month + state, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -58.650  -6.873  -0.527   4.599  96.230 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -2.0271    24.8380  -0.082  0.93500    
## BrandAX            6.1276     9.5040   0.645  0.51949    
## BrandBB            2.5712    10.3337   0.249  0.80364    
## BrandBE           25.7510     9.1478   2.815  0.00514 ** 
## BrandBV            7.2538     9.8473   0.737  0.46181    
## BrandCH           18.2210    12.3096   1.480  0.13966    
## BrandDG            9.2969     9.3754   0.992  0.32202    
## BrandDY           -2.0602     8.8505  -0.233  0.81606    
## BrandEA            3.8759     8.1655   0.475  0.63530    
## BrandHC           17.9172     7.2400   2.475  0.01378 *  
## BrandMK           15.4007     7.2740   2.117  0.03490 *  
## BrandMU           28.2075    16.2832   1.732  0.08404 .  
## BrandOJ            0.6368     9.3583   0.068  0.94579    
## BrandOO           18.2497     7.0704   2.581  0.01023 *  
## BrandOX           16.6811     7.2802   2.291  0.02250 *  
## BrandOY           13.9625     8.0759   1.729  0.08465 .  
## BrandPH           15.8409     8.6224   1.837  0.06698 .  
## BrandPO           41.5900    10.2797   4.046 6.34e-05 ***
## BrandPP            9.9690    13.0342   0.765  0.44485    
## BrandPR           18.0645     7.2789   2.482  0.01351 *  
## BrandPS            8.9873     9.1256   0.985  0.32534    
## BrandRA         -100.7532    21.9942  -4.581 6.32e-06 ***
## BrandRB           19.7317     6.8262   2.891  0.00407 ** 
## BrandRJ            6.8061    13.6416   0.499  0.61813    
## BrandRL            9.9506     9.6307   1.033  0.30217    
## BrandRX           18.2438     6.8730   2.654  0.00829 ** 
## BrandRY            3.5478     7.1285   0.498  0.61900    
## BrandSF           -1.9104    13.1964  -0.145  0.88497    
## BrandTF           13.8926     7.8195   1.777  0.07644 .  
## BrandTY           12.3365     8.0353   1.535  0.12556    
## BrandVA           -9.1514    13.0143  -0.703  0.48238    
## BrandVE           13.0826     7.7991   1.677  0.09429 .  
## BrandVO           12.2923     7.8071   1.575  0.11622    
## promoCODE F       -7.9624    21.7205  -0.367  0.71414    
## promoH15         -11.3711    18.5888  -0.612  0.54110    
## promoLUX40         1.7894    18.5178   0.097  0.92307    
## promoMVE150      -16.2101    20.8636  -0.777  0.43768    
## promoMVENB       -24.1914    23.8723  -1.013  0.31154    
## promoNASC75      -17.4953    20.2637  -0.863  0.38848    
## promoNB            1.9541    21.3986   0.091  0.92729    
## promoOOX40       -12.9253    20.5342  -0.629  0.52944    
## promoOOX60        -2.8152    24.2318  -0.116  0.90757    
## promoOther       -10.1106    18.3122  -0.552  0.58119    
## promoOY12        -13.3079    21.6869  -0.614  0.53983    
## promoPRE-SELL B  -16.0857    26.9046  -0.598  0.55028    
## promoPRE-SELL T  -16.8819    25.0731  -0.673  0.50117    
## promoRB15         -8.1403    19.3365  -0.421  0.67401    
## promoRSD          -8.2395    24.4661  -0.337  0.73648    
## promoSUN1        -13.2789    26.4509  -0.502  0.61595    
## promoSY1          -5.3353    18.8383  -0.283  0.77717    
## promoSY100        -7.3306    18.6147  -0.394  0.69395    
## promoSY1000       93.2378    23.3622   3.991 7.92e-05 ***
## promoSY2           9.9058    19.4176   0.510  0.61025    
## promoSY200         1.1945    19.1970   0.062  0.95042    
## promoSY250        14.8518    19.3964   0.766  0.44434    
## promoSY5          -7.2125    20.1324  -0.358  0.72036    
## promoSYNERGY54   -14.6269    23.8259  -0.614  0.53965    
## promoSYVE         -4.1964    19.4985  -0.215  0.82972    
## promoUP1          -7.0812    18.8553  -0.376  0.70746    
## promoUP3          -5.4731    18.9518  -0.289  0.77290    
## promoVIP         -11.3443    23.9973  -0.473  0.63668    
## promoVP1200      -15.4632    19.9470  -0.775  0.43870    
## promoVPFP150     -28.2500    23.3572  -1.209  0.22724    
## promoVPFP200      38.5838    19.1104   2.019  0.04420 *  
## promoVPNB          3.4045    24.8219   0.137  0.89098    
## promoWC1           6.3420    18.6847   0.339  0.73448    
## promoWC2         -21.6945    24.6035  -0.882  0.37847    
## promoWild Card    -2.0623    19.3295  -0.107  0.91509    
## month2             7.2526     3.1912   2.273  0.02361 *  
## month3             5.5817     3.4056   1.639  0.10206    
## month4             0.1752     3.6563   0.048  0.96180    
## month5            -1.0604     3.6481  -0.291  0.77145    
## month6             4.5362     3.6976   1.227  0.22067    
## month7             1.2923     3.8150   0.339  0.73499    
## month8             6.4958     3.7901   1.714  0.08738 .  
## month9             2.2046     3.6545   0.603  0.54671    
## month10            4.5054     3.7658   1.196  0.23231    
## month11            5.6257     3.7434   1.503  0.13373    
## month12            3.9367     3.7540   1.049  0.29501    
## stateAL           10.6382    18.6311   0.571  0.56835    
## stateAR            4.1500    17.4195   0.238  0.81183    
## stateAZ            8.6466    15.8856   0.544  0.58656    
## stateCA            5.9637    15.4903   0.385  0.70046    
## stateCO            6.6513    16.2777   0.409  0.68306    
## stateCT            8.4799    17.4160   0.487  0.62661    
## stateDE            2.9995    18.6171   0.161  0.87209    
## stateFL            5.2243    15.7274   0.332  0.73994    
## stateGA            4.5722    16.0808   0.284  0.77632    
## stateHI           -2.6937    21.3661  -0.126  0.89974    
## stateIA           -1.5307    21.4072  -0.072  0.94304    
## stateID           -0.6688    18.8024  -0.036  0.97164    
## stateIL           -0.2197    16.0442  -0.014  0.98908    
## stateIN           14.2527    16.2389   0.878  0.38068    
## stateKS            0.7637    17.2332   0.044  0.96467    
## stateKY            3.6630    16.2102   0.226  0.82135    
## stateLA            0.8786    16.5711   0.053  0.95774    
## stateMA           -0.2324    16.3958  -0.014  0.98870    
## stateMD           18.9103    16.3065   1.160  0.24692    
## stateME           -0.6916    21.7265  -0.032  0.97462    
## stateMI            7.6082    15.9714   0.476  0.63409    
## stateMN            3.4418    17.7510   0.194  0.84637    
## stateMO            0.1299    16.9033   0.008  0.99387    
## stateMS            0.8777    17.7718   0.049  0.96064    
## stateNC            0.5107    16.7077   0.031  0.97563    
## stateND          -10.2389    22.1984  -0.461  0.64489    
## stateNE            1.6336    17.6542   0.093  0.92632    
## stateNH          -29.1612    19.1727  -1.521  0.12911    
## stateNJ           19.9607    15.9414   1.252  0.21131    
## stateNM            4.6873    18.8954   0.248  0.80422    
## stateNV           -1.4418    16.1878  -0.089  0.92908    
## stateNY            3.4344    15.6451   0.220  0.82637    
## stateOH            9.6253    15.8424   0.608  0.54385    
## stateOK           12.1529    16.7818   0.724  0.46941    
## stateOR            0.8095    16.2716   0.050  0.96035    
## statePA            3.1171    15.6808   0.199  0.84254    
## statePR            9.1999    16.7608   0.549  0.58340    
## stateRI            4.6064    18.6326   0.247  0.80487    
## stateSC            4.9871    16.4590   0.303  0.76206    
## stateSD           -2.7986    21.4359  -0.131  0.89620    
## stateTN           -0.1352    17.9089  -0.008  0.99398    
## stateTX            8.6111    15.4332   0.558  0.57721    
## stateUT           -0.3484    17.1736  -0.020  0.98382    
## stateVA           11.5649    16.2051   0.714  0.47588    
## stateVT            1.8404    21.7898   0.084  0.93274    
## stateWA            3.4939    17.1316   0.204  0.83851    
## stateWI            2.2197    21.7095   0.102  0.91862    
## stateWY            0.2215    17.8137   0.012  0.99009    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.46 on 373 degrees of freedom
## Multiple R-squared:  0.4545, Adjusted R-squared:  0.2702 
## F-statistic: 2.466 on 126 and 373 DF,  p-value: 1.961e-11

# p-value: 0.08366 
lmdata<-na.omit(lm_cust_s2) 
lm_qbpms <-(lm(formula = Order_Qty ~  Qt + Brand + promo +  `Order Date`,   data = lm_cust_s2))
summary(lm_qbpms)

## 
## Call:
## lm(formula = Order_Qty ~ Qt + Brand + promo + `Order Date`, data = lm_cust_s2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -60.309  -7.377  -1.410   4.684 112.184 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -5.957e+02  4.910e+02  -1.213 0.225743    
## QtQ1_18         -6.035e+00  1.283e+01  -0.470 0.638393    
## QtQ2_17         -3.274e+00  3.927e+00  -0.834 0.404996    
## QtQ2_18         -1.047e+01  1.480e+01  -0.708 0.479512    
## QtQ3_17          8.222e-02  9.081e+00   0.009 0.992781    
## QtQ3_18         -1.332e+01  1.758e+01  -0.757 0.449183    
## QtQ4_17         -5.223e-01  1.052e+01  -0.050 0.960418    
## QtQ4_18         -1.603e+01  1.990e+01  -0.805 0.421056    
## BrandAX          7.969e+00  9.265e+00   0.860 0.390198    
## BrandBB          3.667e+00  9.797e+00   0.374 0.708380    
## BrandBE          2.203e+01  8.528e+00   2.583 0.010128 *  
## BrandBV          4.345e+00  9.791e+00   0.444 0.657445    
## BrandCH          1.568e+01  1.227e+01   1.278 0.202043    
## BrandDG          8.756e+00  9.105e+00   0.962 0.336772    
## BrandDY         -1.922e+00  8.661e+00  -0.222 0.824440    
## BrandEA          7.193e+00  7.902e+00   0.910 0.363195    
## BrandHC          1.727e+01  6.999e+00   2.467 0.014029 *  
## BrandMK          1.422e+01  7.099e+00   2.002 0.045877 *  
## BrandMU          2.832e+01  1.611e+01   1.758 0.079507 .  
## BrandOJ          2.117e+00  8.828e+00   0.240 0.810574    
## BrandOO          1.591e+01  6.899e+00   2.307 0.021564 *  
## BrandOX          1.611e+01  7.061e+00   2.281 0.023035 *  
## BrandOY          1.196e+01  7.642e+00   1.564 0.118470    
## BrandPH          1.361e+01  8.357e+00   1.629 0.104076    
## BrandPO          2.964e+01  9.880e+00   3.000 0.002862 ** 
## BrandPP          1.458e+00  1.223e+01   0.119 0.905188    
## BrandPR          1.523e+01  7.093e+00   2.147 0.032386 *  
## BrandPS          5.409e+00  8.917e+00   0.607 0.544432    
## BrandRA         -8.853e+01  2.172e+01  -4.075 5.49e-05 ***
## BrandRB          1.693e+01  6.649e+00   2.547 0.011233 *  
## BrandRJ         -2.724e+00  1.226e+01  -0.222 0.824180    
## BrandRL          1.054e+01  9.333e+00   1.129 0.259393    
## BrandRX          1.714e+01  6.647e+00   2.579 0.010247 *  
## BrandRY          1.837e+00  6.952e+00   0.264 0.791769    
## BrandSF         -3.040e-01  1.260e+01  -0.024 0.980758    
## BrandTF          1.251e+01  7.505e+00   1.667 0.096212 .  
## BrandTY          9.736e+00  7.690e+00   1.266 0.206183    
## BrandVA         -2.182e+00  1.208e+01  -0.181 0.856715    
## BrandVE          1.274e+01  7.556e+00   1.685 0.092633 .  
## BrandVO          1.204e+01  7.387e+00   1.630 0.103912    
## promoCODE F     -1.926e+01  2.098e+01  -0.918 0.359065    
## promoH15        -1.424e+01  1.825e+01  -0.781 0.435506    
## promoLUX40       2.385e-01  1.819e+01   0.013 0.989544    
## promoMVE150     -1.818e+01  2.066e+01  -0.880 0.379570    
## promoMVENB      -2.461e+01  2.339e+01  -1.052 0.293192    
## promoNASC75     -1.717e+01  2.012e+01  -0.853 0.393878    
## promoNB          1.268e+00  2.096e+01   0.060 0.951810    
## promoOOX40      -1.001e+01  2.008e+01  -0.498 0.618556    
## promoOOX60      -4.214e+00  2.413e+01  -0.175 0.861467    
## promoOther      -1.875e+01  1.921e+01  -0.976 0.329612    
## promoOY12       -1.872e+01  2.085e+01  -0.898 0.369649    
## promoPRE-SELL B -1.410e+01  2.436e+01  -0.579 0.563040    
## promoPRE-SELL T -1.981e+01  2.352e+01  -0.842 0.400236    
## promoRB15       -1.038e+01  1.923e+01  -0.540 0.589690    
## promoRSD        -2.283e+01  2.448e+01  -0.933 0.351606    
## promoSUN1       -7.547e+00  2.546e+01  -0.296 0.767040    
## promoSY1        -1.522e+01  1.961e+01  -0.776 0.438116    
## promoSY100      -1.503e+01  1.940e+01  -0.775 0.439066    
## promoSY1000      8.547e+01  2.426e+01   3.524 0.000472 ***
## promoSY2         5.511e-01  2.029e+01   0.027 0.978338    
## promoSY200      -2.521e+00  1.898e+01  -0.133 0.894410    
## promoSY250       9.060e+00  2.027e+01   0.447 0.655187    
## promoSY5        -1.881e+01  2.073e+01  -0.907 0.364692    
## promoSYNERGY54  -1.993e+01  2.441e+01  -0.816 0.414727    
## promoSYVE       -1.535e+01  2.035e+01  -0.754 0.450998    
## promoUP1        -1.045e+01  1.842e+01  -0.568 0.570625    
## promoUP3        -8.070e+00  1.854e+01  -0.435 0.663508    
## promoVIP        -2.205e+01  2.443e+01  -0.902 0.367390    
## promoVP1200     -1.493e+01  1.951e+01  -0.765 0.444571    
## promoVPFP150    -2.739e+01  2.322e+01  -1.180 0.238848    
## promoVPFP200     3.436e+01  1.899e+01   1.809 0.071139 .  
## promoVPNB        6.542e+00  2.372e+01   0.276 0.782881    
## promoWC1         9.916e-02  1.868e+01   0.005 0.995767    
## promoWC2        -2.926e+01  2.472e+01  -1.183 0.237277    
## promoWild Card  -9.420e+00  2.039e+01  -0.462 0.644306    
## `Order Date`     4.078e-07  3.299e-07   1.236 0.217020    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.61 on 424 degrees of freedom
## Multiple R-squared:  0.367,  Adjusted R-squared:  0.255 
## F-statistic: 3.278 on 75 and 424 DF,  p-value: 1.1e-14

print("Result of Few models:")

## [1] "Result of Few models:"

DT::datatable(result)