Recipe 8
Recipe 8
Matthew Macchi
Rensselaer Polytechnic Institute
11/20/14 Version 1
1. Setting
System under test
This recipe will conduct an experiment on the ecdat dataset. The experiment will attempt to investigate the Prices of Personal Computers from the Computers dataset.
x <- read.csv("~/Desktop/Computers.csv")
x## price speed hd ram screen ads trend
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## 999 2469 1 1 1 1 1 1
## 1000 1495 -1 -1 -1 -1 1 1Factors and Levels
The data being examined has 1000 observations of 7 variables.
head(x)## price speed hd ram screen ads trend
## 1 1499 -1 -1 -1 -1 -1 -1
## 2 1795 -1 -1 -1 -1 -1 -1
## 3 1595 -1 -1 -1 1 -1 -1
## 4 1849 -1 -1 1 -1 -1 -1
## 5 3295 -1 1 1 -1 -1 -1
## 6 3695 1 1 1 -1 -1 -1tail(x)## price speed hd ram screen ads trend
## 995 2744 -1 1 1 -1 1 1
## 996 2444 -1 1 1 -1 1 1
## 997 2090 -1 1 1 1 1 1
## 998 2290 1 -1 -1 -1 1 1
## 999 2469 1 1 1 1 1 1
## 1000 1495 -1 -1 -1 -1 1 1summary(x)## price speed hd ram
## Min. :1049 Min. :-1.000 Min. :-1.00 Min. :-1.000
## 1st Qu.:1975 1st Qu.:-1.000 1st Qu.:-1.00 1st Qu.:-1.000
## Median :2325 Median :-1.000 Median :-1.00 Median : 1.000
## Mean :2382 Mean :-0.028 Mean :-0.43 Mean : 0.046
## 3rd Qu.:2699 3rd Qu.: 1.000 3rd Qu.: 1.00 3rd Qu.: 1.000
## Max. :4494 Max. : 1.000 Max. : 1.00 Max. : 1.000
## screen ads trend
## Min. :-1.0 Min. :-1.000 Min. :-1.000
## 1st Qu.:-1.0 1st Qu.:-1.000 1st Qu.:-1.000
## Median :-1.0 Median :-1.000 Median : 1.000
## Mean :-0.4 Mean :-0.072 Mean : 0.206
## 3rd Qu.: 1.0 3rd Qu.: 1.000 3rd Qu.: 1.000
## Max. : 1.0 Max. : 1.000 Max. : 1.000str(x)## 'data.frame': 1000 obs. of 7 variables:
## $ price : int 1499 1795 1595 1849 3295 3695 1720 1995 2225 2575 ...
## $ speed : int -1 -1 -1 -1 -1 1 -1 1 1 1 ...
## $ hd : int -1 -1 -1 -1 1 1 -1 -1 -1 -1 ...
## $ ram : int -1 -1 -1 1 1 1 -1 -1 1 -1 ...
## $ screen: int -1 -1 1 -1 -1 -1 -1 -1 -1 1 ...
## $ ads : int -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ...
## $ trend : int -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ...Continuous variables (if any)
If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable. The continuous variable in this dataset is price. ### Response variables A response variable is defined as the outcome of a study. It is a variable you would be interested in predicting or forecasting. It is often called a dependent variable or predicted variable. In this instance, a response variable is price. ### The Data: How is it organized and what does it look like? The data is organized initially into a 7 column table: The columns are titled as follows: price, speed, hd, ram, screen, ads, trend. Data in the columns are binary integers.The integers were assigned a 1 if it was greater than the column average and a -1 if the cell was less than the column average.
require(FrF2)## Loading required package: FrF2
## Loading required package: DoE.base
## Loading required package: grid
## Loading required package: conf.design
##
## Attaching package: 'DoE.base'
##
## The following objects are masked from 'package:stats':
##
## aov, lm
##
## The following object is masked from 'package:graphics':
##
## plot.designRandomization
This data comes from a cross-section from 1993-1995. Since this is the only information available in regards to background information about the data collection, it is entirely possible that this data might not be completely randomized or the experiment had a completely randomized design. # 2. (Experimental) Design ### How will the experiment be organized and conducted to test the hypothesis? In order to conduct this experiment, I will conduct an analysis of covariance (ANOVA). After ANOVA has been performed, resampling methods will be implemented to determine their effect on the outcome of the ANOVA test. In this experimental data analysis I will use a fractional factorial design to select 32 out of the total 64 experimental runs to perform data analysis on. I will do this using the R package “FrF2”. Before that I will perform data analysis on the full factorial design to compare to the results of the fractional factorial design.
### What is the rationale for this design? This dataframe in this recipe contains a single factor with multiple levels. Therefore, an ANOVA is the appropriate test to be performed. The resampling techniques performed are conducted due to the fact that ANOVA makes the assumption that all data is normally distributed. While it is possible for this distribution to occur, rarely is any data completely normal. ### Randomize: What is the Randomization Scheme? The experiment was based on observations and therefore was not randomized. ### Replicate: Are there replicates and/or repeated measures? There are no replicates, but repeated measures do occur between the factors and levels. ### Block: Did you use blocking in the design? No blocking was performed in this design. # 3. Statistical Analysis ##Exploratory Data Analysis: Graphics and Decriptive Summary First I must define all six factors as factors up for analysis.
x$speed=as.factor(x$speed)
x$hd=as.factor(x$hd)
x$ram=as.factor(x$ram)
x$screen=as.factor(x$screen)
x$ads=as.factor(x$ads)
x$trend=as.factor(x$trend)Next, I create boxplots to attempt to identify any patterns in the data.
par(mfrow=c(3,2))
plot(x$price~x$speed, xlab="Clock Speed (MHz)", ylab="Price ($)")
plot(x$price~x$hd, xlab="Size of hard drive in MB", ylab="Price ($)")
plot(x$price~x$ram, xlab="Size of Ram in MB", ylab="Price ($)")
plot(x$price~x$screen, xlab="Size of Screen (inches)", ylab="Price ($)")
plot(x$price~x$ads, xlab="Number of price listings for each month", ylab="Price ($)")
plot(x$price~x$trend, xlab="Time trend from Jan 1993 - Nov 1995", ylab="Price ($)")
##ANOVA Testing An analysis of variance (ANOVA) will be used to determine the statistical significance between the individual pricing means. If the null hypothesis is accepted, it can be assumed that there is no variation in price due to the variation of factors. If the null hypothesis is rejected, it can be assumed that the variation of price can be attributed to the variation of factors.
modelx = lm(price~speed+hd+ram+screen+ads+trend, data=x)
anova(modelx)## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## speed 1 6.64e+07 6.64e+07 535.2 < 2e-16 ***
## hd 1 1.22e+08 1.22e+08 987.1 < 2e-16 ***
## ram 1 3.03e+07 3.03e+07 244.1 < 2e-16 ***
## screen 1 4.80e+06 4.80e+06 38.7 7.3e-10 ***
## ads 1 7.47e+06 7.47e+06 60.2 2.1e-14 ***
## trend 1 3.41e+06 3.41e+06 27.5 2.0e-07 ***
## Residuals 993 1.23e+08 1.24e+05
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The results of the ANOVA show that in the full factorial experiment all factors have statistically insignificant effects on the resulting material strength that can likely be attributed to something other than randomization. Thus we reject the null hypothesis which states that a factor does not have an effect on the response variable for those factors. Furthermore we fail to reject this null hypothesis for the factors. ##Fractional Factorial Design The next step is to construct a matrix for a 2^(6-1) experimental design, which equates to one half (32/64) fractional factorial experimental design. I am most interested in observing the main and interaction effects of this factor.
Design2=FrF2(32,nfactors=6, estimable=formula("~speed+hd+ram+screen+ads+trend+screen:(speed+hd+ram+screen+ads+trend)"),factor.names=c("speed", "hd", "ram", "screen", "ads", "trend"), res4=TRUE, clear=FALSE)
Design2## speed hd ram screen ads trend
## 1 1 -1 -1 -1 1 -1
## 2 -1 -1 -1 1 -1 1
## 3 1 1 1 1 1 1
## 4 1 1 1 -1 1 -1
## 5 1 -1 1 1 -1 1
## 6 1 1 -1 -1 1 1
## 7 -1 -1 1 1 -1 -1
## 8 1 -1 -1 -1 -1 1
## 9 -1 -1 1 1 1 1
## 10 -1 1 -1 -1 1 -1
## 11 1 1 -1 1 -1 1
## 12 -1 1 -1 1 -1 -1
## 13 1 -1 -1 1 1 1
## 14 -1 -1 -1 -1 1 1
## 15 -1 -1 1 -1 1 -1
## 16 -1 1 -1 -1 -1 1
## 17 -1 1 1 -1 1 1
## 18 1 1 1 -1 -1 1
## 19 -1 1 1 1 1 -1
## 20 1 -1 1 1 1 -1
## 21 -1 -1 -1 1 1 -1
## 22 1 -1 1 -1 -1 -1
## 23 -1 -1 -1 -1 -1 -1
## 24 1 1 1 1 -1 -1
## 25 -1 -1 1 -1 -1 1
## 26 1 -1 -1 1 -1 -1
## 27 -1 1 -1 1 1 1
## 28 1 1 -1 1 1 -1
## 29 -1 1 1 1 -1 1
## 30 1 1 -1 -1 -1 -1
## 31 -1 1 1 -1 -1 -1
## 32 1 -1 1 -1 1 1
## class=design, type= FrF2.estimablealiasprint(Design2)## $legend
## [1] A=speed B=hd C=ram D=screen E=ads F=trend
##
## [[2]]
## [1] no aliasing among main effects and 2fisNext I create a brand new dataset which only uses the selected experimental runs from the fractional factorial design that I created previously.
Observations=merge(Design2,x,by=c("speed", "hd", "ram", "screen", "ads", "trend"), all=FALSE)
print(Observations)## speed hd ram screen ads trend price
## 1 -1 -1 -1 -1 -1 -1 1945
## 2 -1 -1 -1 -1 -1 -1 1595
## 3 -1 -1 -1 -1 -1 -1 1595
## 4 -1 -1 -1 -1 -1 -1 1495
## 5 -1 -1 -1 -1 -1 -1 1395
## 6 -1 -1 -1 -1 -1 -1 1999
## 7 -1 -1 -1 -1 -1 -1 1895
## 8 -1 -1 -1 -1 -1 -1 1920
## 9 -1 -1 -1 -1 -1 -1 1499
## 10 -1 -1 -1 -1 -1 -1 2799
## 11 -1 -1 -1 -1 -1 -1 2190
## 12 -1 -1 -1 -1 -1 -1 2095
## 13 -1 -1 -1 -1 -1 -1 1495
## 14 -1 -1 -1 -1 -1 -1 1595
## 15 -1 -1 -1 -1 -1 -1 1499
## 16 -1 -1 -1 -1 -1 -1 2099
## 17 -1 -1 -1 -1 -1 -1 1749
## 18 -1 -1 -1 -1 -1 -1 2199
## 19 -1 -1 -1 -1 -1 -1 1395
## 20 -1 -1 -1 -1 -1 -1 2220
## 21 -1 -1 -1 -1 -1 -1 1995
## 22 -1 -1 -1 -1 -1 -1 1699
## 23 -1 -1 -1 -1 -1 -1 2099
## 24 -1 -1 -1 -1 -1 -1 1749
## 25 -1 -1 -1 -1 -1 -1 1499
## 26 -1 -1 -1 -1 -1 -1 2449
## 27 -1 -1 -1 -1 -1 -1 1699
## 28 -1 -1 -1 -1 -1 -1 2449
## 29 -1 -1 -1 -1 -1 -1 1995
## 30 -1 -1 -1 -1 -1 -1 1695
## 31 -1 -1 -1 -1 -1 -1 1895
## 32 -1 -1 -1 -1 -1 -1 2220
## 33 -1 -1 -1 -1 -1 -1 2190
## 34 -1 -1 -1 -1 -1 -1 2499
## 35 -1 -1 -1 -1 -1 -1 1720
## 36 -1 -1 -1 -1 -1 -1 1720
## 37 -1 -1 -1 -1 -1 -1 1975
## 38 -1 -1 -1 -1 -1 -1 1999
## 39 -1 -1 -1 -1 -1 -1 1899
## 40 -1 -1 -1 -1 -1 -1 1775
## 41 -1 -1 -1 -1 -1 -1 1945
## 42 -1 -1 -1 -1 -1 -1 1695
## 43 -1 -1 -1 -1 -1 -1 1895
## 44 -1 -1 -1 -1 -1 -1 1595
## 45 -1 -1 -1 -1 -1 -1 1695
## 46 -1 -1 -1 -1 -1 -1 1995
## 47 -1 -1 -1 -1 -1 -1 2220
## 48 -1 -1 -1 -1 -1 -1 2499
## 49 -1 -1 -1 -1 -1 -1 1775
## 50 -1 -1 -1 -1 -1 -1 1795
## 51 -1 -1 -1 -1 -1 -1 1895
## 52 -1 -1 -1 -1 -1 -1 1795
## 53 -1 -1 -1 -1 -1 -1 1599
## 54 -1 -1 -1 -1 -1 -1 1899
## 55 -1 -1 -1 -1 -1 -1 2195
## 56 -1 -1 -1 -1 -1 -1 1995
## 57 -1 -1 -1 -1 -1 -1 1990
## 58 -1 -1 -1 -1 -1 -1 1795
## 59 -1 -1 -1 -1 -1 -1 1899
## 60 -1 -1 -1 -1 -1 -1 1495
## 61 -1 -1 -1 -1 -1 -1 1995
## 62 -1 -1 -1 -1 -1 -1 2395
## 63 -1 -1 -1 -1 -1 -1 1290
## 64 -1 -1 -1 -1 -1 -1 1995
## 65 -1 -1 -1 -1 -1 -1 1795
## 66 -1 -1 -1 -1 -1 -1 1495
## 67 -1 -1 -1 -1 -1 -1 2095
## 68 -1 -1 -1 -1 -1 -1 1490
## 69 -1 -1 -1 -1 -1 -1 1399
## 70 -1 -1 -1 -1 -1 -1 2090
## 71 -1 -1 -1 -1 -1 -1 1399
## 72 -1 -1 -1 -1 -1 -1 1999
## 73 -1 -1 -1 -1 -1 -1 1499
## 74 -1 -1 -1 -1 -1 -1 1499
## 75 -1 -1 -1 -1 -1 -1 1495
## 76 -1 -1 -1 -1 -1 -1 1799
## 77 -1 -1 -1 -1 -1 -1 1395
## 78 -1 -1 -1 -1 -1 -1 2199
## 79 -1 -1 -1 -1 -1 -1 2199
## 80 -1 -1 -1 -1 -1 -1 1690
## 81 -1 -1 -1 -1 -1 -1 1999
## 82 -1 -1 -1 -1 -1 -1 1920
## 83 -1 -1 -1 -1 -1 -1 2099
## 84 -1 -1 -1 -1 -1 -1 2199
## 85 -1 -1 -1 -1 -1 -1 1720
## 86 -1 -1 -1 -1 -1 -1 1495
## 87 -1 -1 -1 -1 -1 -1 1695
## 88 -1 -1 -1 -1 -1 -1 1920
## 89 -1 -1 -1 -1 -1 -1 1395
## 90 -1 -1 -1 -1 -1 -1 1995
## 91 -1 -1 -1 -1 -1 -1 2095
## 92 -1 -1 -1 -1 -1 -1 1499
## 93 -1 -1 -1 -1 -1 -1 1595
## 94 -1 -1 -1 -1 -1 -1 1590
## 95 -1 -1 -1 -1 -1 -1 1695
## 96 -1 -1 -1 -1 1 1 1994
## 97 -1 -1 -1 -1 1 1 1295
## 98 -1 -1 -1 -1 1 1 1399
## 99 -1 -1 -1 -1 1 1 1575
## 100 -1 -1 -1 -1 1 1 1579
## 101 -1 -1 -1 -1 1 1 1395
## 102 -1 -1 -1 -1 1 1 1699
## 103 -1 -1 -1 -1 1 1 1449
## 104 -1 -1 -1 -1 1 1 1849
## 105 -1 -1 -1 -1 1 1 1799
## 106 -1 -1 -1 -1 1 1 1644
## 107 -1 -1 -1 -1 1 1 1495
## 108 -1 -1 -1 -1 1 1 1990
## 109 -1 -1 -1 -1 1 1 1990
## 110 -1 -1 -1 -1 1 1 1890
## 111 -1 -1 -1 -1 1 1 1720
## 112 -1 -1 -1 -1 1 1 1749
## 113 -1 -1 -1 -1 1 1 1520
## 114 -1 -1 -1 -1 1 1 1694
## 115 -1 -1 -1 -1 1 1 1720
## 116 -1 -1 -1 -1 1 1 1579
## 117 -1 -1 -1 -1 1 1 1848
## 118 -1 -1 -1 -1 1 1 1399
## 119 -1 -1 -1 -1 1 1 2144
## 120 -1 -1 -1 -1 1 1 1950
## 121 -1 -1 -1 -1 1 1 2095
## 122 -1 -1 -1 -1 1 1 1595
## 123 -1 -1 -1 -1 1 1 2090
## 124 -1 -1 -1 -1 1 1 1895
## 125 -1 -1 -1 -1 1 1 1795
## 126 -1 -1 -1 -1 1 1 1795
## 127 -1 -1 -1 -1 1 1 1899
## 128 -1 -1 -1 -1 1 1 1749
## 129 -1 -1 -1 -1 1 1 1899
## 130 -1 -1 -1 -1 1 1 1295
## 131 -1 -1 -1 -1 1 1 1865
## 132 -1 -1 -1 -1 1 1 1399
## 133 -1 -1 -1 -1 1 1 1049
## 134 -1 -1 -1 -1 1 1 1520
## 135 -1 -1 -1 -1 1 1 1599
## 136 -1 -1 -1 -1 1 1 1490
## 137 -1 -1 -1 -1 1 1 1999
## 138 -1 -1 -1 -1 1 1 1695
## 139 -1 -1 -1 -1 1 1 1599
## 140 -1 -1 -1 -1 1 1 1690
## 141 -1 -1 -1 -1 1 1 1695
## 142 -1 -1 -1 -1 1 1 1599
## 143 -1 -1 -1 -1 1 1 1399
## 144 -1 -1 -1 -1 1 1 2044
## 145 -1 -1 -1 -1 1 1 1495
## 146 -1 -1 -1 -1 1 1 1690
## 147 -1 -1 -1 -1 1 1 1844
## 148 -1 -1 -1 -1 1 1 1895
## 149 -1 -1 -1 -1 1 1 1499
## 150 -1 -1 -1 -1 1 1 1944
## 151 -1 -1 -1 -1 1 1 2109
## 152 -1 -1 -1 -1 1 1 1744
## 153 -1 -1 -1 -1 1 1 1495
## 154 -1 -1 -1 -1 1 1 2244
## 155 -1 -1 -1 -1 1 1 1749
## 156 -1 -1 -1 -1 1 1 1399
## 157 -1 -1 -1 -1 1 1 1594
## 158 -1 -1 -1 -1 1 1 1829
## 159 -1 -1 -1 -1 1 1 1499
## 160 -1 -1 -1 -1 1 1 2099
## 161 -1 -1 -1 -1 1 1 1499
## 162 -1 -1 -1 -1 1 1 1799
## 163 -1 -1 -1 -1 1 1 1745
## 164 -1 -1 -1 -1 1 1 1299
## 165 -1 -1 -1 -1 1 1 2099
## 166 -1 -1 -1 -1 1 1 2095
## 167 -1 -1 -1 -1 1 1 1559
## 168 -1 -1 -1 -1 1 1 1795
## 169 -1 -1 -1 -1 1 1 2294
## 170 -1 -1 -1 -1 1 1 1890
## 171 -1 -1 -1 -1 1 1 1744
## 172 -1 -1 -1 -1 1 1 1994
## 173 -1 -1 -1 -1 1 1 2094
## 174 -1 -1 -1 -1 1 1 1725
## 175 -1 -1 -1 -1 1 1 2109
## 176 -1 -1 -1 -1 1 1 2044
## 177 -1 -1 -1 -1 1 1 1899
## 178 -1 -1 -1 -1 1 1 1470
## 179 -1 -1 -1 -1 1 1 1799
## 180 -1 -1 -1 -1 1 1 1544
## 181 -1 -1 -1 -1 1 1 1590
## 182 -1 -1 -1 -1 1 1 1899
## 183 -1 -1 -1 -1 1 1 1999
## 184 -1 -1 -1 -1 1 1 1899
## 185 -1 -1 -1 -1 1 1 2095
## 186 -1 -1 -1 -1 1 1 1295
## 187 -1 -1 -1 -1 1 1 2190
## 188 -1 -1 -1 -1 1 1 1395
## 189 -1 -1 -1 -1 1 1 1495
## 190 -1 -1 -1 -1 1 1 2799
## 191 -1 -1 -1 -1 1 1 2190
## 192 -1 -1 -1 -1 1 1 1499
## 193 -1 -1 -1 -1 1 1 1575
## 194 -1 -1 -1 -1 1 1 1845
## 195 -1 -1 -1 -1 1 1 1390
## 196 -1 -1 -1 -1 1 1 1539
## 197 -1 -1 -1 -1 1 1 1945
## 198 -1 -1 -1 -1 1 1 2799
## 199 -1 -1 -1 -1 1 1 1898
## 200 -1 -1 -1 -1 1 1 2099
## 201 -1 -1 -1 -1 1 1 1299
## 202 -1 -1 -1 -1 1 1 2344
## 203 -1 -1 -1 -1 1 1 1675
## 204 -1 -1 -1 -1 1 1 1495
## 205 -1 -1 -1 -1 1 1 1890
## 206 -1 -1 -1 -1 1 1 1495
## 207 -1 -1 -1 -1 1 1 1899
## 208 -1 -1 -1 -1 1 1 1794
## 209 -1 -1 -1 -1 1 1 2044
## 210 -1 -1 -1 -1 1 1 1449
## 211 -1 -1 -1 -1 1 1 1944
## 212 -1 -1 -1 -1 1 1 1595
## 213 -1 -1 -1 1 -1 1 2190
## 214 -1 -1 -1 1 -1 1 1690
## 215 -1 -1 -1 1 -1 1 1990
## 216 -1 -1 -1 1 -1 1 1490
## 217 -1 -1 -1 1 -1 1 2090
## 218 -1 -1 -1 1 -1 1 1590
## 219 -1 -1 1 -1 -1 1 2035
## 220 -1 -1 1 -1 -1 1 1999
## 221 -1 -1 1 -1 -1 1 2220
## 222 -1 -1 1 -1 -1 1 2155
## 223 -1 -1 1 -1 -1 1 2395
## 224 -1 -1 1 -1 -1 1 1995
## 225 -1 -1 1 1 -1 -1 2345
## 226 -1 -1 1 1 -1 -1 2495
## 227 -1 -1 1 1 -1 -1 2720
## 228 -1 -1 1 1 -1 -1 2599
## 229 -1 -1 1 1 -1 -1 2495
## 230 -1 -1 1 1 -1 -1 2420
## 231 -1 -1 1 1 -1 -1 2785
## 232 -1 -1 1 1 -1 -1 2605
## 233 -1 -1 1 1 -1 -1 2125
## 234 -1 -1 1 1 -1 -1 2825
## 235 -1 -1 1 1 -1 -1 2420
## 236 -1 -1 1 1 -1 -1 2515
## 237 -1 -1 1 1 -1 -1 2420
## 238 -1 -1 1 1 -1 -1 2345
## 239 -1 -1 1 1 -1 -1 2720
## 240 -1 -1 1 1 -1 -1 2635
## 241 -1 -1 1 1 -1 -1 2645
## 242 -1 -1 1 1 -1 -1 2720
## 243 -1 -1 1 1 -1 -1 2195
## 244 -1 -1 1 1 -1 -1 2595
## 245 -1 -1 1 1 -1 -1 2495
## 246 -1 -1 1 1 -1 -1 2195
## 247 -1 -1 1 1 -1 -1 2195
## 248 -1 -1 1 1 -1 -1 2599
## 249 -1 -1 1 1 -1 -1 2195
## 250 -1 -1 1 1 -1 -1 2225
## 251 -1 -1 1 1 -1 -1 2535
## 252 -1 -1 1 1 1 1 2025
## 253 -1 -1 1 1 1 1 1995
## 254 -1 -1 1 1 1 1 2245
## 255 -1 -1 1 1 1 1 2495
## 256 -1 -1 1 1 1 1 2075
## 257 -1 -1 1 1 1 1 1895
## 258 -1 -1 1 1 1 1 2465
## 259 -1 -1 1 1 1 1 2045
## 260 -1 -1 1 1 1 1 2045
## 261 -1 -1 1 1 1 1 1995
## 262 -1 -1 1 1 1 1 2015
## 263 -1 -1 1 1 1 1 2675
## 264 -1 -1 1 1 1 1 2275
## 265 -1 -1 1 1 1 1 1845
## 266 -1 -1 1 1 1 1 2455
## 267 -1 1 -1 -1 -1 1 2049
## 268 -1 1 1 -1 -1 -1 2595
## 269 -1 1 1 -1 -1 -1 3095
## 270 -1 1 1 -1 -1 -1 2895
## 271 -1 1 1 -1 -1 -1 3795
## 272 -1 1 1 -1 -1 -1 2695
## 273 -1 1 1 -1 -1 -1 2690
## 274 -1 1 1 -1 -1 -1 2595
## 275 -1 1 1 -1 -1 -1 2890
## 276 -1 1 1 -1 -1 -1 3295
## 277 -1 1 1 -1 -1 -1 3995
## 278 -1 1 1 -1 -1 -1 2695
## 279 -1 1 1 -1 -1 -1 2695
## 280 -1 1 1 -1 -1 -1 2795
## 281 -1 1 1 -1 -1 -1 3795
## 282 -1 1 1 -1 -1 -1 2795
## 283 -1 1 1 -1 -1 -1 2790
## 284 -1 1 1 -1 -1 -1 2595
## 285 -1 1 1 -1 -1 -1 2795
## 286 -1 1 1 -1 -1 -1 3795
## 287 -1 1 1 -1 1 1 2444
## 288 -1 1 1 -1 1 1 3495
## 289 -1 1 1 -1 1 1 2744
## 290 -1 1 1 -1 1 1 2795
## 291 -1 1 1 -1 1 1 2095
## 292 -1 1 1 -1 1 1 2195
## 293 -1 1 1 -1 1 1 2590
## 294 -1 1 1 -1 1 1 2694
## 295 -1 1 1 -1 1 1 2744
## 296 -1 1 1 -1 1 1 2495
## 297 -1 1 1 -1 1 1 2195
## 298 -1 1 1 -1 1 1 2744
## 299 -1 1 1 -1 1 1 2544
## 300 -1 1 1 -1 1 1 2425
## 301 -1 1 1 -1 1 1 2890
## 302 -1 1 1 -1 1 1 2795
## 303 -1 1 1 -1 1 1 2495
## 304 -1 1 1 -1 1 1 2590
## 305 -1 1 1 -1 1 1 2544
## 306 -1 1 1 -1 1 1 2794
## 307 -1 1 1 -1 1 1 2290
## 308 -1 1 1 -1 1 1 2690
## 309 -1 1 1 -1 1 1 2495
## 310 -1 1 1 -1 1 1 2395
## 311 -1 1 1 -1 1 1 2994
## 312 -1 1 1 -1 1 1 4295
## 313 -1 1 1 -1 1 1 2644
## 314 -1 1 1 -1 1 1 2295
## 315 -1 1 1 -1 1 1 2394
## 316 -1 1 1 -1 1 1 2190
## 317 -1 1 1 -1 1 1 2390
## 318 -1 1 1 -1 1 1 2794
## 319 -1 1 1 -1 1 1 2795
## 320 -1 1 1 -1 1 1 2344
## 321 -1 1 1 -1 1 1 2344
## 322 -1 1 1 -1 1 1 2294
## 323 -1 1 1 -1 1 1 2490
## 324 -1 1 1 -1 1 1 2444
## 325 -1 1 1 -1 1 1 2344
## 326 -1 1 1 -1 1 1 3025
## 327 -1 1 1 -1 1 1 2690
## 328 -1 1 1 -1 1 1 2745
## 329 -1 1 1 -1 1 1 2944
## 330 -1 1 1 -1 1 1 4494
## 331 -1 1 1 -1 1 1 2194
## 332 -1 1 1 -1 1 1 2299
## 333 -1 1 1 -1 1 1 2590
## 334 -1 1 1 -1 1 1 2290
## 335 -1 1 1 -1 1 1 2844
## 336 -1 1 1 -1 1 1 2244
## 337 -1 1 1 -1 1 1 2595
## 338 -1 1 1 -1 1 1 2890
## 339 -1 1 1 -1 1 1 2594
## 340 -1 1 1 -1 1 1 2595
## 341 -1 1 1 -1 1 1 2644
## 342 -1 1 1 -1 1 1 3044
## 343 -1 1 1 1 -1 1 2490
## 344 -1 1 1 1 -1 1 2390
## 345 -1 1 1 1 -1 1 2790
## 346 -1 1 1 1 -1 1 2590
## 347 -1 1 1 1 -1 1 2890
## 348 -1 1 1 1 -1 1 2690
## 349 -1 1 1 1 -1 1 3599
## 350 1 -1 -1 -1 -1 1 2099
## 351 1 -1 -1 -1 -1 1 2499
## 352 1 -1 -1 -1 -1 1 1695
## 353 1 -1 -1 -1 -1 1 2295
## 354 1 -1 -1 -1 -1 1 2025
## 355 1 -1 -1 -1 -1 1 1995
## 356 1 -1 -1 -1 -1 1 2195
## 357 1 -1 -1 -1 -1 1 1999
## 358 1 -1 -1 -1 -1 1 2290
## 359 1 -1 -1 -1 -1 1 2145
## 360 1 -1 -1 -1 -1 1 2195
## 361 1 -1 -1 -1 -1 1 1790
## 362 1 -1 -1 -1 -1 1 2249
## 363 1 -1 -1 -1 -1 1 1795
## 364 1 -1 -1 -1 -1 1 2390
## 365 1 -1 -1 -1 -1 1 1890
## 366 1 -1 -1 -1 -1 1 2075
## 367 1 -1 -1 1 -1 -1 2525
## 368 1 -1 -1 1 -1 -1 2799
## 369 1 -1 -1 1 -1 -1 3065
## 370 1 -1 -1 1 -1 -1 3225
## 371 1 -1 -1 1 -1 -1 2645
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## 482 1 1 1 -1 -1 1 2995
## 483 1 1 1 -1 -1 1 2690
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## 485 1 1 1 -1 -1 1 2990
## 486 1 1 1 -1 -1 1 3595
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## 494 1 1 1 -1 -1 1 2695
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## 497 1 1 1 1 -1 -1 2995
## 498 1 1 1 1 -1 -1 2995
## 499 1 1 1 1 -1 -1 2995
## 500 1 1 1 1 -1 -1 2995
## 501 1 1 1 1 -1 -1 3895
## 502 1 1 1 1 -1 -1 3220
## 503 1 1 1 1 -1 -1 3220
## 504 1 1 1 1 -1 -1 3895
## 505 1 1 1 1 -1 -1 3895
## 506 1 1 1 1 1 1 2845
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## 517 1 1 1 1 1 1 2999
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## 519 1 1 1 1 1 1 2999
## 520 1 1 1 1 1 1 3595
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## 523 1 1 1 1 1 1 2690
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## 525 1 1 1 1 1 1 2595
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## 527 1 1 1 1 1 1 3090
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## 529 1 1 1 1 1 1 2990
## 530 1 1 1 1 1 1 2990
## 531 1 1 1 1 1 1 3899
## 532 1 1 1 1 1 1 2999
## 533 1 1 1 1 1 1 2999
## 534 1 1 1 1 1 1 2690
## 535 1 1 1 1 1 1 3789
## 536 1 1 1 1 1 1 2999
## 537 1 1 1 1 1 1 2790
## 538 1 1 1 1 1 1 2599
## 539 1 1 1 1 1 1 2899
## 540 1 1 1 1 1 1 2469
## 541 1 1 1 1 1 1 2620
## 542 1 1 1 1 1 1 4398
## 543 1 1 1 1 1 1 3398
## 544 1 1 1 1 1 1 3999
## 545 1 1 1 1 1 1 3299
## 546 1 1 1 1 1 1 3595
## 547 1 1 1 1 1 1 2995
## 548 1 1 1 1 1 1 3499
## 549 1 1 1 1 1 1 3789
## 550 1 1 1 1 1 1 2590
## 551 1 1 1 1 1 1 2225I must repeat the ANOVA method on the new dataset which will test the main effects of each factor as well as the interaction effects of Screen with all other factors.
modelb=lm(price~screen*speed+screen*hd+screen*ram+screen*ads+screen*trend, data=Observations)
anova(modelb)## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## screen 1 35753593 35753593 300.63 < 2e-16 ***
## speed 1 28803414 28803414 242.19 < 2e-16 ***
## hd 1 56903596 56903596 478.47 < 2e-16 ***
## ram 1 9787140 9787140 82.29 < 2e-16 ***
## ads 1 3322755 3322755 27.94 1.8e-07 ***
## trend 1 1331195 1331195 11.19 0.00088 ***
## screen:speed 1 300371 300371 2.53 0.11260
## screen:hd 1 1266770 1266770 10.65 0.00117 **
## screen:ram 1 25192 25192 0.21 0.64553
## screen:ads 1 1173403 1173403 9.87 0.00178 **
## screen:trend 1 666454 666454 5.60 0.01827 *
## Residuals 539 64102662 118929
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Again, we can see that most of the factors and the interaction effects are not significant. The only interaction that seems to show significance is the screen:ram interaction with a p-value of 0.646. Every other value falls below the 0.05 alpha value. ##Shapiro-Wilk Here I use the Shapiro-Wilk normality test to determine if the response variable is normally distributed.
shapiro.test(Observations$price)##
## Shapiro-Wilk normality test
##
## data: Observations$price
## W = 0.9682, p-value = 1.494e-09The null of a Shapiro-Wilk test states that the data presented is normally distributed. Unfortunately, the p-value is less than an alpha of 0.05, and therefore, we must reject the null hypothesis. This suggests, instead, that the price of personal computers data is not normally distributed. ##Diagnostics/Model Adequacy Checking Quantile-Quantile (Q-Q) plots are graphs used to verify the distributional assumption for a set of data. Based on the theoretical distribution, the expected value for each datum is determined. If the data values in a set follow the theoretical distribution, then they will appear as a straight line on a Q-Q plot. When an anova is performed, it is done so with the assumption that the test statistic follows a normal distribution. Visualization of a Q-Q plot will further confirm if that assumption is correct for the anova tests that were performed.
qqnorm(residuals(modelb))
qqline(residuals(modelb))
plot(fitted(modelb),residuals(modelb))
A Residuals vs. Fits Plot is a common graph used in residual analysis. It is a scatter plot of residuals as a function of fitted values, or the estimated responses. These plots are used to identify linearity, outliers, and error variances.