Is it statistically significant?

Kiké Hernandez’s Tight Pants: A Longitudinal Study Comparing Performance and Pants Tightness by u/ItalicSlope is the seminal investigation on Enrique “Kiké” Hernandez, utility player for the Los Angeles Dodgers, and his performance with regards to the tightness of his pants. It is shown that the tightness of his pants has a positive correlation with his performance, defined in the original paper as his on-base plus slugging (OPS) percentage. However there is a gap in the literature - it remains to be seen whether or not these differences are statistically significant.

This paper will clarify whether or not the differences in OPS are statistically significant due to the tightness of Hernandez’s pants. Using one-way analysis of variance (ANOVA), it can be shown that these differences are statistically significant.

A short introduction to ANOVA

ANOVA is used to determine whether or not differences among means are statistically significant. The alternative would be to run multiple t-tests, which would increase the likelihood of a Type I error. Therefore this is an ideal situation to use ANOVA to compare the means which in this case are the mean OPS for each different tightness of pants.

ANOVA works by comparing the variance among the means to the variance within the populations from which the sample means were calculated. For example, if Hernandez had great variation in his OPS for each tightness of pants, then ANOVA would fail to reject the null hypothesis that the means are not significantly different from one another. This would make sense since any differences among the means can most likely be attributable to the variation in OPS for a given tightness of pants, rather than to any effects of the pants. Conversely, if Hernandez showed very little change in OPS for any given tightness of pants, any difference among the means is more likely to be statistically significant.

Analysis

This analysis is conducted using the statistical programming language and environment called R. This is because R offers many useful packages for data processing and statistical analysis.

Set-Up and Descriptive Analysis

First the data (kindly provided by u/ItalicSlope) is processed.

library(tidyverse)
library(readxl)

Kike_Pants_Data <- read_excel("Kike-Pants-Data.xlsx")

#The original data file contains some extra numbers
data_clean <- Kike_Pants_Data[1:88,]

data_clean <- select(data_clean, "Pants", "OPS")

Below is a boxplot of the OPS versus different tightness of pants. The distributions of the OPS for the extra tight and tight pants are similar to each other, but the distribution of the OPS for moderately tight pants shows greater spread and skewness.

Assumptions for ANOVA

There are three main assumptions that need to be met in order to use ANOVA. They are 1. Independence of samples 2. Normally distributed populations 3. Equal variances among populations.

Independence of samples

The first requirement is that the samples we use are simple random samples and that they are independent. In this case, that means each and every game that Hernandez plays in is equally likely to be a part of our sample and that the OPS in one sample should not be affected by the OPS in another. If our population is the first half of the 2018 season, then our first subcondition is met since we have OPS for all 81 of the games played. Unfortunately, the second subcondition is not met because each of the samples comes from repeated measures of the same subject. Therefore the samples are dependent and possibly correlated. However, like most elementary statisticians, I will ignore this inconvenience and merely proceed cautiously with the analysis instead of proceeding with a more statistically valid method like the repeated measures ANOVA.

Normally distributed populations

The second requirement is that the populations from which the samples were drawn should approximately follow the normal distribution. The assumption is examined using histograms of the various samples.

For the tight pants, it seems that the population is approximately normal, although it is slightly skewed left by outliers.

The distribution for the moderately tight pants is more problematic since it appears to be bimodal.

Finally, of all three distributions, the distribution for the extra tight pants appears to follow the normal distribution most closely.

The sampling distributions may not be entirely normal but analysis of the distribution of the OPS metric suggests that the distribution is indeed normal. Therefore any deviation can likely be attributed to variations that arise in the moderately small samples that were drawn. Furthermore, the ANOVA test is robust and departures from normality are not too problematic.

Equal variances among populations

Finally, the third requirement is that the populations should have the same variances. For example, the variance in OPS when Hernandez wears tight pants should be roughly equal to the variance in OPS when he wears extra tight pants. In practice, statisticians believe that this assumption is largely met if the largest standard deviation is less than double the smallest standard deviation.

Fortunately, an analysis of the standard deviations in our samples shows that this condition is met. The largest standard deviation, which corresponds to the moderately tight pants sample (0.1081317) is certainly less than double the standard deviation corresponding to the extra tight pants sample (0.06489537).

sd(tight)
## [1] 0.07110156
sd(moderate)
## [1] 0.1081317
sd(extra_tight)
## [1] 0.06489537

Thus this third condition is met. Even though only 1 out of 3 conditions has been fully met (one can argue that 2.5 conditions have been “largely met”), .333 is a fantastic, above-average percentage in baseball. Therefore the analysis will proceed using ANOVA.

Results of ANOVA

R has the capacity to run ANOVA. Running ANOVA shows that the differences in OPS by different tightness of pants are statistically significant at the 0.01 significance level. Thus the conclusions of u/ItalicSlope are supported.

# Compute the analysis of variance
res.aov <- aov(OPS ~ Pants, data = data_clean)
# Summary of the analysis
summary(res.aov)
##             Df Sum Sq Mean Sq F value  Pr(>F)   
## Pants        2 0.0877 0.04383   7.443 0.00105 **
## Residuals   85 0.5006 0.00589                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Discussion

This paper has shown that the differences in OPS are statistically significant. Further investigations may involve uncovering which of the means are significantly different (using Tukey’s HSD analysis) and actually using the more appropriate repeated measures ANOVA test to further reinforce the conclusions in this paper.