### Problem 6.8
CM<-c(rep(1,6),rep(2,6))
Time<-c(rep(12,12),rep(18,12))
obs<-c(21,23,20,22,28,26,25,24,29,26,25,27,37,38,35,39,38,36,31,29,30,34,33,35)
data.frame(CM,Time,obs)
## CM Time obs
## 1 1 12 21
## 2 1 12 23
## 3 1 12 20
## 4 1 12 22
## 5 1 12 28
## 6 1 12 26
## 7 2 12 25
## 8 2 12 24
## 9 2 12 29
## 10 2 12 26
## 11 2 12 25
## 12 2 12 27
## 13 1 18 37
## 14 1 18 38
## 15 1 18 35
## 16 1 18 39
## 17 1 18 38
## 18 1 18 36
## 19 2 18 31
## 20 2 18 29
## 21 2 18 30
## 22 2 18 34
## 23 2 18 33
## 24 2 18 35
dat<-data.frame(CM,Time,obs)
model<-lm(obs~CM*Time,data=dat)
coef(model)
## (Intercept) CM Time CM:Time
## -22.666667 18.333333 3.611111 -1.305556
## halfnormal(model)
plot(model)
## hat values (leverages) are all = 0.1666667
## and there are no factor predictors; no plot no. 5
summary(model)
##
## Call:
## lm.default(formula = obs ~ CM * Time, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.333 -1.500 -0.250 1.208 4.667
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -22.6667 7.4391 -3.047 0.006365 **
## CM 18.3333 4.7049 3.897 0.000896 ***
## Time 3.6111 0.4863 7.426 3.61e-07 ***
## CM:Time -1.3056 0.3076 -4.245 0.000397 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.26 on 20 degrees of freedom
## Multiple R-squared: 0.8713, Adjusted R-squared: 0.852
## F-statistic: 45.12 on 3 and 20 DF, p-value: 4.346e-09
interaction.plot(dat$CM,dat$Time,dat$obs)
## Comments:
## Model is significant p value is less than 0.05
## Factors Time and (Culture Medium: Time) interaction are significant
## The variance is slightly unequal as shown plots in the plot of residuals.
## The F value of F = 45.12 indicates that the model is significant
## Growth rate is affected by factor Time and the CM:Time interaction.
## Problem 6.12
A<-c(-1,1,-1,1)
B<-c(-1,-1,1,1)
R<-c(14.037,13.880,14.821,14.888,
16.165,13.860,14.757,14.921,
13.972,14.032,14.843,14.415,
13.907,13.914,14.878,14.932)
data.frame(A,B,R)
## A B R
## 1 -1 -1 14.037
## 2 1 -1 13.880
## 3 -1 1 14.821
## 4 1 1 14.888
## 5 -1 -1 16.165
## 6 1 -1 13.860
## 7 -1 1 14.757
## 8 1 1 14.921
## 9 -1 -1 13.972
## 10 1 -1 14.032
## 11 -1 1 14.843
## 12 1 1 14.415
## 13 -1 -1 13.907
## 14 1 -1 13.914
## 15 -1 1 14.878
## 16 1 1 14.932
dat1<-data.frame(A,B,R)
model1<-lm(R~A*B,data=dat1)
coef(model1)
## (Intercept) A B A:B
## 14.513875 -0.158625 0.293000 0.140750
halfnormal(model1)
## Warning in halfnormal.lm(model1): halfnormal not recommended for models with
## more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] e7 B A
plot(model1)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
summary(model1)
##
## Call:
## lm.default(formula = R ~ A * B, data = dat1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.61325 -0.14431 -0.00563 0.10188 1.64475
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.5139 0.1412 102.783 <2e-16 ***
## A -0.1586 0.1412 -1.123 0.2833
## B 0.2930 0.1412 2.075 0.0602 .
## A:B 0.1407 0.1412 0.997 0.3386
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5648 on 12 degrees of freedom
## Multiple R-squared: 0.3535, Adjusted R-squared: 0.1918
## F-statistic: 2.187 on 3 and 12 DF, p-value: 0.1425
interaction.plot(dat1$A,dat1$B,dat1$R)
## Comments:
## F-statistic: 2.187 on 3 and 12 DF, p-value: 0.1425 indicate that the model is not significant
## P values are larger than alpha of 0.05. This implies that there are no significant factors in the model.
## Part c: Write down a regression equation that could be used to predict epitaxial layer thickness over the region of arsenic
## flow rate A and deposition time B used in this experiment.
## Thickness = 14.51 - 0.158625*A + 0.293000*B + 0.140750*A*B
## Part d: Analyze the residuals. Are there any residuals that should cause concern?
## There is an outlier falls outside the group in the normal probability plot and the plot of residual.
## Part e: Discuss how you might deal with the potential outlier found in part (d).
## Replace the outlier value with the average of the observations from that experimental design.
R_outlier<-c(14.037,13.880,14.821,14.888,
13.972,13.860,14.757,14.921,
13.972,14.032,14.843,14.415,
13.907,13.914,14.878,14.932)
data.frame(A,B,R_outlier)
## A B R_outlier
## 1 -1 -1 14.037
## 2 1 -1 13.880
## 3 -1 1 14.821
## 4 1 1 14.888
## 5 -1 -1 13.972
## 6 1 -1 13.860
## 7 -1 1 14.757
## 8 1 1 14.921
## 9 -1 -1 13.972
## 10 1 -1 14.032
## 11 -1 1 14.843
## 12 1 1 14.415
## 13 -1 -1 13.907
## 14 1 -1 13.914
## 15 -1 1 14.878
## 16 1 1 14.932
dat2<-data.frame(A,B,R_outlier)
model2<-lm(R_outlier~A*B,data=dat2)
coef(model2)
## (Intercept) A B A:B
## 14.3768125 -0.0215625 0.4300625 0.0036875
halfnormal(model2)
## Warning in halfnormal.lm(model2): halfnormal not recommended for models with
## more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] B e4
plot(model2)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
summary(model2)
##
## Call:
## lm.default(formula = R_outlier ~ A * B, data = dat2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.3740 -0.0465 0.0000 0.0735 0.1430
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.376813 0.033969 423.237 < 2e-16 ***
## A -0.021563 0.033969 -0.635 0.537
## B 0.430062 0.033969 12.661 2.66e-08 ***
## A:B 0.003688 0.033969 0.109 0.915
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1359 on 12 degrees of freedom
## Multiple R-squared: 0.9305, Adjusted R-squared: 0.9131
## F-statistic: 53.57 on 3 and 12 DF, p-value: 3.2e-07
interaction.plot(dat2$A,dat2$B,dat2$R_outlier)
## Thickness = 14.38 - 0.022*A + 0.43*B + 0.0037*A*B
R_outlierr<-c(14.037,13.880,14.821,14.888,
14.165,13.860,14.757,14.921,
13.972,14.032,14.843,14.415,
13.907,13.914,14.878,14.932)
data.frame(A,B,R_outlierr)
## A B R_outlierr
## 1 -1 -1 14.037
## 2 1 -1 13.880
## 3 -1 1 14.821
## 4 1 1 14.888
## 5 -1 -1 14.165
## 6 1 -1 13.860
## 7 -1 1 14.757
## 8 1 1 14.921
## 9 -1 -1 13.972
## 10 1 -1 14.032
## 11 -1 1 14.843
## 12 1 1 14.415
## 13 -1 -1 13.907
## 14 1 -1 13.914
## 15 -1 1 14.878
## 16 1 1 14.932
dat2r<-data.frame(A,B,R_outlierr)
model2r<-lm(R_outlierr~A*B,data=dat2r)
coef(model2r)
## (Intercept) A B A:B
## 14.388875 -0.033625 0.418000 0.015750
halfnormal(model2r)
## Warning in halfnormal.lm(model2r): halfnormal not recommended for models with
## more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] B
plot(model2r)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
summary(model2r)
##
## Call:
## lm.default(formula = R_outlierr ~ A * B, data = dat2r)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.37400 -0.05156 0.00650 0.10188 0.14475
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.38888 0.03605 399.171 < 2e-16 ***
## A -0.03363 0.03605 -0.933 0.369
## B 0.41800 0.03605 11.596 7.08e-08 ***
## A:B 0.01575 0.03605 0.437 0.670
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1442 on 12 degrees of freedom
## Multiple R-squared: 0.9187, Adjusted R-squared: 0.8983
## F-statistic: 45.18 on 3 and 12 DF, p-value: 8.192e-07
interaction.plot(dat2r$A,dat2r$B,dat2r$R_outlierr)
## Thickness = 14.39 - 0.34*A + 0.42*B + 0.016*A*B
## Significant effects (alpha=0.05, Lenth method): [1] B
## Replaced observation 2 with the average of the remaining observations in replicate II.
## another outlier showed up. maybe there is an issue with data recording. New outlier must be investigated.
## Another outlier showed up. Further investigation is required.
## Problem 6.21
Lp<-c(rep(10,1),rep(30,1))
Type<-c(rep("Mallet",2),rep("Cavity",2))
Break<-c(rep("Straight",4),rep("Breaking",4))
Slope<-c(rep("Level",8),rep("Downhill",8))
dis<-c(10.0,0.0,4.0,0.0,0.0,5.0,6.5,16.5,4.5,19.5,15.0,41.5,8.0,21.5,0.0,18.0,
18.0,16.5,6.0,10.0,0.0,20.5,18.5,4.5,18.0,18.0,16.0,39.0,4.5,10.5,0.0,5.0,
14.0,4.5,1.0,34.0,18.5,18.0,7.5,0.0,14.5,16.0,8.5,6.5,6.5,6.5,0.0,7.0,
12.5,17.5,14.5,11.0,19.5,20.0,6.0,23.5,10.0,5.5,0.0,3.5,10.0,0.0,4.5,10.0,
19.0,20.5,12.0,25.5,16.0,29.5,0.0,8.0,0.0,10.0,0.5,7.0,13.0,15.5,1.0,32.5,
16.0,17.5,14.0,21.5,15.0,19.0,10.0,8.0,17.5,7.0,9.0,8.5,41.0,24.0,4.0,18.5,
18.5,33.0,5.0,0.0,11.0,10.0,0.0,8.0,6.0,36.0,3.0,36.0,14.0,16.0,6.5,8.0)
data.frame(Lp,Type,Break,Slope,dis)
## Lp Type Break Slope dis
## 1 10 Mallet Straight Level 10.0
## 2 30 Mallet Straight Level 0.0
## 3 10 Cavity Straight Level 4.0
## 4 30 Cavity Straight Level 0.0
## 5 10 Mallet Breaking Level 0.0
## 6 30 Mallet Breaking Level 5.0
## 7 10 Cavity Breaking Level 6.5
## 8 30 Cavity Breaking Level 16.5
## 9 10 Mallet Straight Downhill 4.5
## 10 30 Mallet Straight Downhill 19.5
## 11 10 Cavity Straight Downhill 15.0
## 12 30 Cavity Straight Downhill 41.5
## 13 10 Mallet Breaking Downhill 8.0
## 14 30 Mallet Breaking Downhill 21.5
## 15 10 Cavity Breaking Downhill 0.0
## 16 30 Cavity Breaking Downhill 18.0
## 17 10 Mallet Straight Level 18.0
## 18 30 Mallet Straight Level 16.5
## 19 10 Cavity Straight Level 6.0
## 20 30 Cavity Straight Level 10.0
## 21 10 Mallet Breaking Level 0.0
## 22 30 Mallet Breaking Level 20.5
## 23 10 Cavity Breaking Level 18.5
## 24 30 Cavity Breaking Level 4.5
## 25 10 Mallet Straight Downhill 18.0
## 26 30 Mallet Straight Downhill 18.0
## 27 10 Cavity Straight Downhill 16.0
## 28 30 Cavity Straight Downhill 39.0
## 29 10 Mallet Breaking Downhill 4.5
## 30 30 Mallet Breaking Downhill 10.5
## 31 10 Cavity Breaking Downhill 0.0
## 32 30 Cavity Breaking Downhill 5.0
## 33 10 Mallet Straight Level 14.0
## 34 30 Mallet Straight Level 4.5
## 35 10 Cavity Straight Level 1.0
## 36 30 Cavity Straight Level 34.0
## 37 10 Mallet Breaking Level 18.5
## 38 30 Mallet Breaking Level 18.0
## 39 10 Cavity Breaking Level 7.5
## 40 30 Cavity Breaking Level 0.0
## 41 10 Mallet Straight Downhill 14.5
## 42 30 Mallet Straight Downhill 16.0
## 43 10 Cavity Straight Downhill 8.5
## 44 30 Cavity Straight Downhill 6.5
## 45 10 Mallet Breaking Downhill 6.5
## 46 30 Mallet Breaking Downhill 6.5
## 47 10 Cavity Breaking Downhill 0.0
## 48 30 Cavity Breaking Downhill 7.0
## 49 10 Mallet Straight Level 12.5
## 50 30 Mallet Straight Level 17.5
## 51 10 Cavity Straight Level 14.5
## 52 30 Cavity Straight Level 11.0
## 53 10 Mallet Breaking Level 19.5
## 54 30 Mallet Breaking Level 20.0
## 55 10 Cavity Breaking Level 6.0
## 56 30 Cavity Breaking Level 23.5
## 57 10 Mallet Straight Downhill 10.0
## 58 30 Mallet Straight Downhill 5.5
## 59 10 Cavity Straight Downhill 0.0
## 60 30 Cavity Straight Downhill 3.5
## 61 10 Mallet Breaking Downhill 10.0
## 62 30 Mallet Breaking Downhill 0.0
## 63 10 Cavity Breaking Downhill 4.5
## 64 30 Cavity Breaking Downhill 10.0
## 65 10 Mallet Straight Level 19.0
## 66 30 Mallet Straight Level 20.5
## 67 10 Cavity Straight Level 12.0
## 68 30 Cavity Straight Level 25.5
## 69 10 Mallet Breaking Level 16.0
## 70 30 Mallet Breaking Level 29.5
## 71 10 Cavity Breaking Level 0.0
## 72 30 Cavity Breaking Level 8.0
## 73 10 Mallet Straight Downhill 0.0
## 74 30 Mallet Straight Downhill 10.0
## 75 10 Cavity Straight Downhill 0.5
## 76 30 Cavity Straight Downhill 7.0
## 77 10 Mallet Breaking Downhill 13.0
## 78 30 Mallet Breaking Downhill 15.5
## 79 10 Cavity Breaking Downhill 1.0
## 80 30 Cavity Breaking Downhill 32.5
## 81 10 Mallet Straight Level 16.0
## 82 30 Mallet Straight Level 17.5
## 83 10 Cavity Straight Level 14.0
## 84 30 Cavity Straight Level 21.5
## 85 10 Mallet Breaking Level 15.0
## 86 30 Mallet Breaking Level 19.0
## 87 10 Cavity Breaking Level 10.0
## 88 30 Cavity Breaking Level 8.0
## 89 10 Mallet Straight Downhill 17.5
## 90 30 Mallet Straight Downhill 7.0
## 91 10 Cavity Straight Downhill 9.0
## 92 30 Cavity Straight Downhill 8.5
## 93 10 Mallet Breaking Downhill 41.0
## 94 30 Mallet Breaking Downhill 24.0
## 95 10 Cavity Breaking Downhill 4.0
## 96 30 Cavity Breaking Downhill 18.5
## 97 10 Mallet Straight Level 18.5
## 98 30 Mallet Straight Level 33.0
## 99 10 Cavity Straight Level 5.0
## 100 30 Cavity Straight Level 0.0
## 101 10 Mallet Breaking Level 11.0
## 102 30 Mallet Breaking Level 10.0
## 103 10 Cavity Breaking Level 0.0
## 104 30 Cavity Breaking Level 8.0
## 105 10 Mallet Straight Downhill 6.0
## 106 30 Mallet Straight Downhill 36.0
## 107 10 Cavity Straight Downhill 3.0
## 108 30 Cavity Straight Downhill 36.0
## 109 10 Mallet Breaking Downhill 14.0
## 110 30 Mallet Breaking Downhill 16.0
## 111 10 Cavity Breaking Downhill 6.5
## 112 30 Cavity Breaking Downhill 8.0
dat_golf<-data.frame(Lp,Type,Break,Slope,dis)
model_golf<-lm(dis~Lp+Type,data=dat_golf)
coef(model_golf)
## (Intercept) Lp TypeMallet
## 4.7142857 0.2861607 3.7232143
## halfnormal(model_golf)
plot(model_golf)
## hat values (leverages) are all = 0.02678571
## and there are no factor predictors; no plot no. 5
summary(model_golf)
##
## Call:
## lm.default(formula = dis ~ Lp + Type, data = dat_golf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.022 -6.632 -1.076 4.701 29.701
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.71429 2.13206 2.211 0.02911 *
## Lp 0.28616 0.08704 3.288 0.00136 **
## TypeMallet 3.72321 1.74082 2.139 0.03469 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.212 on 109 degrees of freedom
## Multiple R-squared: 0.1237, Adjusted R-squared: 0.1076
## F-statistic: 7.692 on 2 and 109 DF, p-value: 0.0007504
## Part a: Analyze the data from this experiment. Which factors significantly affect putting performance?
## The half normal probability plot of effects idicate that only factors Lp and Type are significant
## Length of putt in feets and type of putter Mallet or cavity have significant effect on putting.
## Call:
## lm.default(formula = dis ~ Lp * Type * Break * Slope, data = dat_golf)
## Residuals:
## Min 1Q Median 3Q Max
## -16.786 -6.036 -0.250 4.250 27.143
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.6429 5.5621 -0.655 0.5141
## Lp 0.5929 0.2487 2.383 0.0191 *
## TypeMallet 17.7143 7.8660 2.252 0.0266 *
## BreakStraight 4.6429 7.8660 0.590 0.5564
## SlopeLevel 9.1429 7.8660 1.162 0.2480
## Lp:TypeMallet -0.6143 0.3518 -1.746 0.0840 .
## Lp:BreakStraight 0.0500 0.3518 0.142 0.8873
## TypeMallet:BreakStraight -11.6071 11.1242 -1.043 0.2994
## Lp:SlopeLevel -0.4500 0.3518 -1.279 0.2039
## TypeMallet:SlopeLevel -14.7857 11.1242 -1.329 0.1869
## BreakStraight:SlopeLevel -5.3214 11.1242 -0.478 0.6335
## Lp:TypeMallet:BreakStraight 0.2679 0.4975 0.538 0.5915
## Lp:TypeMallet:SlopeLevel 0.7714 0.4975 1.551 0.1243
## Lp:BreakStraight:SlopeLevel 0.1321 0.4975 0.266 0.7911
## TypeMallet:BreakStraight:SlopeLevel 19.1786 15.7319 1.219 0.2258
## Lp:TypeMallet:BreakStraight:SlopeLevel -0.7393 0.7036 -1.051 0.2960
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Residual standard error: 9.307 on 96 degrees of freedom
## Multiple R-squared: 0.2121, Adjusted R-squared: 0.08898
## F-statistic: 1.723 on 15 and 96 DF, p-value: 0.0589
## There is 0.089% chance that F values is large due to noise in the model.
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.6429 5.5621 -0.655 0.5141
## Lp 0.5929 0.2487 2.383 0.0191 *
## TypeMallet 17.7143 7.8660 2.252 0.0266 *
## P value of 0.0191 and 0.0266 for factor Lp and Type of putter are both less than alpha value 0.05
## This means that Lenght of putt and Type of putter are significant.
## We then pull up all significant factors LP and Type
model_golf1<-aov(dis~Lp+Type,data=dat_golf)
summary(model_golf)
##
## Call:
## lm.default(formula = dis ~ Lp + Type, data = dat_golf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.022 -6.632 -1.076 4.701 29.701
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.71429 2.13206 2.211 0.02911 *
## Lp 0.28616 0.08704 3.288 0.00136 **
## TypeMallet 3.72321 1.74082 2.139 0.03469 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.212 on 109 degrees of freedom
## Multiple R-squared: 0.1237, Adjusted R-squared: 0.1076
## F-statistic: 7.692 on 2 and 109 DF, p-value: 0.0007504
summary(model_golf1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Lp 1 917 917.1 10.809 0.00136 **
## Type 1 388 388.1 4.574 0.03469 *
## Residuals 109 9249 84.9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
interaction.plot(dat_golf$Lp,dat_golf$Type, dat_golf$dis)
## Part b: Analyze the residuals from this experiment. Are there any indications of model inadequacy?
## The normality assumption is violated.
## The variance is slightly inequal.
## Action: Take square root transformation and check for normality and equal variance assumptions again.
## then generate residual plots .
dis_trans<-c(sqrt(dis))
data.frame(Lp,Type,Break,Slope,dis_trans)
## Lp Type Break Slope dis_trans
## 1 10 Mallet Straight Level 3.1622777
## 2 30 Mallet Straight Level 0.0000000
## 3 10 Cavity Straight Level 2.0000000
## 4 30 Cavity Straight Level 0.0000000
## 5 10 Mallet Breaking Level 0.0000000
## 6 30 Mallet Breaking Level 2.2360680
## 7 10 Cavity Breaking Level 2.5495098
## 8 30 Cavity Breaking Level 4.0620192
## 9 10 Mallet Straight Downhill 2.1213203
## 10 30 Mallet Straight Downhill 4.4158804
## 11 10 Cavity Straight Downhill 3.8729833
## 12 30 Cavity Straight Downhill 6.4420494
## 13 10 Mallet Breaking Downhill 2.8284271
## 14 30 Mallet Breaking Downhill 4.6368092
## 15 10 Cavity Breaking Downhill 0.0000000
## 16 30 Cavity Breaking Downhill 4.2426407
## 17 10 Mallet Straight Level 4.2426407
## 18 30 Mallet Straight Level 4.0620192
## 19 10 Cavity Straight Level 2.4494897
## 20 30 Cavity Straight Level 3.1622777
## 21 10 Mallet Breaking Level 0.0000000
## 22 30 Mallet Breaking Level 4.5276926
## 23 10 Cavity Breaking Level 4.3011626
## 24 30 Cavity Breaking Level 2.1213203
## 25 10 Mallet Straight Downhill 4.2426407
## 26 30 Mallet Straight Downhill 4.2426407
## 27 10 Cavity Straight Downhill 4.0000000
## 28 30 Cavity Straight Downhill 6.2449980
## 29 10 Mallet Breaking Downhill 2.1213203
## 30 30 Mallet Breaking Downhill 3.2403703
## 31 10 Cavity Breaking Downhill 0.0000000
## 32 30 Cavity Breaking Downhill 2.2360680
## 33 10 Mallet Straight Level 3.7416574
## 34 30 Mallet Straight Level 2.1213203
## 35 10 Cavity Straight Level 1.0000000
## 36 30 Cavity Straight Level 5.8309519
## 37 10 Mallet Breaking Level 4.3011626
## 38 30 Mallet Breaking Level 4.2426407
## 39 10 Cavity Breaking Level 2.7386128
## 40 30 Cavity Breaking Level 0.0000000
## 41 10 Mallet Straight Downhill 3.8078866
## 42 30 Mallet Straight Downhill 4.0000000
## 43 10 Cavity Straight Downhill 2.9154759
## 44 30 Cavity Straight Downhill 2.5495098
## 45 10 Mallet Breaking Downhill 2.5495098
## 46 30 Mallet Breaking Downhill 2.5495098
## 47 10 Cavity Breaking Downhill 0.0000000
## 48 30 Cavity Breaking Downhill 2.6457513
## 49 10 Mallet Straight Level 3.5355339
## 50 30 Mallet Straight Level 4.1833001
## 51 10 Cavity Straight Level 3.8078866
## 52 30 Cavity Straight Level 3.3166248
## 53 10 Mallet Breaking Level 4.4158804
## 54 30 Mallet Breaking Level 4.4721360
## 55 10 Cavity Breaking Level 2.4494897
## 56 30 Cavity Breaking Level 4.8476799
## 57 10 Mallet Straight Downhill 3.1622777
## 58 30 Mallet Straight Downhill 2.3452079
## 59 10 Cavity Straight Downhill 0.0000000
## 60 30 Cavity Straight Downhill 1.8708287
## 61 10 Mallet Breaking Downhill 3.1622777
## 62 30 Mallet Breaking Downhill 0.0000000
## 63 10 Cavity Breaking Downhill 2.1213203
## 64 30 Cavity Breaking Downhill 3.1622777
## 65 10 Mallet Straight Level 4.3588989
## 66 30 Mallet Straight Level 4.5276926
## 67 10 Cavity Straight Level 3.4641016
## 68 30 Cavity Straight Level 5.0497525
## 69 10 Mallet Breaking Level 4.0000000
## 70 30 Mallet Breaking Level 5.4313902
## 71 10 Cavity Breaking Level 0.0000000
## 72 30 Cavity Breaking Level 2.8284271
## 73 10 Mallet Straight Downhill 0.0000000
## 74 30 Mallet Straight Downhill 3.1622777
## 75 10 Cavity Straight Downhill 0.7071068
## 76 30 Cavity Straight Downhill 2.6457513
## 77 10 Mallet Breaking Downhill 3.6055513
## 78 30 Mallet Breaking Downhill 3.9370039
## 79 10 Cavity Breaking Downhill 1.0000000
## 80 30 Cavity Breaking Downhill 5.7008771
## 81 10 Mallet Straight Level 4.0000000
## 82 30 Mallet Straight Level 4.1833001
## 83 10 Cavity Straight Level 3.7416574
## 84 30 Cavity Straight Level 4.6368092
## 85 10 Mallet Breaking Level 3.8729833
## 86 30 Mallet Breaking Level 4.3588989
## 87 10 Cavity Breaking Level 3.1622777
## 88 30 Cavity Breaking Level 2.8284271
## 89 10 Mallet Straight Downhill 4.1833001
## 90 30 Mallet Straight Downhill 2.6457513
## 91 10 Cavity Straight Downhill 3.0000000
## 92 30 Cavity Straight Downhill 2.9154759
## 93 10 Mallet Breaking Downhill 6.4031242
## 94 30 Mallet Breaking Downhill 4.8989795
## 95 10 Cavity Breaking Downhill 2.0000000
## 96 30 Cavity Breaking Downhill 4.3011626
## 97 10 Mallet Straight Level 4.3011626
## 98 30 Mallet Straight Level 5.7445626
## 99 10 Cavity Straight Level 2.2360680
## 100 30 Cavity Straight Level 0.0000000
## 101 10 Mallet Breaking Level 3.3166248
## 102 30 Mallet Breaking Level 3.1622777
## 103 10 Cavity Breaking Level 0.0000000
## 104 30 Cavity Breaking Level 2.8284271
## 105 10 Mallet Straight Downhill 2.4494897
## 106 30 Mallet Straight Downhill 6.0000000
## 107 10 Cavity Straight Downhill 1.7320508
## 108 30 Cavity Straight Downhill 6.0000000
## 109 10 Mallet Breaking Downhill 3.7416574
## 110 30 Mallet Breaking Downhill 4.0000000
## 111 10 Cavity Breaking Downhill 2.5495098
## 112 30 Cavity Breaking Downhill 2.8284271
dat_golf2<-data.frame(Lp,Type,Break,Slope,dis_trans)
model_golf2<-lm(dis_trans~Lp*Type*Break*Slope,data=dat_golf2)
coef(model_golf2)
## (Intercept) Lp
## -0.15033673 0.12461696
## TypeMallet BreakStraight
## 3.71983177 1.57992513
## SlopeLevel Lp:TypeMallet
## 2.01368366 -0.13282550
## Lp:BreakStraight TypeMallet:BreakStraight
## -0.03575270 -2.78592107
## Lp:SlopeLevel TypeMallet:SlopeLevel
## -0.09379376 -3.34826087
## BreakStraight:SlopeLevel Lp:TypeMallet:BreakStraight
## -1.00747250 0.09285297
## Lp:TypeMallet:SlopeLevel Lp:BreakStraight:SlopeLevel
## 0.16289124 0.02848102
## TypeMallet:BreakStraight:SlopeLevel Lp:TypeMallet:BreakStraight:SlopeLevel
## 4.06457337 -0.16447007
## halfnormal(model_golf2)
plot(model_golf2)
## hat values (leverages) are all = 0.1428571
## and there are no factor predictors; no plot no. 5
summary(model_golf2)
##
## Call:
## lm.default(formula = dis_trans ~ Lp * Type * Break * Slope, data = dat_golf2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5460 -0.9090 0.2177 0.9994 2.9157
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.15034 0.92189 -0.163 0.87080
## Lp 0.12462 0.04123 3.023 0.00321 **
## TypeMallet 3.71983 1.30374 2.853 0.00530 **
## BreakStraight 1.57993 1.30374 1.212 0.22855
## SlopeLevel 2.01368 1.30374 1.545 0.12575
## Lp:TypeMallet -0.13283 0.05831 -2.278 0.02494 *
## Lp:BreakStraight -0.03575 0.05831 -0.613 0.54119
## TypeMallet:BreakStraight -2.78592 1.84377 -1.511 0.13408
## Lp:SlopeLevel -0.09379 0.05831 -1.609 0.11097
## TypeMallet:SlopeLevel -3.34826 1.84377 -1.816 0.07249 .
## BreakStraight:SlopeLevel -1.00747 1.84377 -0.546 0.58605
## Lp:TypeMallet:BreakStraight 0.09285 0.08246 1.126 0.26293
## Lp:TypeMallet:SlopeLevel 0.16289 0.08246 1.975 0.05108 .
## Lp:BreakStraight:SlopeLevel 0.02848 0.08246 0.345 0.73054
## TypeMallet:BreakStraight:SlopeLevel 4.06457 2.60749 1.559 0.12233
## Lp:TypeMallet:BreakStraight:SlopeLevel -0.16447 0.11661 -1.410 0.16165
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.543 on 96 degrees of freedom
## Multiple R-squared: 0.2279, Adjusted R-squared: 0.1073
## F-statistic: 1.889 on 15 and 96 DF, p-value: 0.0335
model_golf2b<-lm(dis_trans~Lp+Type,data=dat_golf2)
model_golf2bb<-aov(dis_trans~Lp+Type,data=dat_golf2)
coef(model_golf2b)
## (Intercept) Lp TypeMallet
## 1.85530859 0.04392853 0.74746603
## halfnormal(model_golf2b)
plot(model_golf2b)
## hat values (leverages) are all = 0.02678571
## and there are no factor predictors; no plot no. 5
summary(model_golf2bb)
## Df Sum Sq Mean Sq F value Pr(>F)
## Lp 1 21.61 21.613 9.109 0.00317 **
## Type 1 15.64 15.644 6.593 0.01159 *
## Residuals 109 258.63 2.373
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model_golf2b)
##
## Call:
## lm.default(formula = dis_trans ~ Lp + Type, data = dat_golf2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9206 -0.7584 0.1492 0.9630 3.3611
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.85531 0.35653 5.204 9.25e-07 ***
## Lp 0.04393 0.01456 3.018 0.00317 **
## TypeMallet 0.74747 0.29110 2.568 0.01159 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.54 on 109 degrees of freedom
## Multiple R-squared: 0.1259, Adjusted R-squared: 0.1099
## F-statistic: 7.851 on 2 and 109 DF, p-value: 0.0006527
## Generate interaction plots
interaction.plot(dat_golf$Lp,dat_golf$Type, dat_golf$dis)
interaction.plot(dat_golf2$Lp,dat_golf2$Type, dat_golf2$dis_trans)
## Conclusion:
## Interaction plot indicate that length of putt of 10 foot and the type cavity-back putter reduce the mean distance after transformation.
## Problem 6.36
A<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
Resistivity<-c(1.92,11.28,1.09,5.75,2.13,9.53,1.03,5.35,1.60,11.73,1.16,4.68,2.16,9.11,1.07,5.30)
data.frame(A,B,C,D,Resistivity)
## A B C D Resistivity
## 1 -1 -1 -1 -1 1.92
## 2 1 -1 -1 -1 11.28
## 3 -1 1 -1 -1 1.09
## 4 1 1 -1 -1 5.75
## 5 -1 -1 1 -1 2.13
## 6 1 -1 1 -1 9.53
## 7 -1 1 1 -1 1.03
## 8 1 1 1 -1 5.35
## 9 -1 -1 -1 1 1.60
## 10 1 -1 -1 1 11.73
## 11 -1 1 -1 1 1.16
## 12 1 1 -1 1 4.68
## 13 -1 -1 1 1 2.16
## 14 1 -1 1 1 9.11
## 15 -1 1 1 1 1.07
## 16 1 1 1 1 5.30
dat_Resistivity<-data.frame(A,B,C,D,Resistivity)
model_Resistivity<-lm(Resistivity~A*B*C*D,data=dat_Resistivity)
coef(model_Resistivity)
## (Intercept) A B C D A:B
## 4.680625 3.160625 -1.501875 -0.220625 -0.079375 -1.069375
## A:C B:C A:D B:D C:D A:B:C
## -0.298125 0.229375 -0.056875 -0.046875 0.029375 0.344375
## A:B:D A:C:D B:C:D A:B:C:D
## -0.096875 -0.010625 0.094375 0.141875
## halfnormal(model_Resistivity)
## plot(model_Resistivity)
summary(model_Resistivity)
##
## Call:
## lm.default(formula = Resistivity ~ A * B * C * D, data = dat_Resistivity)
##
## Residuals:
## ALL 16 residuals are 0: no residual degrees of freedom!
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.68062 NaN NaN NaN
## A 3.16062 NaN NaN NaN
## B -1.50187 NaN NaN NaN
## C -0.22062 NaN NaN NaN
## D -0.07937 NaN NaN NaN
## A:B -1.06938 NaN NaN NaN
## A:C -0.29812 NaN NaN NaN
## B:C 0.22937 NaN NaN NaN
## A:D -0.05687 NaN NaN NaN
## B:D -0.04688 NaN NaN NaN
## C:D 0.02937 NaN NaN NaN
## A:B:C 0.34437 NaN NaN NaN
## A:B:D -0.09688 NaN NaN NaN
## A:C:D -0.01063 NaN NaN NaN
## B:C:D 0.09438 NaN NaN NaN
## A:B:C:D 0.14188 NaN NaN NaN
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 15 and 0 DF, p-value: NA
## Part a: Estimate the factor effects. Plot the effect estimates on a normal probability plot and select a tentative model.
## > halfnormal(model_Resistivity)
## Significant effects (alpha=0.05, Lenth method): [1] A B A:B
## Part b: Fit the model identified in part (a) and analyze the residuals. Is there any indication of model inadequacy?
## The normal probability plot of residuals is not satisfactory. The plots of residual versus predicted,
## residual versus factor A, and the residual versus factor B are funnel shaped indicating non-constant variance.
model_Resistivity1<-lm(Resistivity~A+B+A*B,data=dat_Resistivity)
model_Resistivity2<-aov(Resistivity~A+B+A*B,data=dat_Resistivity)
coef(model_Resistivity1)
## (Intercept) A B A:B
## 4.680625 3.160625 -1.501875 -1.069375
halfnormal(model_Resistivity1)
## Warning in halfnormal.lm(model_Resistivity1): halfnormal not recommended for
## models with more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] A B A:B
plot(model_Resistivity1)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
summary(model_Resistivity2)
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 159.83 159.83 333.09 4.05e-10 ***
## B 1 36.09 36.09 75.21 1.63e-06 ***
## A:B 1 18.30 18.30 38.13 4.76e-05 ***
## Residuals 12 5.76 0.48
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model_Resistivity)
##
## Call:
## lm.default(formula = Resistivity ~ A * B * C * D, data = dat_Resistivity)
##
## Residuals:
## ALL 16 residuals are 0: no residual degrees of freedom!
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.68062 NaN NaN NaN
## A 3.16062 NaN NaN NaN
## B -1.50187 NaN NaN NaN
## C -0.22062 NaN NaN NaN
## D -0.07937 NaN NaN NaN
## A:B -1.06938 NaN NaN NaN
## A:C -0.29812 NaN NaN NaN
## B:C 0.22937 NaN NaN NaN
## A:D -0.05687 NaN NaN NaN
## B:D -0.04688 NaN NaN NaN
## C:D 0.02937 NaN NaN NaN
## A:B:C 0.34437 NaN NaN NaN
## A:B:D -0.09688 NaN NaN NaN
## A:C:D -0.01063 NaN NaN NaN
## B:C:D 0.09438 NaN NaN NaN
## A:B:C:D 0.14188 NaN NaN NaN
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 15 and 0 DF, p-value: NA
## Part c: Repeat the analysis from parts (a) and (b) using ln (y) as the response variable. Is there an indication that the transformation has been useful?
Resistivity_trans<-c(log(Resistivity))
data.frame(A,B,C,D,Resistivity_trans)
## A B C D Resistivity_trans
## 1 -1 -1 -1 -1 0.65232519
## 2 1 -1 -1 -1 2.42303125
## 3 -1 1 -1 -1 0.08617770
## 4 1 1 -1 -1 1.74919985
## 5 -1 -1 1 -1 0.75612198
## 6 1 -1 1 -1 2.25444472
## 7 -1 1 1 -1 0.02955880
## 8 1 1 1 -1 1.67709656
## 9 -1 -1 -1 1 0.47000363
## 10 1 -1 -1 1 2.46214966
## 11 -1 1 -1 1 0.14842001
## 12 1 1 -1 1 1.54329811
## 13 -1 -1 1 1 0.77010822
## 14 1 -1 1 1 2.20937271
## 15 -1 1 1 1 0.06765865
## 16 1 1 1 1 1.66770682
dat_Resistivity_trans<-data.frame(A,B,C,D,Resistivity_trans)
model_Resistivity_trans<-lm(Resistivity_trans~A*B*C*D,data=dat_Resistivity_trans)
coef(model_Resistivity_trans)
## (Intercept) A B C D A:B
## 1.185417116 0.812870345 -0.314277554 -0.006408558 -0.018077390 -0.024684570
## A:C B:C A:D B:D C:D A:B:C
## -0.039723700 -0.004225796 -0.009578245 0.003708723 0.017780432 0.063434408
## A:B:D A:C:D B:C:D A:B:C:D
## -0.029875960 -0.003740235 0.003765760 0.031322043
halfnormal(model_Resistivity_trans)
##
## Significant effects (alpha=0.05, Lenth method):
## [1] A B A:B:C
## plot(model_Resistivity_trans)
## Part d: Fit a model in terms of the coded variables that can be used to predict the resistivity.
model_Resistivity_trans3<-lm(Resistivity_trans~A*B,data=dat_Resistivity_trans)
model_Resistivity_trans4<-aov(Resistivity_trans~A*B,data=dat_Resistivity_trans)
coef(model_Resistivity_trans3)
## (Intercept) A B A:B
## 1.18541712 0.81287034 -0.31427755 -0.02468457
halfnormal(model_Resistivity_trans3)
## Warning in halfnormal.lm(model_Resistivity_trans3): halfnormal not recommended
## for models with more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] A B
plot(model_Resistivity_trans3)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
summary(model_Resistivity_trans4)
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 10.572 10.572 954.04 8.33e-13 ***
## B 1 1.580 1.580 142.61 5.10e-08 ***
## A:B 1 0.010 0.010 0.88 0.367
## Residuals 12 0.133 0.011
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model_Resistivity_trans3)
##
## Call:
## lm.default(formula = Resistivity_trans ~ A * B, data = dat_Resistivity_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.192136 -0.060747 0.005803 0.086805 0.124900
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.18542 0.02632 45.044 9.34e-15 ***
## A 0.81287 0.02632 30.888 8.33e-13 ***
## B -0.31428 0.02632 -11.942 5.10e-08 ***
## A:B -0.02468 0.02632 -0.938 0.367
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1053 on 12 degrees of freedom
## Multiple R-squared: 0.9892, Adjusted R-squared: 0.9865
## F-statistic: 365.8 on 3 and 12 DF, p-value: 4.672e-12
## Problem 6.39 Quality and Reliability Engineering
A<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
E<-c(-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
y<-c(8.11,5.56,5.77,5.82,9.17,7.8,3.23,5.69,8.82,14.23,9.2,8.94,8.68,11.49,6.25,9.12,
7.93,5,7.47,12,9.86,3.65,6.4,11.61,12.43,17.55,8.87, 25.38,13.06,18.85,11.78,26.05)
data.frame(A,B,C,D,E,y)
## A B C D E y
## 1 -1 -1 -1 -1 -1 8.11
## 2 1 -1 -1 -1 -1 5.56
## 3 -1 1 -1 -1 -1 5.77
## 4 1 1 -1 -1 -1 5.82
## 5 -1 -1 1 -1 -1 9.17
## 6 1 -1 1 -1 -1 7.80
## 7 -1 1 1 -1 -1 3.23
## 8 1 1 1 -1 -1 5.69
## 9 -1 -1 -1 1 -1 8.82
## 10 1 -1 -1 1 -1 14.23
## 11 -1 1 -1 1 -1 9.20
## 12 1 1 -1 1 -1 8.94
## 13 -1 -1 1 1 -1 8.68
## 14 1 -1 1 1 -1 11.49
## 15 -1 1 1 1 -1 6.25
## 16 1 1 1 1 -1 9.12
## 17 -1 -1 -1 -1 1 7.93
## 18 1 -1 -1 -1 1 5.00
## 19 -1 1 -1 -1 1 7.47
## 20 1 1 -1 -1 1 12.00
## 21 -1 -1 1 -1 1 9.86
## 22 1 -1 1 -1 1 3.65
## 23 -1 1 1 -1 1 6.40
## 24 1 1 1 -1 1 11.61
## 25 -1 -1 -1 1 1 12.43
## 26 1 -1 -1 1 1 17.55
## 27 -1 1 -1 1 1 8.87
## 28 1 1 -1 1 1 25.38
## 29 -1 -1 1 1 1 13.06
## 30 1 -1 1 1 1 18.85
## 31 -1 1 1 1 1 11.78
## 32 1 1 1 1 1 26.05
dat_Quality<-data.frame(A,B,C,D,E,y)
model_Quality<-lm(y~A*B*C*D*E,data=dat_Quality)
coef(model_Quality)
## (Intercept) A B C D E
## 10.1803125 1.6159375 0.0434375 -0.0121875 2.9884375 2.1878125
## A:B A:C B:C A:D B:D C:D
## 1.2365625 -0.0015625 -0.1953125 1.6665625 -0.0134375 0.0034375
## A:E B:E C:E D:E A:B:C A:B:D
## 1.0271875 1.2834375 0.3015625 1.3896875 0.2503125 -0.3453125
## A:C:D B:C:D A:B:E A:C:E B:C:E A:D:E
## -0.0634375 0.3053125 1.1853125 -0.2590625 0.1709375 0.9015625
## B:D:E C:D:E A:B:C:D A:B:C:E A:B:D:E A:C:D:E
## -0.0396875 0.3959375 -0.0740625 -0.1846875 0.4071875 0.1278125
## B:C:D:E A:B:C:D:E
## -0.0746875 -0.3553125
## halfnormal(model_Quality)
## plot(model_Quality)
## Part a: Analyze the data from this experiment. Identify the significant factors and interactions.
model_Quality1<-lm(y~A+B+D+E+A*B+A*D+A*E+B*E+D*E+ A*B*E+A*D*E,data=dat_Quality)
model_Quality2<-aov(y~A+B+D+E+A*B+A*D+A*E+B*E+D*E+ A*B*E+A*D*E,data=dat_Quality)
coef(model_Quality1)
## (Intercept) A B D E A:B
## 10.1803125 1.6159375 0.0434375 2.9884375 2.1878125 1.2365625
## A:D A:E B:E D:E A:B:E A:D:E
## 1.6665625 1.0271875 1.2834375 1.3896875 1.1853125 0.9015625
halfnormal(model_Quality1)
## Warning in halfnormal.lm(model_Quality1): halfnormal not recommended for models
## with more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):
## [1] D E A:D A D:E B:E A:B A:B:E A:E A:D:E e10
plot(model_Quality1)
## hat values (leverages) are all = 0.375
## and there are no factor predictors; no plot no. 5
summary(model_Quality2)
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 83.56 83.56 51.362 6.10e-07 ***
## B 1 0.06 0.06 0.037 0.849178
## D 1 285.78 285.78 175.664 2.30e-11 ***
## E 1 153.17 153.17 94.149 5.24e-09 ***
## A:B 1 48.93 48.93 30.076 2.28e-05 ***
## A:D 1 88.88 88.88 54.631 3.87e-07 ***
## A:E 1 33.76 33.76 20.754 0.000192 ***
## B:E 1 52.71 52.71 32.400 1.43e-05 ***
## D:E 1 61.80 61.80 37.986 5.07e-06 ***
## A:B:E 1 44.96 44.96 27.635 3.82e-05 ***
## A:D:E 1 26.01 26.01 15.988 0.000706 ***
## Residuals 20 32.54 1.63
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model_Quality1)
##
## Call:
## lm.default(formula = y ~ A + B + D + E + A * B + A * D + A *
## E + B * E + D * E + A * B * E + A * D * E, data = dat_Quality)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.04875 -0.67375 -0.00687 0.65281 2.25375
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.18031 0.22548 45.150 < 2e-16 ***
## A 1.61594 0.22548 7.167 6.10e-07 ***
## B 0.04344 0.22548 0.193 0.849178
## D 2.98844 0.22548 13.254 2.30e-11 ***
## E 2.18781 0.22548 9.703 5.24e-09 ***
## A:B 1.23656 0.22548 5.484 2.28e-05 ***
## A:D 1.66656 0.22548 7.391 3.87e-07 ***
## A:E 1.02719 0.22548 4.556 0.000192 ***
## B:E 1.28344 0.22548 5.692 1.43e-05 ***
## D:E 1.38969 0.22548 6.163 5.07e-06 ***
## A:B:E 1.18531 0.22548 5.257 3.82e-05 ***
## A:D:E 0.90156 0.22548 3.998 0.000706 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.275 on 20 degrees of freedom
## Multiple R-squared: 0.9643, Adjusted R-squared: 0.9447
## F-statistic: 49.15 on 11 and 20 DF, p-value: 5.069e-12
## Part b: Analyze the residuals from this experiment. Are there any indications of model inadequacy or violations of the assumptions?
## The residual plots do not identify any concerns with model adequacy or the violations of the assumptions.
## Part c: One of the factors from this experiment does not seem to be important. If you drop this factor,
## what type of design remains? Analyze the data using the full factorial model for only the four
## active factors. Compare your results with those obtained in part (a).
## The resulting experimental design is a replicated 2^4 full factorial design. The ANOVA is shown below.
## The factor names in the output below were modified to match the factor names in the original problem.
## The same factors are significant below as were significant in the original analysis.
model_Quality3<-lm(y~A+B+D+E+A*B+A*D+A*E+B*D+ B*E+D*E+ A*B*D+A*D*E+A*D*E+B*D*E+A*B*D*E,data=dat_Quality)
model_Quality4<-aov(y~A+B+D+E+A*B+A*D+A*E+B*D+ B*E+D*E+ A*B*D+A*D*E+A*D*E+B*D*E+A*B*D*E,data=dat_Quality)
coef(model_Quality3)
## (Intercept) A B D E A:B
## 10.1803125 1.6159375 0.0434375 2.9884375 2.1878125 1.2365625
## A:D A:E B:D B:E D:E A:B:D
## 1.6665625 1.0271875 -0.0134375 1.2834375 1.3896875 -0.3453125
## A:D:E B:D:E A:B:E A:B:D:E
## 0.9015625 -0.0396875 1.1853125 0.4071875
halfnormal(model_Quality3)
##
## Significant effects (alpha=0.05, Lenth method):
## [1] D E A:D A D:E B:E A:B A:B:E A:E A:D:E e10
plot(model_Quality3)
## hat values (leverages) are all = 0.5
## and there are no factor predictors; no plot no. 5
summary(model_Quality4)
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 83.56 83.56 57.233 1.14e-06 ***
## B 1 0.06 0.06 0.041 0.841418
## D 1 285.78 285.78 195.742 2.16e-10 ***
## E 1 153.17 153.17 104.910 1.97e-08 ***
## A:B 1 48.93 48.93 33.514 2.77e-05 ***
## A:D 1 88.88 88.88 60.875 7.66e-07 ***
## A:E 1 33.76 33.76 23.126 0.000193 ***
## B:D 1 0.01 0.01 0.004 0.950618
## B:E 1 52.71 52.71 36.103 1.82e-05 ***
## D:E 1 61.80 61.80 42.328 7.24e-06 ***
## A:B:D 1 3.82 3.82 2.613 0.125501
## A:D:E 1 26.01 26.01 17.815 0.000650 ***
## B:D:E 1 0.05 0.05 0.035 0.854935
## A:B:E 1 44.96 44.96 30.794 4.40e-05 ***
## A:B:D:E 1 5.31 5.31 3.634 0.074735 .
## Residuals 16 23.36 1.46
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model_Quality3)
##
## Call:
## lm.default(formula = y ~ A + B + D + E + A * B + A * D + A *
## E + B * D + B * E + D * E + A * B * D + A * D * E + A * D *
## E + B * D * E + A * B * D * E, data = dat_Quality)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4750 -0.5637 0.0000 0.5637 1.4750
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.18031 0.21360 47.661 < 2e-16 ***
## A 1.61594 0.21360 7.565 1.14e-06 ***
## B 0.04344 0.21360 0.203 0.841418
## D 2.98844 0.21360 13.991 2.16e-10 ***
## E 2.18781 0.21360 10.243 1.97e-08 ***
## A:B 1.23656 0.21360 5.789 2.77e-05 ***
## A:D 1.66656 0.21360 7.802 7.66e-07 ***
## A:E 1.02719 0.21360 4.809 0.000193 ***
## B:D -0.01344 0.21360 -0.063 0.950618
## B:E 1.28344 0.21360 6.009 1.82e-05 ***
## D:E 1.38969 0.21360 6.506 7.24e-06 ***
## A:B:D -0.34531 0.21360 -1.617 0.125501
## A:D:E 0.90156 0.21360 4.221 0.000650 ***
## B:D:E -0.03969 0.21360 -0.186 0.854935
## A:B:E 1.18531 0.21360 5.549 4.40e-05 ***
## A:B:D:E 0.40719 0.21360 1.906 0.074735 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.208 on 16 degrees of freedom
## Multiple R-squared: 0.9744, Adjusted R-squared: 0.9504
## F-statistic: 40.58 on 15 and 16 DF, p-value: 7.07e-10
## (d) Find the settings of the active factors that maximize the predicted response.
## Cube model can be constructed such that factors E is at +1 and C at 0 and response of other factors, A, B, and D are at +1.