Team 4 Statapult
Final Project IE 5342 - Fall 2021
F. Gersdorff, H. Swerdloff, J. Rosila Mares
Introduction
The following report captures the learnings by Team 4 over our semester in IE5342 with Dr. Matis as our professor. The report is separated into 3 parts which chronologically represent our journey through the course. The report starts with the statistical underpinnings of design of experiments through the execution of designed experiments and culminates with a final model equation explaining the key factors that drive Statapult launch distances.
It was our pleasure to take this journey and it is our hope that the readers of this report enjoy the story as well. ~ Team 4
Part 1
The Siege Setup
We performed a designed experiment on the effect of Release Angle on the distance in which a ball is thrown (measured at its furthest distance). Specifically, we studied whether the settings of 175°, 180°, and 185° significantly differ in their mean distance thrown. Since pulling the lever back takes additional work, we investigated whether this made a significant difference on the mean distance thrown. The other factors were set to constant values of the following:
Fire Angle = 90°
Bungee Position = 200mm
Pin Elevation = 200mm
Cup Elevation = 300mm
To test this hypothesis, we used a completely randomized design with an \(\alpha\) = 0.05.
Sample Size
To save resources during the preparation for the siege, we determined how many samples should be collected to detect a mean difference with a medium effect (i.e. 50% of the standard deviation) with a probability of 75%. Figure 1 below shows the Balanced One-way Analysis of Variance Power Calculation.
Figure 1. Determining Number of Samples Required
The results of the sample size calculation showed the need to run 14 launches per Release Angle.
Trial Run Order & Results
We knew that in order to reduce the margin of error in our sampling process, we would randomize our design for collecting catapult data. In order to achieve this we leveraged the built-in capabilities of R through the design.crd function.
The final randomized run order is shown in Table 1, which contains distance measurements:
| Release Angle | Distance Traveled |
|---|---|
| 180 | 270 |
| 185 | 275 |
| 185 | 281 |
| 175 | 272 |
| 180 | 283 |
| 175 | 269 |
| 180 | 281 |
| 185 | 286 |
| 180 | 266 |
| 175 | 265 |
| 185 | 279 |
| 185 | 282 |
| 175 | 260 |
| 185 | 278 |
| 185 | 281 |
| 175 | 264 |
| 180 | 270 |
| 175 | 267 |
| 180 | 277 |
| 180 | 275 |
| 180 | 267 |
| 185 | 297 |
| 180 | 285 |
| 185 | 278 |
| 175 | 259 |
| 180 | 264 |
| 175 | 268 |
| 185 | 286 |
| 185 | 295 |
| 180 | 263 |
| 185 | 288 |
| 175 | 268 |
| 180 | 250 |
| 180 | 273 |
| 175 | 265 |
| 185 | 273 |
| 175 | 255 |
| 175 | 268 |
| 180 | 268 |
| 175 | 283 |
| 175 | 254 |
| 185 | 275 |
Hypothesis Testing
We needed to then understand if Release Angle plays a significant role in the distance the ball travels. To complete this task, we ran a One-way ANOVA test.
Hypotheses
The hypotheses for the ANOVA were as follows:
\(\quad\)Null:\(\quad\quad\quad H_0 : \mu_{175} = \mu_{180} = \mu_{185}\)
\(\quad\)Alternate:\(\quad H_1 : \mu_{i} \neq \mu{j}\) for at least one pair (\(i\),\(j\))
ANOVA
The ANOVA was calculated in R.
The ANOVA resulted in at least one of the means was significantly different than the others.
| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| angles | 2 | 2096 | 1048 | 16.46 | 6.549e-06 |
| Residuals | 39 | 2483 | 63.66 | NA | NA |
From Table 2, we can see that the p-value is less than 0.001, and with an \(\alpha\) = 0.05, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between at least one of the three sample mean Release Angle launch distances.
Residuals
The residuals were then assessed for normality and constant variance. The charts below show that the residuals are indeed homoscedastic and normal.
Figure 2. Plotting residuals to determine normality and constant variance.
Pairwise Comparisons
Pairwise comparisons were conducted to understand which pairs were statistically significantly different. The following hypotheses were created for a Tukey’s Test.
\(\quad\)Null:\(\quad\quad\quad H_0 : \mu_{i} = \mu{j}\)
\(\quad\)Alternate:\(\quad H_1 : \mu_{i} \neq \mu{j}\) for all \(i \neq j\)
The Tukey’s Test was then run in R, With the following results:
| diff | lwr | upr | p adj | |
|---|---|---|---|---|
| 180-175 | 5.357 | -1.99 | 12.7 | 0.1908 |
| 185-175 | 16.93 | 9.582 | 24.28 | 5.257e-06 |
| 185-180 | 11.57 | 4.224 | 18.92 | 0.001265 |
Figure 3. Tukey plot output from our ANOVA model.
From Table 3 and Figure 3, we can see that the pairwise comparisons resulted in two of the three comparisons being significant to an \(\alpha\) = 0.05.
For the ‘185-175’ and ‘185-180’ comparisons, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between the sample mean distances.
For the ‘180-175’ comparison, we fail to reject the Null hypothesis. There is not enough evidence to support the claim of a significant difference between the sample mean distances.
Conclusions & Recommendations
Release Angle was shown to be a significant factor for some of the pair-wise comparisons but not all. It was not shown to be significant when pulled between 175° to the 180° position.
It is recommended to study other Release Angles to understand the impact they play on ball distance traveled throughout the range of arm motion.
Part 2
The Designed Experiment
In Part 2, we performed a designed experiment on the effect of Pin Elevation and Release Angle on the distance in which a ball is thrown (measured at its furthest distance). Specifically, we studied whether the settings of 100, 150 and 200 mm for Pin Elevation and the settings of 110°, 140° and 170° for Release Angle significantly differ in their mean distance thrown.
The other factors were set to constant values of the following:
Fire Angle = 100°
Bungee Position = 150mm
Cup Elevation = 250mm
The settings of Pin Elevation at 100, 150 and 200 mm were investigated as a fixed effect, as well as settings of the Release Angle corresponding to 110°, 140° and 170° as a random effect. We replicated the design three times.
Model Equation
The model equation is: \(\quad y_{ijk}\) = \(\mu\) + \(\tau_i\) + \(\beta_j\) + \((\tau\beta)_{ij}\) + \(\epsilon_{ijk}\)
Where \(\tau_i\) is the fixed effect Pin Elevation
\(\beta_j\) is the random effect Release Angle
\((\tau\beta)_{ij}\) is the interaction effect of Pin Elevation * Release Angle
and \(\epsilon_{ijk}\) is the error term
Hypotheses
The Hypotheses we tested were:
Pin Elevation Main Fixed Effect:
\(\quad H_0\) : \(\tau_i\) = 0 \(\forall\) i
\(\quad H_a\) : \(\tau_i \neq\) 0 for at least one \(\ i\)
Release Angle Main Random Effect:
\(\quad H_0\) : \(\beta_j\) = 0 \(\forall\) j
\(\quad H_a\) : \(\beta_j \neq\) 0 for at least one \(\ j\)
Pin Elevation * Release Angle Interaction Effect:
\(\quad H_0\) : \((\tau\beta)_{ij}\) = 0 \(\forall\) ij
\(\quad H_a\) : \((\tau\beta)_{ij} \neq\) 0 for at least one \(\ ij\)
\(\quad\)at a significance level of \(\alpha\) = 0.05
Ranomized Layout & Results
The proposed layout with a randomized run order and the recorded observations are found in Table 4:
| pinElevation | releaseAngle | distance |
|---|---|---|
| 200 | 140 | 220 |
| 200 | 170 | 329 |
| 150 | 170 | 271 |
| 200 | 110 | 69 |
| 150 | 110 | 56 |
| 150 | 140 | 183 |
| 150 | 110 | 60 |
| 200 | 110 | 75 |
| 200 | 140 | 215 |
| 150 | 140 | 181 |
| 100 | 110 | 53 |
| 200 | 170 | 340 |
| 150 | 140 | 182 |
| 200 | 140 | 217 |
| 100 | 170 | 209 |
| 150 | 170 | 272 |
| 100 | 110 | 48 |
| 150 | 110 | 64 |
| 100 | 140 | 144 |
| 100 | 170 | 204 |
| 100 | 110 | 50 |
| 100 | 170 | 204 |
| 200 | 110 | 68 |
| 200 | 170 | 334 |
| 150 | 170 | 269 |
| 100 | 140 | 146 |
| 100 | 140 | 144 |
Mixed Effects Model
The result of the General ANOVA Mixed Effects Model shown in Table 5:
| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| pinElevation | 2 | 24569 | 12284 | 5.579 | 0.06964 |
| releaseAngle | 2 | 199787 | 99893 | 10414 | 2.669e-28 |
| pinElevation:releaseAngle | 4 | 8808 | 2202 | 229.6 | 3.528e-15 |
| Residual | 18 | 172.7 | 9.593 | NA | NA |
The interaction effect is significant with a p-value < 0.001 at a significance level of \(\alpha\) = 0.05. We also see that the Release Angle is also significant with a p-value < 0.001 and the Pin Elevation is ‘nearly’ significant to an \(\alpha\) of 0.05, with a p-value = 0.07. We can see the stated impacts in Figure 4 below.
Figure 4. Interaction Plot of Experimental Data.
Residuals
The residuals were then assessed for normality and constant variance. From the charts below the residuals are indeed homoscedastic and normal.
Figure 5. Plotting residuals to determine normality and constant variance.
Part 3
The Designed Experiment
In Part 3, we performed a designed experiment to determine the effect of the available factors of Fire Angle, Bungee Position, Release Angle, Pin Elevation, and Cup Elevation on the distance in which a ball is thrown (measured at its furthest distance). We designed this experiment as a single replicate of a \(2^5\) factorial design with the low and high level of the factors as shown in Table 6.
Table 6. List of factors, levels and settings.
| Factor | Low Level (-1) | High Level (+1) |
|---|---|---|
| Fire Angle | 90 degrees | 110 degrees |
| Bungee Position | 130mm | 150mm |
| Release Angle | 140 degrees | 170 degrees |
| Pin Elevation | 125mm | 175mm |
| Cup Elevation | 225mm | 275mm |
Randomized Layout & Results
The proposed layout (Table 7) with a randomized run order was generated. Trial runs were taken and the data was collected. The observations were recorded and are presented in the table below.
Table 7. Runs, settings, and results
| Fire Angle | Bungee Position | Release Angle | Pin Elevation | Cup Elevation | Response |
|---|---|---|---|---|---|
| 110 | 130 | 170 | 175 | 225 | 237 |
| 110 | 150 | 170 | 175 | 225 | 260 |
| 110 | 150 | 170 | 175 | 275 | 343 |
| 90 | 150 | 170 | 175 | 275 | 189 |
| 110 | 130 | 140 | 175 | 225 | 135 |
| 110 | 150 | 140 | 125 | 275 | 154 |
| 90 | 150 | 140 | 175 | 225 | 121 |
| 90 | 150 | 170 | 175 | 225 | 153 |
| 110 | 130 | 170 | 175 | 275 | 312 |
| 90 | 150 | 140 | 175 | 275 | 152 |
| 110 | 130 | 140 | 175 | 275 | 171 |
| 110 | 150 | 140 | 125 | 225 | 114 |
| 90 | 150 | 170 | 125 | 275 | 169 |
| 90 | 130 | 140 | 175 | 225 | 114 |
| 110 | 150 | 140 | 175 | 225 | 145 |
| 90 | 150 | 170 | 125 | 225 | 135 |
| 90 | 130 | 170 | 175 | 225 | 149 |
| 110 | 150 | 140 | 175 | 275 | 197 |
| 90 | 130 | 140 | 125 | 225 | 98 |
| 90 | 150 | 140 | 125 | 275 | 134 |
| 90 | 130 | 170 | 125 | 275 | 157 |
| 90 | 130 | 140 | 125 | 275 | 125 |
| 110 | 150 | 170 | 125 | 275 | 262 |
| 110 | 130 | 170 | 125 | 225 | 180 |
| 90 | 130 | 170 | 125 | 225 | 124 |
| 90 | 130 | 140 | 175 | 275 | 143 |
| 110 | 130 | 140 | 125 | 225 | 101 |
| 90 | 130 | 170 | 175 | 275 | 183 |
| 110 | 130 | 170 | 125 | 275 | 238 |
| 110 | 150 | 170 | 125 | 225 | 199 |
| 90 | 150 | 140 | 125 | 225 | 104 |
| 110 | 130 | 140 | 125 | 275 | 140 |
Original Model Equation
The original model equation for this experiment is as follows:
Distance = A + B + C + D + E + AB + AC + AD + AE + BC + BD + BE + CD + CE + DE + ABC + ABD + ABE + ACD + ACE + ADE + BCD + BCE + BDE + CDE + ABCD + ABCE + ABDE + ACDE + BCDE + ABCDE
Where: A = Fire Angle; B = Bungee Position; C = Release Angle; D = Pin Elevation; E = Cup Elevation
Significant Factors/Interactions
The half-normal plot for all of the factors and interactions was created which identified the significant factors. 
Figure 6. Half-normal plot of factors from experimental runs.
The significant terms were found to be:
| Signficant Factors/Interactions |
|---|
| Fire_Angle:Release_Angle:Cup_Elevation |
| Fire_Angle:Bungee_Position |
| Fire_Angle:Release_Angle:Pin_Elevation |
| Release_Angle:Cup_Elevation |
| Release_Angle:Pin_Elevation |
| Fire_Angle:Cup_Elevation |
| Bungee_Position |
| Fire_Angle:Pin_Elevation |
| Pin_Elevation |
| Fire_Angle:Release_Angle |
| Cup_Elevation |
| Fire_Angle |
| Release_Angle |
Final ANOVA for Model Equation
As we reduced the model, the insignificant factors/interactions were built into the error term. The following represents the ANOVA with only significant factors and interactions built in (to an \(\alpha\) = 0.05)
| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| Bungee_Position | 1 | 1568 | 1568 | 71.73 | 1.078e-07 |
| Pin_Elevation | 1 | 10153 | 10153 | 464.4 | 2.643e-14 |
| Cup_Elevation | 1 | 15313 | 15313 | 700.4 | 7.297e-16 |
| Fire_Angle | 1 | 27495 | 27495 | 1258 | 4.137e-18 |
| Release_Angle | 1 | 40755 | 40755 | 1864 | 1.246e-19 |
| Fire_Angle:Bungee_Position | 1 | 288 | 288 | 13.17 | 0.001917 |
| Release_Angle:Cup_Elevation | 1 | 544.5 | 544.5 | 24.91 | 9.473e-05 |
| Release_Angle:Pin_Elevation | 1 | 741.1 | 741.1 | 33.9 | 1.624e-05 |
| Fire_Angle:Cup_Elevation | 1 | 1152 | 1152 | 52.7 | 9.502e-07 |
| Fire_Angle:Pin_Elevation | 1 | 2016 | 2016 | 92.22 | 1.659e-08 |
| Fire_Angle:Release_Angle | 1 | 11476 | 11476 | 525 | 9.108e-15 |
| Fire_Angle:Release_Angle:Cup_Elevation | 1 | 264.5 | 264.5 | 12.1 | 0.002682 |
| Fire_Angle:Release_Angle:Pin_Elevation | 1 | 406.1 | 406.1 | 18.58 | 0.0004215 |
| Residuals | 18 | 393.5 | 21.86 | NA | NA |
Final Model Equation
\[ \begin{aligned} \operatorname{\widehat{response}} &= 169.94 + 7(\operatorname{Bungee\_Position}) + 17.81(\operatorname{Pin\_Elevation}) + 21.88(\operatorname{Cup\_Elevation})\ + \\ &\quad 29.31(\operatorname{Fire\_Angle}) + 35.69(\operatorname{Release\_Angle}) + 3(\operatorname{Bungee\_Position}_{\operatorname{Fire\_Angle}} \times \operatorname{Fire\_Angle}_{\operatorname{Bungee\_Position}}) + 4.13(\operatorname{Cup\_Elevation}_{\operatorname{Release\_Angle}} \times \operatorname{Release\_Angle}_{\operatorname{Cup\_Elevation}})\ + \\ &\quad 4.81(\operatorname{Pin\_Elevation}_{\operatorname{Release\_Angle}} \times \operatorname{Release\_Angle}_{\operatorname{Pin\_Elevation}}) + 6(\operatorname{Cup\_Elevation}_{\operatorname{Fire\_Angle}} \times \operatorname{Fire\_Angle}_{\operatorname{Cup\_Elevation}}) + 7.94(\operatorname{Pin\_Elevation}_{\operatorname{Fire\_Angle}} \times \operatorname{Fire\_Angle}_{\operatorname{Pin\_Elevation}}) + 18.94(\operatorname{Fire\_Angle} \times \operatorname{Release\_Angle})\ + \\ &\quad 2.87(\operatorname{Cup\_Elevation}_{\operatorname{Fire\_Angle}} \times \operatorname{Fire\_Angle}_{\operatorname{Release\_Angle}} \times \operatorname{Release\_Angle}_{\operatorname{Cup\_Elevation}}) + 3.56(\operatorname{Pin\_Elevation}_{\operatorname{Fire\_Angle}} \times \operatorname{Fire\_Angle}_{\operatorname{Release\_Angle}} \times \operatorname{Release\_Angle}_{\operatorname{Pin\_Elevation}}) \end{aligned} \]
References
Montgomery, D. C. (2013). Design and analysis of Experiments (8th ed.). John Wiley & Sons, Inc.
Six sigma virtual catapult. SigmaZone. (2020, May 28). Retrieved December 2, 2021, from https://sigmazone.com/catapult/.
Source Code
# Setup Libraries
library(dplyr)
library(tidyr)
library(readr)
library(knitr)
library(agricolae)
library(lawstat)
library(car)
library(GAD)
library(BSDA)
library(pwr)
library(WebPower)
library(ggplot2)
library(ggfortify)
library(ggpubr)
library(SixSigma)
library(DoE.base)
library(FrF2)
library(equatiomatic)
#------------------------------------------------------------------------------
#------------------------------------------------------------------------------
# Part 1
effectiveReleaseAngle = c(175,180,185)
firingAngle = 90 #degrees
bungeePos = 200 #mm
pinElevation = 200 #mm
cupElevation = 300 #mm
alpha = 0.05
#------------------------------------------------------------------------------
# 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%.
power1 = 0.75
#d is the range of means divided by sigma, k is the number of populations
d = mean(diff(effectiveReleaseAngle))
sd1 = sd(effectiveReleaseAngle)
# sigma = 5
# power = .75
# u = k - 1
# u = 2
# a = 0.05
kgroups = length(effectiveReleaseAngle)
#The effect is given by d/2 when k is even
#The effect is given by d*sqrt(k^2-1)/(2*k) when k is odd
fCal = d/sd1*sqrt(kgroups^2-1)/(2*kgroups)
sampleNumber <- pwr.anova.test(n = NULL,
k = kgroups,
f = fCal,
sig.level = alpha,
power = power1)
plot(sampleNumber)#------------------------------------------------------------------------------
# Propose a layout using the number of samples from part (a) with
# randomized run order
n = ceiling(sampleNumber$n)
# set seed here for reproducibility
design <- design.crd(trt = effectiveReleaseAngle,
r = n,
seed = 1234)
design$book
#------------------------------------------------------------------------------
# Collect data and record observations on layout proposed in part (b)
recordedDistance <- c(270,275,281,272,283,269,281,
286,266,265,279,282,260,278,
281,264,270,267,277,275,267,
297,285,278,259,264,268,286,
295,263,288,268,250,273,265,
273,255,268,268,283,254,275)
#------------------------------------------------------------------------------
# Perform hypothesis test and check residuals. Be sure to comment and
# take corrective action if necessary.
dat <- data.frame(angles = design$book$effectiveReleaseAngle,
distance = recordedDistance)
dataov <- aov(distance~angles,data = dat)
summary(dataov)
datHSD <- TukeyHSD(dataov)
datHSD
plot(datHSD)autoplot(dataov)#------------------------------------------------------------------------------
# If the null hypothesis is rejected, investigate pairwise comparisons.
#------------------------------------------------------------------------------
# State conclusions and make recommendation.
#------------------------------------------------------------------------------
#------------------------------------------------------------------------------
# Part 2
# Determine effect of Pin Elevations and Release Angle,
# when Fire Angle at 100 Degrees
#
# Bungee Position is 150m & Cup Elevation is 250mm
#
# Pin Elevation:Fixed Effect, 100, 150, 200 mm.
# Release Angle:Random Effect, 110, 140, 170 degrees.
# model equation:
# distance~pinElevation+releaseAngle+pinElevation*releaseAngle
# init
pinElevation <- as.fixed(c(100,150,200))
releaseAngle <- as.random(c(110,140,170))
# 3x3 factorial with 3 replications
trts<-c(3,3)
design3x3 <- design.ab(trt=trts, r=3, design="crd",seed=1234)
# replace names & values
names(design3x3$book)[names(design3x3$book) == c("A","B")] <- c("pinElevation","releaseAngle")
design3x3$book$pinElevation <- pinElevation[match(design3x3$book$pinElevation,1:3)]
design3x3$book$releaseAngle <- releaseAngle[match(design3x3$book$releaseAngle,1:3)]
design3x3$book
distanceRecorded <- c(220,329,271,69,56,183,60,75,215,
181,53,340,182,217,209,272,48,64,
144,204,50,204,68,334,269,146,144)
dat2 <- data.frame(pinElevation = design3x3$book$pinElevation,
releaseAngle = design3x3$book$releaseAngle,
distance = distanceRecorded)
dat2aov <- aov(distance~pinElevation+releaseAngle+pinElevation*releaseAngle,data = dat2)
gad(dat2aov)
autoplot(dat2aov)#pinElevation vs releaseAngle
interaction.plot(x.factor = dat2$pinElevation, #x-axis variable
trace.factor = dat2$releaseAngle, #variable for lines
response = dat2$distance, #y-axis variable
fun = median, #metric to plot
ylab = "distance",
xlab = "pinElevation",
col = c("green", "blue", "purple"),
lty = 1, #line type
lwd = 2, #line width
trace.label = "releaseAngle")dat2 %>% group_by(pinElevation, releaseAngle) %>%
summarise(
count = n(),
mean = mean(distance, na.rm = TRUE),
sd = sd(distance, na.rm = TRUE))
## Randomized Layout & Results
Fire_Angle <- c(90 ,110) # Degrees
Bungee_Position <- c(130,150) # mm
Release_Angle <- c(140,170) # Degrees
Pin_Elevation <- c(125,175) # mm
Cup_Elevation <- c(225,275) # mm
#(a & b)
trts<-c(2,2,2,2,2)
design2by5 <- design.ab(trt=trts, r=1, design="crd",seed=1234)
#design2by5
# replace names & values
colnames(design2by5$book)[-(1:2)] <- c("Fire_Angle",
"Bungee_Position",
"Release_Angle",
"Pin_Elevation",
"Cup_Elevation")
design2by5_table <- design2by5
design2by5_table$book$Fire_Angle <- Fire_Angle[match(design2by5$book$Fire_Angle,1:2)]
design2by5_table$book$Bungee_Position <- Bungee_Position[match(design2by5$book$Bungee_Position,1:2)]
design2by5_table$book$Release_Angle <- Release_Angle[match(design2by5$book$Release_Angle,1:2)]
design2by5_table$book$Pin_Elevation <- Pin_Elevation[match(design2by5$book$Pin_Elevation,1:2)]
design2by5_table$book$Cup_Elevation <- Cup_Elevation[match(design2by5$book$Cup_Elevation,1:2)]
# Results of the Trials
response <- c(237,260,343,189,135,154,121,153,312,152,
171,114,169,114,145,135,149,197,98,134,157,
125,262,180,124,143,101,183,238,199,104,140)
# Create the Table
design2by5table <- cbind(design2by5_table$book$Fire_Angle,
design2by5_table$book$Bungee_Position,
design2by5_table$book$Release_Angle,
design2by5_table$book$Pin_Elevation,
design2by5_table$book$Cup_Elevation,
response)
colnames(design2by5table) <- c("Fire Angle",
"Bungee Position",
"Release Angle",
"Pin Elevation",
"Cup Elevation",
"Response")
knitr::kable(design2by5table)
#(c)
io <- c(-1,1)
design2by5$book$Fire_Angle <- io[match(design2by5$book$Fire_Angle,1:2)]
design2by5$book$Bungee_Position <- io[match(design2by5$book$Bungee_Position,1:2)]
design2by5$book$Release_Angle <- io[match(design2by5$book$Release_Angle,1:2)]
design2by5$book$Pin_Elevation <- io[match(design2by5$book$Pin_Elevation,1:2)]
design2by5$book$Cup_Elevation <- io[match(design2by5$book$Cup_Elevation,1:2)]
design2by5$book <- cbind(design2by5$book,response)
datmodel <- lm(response~Fire_Angle*
Bungee_Position*
Release_Angle*
Pin_Elevation*
Cup_Elevation,
data = design2by5$book)
modelFactorEff <- coef(datmodel)[-1]*2
halfnormal(datmodel)
halfEff <- halfnormal(datmodel)# significant factors/interactions
knitr::kable(halfEff$signif, col.names = "Signficant Factors/Interactions")
# Final ANOVA Table
datmodelsig <- lm(response~
Fire_Angle:Release_Angle:Cup_Elevation +
Fire_Angle:Bungee_Position +
Fire_Angle:Release_Angle:Pin_Elevation +
Release_Angle:Cup_Elevation +
Release_Angle:Pin_Elevation +
Fire_Angle:Cup_Elevation +
Bungee_Position +
Fire_Angle:Pin_Elevation +
Pin_Elevation +
Fire_Angle:Release_Angle +
Cup_Elevation +
Fire_Angle +
Release_Angle,
data = design2by5$book)
dataovsig <- aov(datmodelsig)
summary(dataovsig)
## Final Model Equation
equatiomatic::extract_eq(datmodelsig, wrap = TRUE, use_coefs = TRUE)