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/cm2)(g/cm2) as the response variable and critical characteristics of this product.
Write the model equation for the full factorial model
\[{y_{ijkl}} = \mu + {\tau _i} + {\beta _j} + {\gamma _k} + {(\tau \beta )_{ij}} + {(\tau \gamma )_{ik}} + {(\beta \gamma )_{jk}} + {(\tau \beta \gamma )_{ijk}} + {\varepsilon _{ijkl}}\]
What factors are deemed significant, using α=.05 as a guide. Report the final p-values of significant factors (and interaction plots if necessary).
library(readr)
PowderProduction <- read_csv("PowderProduction.csv")## Rows: 16 Columns: 4
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (4): Ammonium, StirRate, Temperature, Density
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.#View(PowderProduction)
data.frame(PowderProduction$Ammonium,PowderProduction$StirRate,PowderProduction$Temperature,PowderProduction$Density)## PowderProduction.Ammonium PowderProduction.StirRate
## 1 2 100
## 2 2 100
## 3 30 100
## 4 30 100
## 5 2 150
## 6 2 150
## 7 30 150
## 8 30 150
## 9 2 100
## 10 2 100
## 11 30 100
## 12 30 100
## 13 2 150
## 14 2 150
## 15 30 150
## 16 30 150
## PowderProduction.Temperature PowderProduction.Density
## 1 8 14.68
## 2 8 15.18
## 3 8 15.12
## 4 8 17.48
## 5 8 7.54
## 6 8 6.66
## 7 8 12.46
## 8 8 12.62
## 9 40 10.95
## 10 40 17.68
## 11 40 12.65
## 12 40 15.96
## 13 40 8.03
## 14 40 8.84
## 15 40 14.96
## 16 40 14.96ammonium<-GAD::as.fixed(PowderProduction$Ammonium)
stir_rate<-GAD::as.fixed(PowderProduction$StirRate)
temperature<-GAD::as.fixed(PowderProduction$Temperature)
model_prob1<-aov(PowderProduction$Density~ammonium+stir_rate+temperature+(ammonium*stir_rate)
+(ammonium*temperature)+(stir_rate*temperature)+(ammonium*stir_rate*temperature))
GAD::gad(model_prob1)## $anova
## Analysis of Variance Table
##
## Response: PowderProduction$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## ammonium 1 44.389 44.389 11.1803 0.010175 *
## stir_rate 1 70.686 70.686 17.8037 0.002918 **
## temperature 1 0.328 0.328 0.0826 0.781170
## ammonium:stir_rate 1 28.117 28.117 7.0817 0.028754 *
## ammonium:temperature 1 0.022 0.022 0.0055 0.942808
## stir_rate:temperature 1 10.128 10.128 2.5510 0.148890
## ammonium:stir_rate: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 ' ' 1From the p-values of this output we can see that the ammonium and stir rate are both significant.
A full factorial experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode. (hint: use expand.table() in R to help generate the dataframe and use the package GAD to account for fixed and random effects in the ANOVA model)
p2_temp<-rep(seq(1,3),6)
position<-c(rep(1,9),rep(2,9))
response<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
data.frame(p2_temp,position,response)## p2_temp position response
## 1 1 1 570
## 2 2 1 1063
## 3 3 1 565
## 4 1 1 565
## 5 2 1 1080
## 6 3 1 510
## 7 1 1 583
## 8 2 1 1043
## 9 3 1 590
## 10 1 2 528
## 11 2 2 988
## 12 3 2 526
## 13 1 2 547
## 14 2 2 1026
## 15 3 2 538
## 16 1 2 521
## 17 2 2 1004
## 18 3 2 532Assume that both Temperature and Position are fixed effects. Report p-values
p2_temp<-GAD::as.fixed(p2_temp)
position<-GAD::as.fixed(position)
model<-aov(response~position+p2_temp+(position*p2_temp))
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 15.998 0.001762 **
## p2_temp 2 945342 472671 1056.117 3.25e-14 ***
## position:p2_temp 2 818 409 0.914 0.427110
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 15.998 0.00176 **
## p2_temp 2 945342 472671 1056.117 3.25e-14 ***
## position:p2_temp 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Assume that both Temperature and Position are random effects. Report p-values
p2_temp<-GAD::as.random(p2_temp)
position<-GAD::as.random(position)
model<-aov(response~position+p2_temp+(position*p2_temp))
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 17.504 0.0526583 .
## p2_temp 2 945342 472671 1155.518 0.0008647 ***
## position:p2_temp 2 818 409 0.914 0.4271101
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 15.998 0.00176 **
## p2_temp 2 945342 472671 1056.117 3.25e-14 ***
## position:p2_temp 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Assume the Position effect is fixed and the Temperature effect is random. Report p-values
p2_temp<-GAD::as.random(p2_temp)
position<-GAD::as.fixed(position)
model<-aov(response~position+p2_temp+(position*p2_temp))
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 17.504 0.05266 .
## p2_temp 2 945342 472671 1056.117 3.25e-14 ***
## position:p2_temp 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 15.998 0.00176 **
## p2_temp 2 945342 472671 1056.117 3.25e-14 ***
## position:p2_temp 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment on similarities and/or differences between the p-values in parts a,b,c.
The temperature is always significant no matter whether it is a fixed or random effect. Postion seems to vary in significance depending on whether it is a random or fixed effect.