Final Report
Garrett, Eghorieta, Eswar
12/1/2021
Project For Design of Experments
For our Design of Experiment Project, we created three different parts which looked at three different design of experiments designs. The three different experiment designs that we preformed were a completely randomized design , a factorial design , and \(2^4\) factorial design.
Completely Randomized Design
##
## Balanced one-way analysis of variance power calculation
##
## k = 3
## n = 12.50714
## f = 0.5
## sig.level = 0.05
## power = 0.75
##
## NOTE: n is number in each groupFor this experiment we required 53 samples for each of the 3 different treatment levels. Which resulted in taking 159 total samples.
Layout of Complete Randomized designs
In this experiment, the 3 different treatments are represented by colors yellow, green and blue. the color blue represents the red ball, the color yellow represents the yellow ball and color green represents the green ball that we used in the actual experiment.
| Plot | Replication | Color of Ball |
|---|---|---|
| 101 | 1 | blue |
| 102 | 2 | blue |
| 103 | 3 | blue |
| 104 | 1 | yellow |
| 105 | 1 | green |
| 106 | 2 | green |
| 107 | 4 | blue |
| 108 | 2 | yellow |
| 109 | 3 | yellow |
| 110 | 3 | green |
| 111 | 5 | blue |
| 112 | 4 | green |
| 113 | 5 | green |
| 114 | 6 | green |
| 115 | 7 | green |
| 116 | 8 | green |
| 117 | 6 | blue |
| 118 | 9 | green |
| 119 | 7 | blue |
| 120 | 4 | yellow |
| 121 | 10 | green |
| 122 | 8 | blue |
| 123 | 5 | yellow |
| 124 | 11 | green |
| 125 | 12 | green |
| 126 | 13 | green |
| 127 | 14 | green |
| 128 | 15 | green |
| 129 | 9 | blue |
| 130 | 6 | yellow |
| 131 | 7 | yellow |
| 132 | 10 | blue |
| 133 | 11 | blue |
| 134 | 16 | green |
| 135 | 12 | blue |
| 136 | 8 | yellow |
| 137 | 17 | green |
| 138 | 9 | yellow |
| 139 | 13 | blue |
| 140 | 18 | green |
| 141 | 10 | yellow |
| 142 | 14 | blue |
| 143 | 11 | yellow |
| 144 | 12 | yellow |
| 145 | 19 | green |
| 146 | 20 | green |
| 147 | 13 | yellow |
| 148 | 14 | yellow |
| 149 | 15 | blue |
| 150 | 16 | blue |
| 151 | 17 | blue |
| 152 | 15 | yellow |
| 153 | 21 | green |
| 154 | 18 | blue |
| 155 | 19 | blue |
| 156 | 20 | blue |
| 157 | 16 | yellow |
| 158 | 22 | green |
| 159 | 21 | blue |
| 160 | 17 | yellow |
| 161 | 18 | yellow |
| 162 | 23 | green |
| 163 | 22 | blue |
| 164 | 19 | yellow |
| 165 | 20 | yellow |
| 166 | 24 | green |
| 167 | 23 | blue |
| 168 | 24 | blue |
| 169 | 25 | blue |
| 170 | 25 | green |
| 171 | 26 | green |
| 172 | 26 | blue |
| 173 | 27 | blue |
| 174 | 28 | blue |
| 175 | 27 | green |
| 176 | 28 | green |
| 177 | 21 | yellow |
| 178 | 29 | blue |
| 179 | 30 | blue |
| 180 | 29 | green |
| 181 | 22 | yellow |
| 182 | 23 | yellow |
| 183 | 30 | green |
| 184 | 24 | yellow |
| 185 | 25 | yellow |
| 186 | 26 | yellow |
| 187 | 31 | green |
| 188 | 27 | yellow |
| 189 | 32 | green |
| 190 | 31 | blue |
| 191 | 28 | yellow |
| 192 | 29 | yellow |
| 193 | 30 | yellow |
| 194 | 33 | green |
| 195 | 34 | green |
| 196 | 31 | yellow |
| 197 | 32 | blue |
| 198 | 35 | green |
| 199 | 32 | yellow |
| 200 | 33 | blue |
| 201 | 33 | yellow |
| 202 | 36 | green |
| 203 | 34 | yellow |
| 204 | 37 | green |
| 205 | 34 | blue |
| 206 | 38 | green |
| 207 | 35 | yellow |
| 208 | 39 | green |
| 209 | 40 | green |
| 210 | 35 | blue |
| 211 | 36 | yellow |
| 212 | 37 | yellow |
| 213 | 38 | yellow |
| 214 | 41 | green |
| 215 | 36 | blue |
| 216 | 37 | blue |
| 217 | 42 | green |
| 218 | 38 | blue |
| 219 | 39 | yellow |
| 220 | 39 | blue |
| 221 | 43 | green |
| 222 | 40 | blue |
| 223 | 40 | yellow |
| 224 | 41 | blue |
| 225 | 44 | green |
| 226 | 45 | green |
| 227 | 42 | blue |
| 228 | 46 | green |
| 229 | 43 | blue |
| 230 | 44 | blue |
| 231 | 41 | yellow |
| 232 | 47 | green |
| 233 | 45 | blue |
| 234 | 46 | blue |
| 235 | 48 | green |
| 236 | 47 | blue |
| 237 | 49 | green |
| 238 | 42 | yellow |
| 239 | 43 | yellow |
| 240 | 44 | yellow |
| 241 | 48 | blue |
| 242 | 50 | green |
| 243 | 49 | blue |
| 244 | 45 | yellow |
| 245 | 46 | yellow |
| 246 | 47 | yellow |
| 247 | 51 | green |
| 248 | 48 | yellow |
| 249 | 49 | yellow |
| 250 | 50 | yellow |
| 251 | 52 | green |
| 252 | 53 | green |
| 253 | 51 | yellow |
| 254 | 52 | yellow |
| 255 | 50 | blue |
| 256 | 51 | blue |
| 257 | 52 | blue |
| 258 | 53 | blue |
| 259 | 53 | yellow |
Above is a layout of how we collected the samples for each treatment observation. We saved it in a csv file and used github to read the data into R for further analysis.
Hypothesis test
Ho: \(\mu_1 = \mu_2 = \mu_3\) - Null Hypothesis
Ha: At least 1 differs - Alternative Hypothesis
Boxplot of the experiment
The boxplot reveals that the variation between the red ball, green ball and yellow ball are equal.
Testing normality
The data looks normally distributed with little presence of outliers at the high extreme values of the distance The outliers might be due to excessive force that was applied to the launching process, the ball landing twice , and a misreading of landing position.
Analysis of variance
## Df Sum Sq Mean Sq F value Pr(>F)
## treatments 2 36210 18105 46.77 <2e-16 ***
## Residuals 156 60392 387
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the result fo is 0.783 with a corresponding p-value of 0.465 is significantly greater than \(\alpha\) = 0.05. Therefore we fail to reject Ho that the means are equal, and conclude that none of the means are different.
Conclusion
There seems to be nothing unusual about the plots except for the few outliers as the spread of the data looks constant across all treatment balls
Facotorial Design
Null and Alternative Hypotheses
Ho: \(\alpha_{i} = 0\) - Null Hypothesis
Ha: \(\alpha_{i} \ne 0\) - Alternative Hypothesis
Ho: \(\beta_{i} = 0\) - Null Hypothesis
Ha: \(\beta_{i} \ne 0\) - Alternative Hypothesis
Ho: \(\alpha \beta_{ij} = 0\) - Null Hypothesis
Ha: \(\alpha \beta_{ij} \ne 0\) - Alternative Hypothesis
Level of Significance
\(\alpha\) = 0.05
Model Equation
\(y_{ijk} = \mu + \alpha_{i} + \beta_j + \alpha \beta_{ij} + \epsilon_{ijk}\)
Proposed Layout with a Randomized Run Order
## plots r A B
## 1 101 1 1 2
## 2 102 2 1 2
## 3 103 1 2 3
## 4 104 1 2 1
## 5 105 1 1 1
## 6 106 2 2 1
## 7 107 1 1 3
## 8 108 1 2 2
## 9 109 3 2 1
## 10 110 2 1 3
## 11 111 3 1 2
## 12 112 2 2 3
## 13 113 2 2 2
## 14 114 3 2 2
## 15 115 3 2 3
## 16 116 2 1 1
## 17 117 3 1 1
## 18 118 3 1 3In the layout, factor A(Pin.Location) represents Pin Elevation and it has levels 1 and 2 for settings 1 and 3 respectively. factor B(Angle) represents the Release Angle with levels 1,2 and 3 for corresponding angles 110, 140 and 170 degrees. Number of replications is 3 which gives a total of 18 observations in the experiment
Collected Data on Proposed Layout
## Replication Pin.Location Angle Distance...Inches.
## 1 1 1 140 25
## 2 2 1 140 35
## 3 1 3 170 55
## 4 1 3 110 32
## 5 1 1 110 24
## 6 2 3 110 23
## 7 1 1 170 48
## 8 1 3 140 36
## 9 3 3 110 24
## 10 2 1 170 56
## 11 3 1 140 37
## 12 2 3 170 61
## 13 2 3 140 52
## 14 3 3 140 48
## 15 3 3 170 72
## 16 2 1 110 30
## 17 3 1 110 26
## 18 3 1 170 33Testing the Hypotheses
## Analysis of Variance Table
##
## Response: BungeeEx$Distance...Inches.
## Df Sum Sq Mean Sq F value Pr(>F)
## BungeeEx$Pin.Location 1 440.06 440.06 3.5616 0.1997555
## BungeeEx$Angle 2 2305.33 1152.67 19.4817 0.0001704 ***
## BungeeEx$Pin.Location:BungeeEx$Angle 2 247.11 123.56 2.0883 0.1666387
## Residual 12 710.00 59.17
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Firstly, we tested the interaction hypothesis that the pin location and the angle had an effect on the shooting distance. If we failed to reject the interaction null hypothesis, we tested the main effects the pin location and angle effects on the distance.
From the interaction result, interaction effects has fo value is 2.0883 with a corresponding p-value of 0.1666387 >0.05. Since 0.1666387 >0.05, we failed to reject the interaction null hypothesis that the interaction between pin location and the angle have an effect on the shooting distance.
The next section we removed the interaction effect and tested the main effects.
Model Equation
\(y_{ijk} = \mu + \alpha_{i} + \beta_j + \epsilon_{ijk}\)
model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location+BungeeEx$Angle)
gad(model)## Analysis of Variance Table
##
## Response: BungeeEx$Distance...Inches.
## Df Sum Sq Mean Sq F value Pr(>F)
## BungeeEx$Pin.Location 1 440.06 440.06 6.4368 0.023703 *
## BungeeEx$Angle 2 2305.33 1152.67 16.8605 0.000187 ***
## Residual 14 957.11 68.37
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the pin location result fo value is 6.4368 which corresponds to a p-value of 0.023703. The angle result fo value is 16.8605 which corresponds to a p-value of 0.000187
We concluded that the pin location and angle have an effect on the shooting distance of the ball
Pin.Location: 0.023703 <0.05
Angle: 0.000187 <0.05
ANOVA Test Plots and Interaction Plot
Conclusion
There seems to be nothing unusaual about the plots. the data seems to follow a straight line on the normal probability plot with 2 extreme outliers on the tail ends of the data distribution. Other than that, everything is fairly normal.
We concluded that the pin location and angle have an effect on the shooting distance of the ball.
2^4 Factorial Design Experiment
Data Collection Layout
For \(2^4\) factorial design, we used design.ab to generate one replication of a run order for our \(2^4\) factorial design
## plots r A B C D
## 1 101 1 1 1 2 1
## 2 102 1 1 1 2 2
## 3 103 1 1 2 1 2
## 4 104 1 2 2 2 1
## 5 105 1 2 1 2 1
## 6 106 1 2 2 2 2
## 7 107 1 1 2 2 2
## 8 108 1 1 2 2 1
## 9 109 1 1 2 1 1
## 10 110 1 2 2 1 2
## 11 111 1 2 1 1 2
## 12 112 1 2 1 2 2
## 13 113 1 2 2 1 1
## 14 114 1 1 1 1 2
## 15 115 1 1 1 1 1
## 16 116 1 2 1 1 1Experiment Data and Data Frame
For each of our 4 factors, we had two levels for each factors. They were classified as -1(low) and a +1(high). The different factor levels,and the assigned variables.
| Factor | Low Level(-1) | High Level(+1) | |
|---|---|---|---|
| A | Pin Location | Postion 1 | Postion 3 |
| B | Bungee Position | Position 2 | Position 3 |
| C | Release Angle | 140 degrees | 170 degrees |
| D | Ball Type | Yellow | Red |
Here is our data that we collected from the experiment.
## Pin_Elevation Bungee_Position Release_Angle Ball_Type response
## 1 -1 -1 1 -1 36
## 2 -1 -1 1 1 35
## 3 -1 1 -1 1 34
## 4 1 1 1 -1 60
## 5 1 -1 1 -1 68
## 6 1 1 1 1 60
## 7 -1 1 1 1 37
## 8 -1 1 1 -1 38
## 9 -1 1 -1 -1 33
## 10 1 1 -1 1 41
## 11 1 -1 -1 1 42
## 12 1 -1 1 1 52
## 13 1 1 -1 -1 51
## 14 -1 -1 -1 1 34
## 15 -1 -1 -1 -1 26
## 16 1 -1 -1 -1 47Null and Alternative Hypothesis Testing
Here are the Hypothesis tests that we used in the experiment. We started at the highest order hypothesis test, which was \(\alpha_i\)*\(\beta_j\) hypothesis test.
Ho: \(\alpha_{i} = 0\) - Null Hypothesis
Ha: \(\alpha_{i} \ne 0\) - Alternative Hypothesis
Ho: \(\beta_{j} = 0\) - Null Hypothesis
Ha: \(\beta_{j} \ne 0\) - Alternative Hypothesis
Ho: \(\alpha \beta_{ij} = 0\) - Null Hypothesis
Ha: \(\alpha \beta_{ij} \ne 0\) - Alternative Hypothesis
Half Normal Plot
##
## Significant effects (alpha=0.05, Lenth method):## [1] Pin_Elevation Release_Angle
From the plot, factors Pin Elevation and Release Angle are significant model terms.
Model Equation
\(y_{i} = \beta_{0} + \beta_{1}x_{i1} + \beta_{2}x_{i2} +\epsilon_{i}\)
\(Distance = {43.37} - 9.25x_{i1} + 4.875x_{i2}\)
ANOVA Model
After running the half normal plot , we concluded that Release Angle and Pin Elevation were significant factors. We run the ANOVA model with those factors and generated the following table.
## Df Sum Sq Mean Sq F value Pr(>F)
## Pin_Elevation 1 1369.0 1369.0 51.96 6.86e-06 ***
## Release_Angle 1 380.2 380.2 14.43 0.00221 **
## Residuals 13 342.5 26.3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\(y_{i} = \beta_{0} + \beta_{1}x_{i1} + \beta_{2}x_{i2} +\epsilon_{i}\)
\(Distance = {29.25} - 18.50x_{i1} + 9.75x_{i2}\)
These are model equations wth their respective coeffents.
Conclusion
From the result, values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case Pin Elevation and Release Angle are significant model terms.
Code
### Part 1
library(pwr)
pwr.anova.test(k=3,n=NULL,f=sqrt((.5)^2),sig.level=0.05,power=.75)
library(agricolae)
treatments<-c("green","yellow","blue")
design<-design.crd(trt=treatments,r=13,seed = 12345)
design$book
library(knitr)
F_levels <- cbind(z$plots,z$r,z$treatments)
kable(F_levels,caption = "Completely Randomized Design ", col.names = c("Plot","Replication","Color of Ball"))
z <- read.csv("https://raw.githubusercontent.com/Rusty1299/Projects/main/Part%202%20data%20redoe.csv")
z$treatments <- as.factor(z$treatments)
boxplot(z$distance~z$treatments, col= c("Red","Green","Yellow"), main = "Distance of each ball", xlab = "Treatment balls", ylab = "Distance in inches")
qqnorm(z$distance)
a <- aov(data = z , distance~treatments)
summary(a)
## Part 2
trts<-c(2,3)
design<-design.ab(trt=trts, r=3, design="crd",seed=878900)
design$book
BungeeEx<-read.csv("https://raw.githubusercontent.com/Rusty1299/Projects/main/Factorial%20Design%20Project.csv")
library(GAD)
BungeeEx$Pin.Location<-as.fixed(BungeeEx$Pin.Location)
BungeeEx$Angle<-as.random(BungeeEx$Angle)
model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location*BungeeEx$Angle)
gad(model)
model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location+BungeeEx$Angle)
gad(model)
interaction.plot(BungeeEx$Angle,BungeeEx$Pin.Location,BungeeEx$Distance...Inches., type = "l", col = 5:7 ,main ="Interraction Plot", ylab = "Distance", xlab = "Release Angles", trace.label = "Pin Elevation", lwd = 3, lty = 1)
plot(model)
boxplot(BungeeEx$Distance...Inches.~BungeeEx$Angle, col = 6:9:3, main = "Boxplot for Relaease Angle", xlab = "Release Angle", ylab = "Distance")
boxplot(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location, col = 2:4, main = "Boxplot for Pin Elevation", xlab = "Pin Elevation", ylab = "Distance")
## Part 3
library(agricolae)
#?design.ab
trts<-c(2,2,2,2)
design<-design.ab(trt=trts, r=1, design="crd",seed=878900)
design$book
library(knitr)
A <- c("Pin Location","Postion 1","Postion 3")
B <-c("Bungee Position" ,"Position 2", "Position 3")
C<-c("Release Angle", "140 degrees", "170 degrees")
D<-c("Ball Type", "Yellow", "Red")
F_levels <- rbind(A,B,C,D)
colnames(F_levels)<- c("Factor","Low Level(-1)","High Level(+1)")
kable(F_levels,caption = "Factors and Low and High Levels")
library(DoE.base)
Pin_Elevation<-c(-1,-1,-1,1,1,1,-1,-1,-1,1,1,1,1,-1,-1,1)
Bungee_Position<-c(-1,-1,1,1,-1,1,1,1,1,1,-1,-1,1,-1,-1,-1)
Release_Angle<-c(1,1,-1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1)
Ball_Type<-c(-1,1,1,-1,-1,1,1,-1,-1,1,1,1,-1,1,-1,-1)
response<-c(36,35,34,60,68,60,37,38,33,41,42,52,51,34,26,47)
dat<-data.frame(Pin_Elevation,Bungee_Position,Release_Angle,Ball_Type,response)
dat
model<-lm(response~Pin_Elevation*Bungee_Position*Release_Angle*Ball_Type, data = dat)
#summary(model)
coef(model)
halfnormal(model)
Pin_Elevation<-as.factor(Pin_Elevation)
Bungee_Position<-as.factor(Bungee_Position)
Release_Angle<-as.factor(Release_Angle)
Ball_Type<-as.factor(Ball_Type)
model1<-aov(response~Pin_Elevation+Release_Angle)
summary(model1)
coef(model1)