Analysis of Variance on Tennis Racquet Design
Trevor Manzanares
October 10, 2014
Recipe 4: Example of Descriptive Statistics
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Recipes for the Design of Experiments: Recipe Outline
as of August 28, 2014, superceding the version of August 24. Always use the most recent version.
Design of Experiments
Trevor Manzanares
Rensselaer Polytechnic Institute
10/10/14
1. Setting
System under test
Dataset under analysis includes data on design factors that contribute to reduced stress and improved vibrational dampening of tennis racquets.
remove(list=ls())
test = read.csv("C:/Users/Trevor/Documents/tennis.csv", header=TRUE)
#view first few lines
head(test)## racquet_number material handle_design stiffness_rate
## 1 1 Carbon Fiber Triple core 61
## 2 2 Graphite Dual core 63
## 3 3 Carbon Fiber Triple core 57
## 4 4 Graphite Dual core 64
## 5 5 Graphite Dual core 61
## 6 6 Graphite w basalt planks Hollow 68#observe the structure of the data, ie. how many variables
str(test)## 'data.frame': 10 obs. of 4 variables:
## $ racquet_number: int 1 2 3 4 5 6 7 8 9 10
## $ material : Factor w/ 5 levels "Carbon Fiber",..: 1 2 1 2 2 5 3 2 2 4
## $ handle_design : Factor w/ 3 levels "Dual core","Hollow",..: 3 1 3 1 1 2 2 2 2 2
## $ stiffness_rate: int 61 63 57 64 61 68 67 70 67 65#55 observations of 26 variables
attach(test)Factors of interest and their levels
material (5 levels) and handle_design (3 levels)
#view summary statistics
summary(test)## racquet_number material handle_design
## Min. : 1.00 Carbon Fiber :2 Dual core :3
## 1st Qu.: 3.25 Graphite :5 Hollow :5
## Median : 5.50 Graphite Tungsten :1 Triple core:2
## Mean : 5.50 Graphite Tungsten/Copper/Titanium:1
## 3rd Qu.: 7.75 Graphite w basalt planks :1
## Max. :10.00
## stiffness_rate
## Min. :57.0
## 1st Qu.:61.5
## Median :64.5
## Mean :64.3
## 3rd Qu.:67.0
## Max. :70.0Continuous variables (if any)
the response variable, stiffness_rate, ranged from a rating of 55 to 72 ### The Data: How is it organized and what does it look like? The data are tabluated into 4 columns, with each racquet having a material type, handle design, and stiffness rate ### Randomization The data were collected on ten racquet frame designs from five different manufacturers
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
Analysis of Variance will be used to attempt to reproduce the study performed by Ferrara and Cohen by measuring the effects of material and handle design on stiffness rate
### What is the rationale for this design? It is possible that material by itself may have an effect on the stiffness rate, and it is also possible that the handle design may have an effect on the stiffness rate.
Replicate: Are there replicates and/or repeated measures?
There are no replicates or repeated measures.
### Block: Did you use blocking in the design? Yes, one model was performed with blocking to determine if each of the factors had an effect by themselves.
3. (Statistical) Analysis
(Exploratory Data Analysis) Graphics and descriptive summary
par(mfrow=c(1,2))
plot(stiffness_rate~material,las=3)
plot(stiffness_rate~handle_design,las=3)
### Testing
#set up factorial design
library("DoE.base", lib.loc="~/R/win-library/3.1")## Loading required package: grid
## Loading required package: 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.designfac.design(nlevels=3, nfactors=2)## creating full factorial with 9 runs ...## A B
## 1 1 1
## 2 2 1
## 3 2 2
## 4 3 1
## 5 2 3
## 6 3 2
## 7 1 3
## 8 3 3
## 9 1 2
## class=design, type= full factorial#run analysis of variance for individual factor effects
model1=(aov(stiffness_rate~material,data=test))
anova(model1)## Analysis of Variance Table
##
## Response: stiffness_rate
## Df Sum Sq Mean Sq F value Pr(>F)
## material 4 80.1 20.0 1.73 0.28
## Residuals 5 58.0 11.6#based on material alone, we fail to reject the H0 that material type has no effect on the stiffness rate. Thus, variation in the stiffness rate can be attributed to randomization alone.
model2=(aov(stiffness_rate~handle_design, data=test))
anova(model2)## Analysis of Variance Table
##
## Response: stiffness_rate
## Df Sum Sq Mean Sq F value Pr(>F)
## handle_design 2 112.2 56.1 15.2 0.0028 **
## Residuals 7 25.9 3.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#based on handle design alone, we reject the H0 that handle design has no effect on the stiffness rate. Thus, variation in the stiffness rate can be attributed to something other than randomization, namely handle design.
#run analysis of variance using interaction
interaction=aov(stiffness_rate~material*handle_design, data=test)
anova(interaction)## Analysis of Variance Table
##
## Response: stiffness_rate
## Df Sum Sq Mean Sq F value Pr(>F)
## material 4 80.1 20.0 4.67 0.082 .
## handle_design 1 40.8 40.8 9.51 0.037 *
## Residuals 4 17.2 4.3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#in this particular model, we fail to reject the H0 that material type has no effect on stiffness rate, but reject the H0 that handle design has no effect on stiffness rate.
#run analysis of variance with blocking
blocking=aov(stiffness_rate~material+handle_design, data=test)
anova(blocking)## Analysis of Variance Table
##
## Response: stiffness_rate
## Df Sum Sq Mean Sq F value Pr(>F)
## material 4 80.1 20.0 4.67 0.082 .
## handle_design 1 40.8 40.8 9.51 0.037 *
## Residuals 4 17.2 4.3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#when considering blocking, the result is the same as the interaction effect.Estimation (of Parameters)
# Shapiro-Wilk test of normality. Adequate if p < 0.1
shapiro.test(stiffness_rate)##
## Shapiro-Wilk normality test
##
## data: stiffness_rate
## W = 0.9707, p-value = 0.8977qqnorm(stiffness_rate)
qqline(stiffness_rate)
One of the primary assumptions of the analysis of variance is that the data are normally distributed, and since they are, the results from the analysis of variance are valid. ### Diagnostics/Model Adequacy Checking Describe
qqnorm(residuals(interaction))
qqline(residuals(interaction))
plot(fitted(interaction),residuals(interaction))
par(mfrow=c(1,1))
tukey1 = TukeyHSD(blocking, which="material", ordered = FALSE)
tukey2 = TukeyHSD(blocking, which="handle_design",ordered = FALSE)
plot(tukey1)
#based on this test, there is no significant difference between the mean stiffness rate of any of the materials
plot(tukey2)
#based on this test, there is no significant difference between the mean stiffness rate of any of the handle designs4. References to the literature
This data were originally analyzed by Lisa Ferrara and Anders Cohen in “A mechanical study on tennis racquets to investigate design factors that contribute to reduced stress and improved vibrational dampening”
5. Appendices
A summary of, or pointer to, the raw data
Based on these data, the results were reproducible. There were statistically significant differences in mean stiffness_rate for the handle design. Further, material alone did not have an effect on stiffness, but handle design had an effect on stiffness rate. ### complete and documented R code
Please see above.