Mathematical Foundations
Correlation
Pearson Correlation
$$ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})} {\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} $$
Fisher Z Confidence Interval
$$ z = \tanh^{-1}(r) \newline SE = \frac{1}{\sqrt{n-3}} \newline $$ $$z_{CI} = z \pm z_{\alpha/2} \cdot SE, \quad CI = \tanh(z_{CI})$$
Used for all Pearson correlation CIs.
T-Test (Independent)
$$ t = \frac{\bar{x}_1 - \bar{x}_2} {\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} $$
Degrees of freedom follow Welch correction when variances differ.
Paired T-Test
$$ t = \frac{\bar{d}}{s_d / \sqrt{n}} $$
where $d$ is the difference vector.
ANOVA
$$ F = \frac{MS_{between}}{MS_{within}} $$
Where:
$$ MS = \frac{SS}{df} $$
Post-hoc uses Tukey HSD.
Welch ANOVA
When group normality is acceptable but variances are unequal, Indexly routes to Welch ANOVA with --auto-route.
Welch ANOVA uses group-specific variances and adjusted denominator degrees of freedom rather than assuming a common pooled variance.
Mann–Whitney U
Ranks pooled samples and evaluates difference in rank sums.
Kruskal–Wallis
Nonparametric alternative to ANOVA based on ranked data.
Confidence Interval (Mean)
$$ \bar{x} \pm t_{\alpha/2, df} \cdot \frac{s}{\sqrt{n}} $$
Confidence Interval (Proportion)
$$ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$
Mean Difference CI
$$ (\bar{x}_1 - \bar{x}2) \pm t{\alpha/2} \cdot SE $$
Regression (OLS)
$$ \hat{\beta} = (X^TX)^{-1}X^Ty $$
Standard errors derived from residual variance. When residual diagnostics indicate heteroscedasticity or non-normality and auto-routing is enabled, Indexly reports HC3 robust covariance results and recomputes coefficient confidence intervals from the robust model.
Mixed Effects
$$ y = X\beta + Z\gamma + \epsilon $$
Where:
- $X\beta$ fixed effects
- $Z\gamma$ random effects
Bootstrap
Resampling with replacement:
$$ \hat{\theta}^* = f(X^*) $$
CI derived from empirical percentiles.
Bayesian Independent T-Test
Indexly reports BF10, the Bayes factor for the alternative over the null, using the JZS Cauchy-prior t-test formulation.
Interpretation:
- BF10 < 1 → evidence favors the null
- 1 ≤ BF10 < 3 → anecdotal evidence for the alternative
- 3 ≤ BF10 < 10 → moderate evidence for the alternative
- BF10 ≥ 10 → strong evidence for the alternative
Statistical Power
OLS Regression
Indexly uses Cohen’s (f^2):
$$ f^2 = \frac{R^2}{1 - R^2} $$
Power is computed for the model F-test with numerator degrees of freedom equal to the number of model predictors and denominator degrees of freedom:
$$ df_2 = n - k - 1 $$
ANOVA
ANOVA effect size is reported as eta-squared:
$$ \eta^2 = \frac{SS_{between}}{SS_{total}} $$
Power uses Cohen’s (f), converted from eta-squared:
$$ f = \sqrt{\frac{\eta^2}{1 - \eta^2}} $$
Kruskal–Wallis (Effect Size)
The Kruskal–Wallis test evaluates whether rank distributions differ across groups.
When the test is significant, an effect size should be reported to assess practical significance.
Epsilon-Squared (ε²)
$$ \varepsilon^2 = \frac{H - k + 1}{n - k} $$
Where:
- $H$ = Kruskal–Wallis statistic
- $k$ = number of groups
- $n$ = total sample size
Interpretation Guidelines
- 0.01 → Small
- 0.06 → Medium
- 0.14 → Large
Eta-Squared for Kruskal–Wallis (η²ₕ)
$$ \eta^2_H = \frac{H - k + 1}{n - 1} $$
Where:
- $H$ = Kruskal–Wallis statistic
- $k$ = number of groups
- $n$ = total sample size
Example Calculation
H = 170.014
k = 7
n = 216000
epsilon_sq = (H - k + 1) / (n - k)
eta_sq_h = (H - k + 1) / (n - 1)
print("epsilon^2 =", epsilon_sq)
print("eta^2_H =", eta_sq_h)
Result:
$\varepsilon^2 \approx 0.00076 \quad$ $\eta^2_H \approx 0.00076$
Practical Interpretation
Although the Kruskal–Wallis test may be statistically significant (p < 0.0001), an ε² ≈ 0.00076 indicates a negligible practical effect.
Large sample sizes can produce statistical significance even when the real-world effect is extremely small.
Next
See Developer API for programmatic usage of the Indexly inference engine.