Flipped Assignment 14 - Factorial Designs using R
Forest Kingfisher, Olivia Owens, Kameron Garza
2021-11-07
1.
An article in Quality Progress(May 2011, pp. 42–48) describes the use of factorial experiments to improve a silver powder production process. This product is used in conductive pastes to manufacture a wide variety of products ranging from silicon wafers to elastic membrane switches. We consider powder density(g/cm
# Read in the data table for 14.1
dat141<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
names(dat141)[names(dat141) == "ï..Ammonium"] <- "Ammonium"
dat141## Ammonium StirRate Temperature Density
## 1 2 100 8 14.68
## 2 2 100 8 15.18
## 3 30 100 8 15.12
## 4 30 100 8 17.48
## 5 2 150 8 7.54
## 6 2 150 8 6.66
## 7 30 150 8 12.46
## 8 30 150 8 12.62
## 9 2 100 40 10.95
## 10 2 100 40 17.68
## 11 30 100 40 12.65
## 12 30 100 40 15.96
## 13 2 150 40 8.03
## 14 2 150 40 8.84
## 15 30 150 40 14.96
## 16 30 150 40 14.96The data may be downloaded from here: https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv
Assume Ammonium, Stir Rate, and Temperature are factors with fixed effects, each with two levels, and the design is replicated twice.
(a) Write the model equation for a full factorial model
The following equation describes the components of the full factorial model:
(b) What factors are deemed significant, using a=.05 as a guide. Report final p-values of significant factors (and interaction plots if necessary).
The significant factors are:
- Ammonium, P-value = 0.010175
- StirRate, P-value = 0.002918
- Interaction of Ammonium and StirRate, P-value = 0.028754
Because the interaction of Ammonium and StirRate is significant, we must produce an interaction plot for these factors
# change factor variables into factors
dat141$Ammonium <- as.factor(dat141$Ammonium)
dat141$StirRate <- as.factor(dat141$StirRate)
dat141$Temperature <- as.factor(dat141$Temperature)
str(dat141)## 'data.frame': 16 obs. of 4 variables:
## $ Ammonium : Factor w/ 2 levels "2","30": 1 1 2 2 1 1 2 2 1 1 ...
## $ StirRate : Factor w/ 2 levels "100","150": 1 1 1 1 2 2 2 2 1 1 ...
## $ Temperature: Factor w/ 2 levels "8","40": 1 1 1 1 1 1 1 1 2 2 ...
## $ Density : num 14.68 15.18 15.12 17.48 7.54 ...# Apply analysis of variance model
model141 <- lm(Density ~ Ammonium+StirRate+Temperature+Ammonium:StirRate+Ammonium:Temperature+StirRate:Temperature+Ammonium:StirRate:Temperature, dat141)
anova(model141)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 11.1803 0.010175 *
## StirRate 1 70.686 70.686 17.8037 0.002918 **
## Temperature 1 0.328 0.328 0.0826 0.781170
## Ammonium:StirRate 1 28.117 28.117 7.0817 0.028754 *
## Ammonium:Temperature 1 0.022 0.022 0.0055 0.942808
## StirRate:Temperature 1 10.128 10.128 2.5510 0.148890
## Ammonium:StirRate:Temperature 1 1.519 1.519 0.3826 0.553412
## Residuals 8 31.762 3.970
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Produce an interaction plot
interaction.plot(x.factor = dat141$Ammonium,
trace.factor = dat141$StirRate,
response = dat141$Density,
fun = mean,
type="b", ### Show plot points as symbols
col=c("steelblue","firebrick2"), ### Colors for levels of trace var.
pch=c(19, 17, 15), ### Symbols for levels of trace var.
fixed=TRUE, ### Order by factor order in data
leg.bty = "o") ### Legend box
2.
A full factorial experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode.
# Read in the data table for 14.2
dat142<-read.csv("https://raw.githubusercontent.com/forestwhite/RStatistics/main/142Table.csv")
dat142## Position Temperature Density
## 1 1 800 570
## 2 1 800 565
## 3 1 800 583
## 4 1 825 1063
## 5 1 825 1080
## 6 1 825 1043
## 7 1 850 565
## 8 1 850 510
## 9 1 850 590
## 10 2 800 528
## 11 2 800 547
## 12 2 800 521
## 13 2 825 988
## 14 2 825 1026
## 15 2 825 1004
## 16 2 850 526
## 17 2 850 538
## 18 2 850 532(a) Assume that both Temperature and Position are fixed effects. Report p-values
The P-values for each factor and the interaction are:
- PPosition = 0.001762
- PTemperature = 3.25e-14
- PPosition:Temperature = 0.427110
# designate fixed factors as fixed
library(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.Position <- as.fixed(dat142$Position)
Position## [1] 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
## Levels: 1 2Temperature <- as.fixed(dat142$Temperature)
Temperature## [1] 800 800 800 825 825 825 850 850 850 800 800 800 825 825 825 850 850 850
## Levels: 800 825 850# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)## Analysis of Variance Table
##
## Response: dat142$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Position 1 7160 7160 15.998 0.001762 **
## Temperature 2 945342 472671 1056.117 3.25e-14 ***
## Position:Temperature 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(b) Assume that both Temperature and Position are random effects. Report p-values
The P-values for each factor and the interaction are:
- PPosition = 0.0526583
- PTemperature = 0.0008647
- PPosition:Temperature = 0.4271101
# designate random factors as random
library(GAD)
Position <- as.random(dat142$Position)
Position## [1] 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
## Levels: 1 2Temperature <- as.random(dat142$Temperature)
Temperature## [1] 800 800 800 825 825 825 850 850 850 800 800 800 825 825 825 850 850 850
## Levels: 800 825 850# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)## Analysis of Variance Table
##
## Response: dat142$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Position 1 7160 7160 17.504 0.0526583 .
## Temperature 2 945342 472671 1155.518 0.0008647 ***
## Position:Temperature 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(c) Assume the Position effect is fixed and the Temperature effect is random. Report p-values
The P-values for each factor and the interaction are:
- PPosition = 0.05266
- PTemperature = 3.25e-14
- PPosition:Temperature = 0.42711
# designate fixed factors as fixed
library(GAD)
Position <- as.fixed(dat142$Position)
Position## [1] 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
## Levels: 1 2# designate random factors as random
Temperature <- as.random(dat142$Temperature)
Temperature## [1] 800 800 800 825 825 825 850 850 850 800 800 800 825 825 825 850 850 850
## Levels: 800 825 850# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)## Analysis of Variance Table
##
## Response: dat142$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Position 1 7160 7160 17.504 0.05266 .
## Temperature 2 945342 472671 1056.117 3.25e-14 ***
## Position:Temperature 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(d) Comment on similarities and/or differences between the p-values in parts a,b,c.
The p-values for the interaction stays consistent across parts a, b, and c at a value of 0.427. Interestingly, the p-value for Temperature is the same for parts a and c (in a Temperature is fixed, in c it is random), while it increases in part b (when Temperature is random). The same phenomenon occurs for Position, which has a similar p-value in parts b and c (in part b Position is random, in part c Position is fixed), while it is smaller in part a.
The Temperature variable in all cases is significant, whether it was a random or fixed effect.
The Position variable was significant when it was a fixed effect (part a) and the temperature is a fixed effect but not when it was a random effect (part b) nor when the temperature is a random effect (part c).
The interaction between Temperature and Position is not significant in any case.
The p-values for random factors on their own were generally less definitely significant and less precise than fixed factors, which is reasonable considering random effects are compared to the interaction while fixed effects are compared to error. The p-values for the interactions of the factors was nearly identical whether the factors were random or fixed, which also follows from interactions always being compared to error.
Complete Code
Here we display the complete R code used in this analysis.
# Read in the data table for 14.1
dat141<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
names(dat141)[names(dat141) == "ï..Ammonium"] <- "Ammonium"
dat141
# change factor variables into factors
dat141$Ammonium <- as.factor(dat141$Ammonium)
dat141$StirRate <- as.factor(dat141$StirRate)
dat141$Temperature <- as.factor(dat141$Temperature)
str(dat141)
# Apply analysis of variance model
model141 <- lm(Density ~ Ammonium+StirRate+Temperature+Ammonium:StirRate+Ammonium:Temperature+StirRate:Temperature+Ammonium:StirRate:Temperature, dat141)
anova(model141)
# Produce an interaction plot
interaction.plot(x.factor = dat141$Ammonium,
trace.factor = dat141$StirRate,
response = dat141$Density,
fun = mean,
type="b", ### Show plot points as symbols
col=c("steelblue","firebrick2"), ### Colors for levels of trace var.
pch=c(19, 17, 15), ### Symbols for levels of trace var.
fixed=TRUE, ### Order by factor order in data
leg.bty = "o") ### Legend box
# Read in the data table for 14.2
dat142<-read.csv("https://raw.githubusercontent.com/forestwhite/RStatistics/main/142Table.csv")
dat142
# designate fixed factors as fixed
library(GAD)
Position <- as.fixed(dat142$Position)
Position
Temperature <- as.fixed(dat142$Temperature)
Temperature
# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)
# designate random factors as random
library(GAD)
Position <- as.random(dat142$Position)
Position
Temperature <- as.random(dat142$Temperature)
Temperature
# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)
# designate fixed factors as fixed
library(GAD)
Position <- as.fixed(dat142$Position)
Position
# designate random factors as random
Temperature <- as.random(dat142$Temperature)
Temperature
# Apply analysis of variance model
model142 <- lm(dat142$Density ~ Position+Temperature+Position*Temperature)
gad(model142)