Question9
P<-c(rep(1,9),rep(2,9))
Temp<-rep(c(1,2,3),6)
obs<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
library("GAD")
P<-as.fixed(P)
Temp<-as.fixed(Temp)
(a) Assume that both Temperature and Position are fixed effects.
Report p-values
P<-as.fixed(P)
Temp<-as.fixed(Temp)
model<-aov(obs~P*Temp)
summary(model)
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 15.998 0.00176 **
## Temp 2 945342 472671 1056.117 3.25e-14 ***
## P:Temp 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
we need to check if main effects make sense are not.
model1<-aov(obs~P+Temp)
summary(model1)
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 16.2 0.00125 **
## Temp 2 945342 472671 1069.3 4.92e-16 ***
## Residuals 14 6189 442
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comments: Main effects make sense they are significantly effecting
the model we can reject null hypothesis as we got p-value of position as
0.00125 and p-value of Temperature effect is \(4.92e^{-16}\).firing temperature and
furnace position affects the baked density of a carbon anode.
(b)Assume that both Temperature and Position are random effects.
Report p-values
P<-c(rep(1,9),rep(2,9))
Temp<-rep(c(1,2,3),6)
library("GAD")
P<-as.random(P)
Temp<-as.random(Temp)
model2<-aov(obs~P*Temp)
gad(model2)
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 17.504 0.0526583 .
## Temp 2 945342 472671 1155.518 0.0008647 ***
## P:Temp 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model3<-aov(obs~P+Temp)
gad(model3)
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 16.197 0.001254 **
## Temp 2 945342 472671 1069.257 4.924e-16 ***
## Residual 14 6189 442
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comments; We got P value for Position effect is 0.001254 and for
Temperature effect is \(4.924e^{-16}\)
(C) Assume the Position effect is fixed and the Temperature effect
is random. Report p-values
P<-c(rep(1,9),rep(2,9))
Temp<-rep(c(1,2,3),6)
library("GAD")
P<-as.fixed(P)
Temp<-as.random(Temp)
model4<-aov(obs~P*Temp)
gad(model4)
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 17.504 0.05266 .
## Temp 2 945342 472671 1056.117 3.25e-14 ***
## P:Temp 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model5<-aov(obs~P+Temp)
gad(model5)
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## P 1 7160 7160 16.197 0.001254 **
## Temp 2 945342 472671 1069.257 4.924e-16 ***
## Residual 14 6189 442
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comments: We got P value for Position effect is 0.001254 and for
Temperature effect is \(4.924e^{-16}\)
(d) Comment on similarities and/or differences between the p-values
in parts a,b,c.
Answer; From the final model of all above parts we can conclude by
considering the temperature, position as fixed, random effects , Mixed
there is no change in p values . In all the above parts the interaction
effect is not significant. only the main effects make sense. P values
are same for all the final models of part(a,b,c)
comments: The interaction term is not significant we fail to reject null hypothesis(We got P value(0.42711) as far greater than 0.05(significance level)).