Project Final
Viviana, Dhruv, Gabriel
12/1/2021
This report presents the statistical analysis for the three project parts.
Project Part 1
First, in part 1, to perform an analysis of variance experiment, we collected data for three kinds of balls, keeping the Pin Elevation and Bungee Position both in the highest setting. In Part 1, We had the objective to determine the effect of the type of ball on the distance in which the ball is thrown. Three balls: Red, Yellow, and Mix were used to conduct the experiment.
The analysis for part 1 is illustrated below.
A) Determine how many samples should be collected to detect a mean difference with a medium effect (i.e. 50% of the standard deviation) and a pattern of maximum variability with a probability of 75%.
pwr.anova.test(k=3,n=NULL,f=((0.5*sqrt((3^2)-1))/(2*3)),sig.level = 0.05,power = 0.75)##
## Balanced one-way analysis of variance power calculation
##
## k = 3
## n = 52.55574
## f = 0.2357023
## sig.level = 0.05
## power = 0.75
##
## NOTE: n is number in each groupWe needed to collect 53 samples of each population. In total we need to collected 159 samples for the experiment.
B) Propose a layout using the number of samples from part (a) with randomized run order
trt<-c("yellow","red","mix")
experimentdesign<-design.crd(trt=trt,r= 53,seed=981273)
print(experimentdesign)## $parameters
## $parameters$design
## [1] "crd"
##
## $parameters$trt
## [1] "yellow" "red" "mix"
##
## $parameters$r
## [1] 53 53 53
##
## $parameters$serie
## [1] 2
##
## $parameters$seed
## [1] 981273
##
## $parameters$kinds
## [1] "Super-Duper"
##
## $parameters[[7]]
## [1] TRUE
##
##
## $book
## plots r trt
## 1 101 1 mix
## 2 102 2 mix
## 3 103 1 red
## 4 104 2 red
## 5 105 1 yellow
## 6 106 3 mix
## 7 107 3 red
## 8 108 2 yellow
## 9 109 3 yellow
## 10 110 4 mix
## 11 111 4 yellow
## 12 112 5 mix
## 13 113 6 mix
## 14 114 4 red
## 15 115 5 red
## 16 116 5 yellow
## 17 117 6 yellow
## 18 118 6 red
## 19 119 7 mix
## 20 120 7 red
## 21 121 8 mix
## 22 122 8 red
## 23 123 9 mix
## 24 124 10 mix
## 25 125 9 red
## 26 126 7 yellow
## 27 127 10 red
## 28 128 11 red
## 29 129 8 yellow
## 30 130 9 yellow
## 31 131 12 red
## 32 132 11 mix
## 33 133 10 yellow
## 34 134 12 mix
## 35 135 13 red
## 36 136 14 red
## 37 137 15 red
## 38 138 13 mix
## 39 139 11 yellow
## 40 140 16 red
## 41 141 14 mix
## 42 142 12 yellow
## 43 143 13 yellow
## 44 144 15 mix
## 45 145 16 mix
## 46 146 17 red
## 47 147 14 yellow
## 48 148 18 red
## 49 149 15 yellow
## 50 150 19 red
## 51 151 17 mix
## 52 152 16 yellow
## 53 153 20 red
## 54 154 21 red
## 55 155 22 red
## 56 156 18 mix
## 57 157 23 red
## 58 158 24 red
## 59 159 19 mix
## 60 160 20 mix
## 61 161 17 yellow
## 62 162 21 mix
## 63 163 25 red
## 64 164 18 yellow
## 65 165 22 mix
## 66 166 26 red
## 67 167 19 yellow
## 68 168 23 mix
## 69 169 20 yellow
## 70 170 27 red
## 71 171 24 mix
## 72 172 21 yellow
## 73 173 22 yellow
## 74 174 28 red
## 75 175 25 mix
## 76 176 26 mix
## 77 177 23 yellow
## 78 178 24 yellow
## 79 179 25 yellow
## 80 180 29 red
## 81 181 30 red
## 82 182 26 yellow
## 83 183 31 red
## 84 184 27 mix
## 85 185 32 red
## 86 186 27 yellow
## 87 187 28 yellow
## 88 188 29 yellow
## 89 189 28 mix
## 90 190 29 mix
## 91 191 30 mix
## 92 192 31 mix
## 93 193 30 yellow
## 94 194 31 yellow
## 95 195 32 yellow
## 96 196 33 yellow
## 97 197 32 mix
## 98 198 33 red
## 99 199 33 mix
## 100 200 34 yellow
## 101 201 34 mix
## 102 202 35 yellow
## 103 203 36 yellow
## 104 204 37 yellow
## 105 205 34 red
## 106 206 35 red
## 107 207 36 red
## 108 208 38 yellow
## 109 209 35 mix
## 110 210 39 yellow
## 111 211 40 yellow
## 112 212 36 mix
## 113 213 41 yellow
## 114 214 37 red
## 115 215 37 mix
## 116 216 38 red
## 117 217 42 yellow
## 118 218 38 mix
## 119 219 39 red
## 120 220 39 mix
## 121 221 40 mix
## 122 222 41 mix
## 123 223 43 yellow
## 124 224 42 mix
## 125 225 43 mix
## 126 226 44 mix
## 127 227 44 yellow
## 128 228 40 red
## 129 229 45 yellow
## 130 230 45 mix
## 131 231 41 red
## 132 232 42 red
## 133 233 43 red
## 134 234 44 red
## 135 235 46 mix
## 136 236 46 yellow
## 137 237 47 yellow
## 138 238 47 mix
## 139 239 48 mix
## 140 240 49 mix
## 141 241 50 mix
## 142 242 51 mix
## 143 243 48 yellow
## 144 244 45 red
## 145 245 46 red
## 146 246 47 red
## 147 247 48 red
## 148 248 49 red
## 149 249 49 yellow
## 150 250 52 mix
## 151 251 50 yellow
## 152 252 50 red
## 153 253 51 yellow
## 154 254 53 mix
## 155 255 51 red
## 156 256 52 red
## 157 257 53 red
## 158 258 52 yellow
## 159 259 53 yellowThe sequence above represents the order in which we have to collect the data.
C) Collect data and record observations on layout proposed in part (b)
Data was collected on class.
D)Perform hypothesis test and check residuals.Be sure to comment and take corrective action if necessary.
Null hypothesis: \(H_0: \mu_{1}=\mu_{2}=\mu_{3}=\mu\) Alternative hypothesis: \(H_1:\) at least one \(\mu_{i}\) differs
Nº 1 represents mix ball, Nº 2 represents red ball and Nº 3 represents yellow ball.
obs<-c(71,85,152,104,164,147,161,70,83,168,155,78,87,102,98,130,88,97,157,75,89,78,97,110,139,128,133,94,78,123,93,81,103,138,78,75,139,111,145,75,88,98,149,120,138,91,68,93,120,94,110,87,90,66,113,84,64,80,74,92,82,72,72,88,82,88,75,89,63,88,90,101,83,72,60,98,73,82,70,90,65,95,56,76,78,82,98,74,77,65,63,89,75,90,104,89,65,63,87,76,77,87,90,93,65,84,110,98,107,132,86,125,133,59,113,90,97,86,124,88,130,87,60,98,103,89,78,89,109,120,78,91,124,90,99,102,84,93,74,105,103,87,92,64,98,78,72,99,104,97,67,88,83,81,92,104,82,91,75)
ball<-c(rep(1,53),rep(2,53),rep(3,53))
dat<-data.frame(obs,ball)
dat$ball<-as.factor(dat$ball)
model<-aov(dat$obs~dat$ball, data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## dat$ball 2 20418 10209 23.45 1.25e-09 ***
## Residuals 156 67917 435
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Analysis
The P-value of the test is 1.25e-09 which is less than \(\alpha\) of 0.05. Hence, we reject the Null hypothesis. \(H_1:\) at least one \(\mu_{i}\) differs.
Cheking the residuals
plot(model)
boxplot(dat$obs~dat$ball, data = dat, main="Boxplot of observations")
Analysis
Even if from the Normal plot of residuals we draw that it passes the fat pencil test and it is roughly normally distribute, the plot of residuals versus predicted has a strong outward-opening funnel shape, which indicates the variance of the original observations is not constant. From the boxplot We can see difference in percentile ranges indicating difference in means. Is suggested to use Box-Cox power transformation to stabilize variance.
Using Box-Cox power transformation to stabilize variance.
boxcox(dat$obs~dat$ball, data = dat)
lambda = 0.5
dat$obs<-dat$obs^lambdaVerifying data after transformation
model1<-aov(dat$obs~dat$ball, data = dat)
summary(model1)## Df Sum Sq Mean Sq F value Pr(>F)
## dat$ball 2 50.0 25.00 23.59 1.12e-09 ***
## Residuals 156 165.3 1.06
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1boxplot(dat$obs~dat$ball, data = dat, main="Boxplot of transformed observations")
Analysis
After plotting transformed data, we can see that we corrected the data in “mix”, and “yellow” but the effect on “red” is limited. Overall, we fail to correct the variance and the p-value of the test is 1.12e-09 which is less \(/alpha\) of 0.05. Hence, we reject the Null hypothesis even for the transformed data as well.
E) If the null hypothesis is rejected, investigate pairwise comparisons.
investigating pairwise comparisons
TukeyHSD(model1)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = dat$obs ~ dat$ball, data = dat)
##
## $`dat$ball`
## diff lwr upr p adj
## 2-1 -1.3718948 -1.8450731 -0.8987165 0.0000000
## 3-1 -0.6260351 -1.0992134 -0.1528569 0.0058722
## 3-2 0.7458597 0.2726814 1.2190380 0.0007786plot(TukeyHSD(model1))
Analysis
The Tukey Test indicates there is difference in the pairs of treatments: red-mix, yellow-mix, red-yellow.
F) State all findings and make recommendation.
From the results of the test, we can say that every ball has different mean of distance and the all three pairs differs in mean. It can be recommended that if all the balls have same weight and size than maybe the means of distances can be same.Once the different materials and density of each ball influence the distance that each ball reaches.
Another highlight is the variability in the data collection. For example, in the middle of the experiment, the rope broke, and we needed to fix it and re-start the data collection. Maybe this incident contributed to increasing the non-uniformity of measures for each ball. Moreover, there is the variability of the operator . We could see the operator’s tidiness once we needed to sit on the floor in an uncomfortable position to perform the experiment. Finally, the time was another factor that may have influenced, since there were many groups to collect data during the class period, so, had a concern about collecting the data fast.
This way, to guarantee the accuracy of the data to set up a better-controlled environment to collect data might be a suggestion.
Project Part 2
According Montgomery (2013),factorial designs are efficient in the analysis of experiments that involve the study of effects of two or more factors. To perform the project part 2 we based on the concepts of Factorial Design for Mix effects.
In Part 2 we had the objective to analyze the effect of Pin Elevation and Release Angle on distance in which a red ball is thrown when the Bungee Position is fixed at the second position.In addition, he settings of one and three of Pin Elevation was analyzed as a fixed effect, as well as settings of the Release Angle corresponding to 110, 140, and 170 degrees as a random effect.The design was replicated three times.
The analysis for part 2 is illustrated below.
A) State model equation with the null and alternative hypotheses to be tested. In addition. state the level of significance that will be used in your analysis.
Model equation: \(Y_{ijk}=\mu+\alpha_{i}+\beta_{j}+\alpha\beta_{ij}+e_{ijk}\)
The level of significance for this analysis is \(\alpha=0,05\)
B) Propose a layout with a randomized run order.
library(agricolae)
trts2<-c(2,3)
design.ab(trt=trts2, r=3, design="crd", randomization=TRUE, seed=156812)## $parameters
## $parameters$design
## [1] "factorial"
##
## $parameters$trt
## [1] "1 1" "1 2" "1 3" "2 1" "2 2" "2 3"
##
## $parameters$r
## [1] 3 3 3 3 3 3
##
## $parameters$serie
## [1] 2
##
## $parameters$seed
## [1] 156812
##
## $parameters$kinds
## [1] "Super-Duper"
##
## $parameters[[7]]
## [1] TRUE
##
## $parameters$applied
## [1] "crd"
##
##
## $book
## plots r A B
## 1 101 1 2 3
## 2 102 1 1 3
## 3 103 1 2 2
## 4 104 1 2 1
## 5 105 2 2 3
## 6 106 1 1 2
## 7 107 2 1 3
## 8 108 2 1 2
## 9 109 1 1 1
## 10 110 2 1 1
## 11 111 2 2 2
## 12 112 2 2 1
## 13 113 3 1 1
## 14 114 3 2 2
## 15 115 3 2 3
## 16 116 3 1 3
## 17 117 3 2 1
## 18 118 3 1 2The layout for the design proposed is showed above.
C) Collect data and record observations on the layout proposed in part (A).
Data was collected during the class and the data are showed below:
Angle<-rep(seq(1,3), 3)
Pin<-c(rep(1,9), rep(2,9))
Dist<-c(27,42,49,
27,38,51,
26,40,50,
33,49,79,
29,46,63,
29,43,67)
Dat2<-data.frame(Pin,Angle,Dist)D) Test the hypotheses and state conclusions, determining those effects that are significant. Show any plots that might be useful/necessary to show your findings. You may also show residual plots and make appropriate comments, but do not transform the data (i.e. use the raw data regardless of normality and variance constancy).
Hypothesis for Mixed effects.
Interaction Pin (Fixxed) and Angle (Random) Null hypothesis: \(H_0: \sigma^2_{\alpha\beta}=0\) Alternative hypothesis: \(H_1: \sigma^2_{\alpha\beta}!0\)
Factor Pin (Fixxed) Null hypothesis: \(H_0: \alpha_{i}=0\) Alternative hypothesis: \(H_1: \alpha_{i}!0\)
Factor Angle (Random) Null hypothesis: \(H_0: \sigma^2_{\beta}=0\) Alternative hypothesis: \(H_1: \sigma^2_{\beta}!0\)
Testing the hypothesis
library(GAD)## Loading required package: matrixStats##
## Attaching package: 'matrixStats'## The following object is masked from 'package:dplyr':
##
## count## Loading required package: R.methodsS3## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.Dat2$Pin<-as.fixed(Dat2$Pin)
Dat2$Angle<-as.random(Dat2$Angle)
Model2<-aov(Dist~Angle+Pin+Angle*Pin, data = Dat2)
GAD::gad(Model2)## Analysis of Variance Table
##
## Response: Dist
## Df Sum Sq Mean Sq F value Pr(>F)
## Angle 2 2950.78 1475.39 99.4644 3.391e-08 ***
## Pin 1 430.22 430.22 3.8394 0.189138
## Angle:Pin 2 224.11 112.06 7.5543 0.007524 **
## Residual 12 178.00 14.83
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Analysis
With the p-value=0.007524 < than \(\alpha=0,05\), we reject the Null hypothesis (\(H_0: \sigma^2_{\alpha\beta}=0\)) and conclude the interaction between the factors Pin and Angle are significant and affect the distance in which a red ball is thrown.
With the p-value=0.189138 greater than \(\alpha=0,05\),we fail to reject the null hypothesis (\(H_0: \alpha_{i}=0\)) and conclude the factor Pin by himself has not a significant effect in the distance in which a red ball is thrown.
With the p-value=3.391e-08 very less than \(\alpha=0,05\), we reject the Null Hypothesis (\(H_0: \sigma^2_{\beta}=0\)) and conclude the Factor Angle has a significant effect in the distance in which a red ball is thrown.
verifying plots
plot(Model2)
Graph Analysis
From the Residual vs fitted plot, we can se a outward-opening funnel shape, which indicates the variance of the observations is not constant. From the Normal Q-Q plot, we can say that all the points approximately lies in the straight line, so the data is normally distributed.
In the all plots we also can see the presence of outliers. These outliers may represent the variabilities in the experiment, for example, the operator. In this experiment the operator needed to sit on the floor in an uncomfortable position to perform the experiment. Another possible variability might be the time, since there were many groups to collect data during the class period, so, we had a concern about collecting the data fast.
Interaction Plot
Dat1<-data.frame(Pin,Angle,Dist)
Dat1$Pin<-as.factor(Dat1$Pin)
Dat1$Angle<-as.factor(Dat1$Angle)
interaction.plot(x.factor = Dat1$Angle,
trace.factor = Dat1$Pin,
response = Dat1$Dist,
fun = mean,
type="b",
col=c("black","red","green"),
pch=c(19, 17, 15),
fixed=TRUE,
leg.bty = "o")
Graph Analysis
However in this plot we cannot visualize the crossing of lines, In this interaction plot, we can se the lines are not parallel, indicating the interaction between factors. Furthermore,this interaction effect indicates that the relationship between pin position and distance depends on the value of the angle. This graph agreed with the results of the analysis of variance indicate that the interaction between Pin and Angle is significant.
Project Part 3
In part 3, we generated a 2^4 factorial design with the low and the high levels of the factors given for Pin Elevation, Bungee Position, Release Angle, and Ball Type.
The analysis for part 2 is illustrated below.
a) Propose a data collection layout with a randomized run order
library(agricolae)
trts3<-c(2,2,2,2)
design.ab(trt=trts3, r=1, design="crd",seed=158632)## $parameters
## $parameters$design
## [1] "factorial"
##
## $parameters$trt
## [1] "1 1 1 1" "1 1 1 2" "1 1 2 1" "1 1 2 2" "1 2 1 1" "1 2 1 2" "1 2 2 1"
## [8] "1 2 2 2" "2 1 1 1" "2 1 1 2" "2 1 2 1" "2 1 2 2" "2 2 1 1" "2 2 1 2"
## [15] "2 2 2 1" "2 2 2 2"
##
## $parameters$r
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## $parameters$serie
## [1] 2
##
## $parameters$seed
## [1] 158632
##
## $parameters$kinds
## [1] "Super-Duper"
##
## $parameters[[7]]
## [1] TRUE
##
## $parameters$applied
## [1] "crd"
##
##
## $book
## plots r A B C D
## 1 101 1 1 1 1 1
## 2 102 1 1 1 1 2
## 3 103 1 2 1 2 1
## 4 104 1 2 2 2 2
## 5 105 1 1 2 2 1
## 6 106 1 1 2 2 2
## 7 107 1 1 2 1 1
## 8 108 1 1 1 2 1
## 9 109 1 2 1 1 2
## 10 110 1 2 2 2 1
## 11 111 1 2 1 1 1
## 12 112 1 2 2 1 1
## 13 113 1 2 2 1 2
## 14 114 1 1 2 1 2
## 15 115 1 2 1 2 2
## 16 116 1 1 1 2 2The proposed design is showed above.
b) Collect data and record observations
The data was collected during the class.
c) State model equation and determine what factors/interactions appear to be significant (show any plots that were used in making this determination)
The Model equation for a 2^4 Factorial design is showed below.
\(Y_{ijklm}=\mu+\alpha_{i}+\beta_{j}+\gamma_{k}+\delta_{l}+\alpha\beta_{ij}+\alpha\gamma_{ik}+\alpha\delta_{il}+\beta\gamma_{jk}+\beta\delta_{jl}+\gamma\delta_{kl}+\alpha\beta\gamma_{ijk}+\alpha\beta\delta_{ijl}+\alpha\gamma\delta_{ikl}+\beta\gamma\delta_{jkl}+\alpha\beta\gamma\delta_{ijkl}+\epsilon_{ijklm}\)
Verifying the Halfnormal plot
library(GAD)
library(DoE.base)## Loading required package: grid## Loading required package: conf.design## Registered S3 method overwritten by 'DoE.base':
## method from
## factorize.factor conf.design##
## Attaching package: 'DoE.base'## The following objects are masked from 'package:stats':
##
## aov, lm## The following object is masked from 'package:graphics':
##
## plot.design## The following object is masked from 'package:base':
##
## lengthsA<-c(-1,-1,1,1,-1,-1,-1,-1,1,1,1,1,1,-1,1,-1)
B<-c(-1,-1,-1,1,1,1,1,-1,-1,1,-1,1,1,1,-1,-1)
C<-c(-1,-1,1,1,1,1,-1,1,-1,1,-1,-1,-1,-1,1,1)
D<-c(-1,1,-1,1,-1,1,-1,-1,1,-1,-1,-1,1,1,1,1)
obs3<-c(36,30,75,58,48,41,37,42,43,103,48,40,43,37,54,35)
model3<-lm(obs3~A*B*C*D)
coef(model3)## (Intercept) A B C D A:B
## 48.125 9.875 2.750 8.875 -5.500 0.250
## A:C B:C A:D B:D C:D A:B:C
## 5.625 2.750 -3.000 -0.625 -4.500 2.250
## A:B:D A:C:D B:C:D A:B:C:D
## -1.375 -3.500 -2.375 -1.625halfnormal(model3)##
## Significant effects (alpha=0.05, Lenth method):## [1] A
Analysis From the halfnormal plot we can see that only Factor A (Pin Elevation) appears to be significant.
d) After using insignificant factors/interactions to create an error term, perform ANOVA to determine a final model equation using an alpha = 0.05
A<-as.fixed(A)
model3<-aov(obs3~A)
gad(model3)## Analysis of Variance Table
##
## Response: obs3
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 1560.2 1560.25 6.4104 0.02394 *
## Residual 14 3407.5 243.39
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Analysis The analysis of variance provided the p-value=0.02394 less than alpha=0.05. We conclude that only factor A (Pin Elevation) has a significant effect on the distance in which a ball is thrown.
The Final model equation is: \(Y_{im}=\mu+\alpha_{i}+\epsilon_{im}\)
All Code
#Allcode Part1
knitr::opts_chunk$set(echo = TRUE)
library(pwr)
library(agricolae)
library(dplyr)
library(MASS)
pwr.anova.test(k=3,n=NULL,f=((0.5*sqrt((3^2)-1))/(2*3)),sig.level = 0.05,power = 0.75)
trt<-c("yellow","red","mix")
experimentdesign<-design.crd(trt=trt,r= 53,seed=981273)
print(experimentdesign)
obs<-c(71,85,152,104,164,147,161,70,83,168,155,78,87,102,98,130,88,97,157,75,89,78,97,110,139,128,133,94,78,123,93,81,103,138,78,75,139,111,145,75,88,98,149,120,138,91,68,93,120,94,110,87,90,66,113,84,64,80,74,92,82,72,72,88,82,88,75,89,63,88,90,101,83,72,60,98,73,82,70,90,65,95,56,76,78,82,98,74,77,65,63,89,75,90,104,89,65,63,87,76,77,87,90,93,65,84,110,98,107,132,86,125,133,59,113,90,97,86,124,88,130,87,60,98,103,89,78,89,109,120,78,91,124,90,99,102,84,93,74,105,103,87,92,64,98,78,72,99,104,97,67,88,83,81,92,104,82,91,75)
ball<-c(rep(1,53),rep(2,53),rep(3,53))
dat<-data.frame(obs,ball)
dat$ball<-as.factor(dat$ball)
model<-aov(dat$obs~dat$ball, data = dat)
summary(model)
plot(model)
boxplot(dat$obs~dat$ball, data = dat, main="Boxplot of observations")
TukeyHSD(model1)
plot(TukeyHSD(model1))
boxcox(dat$obs~dat$ball, data = dat)
lambda = 0.5
dat$obs<-dat$obs^lambda
model1<-aov(dat$obs~dat$ball, data = dat)
summary(model1)
boxplot(dat$obs~dat$ball, data = dat, main="Boxplot of transformed observations")
TukeyHSD(model1)
plot(TukeyHSD(model1))
#Allcode Part2
library(agricolae)
trts2<-c(2,3)
design.ab(trt=trts2, r=3, design="crd", randomization=TRUE, seed=156812)
Angle<-rep(seq(1,3), 3)
Pin<-c(rep(1,9), rep(2,9))
Dist<-c(27,42,49,
27,38,51,
26,40,50,
33,49,79,
29,46,63,
29,43,67)
Dat2<-data.frame(Pin,Angle,Dist)
library(GAD)
Dat2$Pin<-as.fixed(Dat2$Pin)
Dat2$Angle<-as.random(Dat2$Angle)
Model2<-aov(Dist~Angle+Pin+Angle*Pin, data = Dat2)
GAD::gad(Model2)
plot(Model2)
Dat1<-data.frame(Pin,Angle,Dist)
Dat1$Pin<-as.factor(Dat1$Pin)
Dat1$Angle<-as.factor(Dat1$Angle)
interaction.plot(x.factor = Dat1$Angle,
trace.factor = Dat1$Pin,
response = Dat1$Dist,
fun = mean,
type="b",
col=c("black","red","green"),
pch=c(19, 17, 15),
fixed=TRUE,
leg.bty = "o")
#Allcode Part3
library(agricolae)
trts3<-c(2,2,2,2)
design.ab(trt=trts3, r=1, design="crd",seed=158632)
library(GAD)
library(DoE.base)
A<-c(-1,-1,1,1,-1,-1,-1,-1,1,1,1,1,1,-1,1,-1)
B<-c(-1,-1,-1,1,1,1,1,-1,-1,1,-1,1,1,1,-1,-1)
C<-c(-1,-1,1,1,1,1,-1,1,-1,1,-1,-1,-1,-1,1,1)
D<-c(-1,1,-1,1,-1,1,-1,-1,1,-1,-1,-1,1,1,1,1)
obs3<-c(36,30,75,58,48,41,37,42,43,103,48,40,43,37,54,35)
model3<-lm(obs3~A*B*C*D)
coef(model3)
halfnormal(model3)
A<-as.fixed(A)
model3<-aov(obs3~A)
gad(model3)