library(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.We want to test the hypotheses that only process is significant:
\[ H_0:\alpha=0\\ H_a:\alpha\neq 0 \]
process <- c(rep(1:3,each=4,3))
batch <- c(rep(1:4,9))
obs <- c(rep(1:3, each=4*3))
response <- c(25,19,15,15,19,23,18,35,14,35,38,25,30,28,17,16,17,24,21,27,15,21,54,29,26,20,14,13,14,21,17,25,20,24,50,33)
dat <- data.frame(process,batch,obs,response)
dat## process batch obs response
## 1 1 1 1 25
## 2 1 2 1 19
## 3 1 3 1 15
## 4 1 4 1 15
## 5 2 1 1 19
## 6 2 2 1 23
## 7 2 3 1 18
## 8 2 4 1 35
## 9 3 1 1 14
## 10 3 2 1 35
## 11 3 3 1 38
## 12 3 4 1 25
## 13 1 1 2 30
## 14 1 2 2 28
## 15 1 3 2 17
## 16 1 4 2 16
## 17 2 1 2 17
## 18 2 2 2 24
## 19 2 3 2 21
## 20 2 4 2 27
## 21 3 1 2 15
## 22 3 2 2 21
## 23 3 3 2 54
## 24 3 4 2 29
## 25 1 1 3 26
## 26 1 2 3 20
## 27 1 3 3 14
## 28 1 4 3 13
## 29 2 1 3 14
## 30 2 2 3 21
## 31 2 3 3 17
## 32 2 4 3 25
## 33 3 1 3 20
## 34 3 2 3 24
## 35 3 3 3 50
## 36 3 4 3 33process <- as.fixed(process)
batch <- as.random(batch)
obs <- as.random(obs)
model <- lm(response~process+batch%in%process)
model##
## Call:
## lm(formula = response ~ process + batch %in% process)
##
## Coefficients:
## (Intercept) process2 process3 process1:batch2
## 27.000 -10.333 -10.667 -4.667
## process2:batch2 process3:batch2 process1:batch3 process2:batch3
## 6.000 10.333 -11.667 2.000
## process3:batch3 process1:batch4 process2:batch4 process3:batch4
## 31.000 -12.333 12.333 12.667summary(model)##
## Call:
## lm(formula = response ~ process + batch %in% process)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.333 -2.083 -0.500 2.333 8.333
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.000 2.511 10.752 1.17e-10 ***
## process2 -10.333 3.551 -2.910 0.00768 **
## process3 -10.667 3.551 -3.004 0.00615 **
## process1:batch2 -4.667 3.551 -1.314 0.20123
## process2:batch2 6.000 3.551 1.690 0.10406
## process3:batch2 10.333 3.551 2.910 0.00768 **
## process1:batch3 -11.667 3.551 -3.285 0.00312 **
## process2:batch3 2.000 3.551 0.563 0.57853
## process3:batch3 31.000 3.551 8.729 6.51e-09 ***
## process1:batch4 -12.333 3.551 -3.473 0.00197 **
## process2:batch4 12.333 3.551 3.473 0.00197 **
## process3:batch4 12.667 3.551 3.567 0.00156 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.349 on 24 degrees of freedom
## Multiple R-squared: 0.8585, Adjusted R-squared: 0.7936
## F-statistic: 13.23 on 11 and 24 DF, p-value: 1.177e-07gad(model)## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## process 2 676.06 338.03 1.4643 0.2815
## process:batch 9 2077.58 230.84 12.2031 5.477e-07 ***
## Residual 24 454.00 18.92
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the anova test, we can conclude that the process is not significant (p=0.2815).