Assignment 14

Q1

# factors A with two levels (2,30)

Factor B Stir Rate with two levels (100,150)

Factor C Temperature with 2 levels (8,40)

a) Write the model equation for a full factorial model

Model equation

\(Y_{ijkl}=\mu+\alpha_{i}+\beta_{j}+\gamma_{k}+\alpha\beta_{ij}+\alpha\gamma_{ik}+\beta\gamma_{jk}+\alpha\beta\gamma_{ijk}+e_{ijkl}\)

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

Dat<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
colnames(Dat)<-c("A", "B", "C", "obs")
library(GAD)

## Loading required package: matrixStats

## 
## Attaching package: 'matrixStats'

## The following object is masked from 'package:dplyr':
## 
##     count

## Loading required package: R.methodsS3

## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.

Dat$A<-as.fixed(Dat$A)
Dat$B<-as.fixed(Dat$B)
Dat$C<-as.fixed(Dat$C)
Model<-aov(obs~A+B+C+A*B+A*C+B*C+A*B*C, data = Dat)
GAD::gad(Model)

## Analysis of Variance Table
## 
## Response: obs
##          Df Sum Sq Mean Sq F value   Pr(>F)   
## A         1 44.389  44.389 11.1803 0.010175 * 
## B         1 70.686  70.686 17.8037 0.002918 **
## C         1  0.328   0.328  0.0826 0.781170   
## A:B       1 28.117  28.117  7.0817 0.028754 * 
## A:C       1  0.022   0.022  0.0055 0.942808   
## B:C       1 10.128  10.128  2.5510 0.148890   
## A:B:C     1  1.519   1.519  0.3826 0.553412   
## Residual  8 31.762   3.970                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

With the p-value = 0.010175 we conclude factor A has a significant effect to improve a silver powder production process.

With the p-value = 0.002918 we conclude factor B has a significant effect to improve a silver powder production process.

With the p-value = 0.028754 we conclude the interaction between factors AB has a significant effect to improve a silver powder production. process.

Model after exclude insignificant factors

Model<-aov(obs~A+B+A*B, data = Dat)
GAD::gad(Model)

## Analysis of Variance Table
## 
## Response: obs
##          Df Sum Sq Mean Sq F value    Pr(>F)    
## A         1 44.389  44.389 12.1727 0.0044721 ** 
## B         1 70.686  70.686 19.3841 0.0008612 ***
## A:B       1 28.117  28.117  7.7103 0.0167511 *  
## Residual 12 43.759   3.647                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q2

Factor A 2 levels, Factor B 3 levels n = 3 replications

Temp<-rep(seq(1,3), 3)
Temp

## [1] 1 2 3 1 2 3 1 2 3

Pos<-c(rep(1,9), rep(2,9))
Pos

##  [1] 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

obs1<-c(570, 1063, 565, 
       565, 1080, 510,
       583, 1043, 590,
       528, 988,  526,
       547, 1026, 538,
       521, 1004, 532)
Dat1<-data.frame(Pos,Temp,obs1)

Dat1$Temp<-as.fixed(Dat1$Temp)
Dat1$Pos<-as.fixed(Dat1$Pos)

Model2<-aov(obs1~Temp+Pos+Temp*Pos, data = Dat1)
GAD::gad(Model2)

## Analysis of Variance Table
## 
## Response: obs1
##          Df Sum Sq Mean Sq  F value   Pr(>F)    
## Temp      2 945342  472671 1056.117 3.25e-14 ***
## Pos       1   7160    7160   15.998 0.001762 ** 
## Temp:Pos  2    818     409    0.914 0.427110    
## Residual 12   5371     448                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

a) Assume that both Temperature and Position are fixed effects. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 3.25e-14.

Factor Pos p-value = 0.001762.

Dat1$Temp<-as.random(Dat1$Temp)
Dat1$Pos<-as.random(Dat1$Pos)

Model3<-aov(obs1~Temp+Pos+Temp*Pos, data = Dat1)
GAD::gad(Model3)

## Analysis of Variance Table
## 
## Response: obs1
##          Df Sum Sq Mean Sq  F value    Pr(>F)    
## Temp      2 945342  472671 1155.518 0.0008647 ***
## Pos       1   7160    7160   17.504 0.0526583 .  
## Temp:Pos  2    818     409    0.914 0.4271101    
## Residual 12   5371     448                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

b) Assume that both Temperature and Position are random effects. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 0.0008647.

Factor Pos p-value = 0.0526583.

Dat1$Temp<-as.random(Dat1$Temp)
Dat1$Pos<-as.fixed(Dat1$Pos)

Model4<-aov(obs1~Temp+Pos+Temp*Pos, data = Dat1)
GAD::gad(Model4)

## Analysis of Variance Table
## 
## Response: obs1
##          Df Sum Sq Mean Sq  F value   Pr(>F)    
## Temp      2 945342  472671 1056.117 3.25e-14 ***
## Pos       1   7160    7160   17.504  0.05266 .  
## Temp:Pos  2    818     409    0.914  0.42711    
## Residual 12   5371     448                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

c) Assume the Position effect is fixed and the Temperature effect is random. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 3.25e-14.

Factor Pos p-value = 0.05266.

d) Comment on similarities and/or differences between the p-values in parts a,b,c.

The interaction between the factor Pos and Temp was not significant in all tests performed and has the same p-value for all conditions.

On the other hand, the factor Temp in part a (as fixed) resulted in the same p-value as part c (as.random). But in part b (both factors as.random) the p-value if factor Temp is bigger than parts a and b.

Assignment 14

Viviana, Dhruv, Gabriel

11/2/2021

Q1

# factors A with two levels (2,30)

Factor B Stir Rate with two levels (100,150)

Factor C Temperature with 2 levels (8,40)

a) Write the model equation for a full factorial model

Model equation

\(Y_{ijkl}=\mu+\alpha_{i}+\beta_{j}+\gamma_{k}+\alpha\beta_{ij}+\alpha\gamma_{ik}+\beta\gamma_{jk}+\alpha\beta\gamma_{ijk}+e_{ijkl}\)

With the p-value = 0.010175 we conclude factor A has a significant effect to improve a silver powder production process.

With the p-value = 0.002918 we conclude factor B has a significant effect to improve a silver powder production process.

With the p-value = 0.028754 we conclude the interaction between factors AB has a significant effect to improve a silver powder production. process.

Model after exclude insignificant factors

Q2

Factor A 2 levels, Factor B 3 levels n = 3 replications

a) Assume that both Temperature and Position are fixed effects. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 3.25e-14.

Factor Pos p-value = 0.001762.

b) Assume that both Temperature and Position are random effects. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 0.0008647.

Factor Pos p-value = 0.0526583.

c) Assume the Position effect is fixed and the Temperature effect is random. Report p-values

Interaction Temp and Position., p-value = 0.427110.

Factor Temp p-value = 3.25e-14.

Factor Pos p-value = 0.05266.

d) Comment on similarities and/or differences between the p-values in parts a,b,c.

The interaction between the factor Pos and Temp was not significant in all tests performed and has the same p-value for all conditions.

On the other hand, the factor Temp in part a (as fixed) resulted in the same p-value as part c (as.random). But in part b (both factors as.random) the p-value if factor Temp is bigger than parts a and b.

For the factor Pos, for parts b and c the p-value resulted is very close, but for part a was very small.