Final Test
Md Ariful Haque Miah
2022-12-08
library(DoE.base)## Loading required package: grid## Loading required package: conf.design## Registered S3 method overwritten by 'DoE.base':
## method from
## factorize.factor 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.design## The following object is masked from 'package:base':
##
## lengthslibrary(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.library(pwr)
library(MASS)
library(agricolae)Question 01
type<-c(rep(1,6),rep(2,6))
temp<-c(rep(seq(900,1000,50),4))
obs<-c(4.6,9.15,12.01,4.4,9.85,11.58,3.2,9.38,10.81,3.5,10.02,10.6)
type<-as.fixed(type)
temp<-as.fixed(temp)
dat<-data.frame(type,temp,obs)
dat## type temp obs
## 1 1 900 4.60
## 2 1 950 9.15
## 3 1 1000 12.01
## 4 1 900 4.40
## 5 1 950 9.85
## 6 1 1000 11.58
## 7 2 900 3.20
## 8 2 950 9.38
## 9 2 1000 10.81
## 10 2 900 3.50
## 11 2 950 10.02
## 12 2 1000 10.60mod<-aov(obs~type*temp,data=dat)
summary(mod)## Df Sum Sq Mean Sq F value Pr(>F)
## type 1 1.39 1.39 13.226 0.0109 *
## temp 2 118.11 59.06 563.062 1.49e-07 ***
## type:temp 2 1.16 0.58 5.546 0.0433 *
## Residuals 6 0.63 0.10
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1qf(0.95,2,6)## [1] 5.143253qf(0.95,1,16)## [1] 4.493998Question 9
pos<-c(rep(1,9),rep(2,9))
temp<-c(rep(seq(800,850,25),6))
obs<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,
1004,532)
pos<-as.fixed(pos)
temp<-as.fixed(temp)
dat<-data.frame(pos,temp,obs)
dat## pos temp obs
## 1 1 800 570
## 2 1 825 1063
## 3 1 850 565
## 4 1 800 565
## 5 1 825 1080
## 6 1 850 510
## 7 1 800 583
## 8 1 825 1043
## 9 1 850 590
## 10 2 800 528
## 11 2 825 988
## 12 2 850 526
## 13 2 800 547
## 14 2 825 1026
## 15 2 850 538
## 16 2 800 521
## 17 2 825 1004
## 18 2 850 532mod1<-aov(obs~pos*temp,data=dat)
gad(mod1)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 15.998 0.001762 **
## temp 2 945342 472671 1056.117 3.25e-14 ***
## pos:temp 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 19-A P values for position, temperature and their interactions are 0.00176, 3.25e-14 and 0.42711
pos<-c(rep(1,9),rep(2,9))
temp<-c(rep(seq(800,850,25),6))
obs<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,
1004,532)
pos<-as.random(pos)
temp<-as.random(temp)
dat<-data.frame(pos,temp,obs)
dat## pos temp obs
## 1 1 800 570
## 2 1 825 1063
## 3 1 850 565
## 4 1 800 565
## 5 1 825 1080
## 6 1 850 510
## 7 1 800 583
## 8 1 825 1043
## 9 1 850 590
## 10 2 800 528
## 11 2 825 988
## 12 2 850 526
## 13 2 800 547
## 14 2 825 1026
## 15 2 850 538
## 16 2 800 521
## 17 2 825 1004
## 18 2 850 532mod2<-aov(obs~pos*temp,data=dat)
gad(mod2)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 17.504 0.0526583 .
## temp 2 945342 472671 1155.518 0.0008647 ***
## pos:temp 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 19-B P values for position, temperature and their interactions are 0.0526583, 0.0008647 and 0.4271101
pos<-c(rep(1,9),rep(2,9))
temp<-c(rep(seq(800,850,25),6))
obs<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,
1004,532)
pos<-as.fixed(pos)
temp<-as.random(temp)
dat<-data.frame(pos,temp,obs)
dat## pos temp obs
## 1 1 800 570
## 2 1 825 1063
## 3 1 850 565
## 4 1 800 565
## 5 1 825 1080
## 6 1 850 510
## 7 1 800 583
## 8 1 825 1043
## 9 1 850 590
## 10 2 800 528
## 11 2 825 988
## 12 2 850 526
## 13 2 800 547
## 14 2 825 1026
## 15 2 850 538
## 16 2 800 521
## 17 2 825 1004
## 18 2 850 532mod3<-aov(obs~pos*temp,data=dat)
gad(mod3)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 17.504 0.05266 .
## temp 2 945342 472671 1056.117 3.25e-14 ***
## pos:temp 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 19-C P values for position, temperature and their interactions are 0.05266, 3.25e-14 and 0.42711
9-D
Similarity: p values and F value for interaction term (pos:temp) is same for part A,B and C also the interaction term is not significant in all the part A,B and C
Difference: p values and F value for the position and the temperature differs in part A,B and C based on whether they are fixed or random. Since in random case, we are dividing mean squares of the factor by their interactions instead of dividing by MSE (if the factor is fixed)
Question 11
A<-c(-1,1)
B<-c(rep(-1,2),rep(1,2))
C<-c(rep(-1,4),rep(1,4))
D<-c(rep(-1,8),rep(1,8))
obs<-c(12,18,13,20,17,25,15,25,10,24,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,obs)
dat## A B C D obs
## 1 -1 -1 -1 -1 12
## 2 1 -1 -1 -1 18
## 3 -1 1 -1 -1 13
## 4 1 1 -1 -1 20
## 5 -1 -1 1 -1 17
## 6 1 -1 1 -1 25
## 7 -1 1 1 -1 15
## 8 1 1 1 -1 25
## 9 -1 -1 -1 1 10
## 10 1 -1 -1 1 24
## 11 -1 1 -1 1 13
## 12 1 1 -1 1 24
## 13 -1 -1 1 1 19
## 14 1 -1 1 1 21
## 15 -1 1 1 1 17
## 16 1 1 1 1 23mod4<-lm(obs~A*B*C*D,data=dat)
halfnormal(mod4)##
## Significant effects (alpha=0.05, Lenth method):## [1] A C A:C:D
Question 11-A
A,C and A:C:D factors are significant
mod5<-lm(obs~A+C+D+A:C:D,data=dat)
summary(mod5)##
## Call:
## lm.default(formula = obs ~ A + C + D + A:C:D, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.250 -1.000 -0.250 1.062 2.500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.5000 0.4024 45.973 6.31e-14 ***
## A 4.0000 0.4024 9.940 7.85e-07 ***
## C 1.7500 0.4024 4.349 0.00116 **
## D 0.3750 0.4024 0.932 0.37140
## A:C:D -1.3750 0.4024 -3.417 0.00575 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.61 on 11 degrees of freedom
## Multiple R-squared: 0.9221, Adjusted R-squared: 0.8938
## F-statistic: 32.57 on 4 and 11 DF, p-value: 4.851e-06Question 11-B Hypothesis H_0: (alphagammadelta)_ikl = 0 H_a: (alphagammadelta)_ikl != 0
From the summary, we see that the interaction term A:C:D is significant as the p value is less than alpha= 0.05 level of significance. so we reject the null hypothesis.
Question 13
process<-c(rep(1,12),rep(2,12),rep(3,12))
batch<-rep(c(rep(1,3),rep(2,3),rep(3,3),rep(4,3)),3)
obs<-c(25,30,26,19,28,20,15,17,14,15,16,13,29,27,24,23,24,21,28,21,27,35,27,25,24,25,20,35,21,24,38,34,30,25,29,33)
process<-as.fixed(process)
batch<-as.random(batch)
dat<-data.frame(process,batch,obs)
dat## process batch obs
## 1 1 1 25
## 2 1 1 30
## 3 1 1 26
## 4 1 2 19
## 5 1 2 28
## 6 1 2 20
## 7 1 3 15
## 8 1 3 17
## 9 1 3 14
## 10 1 4 15
## 11 1 4 16
## 12 1 4 13
## 13 2 1 29
## 14 2 1 27
## 15 2 1 24
## 16 2 2 23
## 17 2 2 24
## 18 2 2 21
## 19 2 3 28
## 20 2 3 21
## 21 2 3 27
## 22 2 4 35
## 23 2 4 27
## 24 2 4 25
## 25 3 1 24
## 26 3 1 25
## 27 3 1 20
## 28 3 2 35
## 29 3 2 21
## 30 3 2 24
## 31 3 3 38
## 32 3 3 34
## 33 3 3 30
## 34 3 4 25
## 35 3 4 29
## 36 3 4 33mod6<-lm(obs~process+batch%in%process)
gad(mod6)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## process 2 446.06 223.028 3.5365 0.073563 .
## process:batch 9 567.58 63.065 4.1965 0.002349 **
## Residual 24 360.67 15.028
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 113-A Model equation \(Y_{ijk}\) = \(\mu + \alpha_i +\beta_{j(i)} + \epsilon_{ijk}\)
13-B
Hypothesis: \(H_0\) = \(\alpha_i = 0\)
\(H_a\) = \(\alpha_i \neq 0\)
\(H_0\) = \(\beta_{j(i)} = 0\)
\(H_a\) = \(\beta_{j(i)} \neq 0\)
13-C
From the summary, we see that the process factor is fixed and p value is higher than \(\alpha = 0.05\) so we fail to reject \(H_0\) and the factor process is not significant. But the nested factor batch is within the factor process, p value is less than \(\alpha = 0.05\) so we reject \(H_0\) and the nested factor batch is within the factor process is significant.