Project Part 3
Ben Herndon, Gabriel Garze, Jeffrey Goens, Scott Michaelis
11/21/2021
Perform a designed experiment to determine the effect of the available factors of Fire Angle, Bungee Position, Release Angle, Pin Elevation, and Cup Elevation on distance in which a ball is thrown. Design this experiment as a single replicate of a 2^5 factorial design with the low and high level of the factors being as follows
image: 
Part A
Propose a data collection layout with a randomized run order
trts <- c(2,2,2,2,2)
reps <- 1
seedNum <- 1234567
experiment <- design.ab(trt = trts, r=reps,design="crd",seed = seedNum)
experiment$book## plots r A B C D E
## 1 101 1 1 1 2 1 1
## 2 102 1 1 2 1 1 1
## 3 103 1 2 2 1 1 1
## 4 104 1 2 1 2 2 1
## 5 105 1 2 2 2 1 2
## 6 106 1 2 2 1 1 2
## 7 107 1 2 1 1 1 1
## 8 108 1 1 2 2 2 1
## 9 109 1 1 2 1 1 2
## 10 110 1 1 1 2 1 2
## 11 111 1 1 1 1 1 1
## 12 112 1 2 1 1 1 2
## 13 113 1 2 2 2 2 2
## 14 114 1 2 1 1 2 2
## 15 115 1 1 1 1 2 2
## 16 116 1 2 1 2 2 2
## 17 117 1 2 1 2 1 1
## 18 118 1 1 2 1 2 1
## 19 119 1 1 1 1 1 2
## 20 120 1 2 1 1 2 1
## 21 121 1 2 2 2 2 1
## 22 122 1 2 2 1 2 1
## 23 123 1 2 2 1 2 2
## 24 124 1 1 1 2 2 1
## 25 125 1 1 2 1 2 2
## 26 126 1 1 2 2 1 1
## 27 127 1 1 1 1 2 1
## 28 128 1 2 1 2 1 2
## 29 129 1 1 1 2 2 2
## 30 130 1 2 2 2 1 1
## 31 131 1 1 2 2 1 2
## 32 132 1 1 2 2 2 2Part B
Collect the data and record observations
p3data## ï..Fire_Ang Bungee Release Pin_El Cup_El distance
## 1 -1 -1 1 -1 -1 123.0
## 2 -1 1 -1 -1 -1 101.0
## 3 1 1 -1 -1 -1 112.0
## 4 1 -1 1 1 -1 232.0
## 5 1 1 1 -1 1 261.0
## 6 1 1 -1 -1 1 148.5
## 7 1 -1 -1 -1 -1 100.0
## 8 -1 1 1 1 -1 151.0
## 9 -1 1 -1 -1 1 126.5
## 10 -1 -1 1 -1 1 155.0
## 11 -1 -1 -1 -1 -1 96.0
## 12 1 -1 -1 -1 1 138.0
## 13 1 1 1 1 1 342.0
## 14 1 -1 -1 1 1 173.5
## 15 -1 -1 -1 1 1 140.5
## 16 1 -1 1 1 1 310.0
## 17 1 -1 1 -1 -1 180.5
## 18 -1 1 -1 1 -1 118.0
## 19 -1 -1 -1 -1 1 120.5
## 20 1 -1 -1 1 -1 130.0
## 21 1 1 1 1 -1 261.0
## 22 1 1 -1 1 -1 142.0
## 23 1 1 -1 1 1 196.0
## 24 -1 -1 1 1 -1 143.5
## 25 -1 1 -1 1 1 152.5
## 26 -1 1 1 -1 -1 127.0
## 27 -1 -1 -1 1 -1 112.0
## 28 1 -1 1 -1 1 233.0
## 29 -1 -1 1 1 1 178.0
## 30 1 1 1 -1 -1 195.0
## 31 -1 1 1 -1 1 173.0
## 32 -1 1 1 1 1 196.0Part C
State model equation and determine what factors and interactions appear to be significant
distance<-(p3data$distance)
A<-as.factor(p3data$ï..Fire_Ang)
B<-as.factor(p3data$Bungee)
C<-as.factor(p3data$Release)
D<-as.factor(p3data$Pin_El)
E<-as.factor(p3data$Cup_El)
p3m <- data.frame(A,B,C,D,E,distance)
p3mod<-lm(distance~A*B*C*D*E,data=p3m)
halfnormal(p3mod,ME.partial = TRUE)##
## Creation of A## Projected out: (Intercept)##
## Creation of B## Projected out: (Intercept),A##
## Creation of C## Projected out: (Intercept),A,B##
## Creation of D## Projected out: (Intercept),A,B,C##
## Creation of E## Projected out: (Intercept),A,B,C,D##
## Creation of AB## Projected out: (Intercept),A,B,C,D,E##
## Creation of AC## Projected out: (Intercept),A,B,C,D,E,AB##
## Creation of BC## Projected out: (Intercept),A,B,C,D,E,AB,AC##
## Creation of AD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC##
## Creation of BD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD##
## Creation of CD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD##
## Creation of AE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD##
## Creation of BE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE##
## Creation of CE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE##
## Creation of DE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE##
## Creation of ABC## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE##
## Creation of ABD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC##
## Creation of ACD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD##
## Creation of BCD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD##
## Creation of ABE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD##
## Creation of ACE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE##
## Creation of BCE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE##
## Creation of ADE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE##
## Creation of BDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE##
## Creation of CDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE##
## Creation of ABCD## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE##
## Creation of ABCE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE,ABCD##
## Creation of ABDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE,ABCD,ABCE##
## Creation of ACDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE,ABCD,ABCE,ABDE##
## Creation of BCDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE,ABCD,ABCE,ABDE,ACDE##
## Creation of ABCDE## Projected out: (Intercept),A,B,C,D,E,AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,ABC,ABD,ACD,BCD,ABE,ACE,BCE,ADE,BDE,CDE,ABCD,ABCE,ABDE,ACDE,BCDE##
## Significant effects (alpha=0.05, Lenth method):## [1] C1 A1 E1 A1:C1 D1 A1:D1 B1 A1:E1
##
## [9] C1:E1 C1:D1 A1:C1:D1
coef(p3mod)## (Intercept) A1 B1 C1 D1
## 96.0 4.0 5.0 27.0 16.0
## E1 A1:B1 A1:C1 B1:C1 A1:D1
## 24.5 7.0 53.5 -1.0 14.0
## B1:D1 C1:D1 A1:E1 B1:E1 C1:E1
## 1.0 4.5 13.5 1.0 7.5
## D1:E1 A1:B1:C1 A1:B1:D1 A1:C1:D1 B1:C1:D1
## 4.0 3.5 -1.0 17.0 2.5
## A1:B1:E1 A1:C1:E1 B1:C1:E1 A1:D1:E1 B1:D1:E1
## -2.5 7.0 13.0 1.5 5.0
## C1:D1:E1 A1:B1:C1:D1 A1:B1:C1:E1 A1:B1:D1:E1 A1:C1:D1:E1
## -1.5 12.0 2.0 7.0 21.5
## B1:C1:D1:E1 A1:B1:C1:D1:E1
## -8.5 -14.0The following effects appear to be significant: A, B, C, D, E (All of them)
The following interactions appear to be significant: A:C, A:D, A:E, C:E, C:D and A:C:D
Based on these estimates,
Initial Model Equation:
\[y= A+B+C+D+E+A:C+A:D+A:E+C:E+C:D+A:C:D\]
Initial Regression Function:
\[ y=96.0+4.0A+5.0B+27.0C+16.0D+24.5E+53.5A:C+14.0A:D+13.5A:E+7.5C:E+4.5C:D+17.0A:C:D \]
Part D
After using insignificant factors/interactions to create an error term, perform ANOVA to determine a final model equation using an alpha = 0.05
The ANOVA will now be conducted on the initial model equation and all insignificant factors will be eliminated, if any remain.
p3mod2 = lm(distance~A+B+C+D+E+A*C+A*D+A*D+A*E+C*E+C*D+A*C*D,data=p3m)
anova(p3mod2)## Analysis of Variance Table
##
## Response: distance
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 27671 27671 544.677 5.564e-16 ***
## B 1 1755 1755 34.551 9.477e-06 ***
## C 1 41616 41616 819.165 < 2.2e-16 ***
## D 1 10805 10805 212.674 4.041e-12 ***
## E 1 16200 16200 318.878 9.286e-14 ***
## A:C 1 11063 11063 217.768 3.251e-12 ***
## A:D 1 1938 1938 38.138 4.936e-06 ***
## A:E 1 1001 1001 19.709 0.0002519 ***
## C:E 1 703 703 13.840 0.0013520 **
## C:D 1 648 648 12.755 0.0019110 **
## A:C:D 1 458 458 9.006 0.0070601 **
## Residuals 20 1016 51
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coef(p3mod2)## (Intercept) A1 B1 C1 D1 E1
## 91.3750 2.4375 14.8125 24.1250 19.7500 24.4375
## A1:C1 A1:D1 A1:E1 C1:E1 C1:D1 A1:C1:D1
## 59.2500 16.0000 22.3750 18.7500 2.8750 30.2500All factors in the model are significant at an \(\alpha = 0.05\) level.
The model will be checked for adequacy.
plot(p3mod2)
Measurements 4, 28 and 13 residuals are a little excessive. The data points were checked and were collected correctly. The model looks pretty linear and nothing too alarming in the residuals.
Final Model Equation:
\[y= A+B+C+D+E+A:C+A:D+A:E+C:E+C:D+A:C:D\] Final Regression Function:
\[y= 91.4+2.4A+14.8B+24.1C+19.8D+24.4E+59.3A:C+16.0A:D+22.3A:E+18.8C:E+2.9C:D+30.3A:C:D\]
All Code Used
Below you will find a block containing all of the code used to generate this document.
library(agricolae)
library(DoE.base)
#Part A
trts <- c(2,2,2,2,2)
reps <- 1
seedNum <- 1234567
experiment <- design.ab(trt = trts, r=reps,design="crd",seed = seedNum)
experiment$book
#Part B
p3data<-read.csv("C:/Users/.../Part 3 csv.csv")
p3data
#Part C
distance<-(p3data$distance)
A<-as.factor(p3data$ï..Fire_Ang)
B<-as.factor(p3data$Bungee)
C<-as.factor(p3data$Release)
D<-as.factor(p3data$Pin_El)
E<-as.factor(p3data$Cup_El)
p3m <- data.frame(A,B,C,D,E,distance)
p3mod<-lm(distance~A*B*C*D*E,data=p3m)
halfnormal(p3mod,ME.partial = TRUE)
coef(p3mod)
#Part D
p3mod2 = lm(distance~A+B+C+D+E+A*C+A*D+A*D+A*E+C*E+C*D+A*C*D,data=p3m)
anova(p3mod2)
coef(p3mod2)
plot(p3mod2)