1. Neyman-Pearson Decision Framework vs. Fisherian Significance

Modern inferential testing synthesizes two distinct historical methodologies:

• Ronald Fisher’s P-Value Approach: Focuses on a single hypothesis—the Null Hypothesis (H₀). The p-value represents the conditional probability P(D ≥ d_obs | H₀) of observing data at least as extreme as the current sample, assuming H₀ is true. It serves as an index of evidence against the null, with no explicit alternative hypothesis specified.
• Jerzy Neyman & Egon Pearson’s Decision Theory: Establishes two competing statistical models: H₀ versus an Alternative Hypothesis (Ha). Before collecting data, the experimenter selects an acceptable false-positive risk α (Type I error) and designs the trial to minimize false negatives β (Type II error), maximizing Statistical Power (1 - β).

2. Student’s t-Distribution & Degrees of Freedom

When population variance σ² is unknown and must be estimated via sample variance s², the standardized variable follows William Sealy Gosset’s Student’s t-distribution rather than a normal distribution:

The t-distribution has heavier tails than the normal curve to compensate for the additional uncertainty of estimating σ. As sample size n grows large, the degrees of freedom df grow, and the t-distribution converges asymptotically to the standard Gaussian normal distribution N(0, 1).

For comparing two independent groups without assuming equal variances, Welch’s t-test must be used instead of Student’s pooled t-test. The Welch-Satterthwaite approximation computes degrees of freedom as:

3. Chi-Square (χ²) Test of Independence

Used to evaluate associations between categorical variables arranged in an r × c contingency table. Under the null hypothesis of independence, expected frequencies are computed as E_ij = (R_i × C_j) / N. The Pearson Chi-Square test statistic is:

Cochran’s Criteria: The asymptotic Chi-Square approximation is mathematically valid only when no cell has an expected frequency less than 1, and no more than 20% of cells have expected counts less than 5. If violated, Fisher’s Exact Test must be employed.

4. One-Way Analysis of Variance (ANOVA)

When comparing means across k ≥ 3 groups, conducting pairwise t-tests causes catastrophic inflation of the family-wise error rate (α_family = 1 - (1 - α)^m). ANOVA resolves this by partitioning total sum of squares (SST) into between-group variance (SSB) and within-group error variance (SSW):

If between-group variation significantly exceeds within-group random error (F > F_critical), the null hypothesis that all group means are identical is rejected. Post-hoc testing (Tukey’s HSD or Bonferroni correction) is subsequently conducted to locate specific pair differences.

5. Statistical Power & Sample Size Determination

Underpowered studies fail to detect true clinical or scientific effects, wasting capital and patient trials. Sample size sizing via Cochran’s formula establishes minimum required participants:

Where Z corresponds to desired confidence (1.96 for 95%), p is the estimated proportion (0.50 provides maximum variance and the most conservative sample estimate), and E is allowable margin of error. If sampling from a known finite population N, the Finite Population Correction (FPC) prevents excessive oversampling.

1. Neyman-Pearson Decision Framework vs. Fisherian Significance

Modern inferential testing synthesizes two distinct historical methodologies:

• Ronald Fisher’s P-Value Approach: Focuses on a single hypothesis—the Null Hypothesis (H₀). The p-value represents the conditional probability P(D ≥ d_obs | H₀) of observing data at least as extreme as the current sample, assuming H₀ is true. It serves as an index of evidence against the null, with no explicit alternative hypothesis specified.
• Jerzy Neyman & Egon Pearson’s Decision Theory: Establishes two competing statistical models: H₀ versus an Alternative Hypothesis (Ha). Before collecting data, the experimenter selects an acceptable false-positive risk α (Type I error) and designs the trial to minimize false negatives β (Type II error), maximizing Statistical Power (1 - β).

2. Student’s t-Distribution & Degrees of Freedom

When population variance σ² is unknown and must be estimated via sample variance s², the standardized variable follows William Sealy Gosset’s Student’s t-distribution rather than a normal distribution:

The t-distribution has heavier tails than the normal curve to compensate for the additional uncertainty of estimating σ. As sample size n grows large, the degrees of freedom df grow, and the t-distribution converges asymptotically to the standard Gaussian normal distribution N(0, 1).

For comparing two independent groups without assuming equal variances, Welch’s t-test must be used instead of Student’s pooled t-test. The Welch-Satterthwaite approximation computes degrees of freedom as:

3. Chi-Square (χ²) Test of Independence

Used to evaluate associations between categorical variables arranged in an r × c contingency table. Under the null hypothesis of independence, expected frequencies are computed as E_ij = (R_i × C_j) / N. The Pearson Chi-Square test statistic is:

Cochran’s Criteria: The asymptotic Chi-Square approximation is mathematically valid only when no cell has an expected frequency less than 1, and no more than 20% of cells have expected counts less than 5. If violated, Fisher’s Exact Test must be employed.

4. One-Way Analysis of Variance (ANOVA)

When comparing means across k ≥ 3 groups, conducting pairwise t-tests causes catastrophic inflation of the family-wise error rate (α_family = 1 - (1 - α)^m). ANOVA resolves this by partitioning total sum of squares (SST) into between-group variance (SSB) and within-group error variance (SSW):

If between-group variation significantly exceeds within-group random error (F > F_critical), the null hypothesis that all group means are identical is rejected. Post-hoc testing (Tukey’s HSD or Bonferroni correction) is subsequently conducted to locate specific pair differences.

5. Statistical Power & Sample Size Determination

Underpowered studies fail to detect true clinical or scientific effects, wasting capital and patient trials. Sample size sizing via Cochran’s formula establishes minimum required participants:

Where Z corresponds to desired confidence (1.96 for 95%), p is the estimated proportion (0.50 provides maximum variance and the most conservative sample estimate), and E is allowable margin of error. If sampling from a known finite population N, the Finite Population Correction (FPC) prevents excessive oversampling.