library(GAD)## Loading required package: matrixStats## Warning: package 'matrixStats' was built under R version 4.2.2## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.pos <- c(rep(1,9),rep(2,9))
temp <- c(rep(seq(1,3),6))
obs <- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
dat <- data.frame(pos,temp,obs)
dat## pos temp obs
## 1 1 1 570
## 2 1 2 1063
## 3 1 3 565
## 4 1 1 565
## 5 1 2 1080
## 6 1 3 510
## 7 1 1 583
## 8 1 2 1043
## 9 1 3 590
## 10 2 1 528
## 11 2 2 988
## 12 2 3 526
## 13 2 1 547
## 14 2 2 1026
## 15 2 3 538
## 16 2 1 521
## 17 2 2 1004
## 18 2 3 532pos <- as.fixed(pos)
temp <- as.fixed(temp)
dat1 <- data.frame(pos,temp,obs)
model1 <- aov(obs~pos+temp+pos*temp,data = dat1)
gad(model1)## 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 ' ' 1The p-value for factor A (position) and factor B (temperature) was respectively 0.001762 and 3.25e-14 which are low. However the p-value of the interaction term was rather high (0.427110).
pos <- as.random(pos)
temp <- as.random(temp)
dat2 <- data.frame(pos,temp,obs)
model2 <- aov(obs~pos+temp+pos*temp,data = dat2)
gad(model2)## 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 ' ' 1The p-value for factor A (position) and factor B (temperature) was respectively 0.0526583 and 0.0008647 whereas the p-value of the interaction term was rather high (0.4271101).
pos <- as.fixed(pos)
temp <- as.random(temp)
dat2 <- data.frame(pos,temp,obs)
model2 <- aov(obs~pos+temp+pos*temp,data = dat2)
gad(model2)## 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 ' ' 1The p-value for factor A (position) and factor B (temperature) was respectively 0.0526583 and 3.25e-14 whereas the p-value of the interaction term was rather high (0.4271101).
In the part a), where the both effects are fixed, we get very low p-values for both factors. However, in par b), both p-values do increase significantly. Finally in part c), the p-value of factor A remains the same as in part b), but the p-value of factor B reverts back to the part a) value. Meanwhile the p-value of the interaction term remains unchanged. As we know, when we have fixed effects for both factors, the f-statistic is calculated by dividing the MS of the factor by MSE, however in a random effect model, the factors MS will be divided by MS of interaction term (MSAB). As the MSAB here is larger than the MSE doing so will reduce the the f value and increase the p-value. As in a mixed effect model (like part c), only one of the factors MS is divided by MSAB. Furthermore, the f value for the interaction term is always calculated by diving it by the MSE, regardless of random/fixed/mixed effect model. That is why, the interaction terms p-value doesn’t change here.