Week 7 - Data Dive - Hypothesis Testing
Raghuveer Venkatesh
2024-10-16
Introduction
In this analysis, we will explore FIFA player data to test two hypotheses. First, we’ll examine whether there’s a difference in the mean overall_rating between players with different preferred_foot (left vs. right)?. For the second hypothesis, we’ll assess whether there is a difference in overall ratings between players from different nationalities?
Hypothesis 1 (Neyman-Pearson Framework):
Is there a difference in the mean overall_rating between players with different preferred_foot (left vs. right)?
Null Hypothesis (H₀):
There is no significant difference in the mean overall_rating between left-footed and right-footed players.
Alternative Hypothesis (H₁):
There is a significant difference in the mean overall_rating between left-footed and right-footed players.
Checking descriptive statistics
# Descriptive statistics for finishing ability by preferred foot
summary_stats <- data |>
group_by(preferred_foot) |>
summarize(
mean_finishing = mean(finishing, na.rm = TRUE),
sd_finishing = sd(finishing, na.rm = TRUE),
n = n()
)
print(summary_stats)## # A tibble: 2 × 4
## preferred_foot mean_finishing sd_finishing n
## <chr> <dbl> <dbl> <int>
## 1 Left 47.0 17.5 4173
## 2 Right 44.9 20.2 13781Average Finishing Ability: Left-footed players have a higher average finishing ability score (46.97) compared to right-footed players (44.87). This suggests that, on average, left-footed players may have a slight edge in finishing ability based on the data you have analyzed.
Significance Level (Alpha, α)
We choose a significance level of 0.05. This means we are willing to accept a 5% chance of rejecting the null hypothesis when it is actually true (Type I error).
Power Level:
We aim for a power of 0.8, meaning we have an 80% chance of correctly rejecting the null hypothesis if there is indeed a significant effect (Type II error of 20%).
Calculating Effect Size: Cohen-D
# Filter data for left and right footed players
left_footed <- data$finishing[data$preferred_foot == "Left"]
right_footed <- data$finishing[data$preferred_foot == "Right"]
# Calculate means
mean_left <- mean(left_footed)
mean_right <- mean(right_footed)
# Calculate standard deviations
sd_left <- sd(left_footed)
sd_right <- sd(right_footed)
# Calculate sample sizes
n_left <- length(left_footed)
n_right <- length(right_footed)
# Calculate pooled standard deviation
s_pooled <- sqrt(((n_left - 1) * sd_left^2 + (n_right - 1) * sd_right^2) / (n_left + n_right - 2))
# Calculate Cohen's d
cohen_d <- (mean_left - mean_right) / s_pooled
cat("Calculated effect size is: ", cohen_d)## Calculated effect size is: 0.1072427This suggests that there is a small difference in finishing ability between left-footed and right-footed players
Sample size calculation
Effect size d = 0.1073, alpha = 0.05, power = 0.8
sample_size <- pwr.t.test(d = 0.5, sig.level = 0.05, power = 0.8, type = "two.sample")
sample_size##
## Two-sample t test power calculation
##
## n = 63.76561
## d = 0.5
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n is number in *each* groupConclusion
This output tells us that we need approximately 64 players per group with 80% power at a 0.05 significance level.
Testing the Hypothesis
Perform two-sample t-test
# Define parameters based on your analysis
mu1 <- 46.97340 # Mean finishing ability of left-footed players
sd1 <- 17.52159 # SD of finishing ability of left-footed players
n_left <- 4173 # Sample size of left-footed players
n_right <- 13781 # Sample size of right-footed players
# Calculate kappa (ratio of sample sizes) - Because sample sizes are unequal
kappa <- n_left / n_right
# Perform the power analysis for the hypothesis test
test <- pwrss.t.2means(mu1 = mu1,
sd1 = sd1,
kappa = kappa,
power = 0.8,
alpha = 0.1,
alternative = "not equal")## Difference between Two means
## (Independent Samples t Test)
## H0: mu1 = mu2
## HA: mu1 != mu2
## ------------------------------
## Statistical power = 0.8
## n1 = 2
## n2 = 6
## ------------------------------
## Alternative = "not equal"
## Degrees of freedom = 6
## Non-centrality parameter = 3.283
## Type I error rate = 0.1
## Type II error rate = 0.2# Print the results
print(test)## $parms
## $parms$mu1
## [1] 46.9734
##
## $parms$mu2
## [1] 0
##
## $parms$sd1
## [1] 17.52159
##
## $parms$sd2
## [1] 17.52159
##
## $parms$kappa
## [1] 0.3028082
##
## $parms$welch.df
## [1] FALSE
##
## $parms$paired
## [1] FALSE
##
## $parms$paired.r
## [1] 0.5
##
## $parms$alpha
## [1] 0.1
##
## $parms$margin
## [1] 0
##
## $parms$alternative
## [1] "not equal"
##
## $parms$verbose
## [1] TRUE
##
##
## $test
## [1] "t"
##
## $df
## [1] 6
##
## $ncp
## [1] 3.283402
##
## $power
## [1] 0.8
##
## $n
## n1 n2
## 2 6
##
## attr(,"class")
## [1] "pwrss" "t" "2means"# Plot the results
plot(test)
### Interpretation of results
Sample size n = 63.76561 This output tells us that we need approximately 64 players per group with 80% power at a 0.05 significance level.
The calculated Co-hen’s d of 0.107 indicates a small effect size. This means that while the difference is statistically significant, it may not be large enough to warrant major changes in coaching or player development strategies
Alpha = 0.1 Beta = 0.11
Hypothesis tests revealed a small but statistically significant difference in finishing ability between left-footed and right-footed players. The significance level (alpha) was set to 0.1, indicating that you were willing to accept a 10% chance of incorrectly rejecting the null hypothesis (Type I error).
With a desired power level (1 - beta) of 0.89, we set the acceptable Type II error rate (beta) to 0.11.
The observed difference in means is 2.10423
Overall, the small effect size suggests that any difference in training focus based on preferred foot may need to be carefully considered.
Fisher’s Significance Testing
Hypothesis 2: Is there a difference in overall ratings between players from different nationalities?
Null Hypothesis: There is no association between a player’s nationality and their overall rating category (high vs low). Alternative Hypothesis: There is an association between a player’s nationality and their overall rating category (high vs low).
For this hypothesis, we will perform Fisher’s exact test to evaluate the relationship between nationality and overall rating.
Fishers’s exact test
# List of main football countries
main_countries <- c("Brazil", "Argentina", "Germany")
# Filter the dataset for main football countries
main_football_data <- data |>
filter(nationality %in% main_countries)
# Categorize overall ratings into 'High' (>=70) and 'Low' (<70)
main_football_data$rating_category <- ifelse(main_football_data$overall_rating >= 70, "High", "Low")
# Create a contingency table between nationality and rating category
contingency_table <- table(main_football_data$nationality, main_football_data$rating_category)
# View the table to see the data distribution
print(head(contingency_table))##
## High Low
## Argentina 434 470
## Brazil 511 321
## Germany 310 889Perform Fisher’s Exact Test
# Increase the workspace and rerun Fisher's Exact Test
fisher_test_result <- fisher.test(contingency_table, workspace = 2e8) # Increase workspace size
# View the result
fisher_test_result##
## Fisher's Exact Test for Count Data
##
## data: contingency_table
## p-value < 2.2e-16
## alternative hypothesis: two.sidedResults
P-Value: The p-value of 2.2e-16 is extremely small, indicating a very strong statistical significance.
Because p-value is so small, you reject the null hypothesis. This means there is strong evidence to suggest that there are significant differences in overall ratings among players from different nationalities.
Visualisation
Since there are too many nationality, we take the top 5 by average rating
# Create a bar plot showing the distribution of rating categories by nationality
ggplot(main_football_data, aes(x = nationality, fill = rating_category)) +
geom_bar(position = "fill") +
labs(title = "Distribution of Overall Ratings by Nationality",
x = "Nationality", y = "Proportion",
fill = "Rating Category") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
scale_fill_manual(values = c("High" = "darkgreen", "Low" = "red"))
Further questions
Which specific nationalities show the most significant differences in overall ratings? Are there patterns in performance based on geography or cultural factors?
Are there other player attributes (like age, height, or position) that correlate with overall ratings? How do these factors interact with nationality in influencing player performance?