Recipe 7: Resampling Techniques
Weights of Football Players
Max Winkelman
Rensselaer Polytechnic Institute
November 11, 2014
Version 1
1. Setting
Football Player Weights
The data analyzed in this recipe is a csv file that contains the weights of football players from five NFL teams.
Install the ‘FootballPlayerWeights.csv’ file
weight <- read.csv("~/RPI/Classes/Design of Experiments/R/FootballPlayerWeights.csv", header=TRUE)
#reads in the data from the csv file 'FootballPlayerWeights.csv' and assigns it to the dataframe weight
weight$Team = as.factor(weight$Team)Factors and Levels
Factor: NFL Team
Levels: Cowboys, Packers, Broncos, Dolphins, Forty Niners
#Summary of Data
head(weight)## Team Weight
## 1 Cowboys 250
## 2 Cowboys 255
## 3 Cowboys 255
## 4 Cowboys 264
## 5 Cowboys 250
## 6 Cowboys 265#displays the first 6 sets of variables
tail(weight)## Team Weight
## 80 Forty Niners 253
## 81 Forty Niners 249
## 82 Forty Niners 223
## 83 Forty Niners 221
## 84 Forty Niners 228
## 85 Forty Niners 271#displays the last 6 sets of variables
summary(weight)## Team Weight
## Broncos :17 Min. :208
## Cowboys :17 1st Qu.:236
## Dolphins :17 Median :252
## Forty Niners:17 Mean :248
## Packers :17 3rd Qu.:260
## Max. :281#displays a summary of the variablesContinuous variables:
In the csv file, ‘FootballPlayerWeights,’ the continous variable is ‘Weight’ and is measured in pounds.
Response variables:
TRhe response variable in this recipe will be football player weight (lbs).
The Data: How is it organized and what does it look like?
The data in this recipe was taken from The Sports Encyclopedia Pro Football can is organized into two columns; ‘Team,’ which displays the NFL team that a player is from, and ‘Weight,’ which is the weight of the player in pounds.
Randomization
It can be assumed that the original data was gathered with proper randomization methods.
2. Experimental Design
How will the experiment be organized and conducted to test the hypothesis?
The experiment that will be conducted in this recipe will be to determine if the variation of NFL player weight can be attributed to the variation of NFL teams. An analysis of variance with a confidence interval of 95% will be performed to determine if the team that a player is a member of will have an effect on his weight. The null hypothesis for this ANOVA will be that the means of the player weights will be equal for all NFL teams. If this is rejected, the alternative hypothesis, which states that the mean weights are different between NFL teams, will be accepted. After the ANOVA has been performed, resampling methods will be implemented to determine their effect of the outcome of the ANOVA test.
What is the rationale for this design?
This dataframe in this recipe contains a single factor with multiple levels. Therefore, an ANOVA is the appropriate test to be performed. The resampling techniques performed are conducted due to the fact that ANOVA makes the assumption that all data is normally distributed. While it is possible for this distribution to occur, rarely is any data completely normal.
Randomize: What is the Randomization Scheme?
No randomization scheme was used in this recipe because the data used is not original content.
Replicate: Are there replicates and/or repeated measures?
There are no replicates or repeated measures in this data set.
Block: Did you use blocking in the design?
There will be no blocking in this recipe.
3. Statistical Analysis
Exploratory Data Analysis: Graphics and Descriptive Summary
boxplot(Weight~Team,data=weight, xlab="NFL Team", ylab="Weight (lbs)")
title("NFL Player Weights")
The boxplot above shows the distribution of player weights from five NFL teams. All medians fall in between 240 and 260 pounds. The Broncos and the Cowboys are the only teams that have outliers. No statistical inference can be drawn from this plot.
ANOVA Testing
An analysis of variance (ANOVA) will be used to determine the statistical significance between the player weight means. If the null hypothesis is accepted, it can be assumed that there is no variation in player weight due to the variation of NFL teams. If the null hypothesis is rejected, it can be assumed that the variation of player weights can be attributed to the variation of teams.
# ANOVA
#Player Weight
model = aov(Weight~Team,data=weight)
anova(model)## Analysis of Variance Table
##
## Response: Weight
## Df Sum Sq Mean Sq F value Pr(>F)
## Team 4 1714 428 1.58 0.19
## Residuals 80 21761 272#performs an anova testANOVA Results: The anova test that analyzed the variation in player weight as a result of the variation in player team produced a p-value of 0.19. This indicates that there is a high probability that the variation of player weight can be attributed to solely randomization. It is highly likely that a player’s team has no effect on the player’s weight. The mean square of the residuals is 272.
Post-Hoc Analysis
Tukey’s Honestly Significantly Difference is a multiple comparison procedure that is used after an ANOVA to determine which specific sample means are significantly different from the others. In this recipe, Tukey’s HSD is not necessary because the ANOVA indicated that there was no variation in weight means.
Resampling Methods
Monte Carlo Estimation of Power
If resampling is performed from a known theoretical distribution, it is refered to as a “Monte Carlo” simulation. Monte Carlo simulations are algorithms that utilize repeated random sampling techniques to determine the distribution of a data set.
meanstar=with(weight,tapply(Weight,Team,mean))
#creates an array of the means of the levels
with(weight,tapply(Weight,Team,var))## Broncos Cowboys Dolphins Forty Niners Packers
## 263.6 235.3 281.1 288.5 291.6#creates an array of the variances of the levels
with(weight,tapply(Weight,Team,length))## Broncos Cowboys Dolphins Forty Niners Packers
## 17 17 17 17 17#creates an array of the lengths of the levels
summary(aov(Weight~Team,data=weight))## Df Sum Sq Mean Sq F value Pr(>F)
## Team 4 1714 428 1.58 0.19
## Residuals 80 21761 272#summary of the avona model for reference
meanWeight = mean(weight$Weight)
#mean of all the weights in the data set
sqrtMS = sqrt(272)
#squareroot of the mean square of the residuals
simTeam = weight$Team
#assigns the Teams to one data set
Runs = 1000
#number of random simulations that will be run
Fstar = numeric(Runs)
for (i in 1:Runs)
{A = rnorm(17, mean=meanWeight, sd=sqrtMS)
B = rnorm(17, mean=meanWeight, sd=sqrtMS)
C = rnorm(17, mean=meanWeight, sd=sqrtMS)
D = rnorm(17, mean=meanWeight, sd=sqrtMS)
E = rnorm(17, mean=meanWeight, sd=sqrtMS)
simWeight = c(A,B,C,D,E)
simdata = data.frame(simWeight,simTeam)
Fstar[i] = oneway.test(simWeight~simTeam, var.equal=T, data=simdata)$statistic}The code above creates a random normal distributed with 85 observations and performs an ANOVA on the resultant dataframe. The result is the theoretical F-distribution, show in the histogram with 4 degrees of freedom for the NFL Teams and 80 degrees of freedom for the residuals.
hist(Fstar, prob=TRUE, main ="Theoretical F Distribution", ylim=c(0,0.8), xlim=c(0,7))
x=seq(.25,7,.25)
#4=5levels-1mean and 80=85values-5means
points(x,y=df(x,4,80),type="b",col="blue")
The plotted points of the theoretical distribution match up relatively nicely with the Monte Carlo F-distribution. This indicates that the original data set is close to being normally distributed.
Bootstrap Method
This will be the second resampling method performed. It is completely analytical and there are no assumtptions about the distribution of the data. The results will be inputed into a second ANOVA.
# Bootstrap version
#rm(list=ls())
grpB = weight$Weight[weight$Team=="Broncos"] - meanstar[1]
grpC = weight$Weight[weight$Team=="Cowboys"] - meanstar[2]
grpD = weight$Weight[weight$Team=="Dolphins"] - meanstar[3]
grpFN = weight$Weight[weight$Team=="Forty Niners"] - meanstar[4]
grpP = weight$Weight[weight$Team=="Packers"] - meanstar[5]
newsimTeam = weight$Team
newFstar = numeric(Runs)
for (i in 1:Runs)
{groupA = sample(grpB, size=17, replace=TRUE)
groupB = sample(grpC, size=17, replace=TRUE)
groupC = sample(grpD, size=17, replace=TRUE)
groupD = sample(grpFN, size=17, replace=TRUE)
groupE = sample(grpP, size=17, replace=TRUE)
newsimweight = c(groupA,groupB,groupC,groupD,groupE)
newsimdata = data.frame(newsimweight,newsimTeam)
newFstar[i] = oneway.test(newsimweight~newsimTeam, var.equal=T, data=newsimdata)$statistic}Bootstrap F-distribution Histogram
hist(newFstar, prob=TRUE, main ="BootStrap F-Distribution", ylim=c(0,0.8), xlim=c(0,7))
x=seq(.25,7,.5)
#4=5levels-1mean and 80=85values-5means
points(x,y=df(x,4,80),type="b",col="blue")
The bootstrapping histogram is very similar to the theoretical distribution from the first histogram, confirming that the original data set is very close to being normally distributed.
Estimatation of the Alpha Level
# Alpha level (Bootstrapped Distribution)
print(realFstar<-oneway.test(Weight~Team, var.equal=T, data=weight)$statistic)## F
## 1.575#Display the probability
mean(newFstar>=realFstar)## [1] 0.16#Quantiles
#Analytic F distribution
qf(.95,4,80)## [1] 2.486#Bootstrapped F distribution (alpha= 0.05)
quantile(newFstar,.95)## 95%
## 2.231Repeat ANOVA Test with Resampling
newmodel=aov(newsimweight~newsimTeam, data=newsimdata)
anova(newmodel)## Analysis of Variance Table
##
## Response: newsimweight
## Df Sum Sq Mean Sq F value Pr(>F)
## newsimTeam 4 1575 394 1.55 0.2
## Residuals 80 20293 254ANOVA Results: The second anova test that analyzed the variation in player weight as a result of the variation in player team produced a p-value greater than 0.19. Again, this indicates that there is a high probability that the variation of player weight can be attributed to solely randomization.
Estimation of Parameters
Summary of all factors and levels
#Summaries
#Broncos
broncos<-weight$Team=="Broncos"
#assigns all bronco data a vector
summary(weight[broncos,"Weight"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 216 250 256 254 265 281#displays a summary of the weight data for the Broncos
#Packers
packers<-weight$Team=="Packers"
#assigns all packer data a vector
summary(weight[packers,"Weight"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 222 240 254 251 263 275#displays a summary of the weight data for the Packers
#Dolphins
dolphins<-weight$Team=="Dolphins"
#assigns all packer data a vector
summary(weight[dolphins,"Weight"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 210 236 253 249 263 268#displays a summary of the weight data for the Dolphins
#49ers
fortyniners<-weight$Team=="Forty Niners"
#assigns all packer data a vector
summary(weight[fortyniners,"Weight"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 208 228 247 241 249 271#displays a summary of the weight data for the 49ers
#Cowboys
cowboys<-weight$Team=="Cowboys"
#assigns all packer data a vector
summary(weight[cowboys,"Weight"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 220 245 250 247 255 266#displays a summary of the weight data for the CowboysDiagnostics/Model Adequacy Checking
Quantile-Quantile (Q-Q) plots are graphs used to verify the distributional assumption for a set of data. Based on the theoretical distribution, the expected value for each datum is determined. If the data values in a set follow the theoretical distribution, then they will appear as a straight line on a Q-Q plot. When an anova is performed, it is done so with the assumption that the test statistic follows a normal distribution. Visualization of a Q-Q plot will further confirm if that assumption is correct for the anova tests that were performed.
#Q-Q Plots
#Player Weight
qqnorm(residuals(model), main="Normal Q-Q Plot for NFL Player Weight", ylab="Weight (lbs) Residuals")
qqline(residuals(model))
#produces a Q-Q normal plot for the anova model
qqnorm(residuals(newmodel), main="Normal Q-Q Plot for NFL Player Weight (Resampling)", ylab="Weight (lbs) Residuals")
qqline(residuals(newmodel))
#produces a Q-Q normal plot for the anova model with resamplingThe Normal Q-Q plots for both anova models produced relatively linear relationships between the residual and theoretical values, indicating that the use of ANOVA in this recipe was appropriate.
A Residuals vs. Fits Plot is a common graph used in residual analysis. It is a scatter plot of residuals as a function of fitted values, or the estimated responses. These plots are used to identify linearity, outliers, and error variances.
#Player Weights
plot(fitted(model),residuals(model), main="Residual vs Fitted Plot for NFL Player Weights") 
plot(fitted(newmodel),residuals(newmodel), main="Residual vs Fitted Plot for NFL Player Weights (Resampling)") 
#produces a Residual vs Fits Plot for the first and second ANOVA