HW8
Hongyu Chen
Tuesday, November 18, 2014
Fractional Factorial Design Practice
Hongyu Chen
RPI, RIN:661405156
Nov.17 V1.0
1. Setting
System under test
This analysis is going to analyze sales prices of houses in city of Windsor, Canada. Data set is originally from ‘housing’ in package ‘Ecdat’. To practice 2^k design where there are 6 factors each with only 2 levels, I rearranged raw data and formed a CSV file.
Below is CSV file reading and quick view of data
housing<-read.csv("housingprice.csv", header=TRUE)
head(housing)## price driveway recroom fullbase gashw airco prefarea garagepl lotsize
## 1 30500 -1 -1 -1 -1 -1 -1 0 3000
## 2 42300 1 -1 -1 -1 -1 -1 0 3000
## 3 35000 -1 1 -1 -1 -1 -1 1 3240
## 4 54500 -1 -1 1 -1 -1 -1 0 3150
## 5 40500 -1 -1 -1 1 -1 -1 1 4350
## 6 44500 -1 -1 -1 -1 1 -1 0 3000
## bedrooms bathrms stories
## 1 2 1 1
## 2 2 1 2
## 3 2 1 1
## 4 2 2 1
## 5 3 1 2
## 6 3 1 1str(housing)## 'data.frame': 64 obs. of 12 variables:
## $ price : int 30500 42300 35000 54500 40500 44500 35500 35000 51000 56000 ...
## $ driveway: int -1 1 -1 -1 -1 -1 -1 1 1 1 ...
## $ recroom : int -1 -1 1 -1 -1 -1 -1 1 -1 -1 ...
## $ fullbase: int -1 -1 -1 1 -1 -1 -1 -1 1 -1 ...
## $ gashw : int -1 -1 -1 -1 1 -1 -1 -1 -1 1 ...
## $ airco : int -1 -1 -1 -1 -1 1 -1 -1 -1 -1 ...
## $ prefarea: int -1 -1 -1 -1 -1 -1 1 -1 -1 -1 ...
## $ garagepl: int 0 0 1 0 1 0 0 0 0 1 ...
## $ lotsize : int 3000 3000 3240 3150 4350 3000 3000 3500 3150 3290 ...
## $ bedrooms: int 2 2 2 2 3 3 3 2 3 2 ...
## $ bathrms : int 1 1 1 2 1 1 1 1 1 1 ...
## $ stories : int 1 2 1 1 2 1 2 1 2 1 ...summary(housing)## price driveway recroom fullbase gashw
## Min. : 25000 Min. :-1.0000 Min. :-1 Min. :-1 Min. :-1
## 1st Qu.: 41625 1st Qu.:-1.0000 1st Qu.:-1 1st Qu.:-1 1st Qu.:-1
## Median : 51500 Median : 1.0000 Median : 0 Median : 0 Median : 0
## Mean : 54527 Mean : 0.0625 Mean : 0 Mean : 0 Mean : 0
## 3rd Qu.: 62000 3rd Qu.: 1.0000 3rd Qu.: 1 3rd Qu.: 1 3rd Qu.: 1
## Max. :138300 Max. : 1.0000 Max. : 1 Max. : 1 Max. : 1
## airco prefarea garagepl lotsize bedrooms
## Min. :-1 Min. :-1 Min. :0.000 Min. :1905 Min. :2.00
## 1st Qu.:-1 1st Qu.:-1 1st Qu.:0.000 1st Qu.:3000 1st Qu.:2.00
## Median : 0 Median : 0 Median :0.000 Median :3415 Median :3.00
## Mean : 0 Mean : 0 Mean :0.453 Mean :3654 Mean :2.92
## 3rd Qu.: 1 3rd Qu.: 1 3rd Qu.:1.000 3rd Qu.:4081 3rd Qu.:3.00
## Max. : 1 Max. : 1 Max. :2.000 Max. :7686 Max. :5.00
## bathrms stories
## Min. :1.00 Min. :1.00
## 1st Qu.:1.00 1st Qu.:1.00
## Median :1.00 Median :2.00
## Mean :1.30 Mean :1.62
## 3rd Qu.:1.25 3rd Qu.:2.00
## Max. :3.00 Max. :3.00Factors and Levels
In this study we focus on six factors that possibly influence housing price, including driveway, recreational room, full finished basement, gas for heating, central air conditioning and whether located in preffered neighbourhood of the city. Each of them has 2 levels, namely ‘yes’ or ‘no’. In data set ‘1’ represents ‘yes’, no represents ‘no’.
Response variables
Housing price is the response variable in this analysis, which is column ‘price’ in the data set.
The Data: How is it organized and what does it look like?
The dataframe contains columns about factors that might influence housing price. For example size in square feet, number of bedrooms and so on. What we are interested in are 6 factors, namely whether the house has a driveway, a recreational room, a full finished basement, gas for water heating, central air conditioning and preferred neighbourhood.
Randomization
It can be assumed that all data were randomly collected.
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
Null hypothesis H0:None of these six factors could explain variance of housing price.
Alternative hypothesis H1: variance of housing price is due to something other than randomizaion.
First set up complete factorial design and conduct exploratory data analysis, analysis of variance and model adequacy checking; then use ‘FrF2’ function to generate selected half data and set up factional factorial design. Conduct analysis of variance, model adequacy checking and then compare results of complete and fractional factorial design in terms of main effect and low order interations to see whether fractional factorial design could successfully replace complete factorial design.
What is the rationale for this design?
In fractional factorial design, you do not have to take measures on every possible combination of factor levels. Just carefully choose a few to ensure main effect and low-order interation effect can be estimated, therefore helps to save time and energy.
Randomize:What is the Randomization Scheme?
Randomization depends on the way data collected which can be assumed as randomly gethered.
Replicate:Are there replicates and/or repeated measures?
There are no replicates or repeated measures in this test.
3. Analysis of complete factorial design
(Exploratory Data Analysis) Graphics and descriptive summary
#Factors in dataset
housing$driveway=as.character(housing$driveway)
housing$recroom=as.character(housing$recroom)
housing$fullbase=as.character(housing$fullbase)
housing$gashw=as.character(housing$gashw)
housing$airco=as.character(housing$airco)
housing$prefarea=as.character(housing$prefarea)
#Boxplot
par(mfrow=c(2,3))
boxplot(housing$price~housing$driveway,xlab="driveway (1=yes,-1=no)",ylab="housing price")
boxplot(housing$price~housing$recroom,xlab="recroom (1=yes,-1=no)",ylab="housing price")
boxplot(housing$price~housing$fullbase,xlab="fullbase (1=yes,-1=no)",ylab="housing price")
boxplot(housing$price~housing$gashw,xlab="gashw (1=yes,-1=no)",ylab="housing price")
boxplot(housing$price~housing$airco,xlab="airco (1=yes,-1=no)",ylab="housing price")
boxplot(housing$price~housing$prefarea,xlab="prefarea (1=yes,-1=no)",ylab="housing price")
From plots above, it seems all six factors could influence the price of a house. Especially driveway, recreational room and gas supply, means are different whether these are provided or not. However for other 3 factors means are only slightly different.
Testing
Analysis of variance (ANOVA):
# Analysis of variance
aov=lm(price~driveway+recroom+fullbase+gashw+airco+prefarea, data=housing)
anova(aov)## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## driveway 1 2.16e+09 2.16e+09 7.20 0.0095 **
## recroom 1 1.36e+09 1.36e+09 4.54 0.0375 *
## fullbase 1 1.91e+08 1.91e+08 0.64 0.4279
## gashw 1 2.09e+09 2.09e+09 6.98 0.0106 *
## airco 1 8.58e+07 8.58e+07 0.29 0.5949
## prefarea 1 1.17e+09 1.17e+09 3.90 0.0533 .
## Residuals 57 1.71e+10 3.00e+08
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ANOVA result is as expected. Only ‘driveway’, ‘recroom’ and ‘gashw’ return P-values smaller than 0.05, which indicates variation of these three variables may explain variance of price.
From this ANOVA we could reject the null hypothesis and accept the alternative hypothesis that variation of price of house can be explained by something other than randomization.
Estimation (of parameters)
# Summary of linear model
summodel=lm(price~driveway+recroom+fullbase+gashw+airco+prefarea, data=housing)
summary(summodel)##
## Call:
## lm(formula = price ~ driveway + recroom + fullbase + gashw +
## airco + prefarea, data = housing)
##
## Residuals:
## Min 1Q Median 3Q Max
## -26545 -10533 -1595 8219 61315
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31408 5735 5.48 1e-06 ***
## driveway1 10559 4365 2.42 0.019 *
## recroom1 9274 4339 2.14 0.037 *
## fullbase1 3419 4339 0.79 0.434
## gashw1 11462 4339 2.64 0.011 *
## airco1 2316 4330 0.53 0.595
## prefarea1 8547 4330 1.97 0.053 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17300 on 57 degrees of freedom
## Multiple R-squared: 0.292, Adjusted R-squared: 0.218
## F-statistic: 3.92 on 6 and 57 DF, p-value: 0.00238confint(summodel)## 2.5 % 97.5 %
## (Intercept) 19923.9 42892
## driveway1 1818.7 19299
## recroom1 585.8 17963
## fullbase1 -5269.4 12108
## gashw1 2773.3 20151
## airco1 -6355.8 10987
## prefarea1 -124.6 17218Result corresponds to ANOVA that ‘driveway’ ‘recroom’ and ‘gashw’ have a P-value smaller than 0.05.
Diagnostics/Model adequacy checking for complete design
#Q-Q norm plot
qqnorm(residuals(summodel))
qqline(residuals(summodel))
#Plot of fitted and residuals
plot(fitted(summodel),residuals(summodel))
Q-Q norm shows a linear pattern. However points are not well distributed on each side of zero in residuals-fitted plot, indicating the complete design model used previously is not well fitted, further effort should be made.
4. Analysis of fractional factorial design
Construct a design matrix
library(FrF2)## Warning: package 'FrF2' was built under R version 3.1.2## 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.designmatrix = FrF2(32, nfactors=6, estimable=formula("~driveway+recroom+fullbase+gashw+airco+prefarea+driveway:(recroom+fullbase+gashw+airco+prefarea)"),factor.names=c("driveway","recroom","fullbase","gashw","airco","prefarea"),res4=TRUE,clear=FALSE,data=housing)
matrix## driveway recroom fullbase gashw airco prefarea
## 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(matrix)## $legend
## [1] A=driveway B=recroom C=fullbase D=gashw E=airco F=prefarea
##
## [[2]]
## [1] no aliasing among main effects and 2fisDesign matrix is generated using FrF2() function. Since we do not want main effect aliased with any other main effect or any 2 factor interactions besides main effect of driveway and interation effect with others, we set resolution as 4, which is effective and enough.
Select exprimental runs based on design matrix
sample=merge(matrix,housing,by=c("driveway","recroom","fullbase","gashw","airco","prefarea"),all=FALSE)
sample## driveway recroom fullbase gashw airco prefarea price garagepl lotsize
## 1 -1 -1 -1 -1 -1 -1 30500 0 3000
## 2 -1 -1 -1 -1 1 1 61000 0 2175
## 3 -1 -1 -1 1 -1 1 44000 1 4500
## 4 -1 -1 -1 1 1 -1 51000 0 4500
## 5 -1 -1 1 -1 -1 1 51000 2 4500
## 6 -1 -1 1 -1 1 -1 40000 1 2650
## 7 -1 -1 1 1 -1 -1 56000 0 3000
## 8 -1 -1 1 1 1 1 38000 0 2800
## 9 -1 1 -1 -1 -1 1 31900 0 5300
## 10 -1 1 -1 -1 1 -1 35000 1 3240
## 11 -1 1 -1 1 -1 -1 38000 0 3630
## 12 -1 1 -1 1 1 1 46000 1 2684
## 13 -1 1 1 -1 -1 -1 72000 0 3540
## 14 -1 1 1 -1 1 1 48000 0 3100
## 15 -1 1 1 1 -1 1 52500 0 3630
## 16 -1 1 1 1 1 -1 32000 0 1950
## 17 1 -1 -1 -1 -1 1 62900 0 2880
## 18 1 -1 -1 -1 1 -1 37900 0 3185
## 19 1 -1 -1 1 -1 -1 56000 1 3290
## 20 1 -1 -1 1 1 1 38000 0 2430
## 21 1 -1 1 -1 -1 -1 51000 0 3150
## 22 1 -1 1 -1 1 1 46000 1 4320
## 23 1 -1 1 1 -1 1 50000 0 3036
## 24 1 -1 1 1 1 -1 57500 2 3630
## 25 1 1 -1 -1 -1 -1 35000 0 3500
## 26 1 1 -1 -1 1 1 70000 0 5400
## 27 1 1 -1 1 -1 1 69900 2 3420
## 28 1 1 -1 1 -1 1 130000 2 6000
## 29 1 1 -1 1 1 -1 63900 1 3162
## 30 1 1 -1 1 1 -1 74500 2 3180
## 31 1 1 1 -1 -1 1 42000 0 3660
## 32 1 1 1 -1 1 -1 46500 0 3930
## 33 1 1 1 1 -1 -1 52000 0 3570
## 34 1 1 1 1 1 1 138300 2 6000
## bedrooms bathrms stories
## 1 2 1 1
## 2 3 1 2
## 3 2 1 2
## 4 2 1 1
## 5 4 2 2
## 6 3 1 2
## 7 3 1 2
## 8 3 1 1
## 9 3 1 1
## 10 2 1 1
## 11 3 3 2
## 12 2 1 1
## 13 2 1 1
## 14 3 1 2
## 15 2 1 1
## 16 3 1 1
## 17 3 1 2
## 18 2 1 1
## 19 2 1 1
## 20 3 1 1
## 21 3 1 2
## 22 3 1 1
## 23 3 1 2
## 24 3 2 2
## 25 2 1 1
## 26 4 1 2
## 27 4 2 2
## 28 4 1 2
## 29 3 1 2
## 30 3 2 2
## 31 4 1 2
## 32 2 1 1
## 33 3 1 2
## 34 4 3 2#There supposed to be 32 runs but somehow there are 34 in sample. Problem will be solved ASAP.#Computing main effect and interation effect through ANOVA
aovsample=lm(price~driveway*recroom+driveway*fullbase+driveway*gashw+driveway*airco+driveway*prefarea,data=sample)
anova(aovsample)## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## driveway 1 2.41e+09 2.41e+09 5.54 0.028 *
## recroom 1 9.82e+08 9.82e+08 2.26 0.147
## fullbase 1 2.94e+07 2.94e+07 0.07 0.797
## gashw 1 1.12e+09 1.12e+09 2.58 0.122
## airco 1 3.56e+04 3.56e+04 0.00 0.993
## prefarea 1 1.07e+09 1.07e+09 2.46 0.131
## driveway:recroom 1 1.16e+09 1.16e+09 2.66 0.117
## driveway:fullbase 1 9.40e+07 9.40e+07 0.22 0.647
## driveway:gashw 1 1.17e+09 1.17e+09 2.68 0.116
## driveway:airco 1 1.04e+08 1.04e+08 0.24 0.629
## driveway:prefarea 1 6.45e+08 6.45e+08 1.48 0.236
## Residuals 22 9.57e+09 4.35e+08
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From ANOVA for fractional design, only ‘driveway’ returns a P-value smaller than 0.05. Based on this result we can also reject null hypothesis and accept alternative hypothesis that variance of price can be explained by something other than randomization (driveway).
Comparing ANOVA results of complete and fractional factorial design, it seems that fractional design is able to reflect result of complete design to some extent, however it cannot totally replace complete design, at least in this case.
Model adequacy checking for fractional design
#Shapiro.test
shapiro.test(sample$price)##
## Shapiro-Wilk normality test
##
## data: sample$price
## W = 0.7431, p-value = 2.421e-06#Q-Q norm plot
qqnorm(residuals(aovsample))
qqline(residuals(aovsample))
#Plot of fitted and residuals
plot(fitted(aovsample),residuals(aovsample))
Shapiro test returns a very small P-value, indicating response variables in fractional design is not normally distributed.
Q-Q norm plot has a good linear pattern. Though points are not evenly distributed on each side of zero in residuals-fitted plot, the plot has a similar pattern compared with residuals-fitted plot of complete design, indicating fractional design could well simulated complete design in this aspect.
4. References to the literature
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