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. The combination of ammonium and stir rate would also be significant to this process.
A horticultural researcher wants to investigate how three controllable factors affect tomato yield in a greenhouse environment. You will use R and the agricolae package to create the experimental layout for this study.
Experimental Factors:
Factor Description Levels A Fertilizer Type A₁ = Organic compost, A₂ = Chemical fertilizer B Irrigation Frequency B₁ = Once per day, B₂ = Twice per day C Temperature Setting C₁ = 22°C, C₂ = 28°C
The experimenter wants to run a full factorial design (2³ = 8 treatments) with three replications arranged in a randomized complete block design (RCBD) to control for variation between blocks (e.g., differences among greenhouse benches). Use the design.ab() function from the agricolae package to create a 2³ factorial design with a unique seed. Display the layout with $book.
# factorial 2 x 2 x 2 with 5 replications in completely randomized design.
trt<-c(2,2,2)
outdesign<-agricolae::design.ab(trt, r=3, seed=23, design="rcbd")
book<-outdesign$book
print(book)## plots block A B C
## 1 101 1 1 2 1
## 2 102 1 1 2 2
## 3 103 1 2 2 2
## 4 104 1 2 1 2
## 5 105 1 2 2 1
## 6 106 1 1 1 1
## 7 107 1 1 1 2
## 8 108 1 2 1 1
## 9 109 2 1 2 2
## 10 110 2 1 2 1
## 11 111 2 2 1 1
## 12 112 2 1 1 2
## 13 113 2 2 1 2
## 14 114 2 2 2 1
## 15 115 2 2 2 2
## 16 116 2 1 1 1
## 17 117 3 2 1 1
## 18 118 3 1 2 2
## 19 119 3 2 2 1
## 20 120 3 1 2 1
## 21 121 3 2 1 2
## 22 122 3 1 1 1
## 23 123 3 1 1 2
## 24 124 3 2 2 2