Response Surfact Methods Practice
Hongyu Chen
RPI, RIN:661405156
Nov.28 V1.0
1. Setting
System under test
This analysis is going to analyze prices of personal computers in the US from 1993-1995. Dataset is from ‘Computers’ in package ‘Ecdat’, which contains 6259 observations in total. Dataset contains several factors that may explain difference in computers price.
Below is the quick view of data
library(Ecdat)## Loading required package: Ecfun
##
## Attaching package: 'Ecdat'
##
## The following object is masked from 'package:datasets':
##
## Orange#First and last several lines of data set
head(Computers)## price speed hd ram screen cd multi premium ads trend
## 1 1499 25 80 4 14 no no yes 94 1
## 2 1795 33 85 2 14 no no yes 94 1
## 3 1595 25 170 4 15 no no yes 94 1
## 4 1849 25 170 8 14 no no no 94 1
## 5 3295 33 340 16 14 no no yes 94 1
## 6 3695 66 340 16 14 no no yes 94 1tail(Computers)## price speed hd ram screen cd multi premium ads trend
## 6254 2154 66 850 16 15 yes no yes 39 35
## 6255 1690 100 528 8 15 no no yes 39 35
## 6256 2223 66 850 16 15 yes yes yes 39 35
## 6257 2654 100 1200 24 15 yes no yes 39 35
## 6258 2195 100 850 16 15 yes no yes 39 35
## 6259 2490 100 850 16 17 yes no yes 39 35#Summary and structure of data
summary(Computers)## price speed hd ram
## Min. : 949 Min. : 25 Min. : 80 Min. : 2.00
## 1st Qu.:1794 1st Qu.: 33 1st Qu.: 214 1st Qu.: 4.00
## Median :2144 Median : 50 Median : 340 Median : 8.00
## Mean :2220 Mean : 52 Mean : 417 Mean : 8.29
## 3rd Qu.:2595 3rd Qu.: 66 3rd Qu.: 528 3rd Qu.: 8.00
## Max. :5399 Max. :100 Max. :2100 Max. :32.00
## screen cd multi premium ads
## Min. :14.0 no :3351 no :5386 no : 612 Min. : 39
## 1st Qu.:14.0 yes:2908 yes: 873 yes:5647 1st Qu.:162
## Median :14.0 Median :246
## Mean :14.6 Mean :221
## 3rd Qu.:15.0 3rd Qu.:275
## Max. :17.0 Max. :339
## trend
## Min. : 1.0
## 1st Qu.:10.0
## Median :16.0
## Mean :15.9
## 3rd Qu.:21.5
## Max. :35.0str(Computers)## 'data.frame': 6259 obs. of 10 variables:
## $ price : num 1499 1795 1595 1849 3295 ...
## $ speed : num 25 33 25 25 33 66 25 50 50 50 ...
## $ hd : num 80 85 170 170 340 340 170 85 210 210 ...
## $ ram : num 4 2 4 8 16 16 4 2 8 4 ...
## $ screen : num 14 14 15 14 14 14 14 14 14 15 ...
## $ cd : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 2 1 1 1 ...
## $ multi : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
## $ premium: Factor w/ 2 levels "no","yes": 2 2 2 1 2 2 2 2 2 2 ...
## $ ads : num 94 94 94 94 94 94 94 94 94 94 ...
## $ trend : num 1 1 1 1 1 1 1 1 1 1 ...Factors and Levels
In this study we focus on four factors that possibly influence computers price, including clock speed in MHz (speed), size of hard drive in MB (hd), size of Ram in MB (ram) and size of screen in inches (screen).
# Levels of factors
speed<-factor(Computers$speed)
hd<-factor(Computers$hd)
ram<-factor(Computers$ram)
screen<-factor(Computers$screen)
levels(speed)## [1] "25" "33" "50" "66" "75" "100"levels(hd)## [1] "80" "85" "100" "107" "120" "125" "128" "130" "170" "200"
## [11] "210" "212" "213" "214" "230" "240" "245" "250" "256" "260"
## [21] "270" "320" "330" "340" "345" "364" "365" "405" "420" "424"
## [31] "425" "426" "428" "450" "452" "470" "500" "520" "525" "527"
## [41] "528" "530" "540" "545" "630" "720" "728" "730" "810" "850"
## [51] "1000" "1060" "1080" "1100" "1200" "1260" "1370" "1600" "2100"levels(ram)## [1] "2" "4" "8" "16" "24" "32"levels(screen)## [1] "14" "15" "17"Response variables
Price of personal computer (dollars) is the response variable in this analysis, which is column ‘price’ in the dataset.
The Data: How is it organized and what does it look like?
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 4 factors above could explain variance of computers price.
Alternative hypothesis H1: variance of computers price can be explained by something other than randomizaion.
Set up columns which will be under investigation; run exploratory data analysis for each factor; construct a linear model to conduct response surface methods, second order effects are included; use ‘rsm’ for parameter estimation and draw response surface plots.
What is the rationale for this design?
In statistics, response surface method (RSM) explores the relationships between several explanatory variables and one or more response variables. Through estimation and optimization, it is possible to find certain combination of factors that can have the greatest influence to response variable.
Randomize: What is the Randomization Scheme?
Randomization depends on the way data collected which can be assumed as complete randomization.
Replicate: Are there replicates and/or repeated measures?
There are no replicates or repeated measures in this test.
Block
There is no block used in this recipe.
3. Analysis and testing
Exploratory data analysis
#Boxplot
boxplot(Computers$price~Computers$speed,xlab="Clock speed in MHz",ylab="Computer price")
boxplot(Computers$price~Computers$hd,xlab="Size of hard drive in MB",ylab="Computer price")
boxplot(Computers$price~Computers$ram,xlab="Size of Ram in MB",ylab="Computer price")
boxplot(Computers$price~Computers$screen,xlab="Size of screen in inches",ylab="Computer price")
From plots above, it seems all 4 factors could influence the price of a computer. For Ram size and screen size, computer price increases as they increase. However for clock speed, the means of price among computers with speed from 25MHz to 66 MHz are different, while there is no obvious difference when speed changes from 66MHz to 100MHz. There is no clear relationship between computer price and size of hard drive.
Response surface method using rsm:
#Construct linear model and run rsm
Cmpt.lm=lm(price~speed+hd+ram+screen,data=Computers)
anova(Cmpt.lm)## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## speed 1 1.91e+08 1.91e+08 1046 <2e-16 ***
## hd 1 2.48e+08 2.48e+08 1358 <2e-16 ***
## ram 1 4.76e+08 4.76e+08 2603 <2e-16 ***
## screen 1 5.31e+07 5.31e+07 291 <2e-16 ***
## Residuals 6254 1.14e+09 1.83e+05
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1library(rsm)## Warning: package 'rsm' was built under R version 3.1.2Cmpt.rsm=rsm(price~SO(speed,hd,ram,screen),data=Computers)
summary(Cmpt.rsm)##
## Call:
## rsm(formula = price ~ SO(speed, hd, ram, screen), data = Computers)
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.34e+04 1.40e+03 9.53 < 2e-16 ***
## speed 1.04e+01 4.56e+00 2.28 0.02243 *
## hd 3.71e+00 5.51e-01 6.72 1.9e-11 ***
## ram 1.07e+02 2.28e+01 4.70 2.7e-06 ***
## screen -1.79e+03 1.84e+02 -9.72 < 2e-16 ***
## speed:hd -1.41e-02 1.72e-03 -8.20 2.9e-16 ***
## speed:ram 2.57e-01 7.15e-02 3.59 0.00033 ***
## speed:screen 6.24e-01 3.13e-01 1.99 0.04641 *
## hd:ram -5.26e-02 7.97e-03 -6.60 4.4e-11 ***
## hd:screen -2.89e-01 3.81e-02 -7.59 3.7e-14 ***
## ram:screen 5.50e-01 1.56e+00 0.35 0.72370
## speed^2 -8.41e-02 1.14e-02 -7.41 1.5e-13 ***
## hd^2 1.11e-03 8.23e-05 13.52 < 2e-16 ***
## ram^2 -6.86e-01 3.15e-01 -2.18 0.02932 *
## screen^2 6.44e+01 6.04e+00 10.66 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Multiple R-squared: 0.522, Adjusted R-squared: 0.521
## F-statistic: 487 on 14 and 6244 DF, p-value: <2e-16
##
## Analysis of Variance Table
##
## Response: price
## Df Sum Sq Mean Sq F value Pr(>F)
## FO(speed, hd, ram, screen) 4 9.68e+08 2.42e+08 1497.7 <2e-16
## TWI(speed, hd, ram, screen) 6 6.96e+07 1.16e+07 71.8 <2e-16
## PQ(speed, hd, ram, screen) 4 6.41e+07 1.60e+07 99.1 <2e-16
## Residuals 6244 1.01e+09 1.62e+05
## Lack of fit 547 5.61e+08 1.03e+06 13.0 <2e-16
## Pure error 5697 4.48e+08 7.87e+04
##
## Stationary point of response surface:
## speed hd ram screen
## 41.85 1583.28 32.01 17.13
##
## Eigenanalysis:
## $values
## [1] 64.369987 0.003605 -0.061810 -0.713299
##
## $vectors
## [,1] [,2] [,3] [,4]
## speed 0.004849 -0.165577 0.966058 0.198225
## hd -0.002250 0.983905 0.175515 -0.033471
## ram 0.004236 -0.067114 0.189475 -0.979580
## screen 0.999977 0.003301 -0.005093 0.003113From anova and summary of Cmpt.rsm, we can find out that all four factors alone, combinations of any two factors except for ram:screen, and pure quadratic of all four factors are statistically significant and probably explain variance of computers price.
Therefore we can reject the null hypothesis and accept the alternative hypothesis that variance of computers price can be explained by something other than randomizaion.
For second order effects, FO, TWI and PQ all return a p-value equals 0, indicating statistical significance.
Stationary point of response surface: speed(41.85), hd(1583.28), ram(32.01) and screen(17.13).
Response surface characterization through eigenanalysis values:
speed(64.37), hd(0.004), ram(-0.062) and screen(17.13)
Contour plots of response surface
par(mfrow=c(2,3))
contour(Cmpt.rsm, ~speed+hd+ram+screen, image=TRUE, at=summary(Cmpt.rsm$canonical$xs))
From plots above, we can find out how combinations of any two factors effect computer price. It is important to point out that from the ram:screen plot, it seems that only ram plays a role in determination of price, which corresponds analysis of variance above. However from the speed:ram plot, it seems speed plays a more significant role than ram.
3D plots are as below:
library(rgl)## Warning: package 'rgl' was built under R version 3.1.2par(mfrow=c(1,1))
persp(Cmpt.rsm, ~ speed+hd, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
persp(Cmpt.rsm, ~ speed+ram, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
persp(Cmpt.rsm, ~ speed+screen, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
persp(Cmpt.rsm, ~ hd+ram, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
persp(Cmpt.rsm, ~ hd+screen, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
persp(Cmpt.rsm, ~ ram+screen, image = TRUE,
at = c(summary(Cmpt.rsm)$canonical$xs),zlab="Computer price",col.lab=33,contour="colors")## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
## Warning: "image"不是图形参数
For hd:speed plot, stationary point (1583.28, 41.85) seems to be a ridge point. For ram:speed plot, stationary point (32.01, 41.85) seems to be a ridge point. For screen:speed plot, stationary point (17.13, 41.85) seems to be a saddle point . For ram:hd plot, stationary point (32.01, 1583.28) seems to be a saddle point. For screen:hd plot, stationary point (17.13, 1583.28) seems to be a minima point. For screen:ram plot, stationary point (17.13, 32.01) might be a maxima or ridge point.
Parameters estimation
Shapiro test for normality
#shapiro.test(Computers$price)Shapiro test cannot be performed since there are more than 5000 observations in the dataset.
Diagnostics/Model adequacy checking for complete design
#Q-Q norm plot
qqnorm(residuals(Cmpt.rsm))
qqline(residuals(Cmpt.rsm))
#Plot of fitted and residuals
plot(fitted(Cmpt.rsm),residuals(Cmpt.rsm))