\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \] Where…

• i = contestant
• \(x_i\) = number of episode wins for contestant i
• \(y_i\) = whether contestant i won the season (1=TRUE and 0=FALSE)
• \(\bar{x}\) = average number of episode wins across all contestants
• \(\bar{y}\) = average number of winners per season (1 per season)

The value of \(r\) gets closer to 1 as a contestant wins more episodes, indicating that contestants who win more episodes are more likely to win the season

\[ D_i = \frac{r_i^2}{p \cdot \hat{\sigma}^2} \cdot \frac{h_{ii}}{(1 - h_{ii})^2} \]

Where…

• \(r_i\) = residual
• \(h_{ii}\) = leverage
• \(p\) = number of variables (in this case, 1)
• \(\hat{\sigma}^2\) = residual variance

model <- glm(won_season ~ ep_wins, data = queen_stats, family = "binomial")

influence <- data.frame(
  residuals = residuals(model, type = "pearson"),
  leverage = hatvalues(model),
  distance = cooks.distance(model),
  name = queen_stats$contestant)

ggplot(influence, aes(x=leverage, y=residuals)) +
  geom_point(aes(size=distance, color=distance)) +
  geom_text(aes(label=ifelse(distance > 0.1, name, "")), 
            hjust=-0.15, vjust=0.3, size=3) +
  scale_color_gradient(low="skyblue", high="red") +
  labs(title="Contestant Influence on Season Outcome",
       x="Leverage",
       y="Residuals",
       color="Influence",
       size="Influence") +
  theme_minimal()

queen_stats <- queens %>%
  left_join(ep_wins, by = "contestant") %>%
  mutate(ep_wins = ifelse(is.na(ep_wins), 0, ep_wins))

plot_ly(queen_stats,
        x = ~age,
        y = ~ep_wins,
        z = ~placement,
        type = "scatter3d",
        mode = "markers",
        marker=list(size=4, color=~placement, colorscale='Viridis',
                    showscale=TRUE)) %>%
        layout(scene=list(
            xaxis=list(title="Age"),
            yaxis=list(title="Episode Wins"),
            zaxis=list(title="Season Placement")),
          title="Age vs Episode Wins vs Season Placement")