11.1 Inferential Statistics, Correlation, Regression, ANOVA, and Effect Sizes

11.1 Inferential Statistics, Correlation, Regression, ANOVA, and Effect Sizes

Quick Answer: Choosing the valid statistical model in sport science depends on four structural factors: the measurement scale and distribution (parametric continuous vs. non-parametric ordinal/skewed), the sampling design (independent groups vs. repeated within-athlete measures), the number of conditions ($2$ vs. $\ge 3$), and whether longitudinal observations contain missing data and nested random effects (requiring Linear Mixed-Effects Models [LMMs]). Because $p\text{-values}$ confound effect magnitude with sample size, inferential tests must always be paired with standardized effect sizes (Cohen's $d$, Hedges' $g$, Partial Eta-Squared [$\eta_p^2$], Correlation [$r$], or Odds Ratios [$\text{OR}$]) and confidence intervals.

In high-performance sport science, sample sizes are naturally constrained by roster limits ($N = 15\text{–}60$ athletes), and datasets frequently exhibit skewness, outliers, or missing observations due to injury and international travel. Mastering assumption diagnostics, test selection, regression modeling, and effect-size scaling ensures that performance conclusions rest on sound mathematical foundations.

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Evaluating Parametric Assumptions: Normality, Homoscedasticity, and Sphericity

Before applying parametric models that rely on sample means and standard deviations, the sport scientist must audit three distributional assumptions:

1. Normality of Residuals / Distributions:
   • Formal Hypothesis Test: The Shapiro-Wilk test is preferred for typical squad sample sizes ($n < 50$). The null hypothesis ($H_0$) states that the data follow a normal Gaussian distribution. Therefore, $p > 0.05$ satisfies the normality assumption, whereas $p \le 0.05$ indicates a statistically significant departure from normality (skewness or heavy tails).
   • Visual & Descriptive Diagnostics: Because the Shapiro-Wilk test can be underpowered in tiny samples ($n < 12$) and overly sensitive in huge wearables datasets ($n > 1{,}000$), practitioners always inspect Quantile-Quantile (Q-Q) plots (checking that standardized residuals adhere closely to the $45^\circ$ diagonal line), histograms, and standardized skewness and kurtosis ratios.
2. Homogeneity of Variance (Homoscedasticity):
   • Tested across independent groups via Levene's test (or Bartlett's test). A result of $p > 0.05$ confirms equal variances across position groups or cohorts. If Levene's test is violated ($p \le 0.05$) in a two-group comparison, the practitioner substitutes Welch's unpooled $t\text{-test}$, which adjusts the degrees of freedom via the Welch-Satterthwaite equation to prevent Type I error inflation.
3. Sphericity in Repeated-Measures Designs:
   • In repeated-measures Analysis of Variance ($\text{ANOVA}$) across $\ge 3$ timepoints, sphericity requires that the variances of the differences between all possible pairs of within-subject conditions are equal ($\sigma_{T1-T2}^2 \approx \sigma_{T1-T3}^2 \approx \sigma_{T2-T3}^2$).
   • Mauchly's Test of Sphericity: Evaluates whether sphericity holds ($p > 0.05$). When Mauchly's test is significant ($p \le 0.05$, indicating sphericity is violated), the standard unadjusted $F\text{-test}$ overestimates degrees of freedom and inflates the Type I (false positive) error rate.
   • Epsilon ($\epsilon$) Degrees-of-Freedom Corrections: When sphericity is violated, practitioners multiply both numerator and denominator degrees of freedom by an epsilon adjustment factor ($0 < \epsilon \le 1$), which raises the critical $F\text{-value}$ required for significance: the Greenhouse-Geisser correction (more conservative; recommended when $\epsilon < 0.75$) or the Huynh-Feldt correction (less conservative; preferred when $\epsilon \ge 0.75$).

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Parametric vs. Non-Parametric Statistical Test Selection Matrix

When data are ordinal (such as raw $1\text{–}5$ Likert soreness scores) or remain severely skewed after logarithmic transformation, non-parametric rank-based equivalents must be selected:

ANOVA Familywise Error Control and Post-Hoc Testing

In a One-Way $\text{ANOVA}$, the omnibus $F\text{-ratio}$ divides systematic between-group variance by unsystematic within-group error variance:

$$F = \frac{\text{MS}_{\text{between}}}{\text{MS}_{\text{within}}}$$

A significant omnibus $F\text{-test}$ ($p < 0.05$) indicates only that at least one group mean differs from another, without specifying which pair differs. Running multiple unadjusted $t\text{-tests}$ across $k$ pairs inflates the Familywise Type I Error Rate ($\alpha_{\text{FW}} = 1 - (1 - \alpha)^c$, rising to $14.3\%$ across $3$ comparisons and $40.1\%$ across $10$ comparisons at $\alpha = 0.05$). Practitioners control $\alpha_{\text{FW}}$ using:

• Tukey's Honestly Significant Difference ($\text{HSD}$): Optimal for all-pairwise comparisons among balanced independent groups.
• Bonferroni Correction: Divides the alpha threshold by the number of comparisons ($\alpha_{\text{adjusted}} = \alpha / c$, e.g., $0.05 / 6 = 0.0083$). Highly conservative, increasing Type II (false negative) error risk when $c$ is large.
• Holm-Bonferroni Step-Down Procedure: Ranks $p\text{-values}$ from smallest to largest and applies sequentially less restrictive denominators ($\alpha / c, \alpha / [c-1], \dots$), preserving greater statistical power than classical Bonferroni while strictly controlling $\alpha_{\text{FW}}$.

Why Linear Mixed-Effects Models (LMMs) Have Replaced Repeated-Measures ANOVA

Classical repeated-measures $\text{ANOVA}$ suffers from a fatal flaw in applied sport settings: listwise deletion. If an athlete misses even a single testing timepoint out of $10$ due to national team call-up or minor illness, repeated-measures $\text{ANOVA}$ discards that athlete's entire longitudinal record. Linear Mixed-Effects Models ($\text{LMMs}$) overcome this via Restricted Maximum Likelihood ($\text{REML}$) estimation:

• Retaining Unbalanced & Missing Data: $\text{LMMs}$ incorporate every valid observation from every athlete under Missing at Random ($\text{MAR}$) conditions without listwise deletion.
• Fixed Effects vs. Random Effects: $\text{LMMs}$ simultaneously estimate population-level fixed effects (such as playing position, match location, or environmental temperature) alongside subject-specific random intercepts (accounting for each athlete's unique baseline physiological ceiling) and random slopes (modeling how individual athletes adapt or fatigue at different rates in response to the same training load).

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Correlation, Regression, and Effect Size Magnitude Scales

Bivariate Correlation and the Coefficient of Determination ($R^2$)

• Pearson Product-Moment Correlation ($r$): Quantifies the linear relationship between two continuous, normally distributed variables ($-1.0 \le r \le +1.0$).
• Spearman's Rank-Order Correlation ($\rho$ or $r_s$): Quantifies the monotonic relationship between ranked, ordinal, or skewed variables, reducing distortion from extreme outliers.
• Hopkins's Correlation Magnitude Scale:
  • $|r| < 0.10$: Trivial
  • $0.10\text{–}0.29$: Small
  • $0.30\text{–}0.49$: Moderate
  • $0.50\text{–}0.69$: Large
  • $0.70\text{–}0.89$: Very Large
  • $|r| \ge 0.90$: Nearly Perfect
• Coefficient of Determination ($R^2 = r^2$): Squaring the correlation coefficient reveals the exact proportion of variance in the outcome variable ($Y$) explained by the predictor ($X$). For example, a "large" correlation of $r = 0.60$ between isometric mid-thigh pull peak force and sprint acceleration yields $R^2 = 0.60^2 = 0.36$, meaning $36\%$ of the variance in sprint time is explained by isometric force, while $64\%$ ($1 - R^2$) remains unexplained (attributable to technique, rate of force development, and measurement error).

Ordinary Least Squares (OLS) Multiple Regression and Binary Logistic Regression

In Linear and Multiple Regression ($Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \epsilon$), practitioners evaluate three model diagnostics:

1. Adjusted $R^2$: Unlike raw $R^2$—which artificially increases every time any random variable is added to a model—Adjusted $R^2$ penalizes the model for adding predictors that do not meaningfully improve fit relative to the lost degrees of freedom.
2. Standard Error of the Estimate ($\text{SEE}$): Quantifies the standard deviation of the residuals around the regression line in the original measurement units of $Y$ (e.g., $\pm 2.4\text{ kg}$ for a $1\text{RM}$ prediction equation from barbell velocity).
3. Variance Inflation Factor ($\text{VIF} = \frac{1}{1 - R_j^2}$): Diagnoses multicollinearity (entering highly redundant predictors into the same regression model, such as including both Total Distance and Meters per Minute). A $\text{VIF} > 5\text{ to }10$ signals severe multicollinearity, which inflates standard errors and makes individual $\beta$ coefficients unstable and uninterpretable.

When the dependent outcome is dichotomous ($0 = \text{Uninjured}, 1 = \text{Injured}$; or $\text{Win vs. Loss}$), ordinary linear regression is invalid because predicted probabilities can fall below $0$ or above $1$ and residuals are heteroscedastic. Practitioners instead fit a Binary Logistic Regression modeling the natural log-odds ($\text{logit}$) of the event:

$$\ln\left(\frac{P}{1 - P}\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k$$

Exponentiating a predictor's regression slope ($\beta$) yields the Odds Ratio ($\text{OR} = e^{\beta}$):

• $\text{OR} = 1.00$: Predictor has no association with the outcome odds.
• $\text{OR} > 1.00$: Each $1\text{-unit}$ increase in $X$ multiplies the odds of the outcome (e.g., $\text{OR} = 2.40$ means $2.4\times$ higher odds, or $+140\%$ increased odds).
• $\text{OR} < 1.00$: Predictor is protective (e.g., $\text{OR} = 0.55$ indicates a $45\%$ reduction in the odds of injury per unit increase in eccentric hamstring strength).

Standardized Effect Sizes for Group and ANOVA Comparisons

• Cohen's $d$ and Hedges' $g$: Standardize the difference between two means in standard deviation units ($d = \frac{\bar{X}_1 - \bar{X}_2}{SD_{\text{pooled}}}$). When sample sizes are small ($n < 20$ per group), Cohen's $d$ slightly overestimates the population effect; Hedges' $g$ multiplies $d$ by a degrees-of-freedom correction factor ($J \approx 1 - \frac{3}{4\text{df} - 1}$) to eliminate small-sample bias. On Hopkins's applied sport science scale, $|d| < 0.20$ is trivial, $0.20\text{–}0.59$ is small, $0.60\text{–}1.19$ is moderate, $1.20\text{–}1.99$ is large, and $\ge 2.00$ is very large (whereas Cohen's classic behavioral thresholds are $0.20$ small, $0.50$ medium, and $0.80$ large).
• Eta-Squared ($\eta^2$) and Partial Eta-Squared ($\eta_p^2$): Quantify the proportion of variance explained by an $\text{ANOVA}$ main effect or interaction:

$$\eta_p^2 = \frac{\text{SS}_{\text{effect}}}{\text{SS}_{\text{effect}} + \text{SS}_{\text{error}}}$$

Conventional $\eta_p^2$ benchmarks classify $0.01$ as a small effect, $0.06$ as a medium effect, and $\ge 0.14$ as a large effect.