HW7

Directions: In this assignment you will get to practice more with inference from a categorical predictor. Note: You don’t have to do these in any particular order. You might want to do Part II first to help you with Part I.

Part I: Use the your regression project dataset to explore models that incorporate categorical predictors.

B) Fit a simple linear model with response variable and the numeric predictor that you chose. Does the relationship appear to be significant? Make sure to also include a graphic.

library(tidyverse)

## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.0 ──

## ✓ ggplot2 3.2.1     ✓ purrr   0.3.3
## ✓ tibble  2.1.3     ✓ dplyr   0.8.3
## ✓ tidyr   1.0.2     ✓ stringr 1.4.0
## ✓ readr   1.3.1     ✓ forcats 0.4.0

## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(ggplot2)
library(readxl)
bb <- read_excel("basketball.xlsx")

## New names:
## * `` -> ...1

str(bb$Country)

##  chr [1:559] "USA" "USA" "USA" "USA" "USA" "Serbia" "Ukraine" "USA" "Spain" ...

str(bb$Position)

##  chr [1:559] "Point Guard" "Power Forward" "Power Forward" "Small Forward" ...

Categorical variables that might be related to the response variable \(Y_{Salary}\) are Country and Position. Country variable(categorical) is used for dummy variable in the model, and levels for Position variables are “Point Guard” “Power Forward” “Power Forward” “Small Forward” and “Center”

modbb2 <- lm(Salary~Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player, data=bb)
modbb2

## 
## Call:
## lm(formula = Salary ~ Age + Guaranteed + Player_Efficiency_Rating + 
##     True_Shooting_Percentage + Three_Point_Field_Goal_Percentage + 
##     Free_Throw_Percentage + Offensive_Rebound_Percentage + Defensive_Rebound_Percentage + 
##     Total_Rebound_Percentage + Assist_Percentage + Steal_Percentage + 
##     Block_Percentage + Turnover_Percentage + Usage_Percentage + 
##     Offensive_Win_Shares + Defensive_Win_Shares + Win_Shares + 
##     Win_Shares_Per_48_Minutes + Offense_Box_Plus_Minus + Defense_Box_Plus_Minus + 
##     Box_Plus_Minus + Value_Over_Replacement_Player, data = bb)
## 
## Coefficients:
##                       (Intercept)                                Age  
##                        -7.300e+06                          4.011e+05  
##                        Guaranteed           Player_Efficiency_Rating  
##                         1.774e-01                          1.384e+05  
##          True_Shooting_Percentage  Three_Point_Field_Goal_Percentage  
##                        -2.283e+06                         -1.546e+04  
##             Free_Throw_Percentage       Offensive_Rebound_Percentage  
##                        -3.037e+03                         -1.045e+06  
##      Defensive_Rebound_Percentage           Total_Rebound_Percentage  
##                        -8.493e+05                          1.911e+06  
##                 Assist_Percentage                   Steal_Percentage  
##                        -5.664e+04                         -2.103e+05  
##                  Block_Percentage                Turnover_Percentage  
##                        -1.739e+05                          1.448e+04  
##                  Usage_Percentage               Offensive_Win_Shares  
##                         5.667e+04                          6.708e+04  
##              Defensive_Win_Shares                         Win_Shares  
##                         4.394e+05                          2.403e+05  
##         Win_Shares_Per_48_Minutes             Offense_Box_Plus_Minus  
##                        -1.330e+07                         -2.668e+06  
##            Defense_Box_Plus_Minus                     Box_Plus_Minus  
##                        -2.804e+06                          2.954e+06  
##     Value_Over_Replacement_Player  
##                        -1.550e+05

summary(modbb2)

## 
## Call:
## lm(formula = Salary ~ Age + Guaranteed + Player_Efficiency_Rating + 
##     True_Shooting_Percentage + Three_Point_Field_Goal_Percentage + 
##     Free_Throw_Percentage + Offensive_Rebound_Percentage + Defensive_Rebound_Percentage + 
##     Total_Rebound_Percentage + Assist_Percentage + Steal_Percentage + 
##     Block_Percentage + Turnover_Percentage + Usage_Percentage + 
##     Offensive_Win_Shares + Defensive_Win_Shares + Win_Shares + 
##     Win_Shares_Per_48_Minutes + Offense_Box_Plus_Minus + Defense_Box_Plus_Minus + 
##     Box_Plus_Minus + Value_Over_Replacement_Player, data = bb)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -22609397  -1963642   -455650   1548429  14898466 
## 
## Coefficients:
##                                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                       -7.300e+06  2.849e+06  -2.562   0.0107 *  
## Age                                4.011e+05  3.873e+04  10.356   <2e-16 ***
## Guaranteed                         1.774e-01  7.314e-03  24.255   <2e-16 ***
## Player_Efficiency_Rating           1.384e+05  1.887e+05   0.734   0.4635    
## True_Shooting_Percentage          -2.283e+06  3.018e+06  -0.756   0.4498    
## Three_Point_Field_Goal_Percentage -1.546e+04  1.491e+04  -1.037   0.3002    
## Free_Throw_Percentage             -3.037e+03  6.738e+03  -0.451   0.6524    
## Offensive_Rebound_Percentage      -1.045e+06  6.002e+05  -1.741   0.0823 .  
## Defensive_Rebound_Percentage      -8.493e+05  5.921e+05  -1.434   0.1520    
## Total_Rebound_Percentage           1.911e+06  1.186e+06   1.611   0.1078    
## Assist_Percentage                 -5.664e+04  2.986e+04  -1.897   0.0584 .  
## Steal_Percentage                  -2.103e+05  2.870e+05  -0.733   0.4640    
## Block_Percentage                  -1.739e+05  2.128e+05  -0.817   0.4143    
## Turnover_Percentage                1.448e+04  3.311e+04   0.437   0.6620    
## Usage_Percentage                   5.667e+04  7.048e+04   0.804   0.4217    
## Offensive_Win_Shares               6.708e+04  3.083e+06   0.022   0.9827    
## Defensive_Win_Shares               4.394e+05  3.083e+06   0.143   0.8867    
## Win_Shares                         2.403e+05  3.086e+06   0.078   0.9380    
## Win_Shares_Per_48_Minutes         -1.330e+07  6.777e+06  -1.962   0.0503 .  
## Offense_Box_Plus_Minus            -2.668e+06  3.304e+06  -0.807   0.4199    
## Defense_Box_Plus_Minus            -2.804e+06  3.257e+06  -0.861   0.3896    
## Box_Plus_Minus                     2.954e+06  3.268e+06   0.904   0.3665    
## Value_Over_Replacement_Player     -1.550e+05  3.718e+05  -0.417   0.6770    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3721000 on 536 degrees of freedom
## Multiple R-squared:  0.7453, Adjusted R-squared:  0.7349 
## F-statistic: 71.31 on 22 and 536 DF,  p-value: < 2.2e-16

anova(modbb2)

## Analysis of Variance Table
## 
## Response: Salary
##                                    Df     Sum Sq    Mean Sq   F value    Pr(>F)
## Age                                 1 3.1939e+15 3.1939e+15  230.6436 < 2.2e-16
## Guaranteed                          1 1.7116e+16 1.7116e+16 1235.9803 < 2.2e-16
## Player_Efficiency_Rating            1 2.1420e+14 2.1420e+14   15.4684 9.492e-05
## True_Shooting_Percentage            1 5.9988e+12 5.9988e+12    0.4332  0.510708
## Three_Point_Field_Goal_Percentage   1 4.1604e+13 4.1604e+13    3.0044  0.083614
## Free_Throw_Percentage               1 9.5711e+11 9.5711e+11    0.0691  0.792728
## Offensive_Rebound_Percentage        1 3.6157e+11 3.6157e+11    0.0261  0.871692
## Defensive_Rebound_Percentage        1 3.4278e+14 3.4278e+14   24.7534 8.790e-07
## Total_Rebound_Percentage            1 6.2076e+13 6.2076e+13    4.4827  0.034699
## Assist_Percentage                   1 2.1969e+13 2.1969e+13    1.5865  0.208376
## Steal_Percentage                    1 5.4734e+10 5.4734e+10    0.0040  0.949894
## Block_Percentage                    1 5.8237e+12 5.8237e+12    0.4206  0.516938
## Turnover_Percentage                 1 5.7151e+12 5.7151e+12    0.4127  0.520871
## Usage_Percentage                    1 1.3529e+14 1.3529e+14    9.7696  0.001870
## Offensive_Win_Shares                1 3.2995e+14 3.2995e+14   23.8270 1.393e-06
## Defensive_Win_Shares                1 1.4424e+14 1.4424e+14   10.4164  0.001325
## Win_Shares                          1 1.5392e+10 1.5392e+10    0.0011  0.973417
## Win_Shares_Per_48_Minutes           1 4.2972e+13 4.2972e+13    3.1032  0.078708
## Offense_Box_Plus_Minus              1 4.1327e+13 4.1327e+13    2.9844  0.084649
## Defense_Box_Plus_Minus              1 5.8863e+12 5.8863e+12    0.4251  0.514697
## Box_Plus_Minus                      1 1.1415e+13 1.1415e+13    0.8243  0.364325
## Value_Over_Replacement_Player       1 2.4060e+12 2.4060e+12    0.1737  0.676970
## Residuals                         536 7.4224e+15 1.3848e+13                    
##                                      
## Age                               ***
## Guaranteed                        ***
## Player_Efficiency_Rating          ***
## True_Shooting_Percentage             
## Three_Point_Field_Goal_Percentage .  
## Free_Throw_Percentage                
## Offensive_Rebound_Percentage         
## Defensive_Rebound_Percentage      ***
## Total_Rebound_Percentage          *  
## Assist_Percentage                    
## Steal_Percentage                     
## Block_Percentage                     
## Turnover_Percentage                  
## Usage_Percentage                  ** 
## Offensive_Win_Shares              ***
## Defensive_Win_Shares              ** 
## Win_Shares                           
## Win_Shares_Per_48_Minutes         .  
## Offense_Box_Plus_Minus            .  
## Defense_Box_Plus_Minus               
## Box_Plus_Minus                       
## Value_Over_Replacement_Player        
## Residuals                            
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

ggplot(bb, aes(x=Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player, y=Salary))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "lm", se = FALSE) #regression line

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

C) Now, write the “dummy” variable coding for your categorical variable. (Hint: the contrasts() function might help).

positionvector <- factor(bb$Position)
contrasts(positionvector)

##                Point Guard Power Forward Shooting Guard Small Forward
## Center                   0             0              0             0
## Point Guard              1             0              0             0
## Power Forward            0             1              0             0
## Shooting Guard           0             0              1             0
## Small Forward            0             0              0             1

Countrydummy <- factor(bb$Country_Dummy)
contrasts(Countrydummy)

##   1
## 0 0
## 1 1

str(bb$Country)

##  chr [1:559] "USA" "USA" "USA" "USA" "USA" "Serbia" "Ukraine" "USA" "Spain" ...

str(bb$Country_Dummy)

##  num [1:559] 1 1 1 1 1 0 0 1 0 0 ...

D) Fit a linear model with response variable and the categorical variable. Does it appear that there are differences among the means of levels of the categorical variable? (Hint: Look at the ANOVA F-test). Be sure to include an appropriate graphic (i.e. side-by-side boxplot)

categoricalmodbb <- lm(Salary~Country_Dummy+positionvector, data=bb)
categoricalmodbb

## 
## Call:
## lm(formula = Salary ~ Country_Dummy + positionvector, data = bb)
## 
## Coefficients:
##                  (Intercept)                 Country_Dummy  
##                      7632245                        100392  
##    positionvectorPoint Guard   positionvectorPower Forward  
##                     -1610166                        -65319  
## positionvectorShooting Guard   positionvectorSmall Forward  
##                     -1176955                      -1790813

summary(categoricalmodbb)

## 
## Call:
## lm(formula = Salary ~ Country_Dummy + positionvector, data = bb)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -7632637 -5051122 -2913414  3027783 28560079 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                   7632245     814911   9.366   <2e-16 ***
## Country_Dummy                  100392     748992   0.134   0.8934    
## positionvectorPoint Guard    -1610166     960240  -1.677   0.0941 .  
## positionvectorPower Forward    -65319     971811  -0.067   0.9464    
## positionvectorShooting Guard -1176955     952244  -1.236   0.2170    
## positionvectorSmall Forward  -1790813     997190  -1.796   0.0731 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7221000 on 553 degrees of freedom
## Multiple R-squared:  0.01064,    Adjusted R-squared:  0.001694 
## F-statistic: 1.189 on 5 and 553 DF,  p-value: 0.313

ggplot(bb, aes(x=Country_Dummy+positionvector, y=Salary))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "loess", se = FALSE) #regression line

## Warning in Ops.factor(Country_Dummy, positionvector): '+' not meaningful for
## factors

## Warning in Ops.factor(Country_Dummy, positionvector): '+' not meaningful for
## factors

## Warning in Ops.factor(Country_Dummy, positionvector): '+' not meaningful for
## factors

## Warning in Ops.factor(Country_Dummy, positionvector): '+' not meaningful for
## factors

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

ggplot(bb, aes(y=Salary, x=Countrydummy, fill=Countrydummy))+
  geom_boxplot()

ggplot(bb, aes(y=Salary, x=positionvector, fill=positionvector))+
  geom_boxplot()

ggplot(bb, aes(x=Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player, y=Salary, color=positionvector))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "lm", se = FALSE) #regression line

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

#categorical 

ggplot(bb, aes(x=Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player+Country_Dummy, y=Salary, color=Countrydummy))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "lm", se = FALSE) #regression line

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

#Dummy

E) Now fit a multiple linear model that combines parts (b) and (d), with both the numeric and categorical variables. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

modbb3 <- lm(Salary~Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player+Country_Dummy+positionvector, data=bb)
modbb3

## 
## Call:
## lm(formula = Salary ~ Age + Guaranteed + Player_Efficiency_Rating + 
##     True_Shooting_Percentage + Three_Point_Field_Goal_Percentage + 
##     Free_Throw_Percentage + Offensive_Rebound_Percentage + Defensive_Rebound_Percentage + 
##     Total_Rebound_Percentage + Assist_Percentage + Steal_Percentage + 
##     Block_Percentage + Turnover_Percentage + Usage_Percentage + 
##     Offensive_Win_Shares + Defensive_Win_Shares + Win_Shares + 
##     Win_Shares_Per_48_Minutes + Offense_Box_Plus_Minus + Defense_Box_Plus_Minus + 
##     Box_Plus_Minus + Value_Over_Replacement_Player + Country_Dummy + 
##     positionvector, data = bb)
## 
## Coefficients:
##                       (Intercept)                                Age  
##                        -6.382e+06                          3.914e+05  
##                        Guaranteed           Player_Efficiency_Rating  
##                         1.770e-01                          1.944e+05  
##          True_Shooting_Percentage  Three_Point_Field_Goal_Percentage  
##                        -3.044e+06                         -1.198e+04  
##             Free_Throw_Percentage       Offensive_Rebound_Percentage  
##                        -2.272e+03                         -1.128e+06  
##      Defensive_Rebound_Percentage           Total_Rebound_Percentage  
##                        -9.413e+05                          2.019e+06  
##                 Assist_Percentage                   Steal_Percentage  
##                        -3.155e+04                         -1.703e+05  
##                  Block_Percentage                Turnover_Percentage  
##                        -2.463e+05                          2.502e+04  
##                  Usage_Percentage               Offensive_Win_Shares  
##                         3.278e+04                          3.190e+05  
##              Defensive_Win_Shares                         Win_Shares  
##                         8.323e+05                         -2.524e+04  
##         Win_Shares_Per_48_Minutes             Offense_Box_Plus_Minus  
##                        -1.419e+07                         -2.561e+06  
##            Defense_Box_Plus_Minus                     Box_Plus_Minus  
##                        -2.701e+06                          2.791e+06  
##     Value_Over_Replacement_Player                      Country_Dummy  
##                        -1.972e+05                          4.632e+05  
##         positionvectorPoint Guard        positionvectorPower Forward  
##                        -1.746e+06                         -1.050e+05  
##      positionvectorShooting Guard        positionvectorSmall Forward  
##                        -1.102e+06                         -1.288e+06

summary(modbb3)

## 
## Call:
## lm(formula = Salary ~ Age + Guaranteed + Player_Efficiency_Rating + 
##     True_Shooting_Percentage + Three_Point_Field_Goal_Percentage + 
##     Free_Throw_Percentage + Offensive_Rebound_Percentage + Defensive_Rebound_Percentage + 
##     Total_Rebound_Percentage + Assist_Percentage + Steal_Percentage + 
##     Block_Percentage + Turnover_Percentage + Usage_Percentage + 
##     Offensive_Win_Shares + Defensive_Win_Shares + Win_Shares + 
##     Win_Shares_Per_48_Minutes + Offense_Box_Plus_Minus + Defense_Box_Plus_Minus + 
##     Box_Plus_Minus + Value_Over_Replacement_Player + Country_Dummy + 
##     positionvector, data = bb)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -21928752  -1875774   -342348   1524878  14084130 
## 
## Coefficients:
##                                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                       -6.382e+06  3.016e+06  -2.116   0.0348 *  
## Age                                3.914e+05  3.914e+04  10.000   <2e-16 ***
## Guaranteed                         1.770e-01  7.333e-03  24.141   <2e-16 ***
## Player_Efficiency_Rating           1.944e+05  1.919e+05   1.013   0.3115    
## True_Shooting_Percentage          -3.044e+06  3.045e+06  -1.000   0.3179    
## Three_Point_Field_Goal_Percentage -1.198e+04  1.505e+04  -0.796   0.4263    
## Free_Throw_Percentage             -2.272e+03  6.756e+03  -0.336   0.7368    
## Offensive_Rebound_Percentage      -1.128e+06  6.007e+05  -1.878   0.0610 .  
## Defensive_Rebound_Percentage      -9.413e+05  5.926e+05  -1.588   0.1128    
## Total_Rebound_Percentage           2.019e+06  1.186e+06   1.703   0.0892 .  
## Assist_Percentage                 -3.155e+04  3.622e+04  -0.871   0.3842    
## Steal_Percentage                  -1.703e+05  2.882e+05  -0.591   0.5548    
## Block_Percentage                  -2.463e+05  2.177e+05  -1.132   0.2582    
## Turnover_Percentage                2.502e+04  3.377e+04   0.741   0.4592    
## Usage_Percentage                   3.278e+04  7.160e+04   0.458   0.6473    
## Offensive_Win_Shares               3.190e+05  3.084e+06   0.103   0.9177    
## Defensive_Win_Shares               8.323e+05  3.086e+06   0.270   0.7875    
## Win_Shares                        -2.524e+04  3.087e+06  -0.008   0.9935    
## Win_Shares_Per_48_Minutes         -1.419e+07  6.941e+06  -2.045   0.0413 *  
## Offense_Box_Plus_Minus            -2.561e+06  3.300e+06  -0.776   0.4380    
## Defense_Box_Plus_Minus            -2.701e+06  3.251e+06  -0.831   0.4065    
## Box_Plus_Minus                     2.791e+06  3.264e+06   0.855   0.3928    
## Value_Over_Replacement_Player     -1.972e+05  3.757e+05  -0.525   0.5999    
## Country_Dummy                      4.632e+05  3.969e+05   1.167   0.2438    
## positionvectorPoint Guard         -1.746e+06  8.793e+05  -1.986   0.0476 *  
## positionvectorPower Forward       -1.050e+05  5.717e+05  -0.184   0.8544    
## positionvectorShooting Guard      -1.102e+06  7.658e+05  -1.439   0.1506    
## positionvectorSmall Forward       -1.288e+06  7.124e+05  -1.808   0.0712 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3712000 on 531 degrees of freedom
## Multiple R-squared:  0.749,  Adjusted R-squared:  0.7362 
## F-statistic: 58.68 on 27 and 531 DF,  p-value: < 2.2e-16

ggplot(bb, aes(x=Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player+Country_Dummy, y=Salary, color = positionvector))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "lm", se = FALSE) #regression line

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

ggplot(bb, aes(x=Age+Guaranteed+Player_Efficiency_Rating+True_Shooting_Percentage+Three_Point_Field_Goal_Percentage+Free_Throw_Percentage+Offensive_Rebound_Percentage+Defensive_Rebound_Percentage+Total_Rebound_Percentage+Assist_Percentage+Steal_Percentage+Block_Percentage+Turnover_Percentage+Usage_Percentage+Offensive_Win_Shares+Defensive_Win_Shares+Win_Shares+Win_Shares_Per_48_Minutes+Offense_Box_Plus_Minus+Defense_Box_Plus_Minus+Box_Plus_Minus+Value_Over_Replacement_Player+Country_Dummy, y=Salary, color = Countrydummy))+
  geom_jitter()+
  geom_smooth(col = "orange")+ #least square line 
  geom_smooth(method = "lm", se = FALSE) #regression line

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

F) Finally, fit a multiple linear model that includes also the interaction between the numeric and categorical variables, which allows for different slopes. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

G) Conclusion: What did you learn from this exercise? Were any of the relationships significant? (Note: This would be great to include in your final project write up!)

Part II:

This question involves the use of multiple linear regression with categorical variables on the Carseat dataset from the textbook (i.e. using library(ISLR))]

A) Describe the variables Sales, Price, Urban, and US. Are the variables numeric or categorical? If they are categorical describe the levels.

library(ISLR)
data(Carseats) 
str(Carseats)

## 'data.frame':    400 obs. of  11 variables:
##  $ Sales      : num  9.5 11.22 10.06 7.4 4.15 ...
##  $ CompPrice  : num  138 111 113 117 141 124 115 136 132 132 ...
##  $ Income     : num  73 48 35 100 64 113 105 81 110 113 ...
##  $ Advertising: num  11 16 10 4 3 13 0 15 0 0 ...
##  $ Population : num  276 260 269 466 340 501 45 425 108 131 ...
##  $ Price      : num  120 83 80 97 128 72 108 120 124 124 ...
##  $ ShelveLoc  : Factor w/ 3 levels "Bad","Good","Medium": 1 2 3 3 1 1 3 2 3 3 ...
##  $ Age        : num  42 65 59 55 38 78 71 67 76 76 ...
##  $ Education  : num  17 10 12 14 13 16 15 10 10 17 ...
##  $ Urban      : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 1 2 2 1 1 ...
##  $ US         : Factor w/ 2 levels "No","Yes": 2 2 2 2 1 2 1 2 1 2 ...

# Sales and Price variables are numeric and Urban and US variables are categotical("Yes" or "No")

library(ggplot2)
ggplot(Carseats, aes(y=Sales, x=US, fill=US))+
  geom_boxplot()

ggplot(Carseats, aes(y=Sales, x=Urban, fill=Urban))+
  geom_boxplot()

B) Fit a multiple regression model to predict Sales using Price, Urban, and US.

modcarseats1 <- lm(Sales~ Price+Urban+US, data=Carseats)
modcarseats1

## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
## 
## Coefficients:
## (Intercept)        Price     UrbanYes        USYes  
##    13.04347     -0.05446     -0.02192      1.20057

summary(modcarseats1)

## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9206 -1.6220 -0.0564  1.5786  7.0581 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
## Price       -0.054459   0.005242 -10.389  < 2e-16 ***
## UrbanYes    -0.021916   0.271650  -0.081    0.936    
## USYes        1.200573   0.259042   4.635 4.86e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2335 
## F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

C) Provide an interpretation of each coefficient in the model. Be careful - some of the variables in the model are categorical!

ggplot(data=Carseats, aes(x = Price+Urban+US , y = Sales, color=Urban))+
  geom_jitter()+
  geom_abline(intercept = modcarseats1$coefficients[1], slope=modcarseats1$coefficients[4],
              color="red", lwd=1)+ #non US and non Urban
  geom_abline(intercept = modcarseats1$coefficients[1]+modcarseats1$coefficients[2], slope=modcarseats1$coefficients[4],
              color="forestgreen", lwd=1)+ #Urban Yes
  geom_abline(intercept = modcarseats1$coefficients[1]+modcarseats1$coefficients[3], slope=modcarseats1$coefficients[4], 
              color="blue", lwd=1) #US Yes

## Warning in Ops.factor(Price, Urban): '+' not meaningful for factors

## Warning in Ops.factor(Price + Urban, US): '+' not meaningful for factors

## Warning in Ops.factor(Price, Urban): '+' not meaningful for factors

## Warning in Ops.factor(Price + Urban, US): '+' not meaningful for factors

ggplot(data=Carseats, aes(x = Price+Urban+US , y = Sales, color=US))+
  geom_jitter()+
  geom_abline(intercept = modcarseats1$coefficients[1], slope=modcarseats1$coefficients[4],
              color="red", lwd=1)+ #non US and non Urban
  geom_abline(intercept = modcarseats1$coefficients[1]+modcarseats1$coefficients[2], slope=modcarseats1$coefficients[4],
              color="forestgreen", lwd=1)+ #Urban Yes
  geom_abline(intercept = modcarseats1$coefficients[1]+modcarseats1$coefficients[3], slope=modcarseats1$coefficients[4], 
              color="blue", lwd=1) #US Yes

## Warning in Ops.factor(Price, Urban): '+' not meaningful for factors

## Warning in Ops.factor(Price, Urban): '+' not meaningful for factors

## Warning in Ops.factor(Price, Urban): '+' not meaningful for factors

## Warning in Ops.factor(Price + Urban, US): '+' not meaningful for factors

D) Write out the model in equation form, being careful to handle the qualitative variables properly. (Hint: You can write separate equations for each combination of Urban and US groups).

\(Y_{Sales}\) = \(\beta_0\) + \(\beta_1\) \(X_{Price}\) + \(\beta_2\) \(X_{Urban}\), where \(X_1\) is estimated coefficients of Price and \(X_{Urban}\) is equal to 1 if Urban carsales is yes and is equal to 0 if Urban carsales is no.

\(Y_{Sales}\) = \(\beta_0\) + \(\beta_1\) \(X_{Price}\) + \(\beta_2\) \(X_{US}\), where \(X_1\) is estimated coefficients of Price and \(X_2\) is equal to 1 if US carsales is yes and equal to 0 if US carsales is no.

E) For which of the predictors can you reject the null hypothesis H0 : βj = 0 ?

F) On the basis of your response in the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome. (i.e. keep only the predictors that are significant)

G) How well do the model in (a) and (f) fit the data? (Hint: Use the MSE)

H) Using the model from (a), obtain 95% confidence intervals for the coefficient(s). Discuss what the confidence intervals for the coefficients tell us. (Hint: confint()).

HW7

Takuma Mimura

2/27/2020

Directions: In this assignment you will get to practice more with inference from a categorical predictor. Note: You don’t have to do these in any particular order. You might want to do Part II first to help you with Part I.

Part I: Use the your regression project dataset to explore models that incorporate categorical predictors.

B) Fit a simple linear model with response variable and the numeric predictor that you chose. Does the relationship appear to be significant? Make sure to also include a graphic.

C) Now, write the “dummy” variable coding for your categorical variable. (Hint: the contrasts() function might help).

D) Fit a linear model with response variable and the categorical variable. Does it appear that there are differences among the means of levels of the categorical variable? (Hint: Look at the ANOVA F-test). Be sure to include an appropriate graphic (i.e. side-by-side boxplot)

E) Now fit a multiple linear model that combines parts (b) and (d), with both the numeric and categorical variables. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

F) Finally, fit a multiple linear model that includes also the interaction between the numeric and categorical variables, which allows for different slopes. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

G) Conclusion: What did you learn from this exercise? Were any of the relationships significant? (Note: This would be great to include in your final project write up!)

Part II:

This question involves the use of multiple linear regression with categorical variables on the Carseat dataset from the textbook (i.e. using library(ISLR))]

A) Describe the variables Sales, Price, Urban, and US. Are the variables numeric or categorical? If they are categorical describe the levels.

B) Fit a multiple regression model to predict Sales using Price, Urban, and US.

C) Provide an interpretation of each coefficient in the model. Be careful - some of the variables in the model are categorical!

D) Write out the model in equation form, being careful to handle the qualitative variables properly. (Hint: You can write separate equations for each combination of Urban and US groups).

E) For which of the predictors can you reject the null hypothesis H0 : βj = 0 ?

F) On the basis of your response in the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome. (i.e. keep only the predictors that are significant)

G) How well do the model in (a) and (f) fit the data? (Hint: Use the MSE)

H) Using the model from (a), obtain 95% confidence intervals for the coefficient(s). Discuss what the confidence intervals for the coefficients tell us. (Hint: confint()).

HW7

Takuma Mimura

2/27/2020

Directions: In this assignment you will get to practice more with inference from a categorical predictor. Note: You don’t have to do these in any particular order. You might want to do Part II first to help you with Part I.

Part I: Use the your regression project dataset to explore models that incorporate categorical predictors.

A) Identify your response variable, a categorical predictor, and a numeric predictor (that you suspect might be related to your response). Describe the units for these variables and for the categorical variable describe the levels.

B) Fit a simple linear model with response variable and the numeric predictor that you chose. Does the relationship appear to be significant? Make sure to also include a graphic.

C) Now, write the “dummy” variable coding for your categorical variable. (Hint: the contrasts() function might help).

D) Fit a linear model with response variable and the categorical variable. Does it appear that there are differences among the means of levels of the categorical variable? (Hint: Look at the ANOVA F-test). Be sure to include an appropriate graphic (i.e. side-by-side boxplot)

E) Now fit a multiple linear model that combines parts (b) and (d), with both the numeric and categorical variables. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

F) Finally, fit a multiple linear model that includes also the interaction between the numeric and categorical variables, which allows for different slopes. What are the estimated models for the different levels? Include a graphic of the scatter plot with lines overlaid for each level.

G) Conclusion: What did you learn from this exercise? Were any of the relationships significant? (Note: This would be great to include in your final project write up!)

Part II:

This question involves the use of multiple linear regression with categorical variables on the Carseat dataset from the textbook (i.e. using library(ISLR) )]

A) Describe the variables Sales , Price , Urban , and US . Are the variables numeric or categorical? If they are categorical describe the levels.

B) Fit a multiple regression model to predict Sales using Price , Urban , and US.

C) Provide an interpretation of each coefficient in the model. Be careful - some of the variables in the model are categorical!

D) Write out the model in equation form, being careful to handle the qualitative variables properly. (Hint: You can write separate equations for each combination of Urban and US groups).

E) For which of the predictors can you reject the null hypothesis H0 : βj = 0 ?

F) On the basis of your response in the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome. (i.e. keep only the predictors that are significant)

G) How well do the model in (a) and (f) fit the data? (Hint: Use the MSE)

H) Using the model from (a), obtain 95% confidence intervals for the coefficient(s). Discuss what the confidence intervals for the coefficients tell us. (Hint: confint() ).

This question involves the use of multiple linear regression with categorical variables on the Carseat dataset from the textbook (i.e. using library(ISLR))]

A) Describe the variables Sales, Price, Urban, and US. Are the variables numeric or categorical? If they are categorical describe the levels.

B) Fit a multiple regression model to predict Sales using Price, Urban, and US.

H) Using the model from (a), obtain 95% confidence intervals for the coefficient(s). Discuss what the confidence intervals for the coefficients tell us. (Hint: confint()).