Summary of "Parametric Test | Non parametric test | T-test | ANOVA | wilcoxon rank sum test | Friedman test"

Big picture: parametric vs non‑parametric tests

• Parametric tests

  • Assume data follow a known distribution (usually normal / bell‑shaped).
  • Work with quantitative data and focus on means and standard deviations.
  • Generally more powerful when assumptions are met.
  • Typical assumptions: normality (or sampling distribution of the mean approximately normal), random independent sampling, homogeneity of variances. Often suitable for larger samples; t‑tests are commonly used for smaller samples when population SD is unknown.
  • Examples: Student’s t‑test, ANOVA.

• Non‑parametric tests

  • Distribution‑free (do not assume a specific underlying distribution); useful for skewed data or when population parameters are unknown.
  • Often applied to ordinal data, ranks, medians, percentages/proportions, or qualitative variables.
  • Generally less powerful than parametric tests but more robust when assumptions fail.
  • Examples: Mann–Whitney U (Wilcoxon rank‑sum), Wilcoxon signed‑rank, Kruskal–Wallis, Friedman, Chi‑square, sign test, median test.

General hypothesis testing procedure (applies to parametric and non‑parametric)

1. State null hypothesis H0 and alternative hypothesis Ha.
2. Compute the appropriate test statistic from the sample data.
3. Compare the test statistic to a critical/table value (or compare the p‑value to significance level α).
   • If the test statistic exceeds the critical value (or p < α), reject H0.
   • Otherwise, fail to reject H0.
4. Report the result (state decision and brief conclusion).

Decision can be phrased either by comparing the test statistic to a table/critical value or by comparing the p‑value to α.

Parametric tests — details and when to use

Student’s t‑test

• Purpose: test means when population SD is unknown; commonly used with small samples.
• Typical assumptions:
  • Data (or sampling distribution of the mean) approximately normal.
  • Random independent sampling.
  • Population SD unknown; commonly used when sample size < 30 (guideline).
  • Homogeneity of variance when comparing two groups (depends on variant).
• Common variants:
  • One‑sample t‑test: compare sample mean x̄ to known population mean μ.
    • Formula: t = (x̄ − μ) / (s / √n)
  • Independent (two‑sample / unpaired) t‑test: compare means of two independent samples.
    • General idea: t = (x̄1 − x̄2) / SE where SE depends on sample SDs and n1, n2 (pooled or unpooled).
  • Paired t‑test: compare paired observations (e.g., before vs after); analyze differences within pairs.
• t‑distribution notes:
  • Bell‑shaped and symmetric like the normal distribution but with heavier tails at low degrees of freedom.
  • As degrees of freedom increase, the t‑distribution approaches the normal distribution.

ANOVA (Analysis of Variance)

• Purpose: compare means across more than two groups (or across groups defined by factors).
• Core idea: partition variance into between‑group and within‑group components and test whether between‑group variance is large relative to within‑group variance.
• Assumptions: normality, independent random samples, homogeneity of variances.
• Common types:
  • One‑way ANOVA: compare means across three or more groups for one factor.
  • Two‑way ANOVA: compare means based on two factors; can test interaction effects.
• Decision rule: compare the F statistic to critical value (or p < α). If significant, conclude that not all group means are equal.
• Post‑hoc: when ANOVA rejects H0, use post‑hoc comparisons to identify which pairs differ.

Least Significant Difference (LSD) — simple post‑hoc procedure

• Purpose: identify which pairs of group means differ after a significant ANOVA.
• Basic steps:
  1. Compute group means.
  2. Compute LSD. For equal group sizes:
     • LSD = t_{α/2, df_error} × sqrt(2 × MSE / n)
       • MSE = mean square error from ANOVA, df_error = error degrees of freedom, n = group sample size (adjust formula for unequal n).
  3. Compute absolute pairwise differences between group means.
  4. A pairwise difference ≥ LSD is considered statistically significant.

Non‑parametric tests — details and when to use

Mann–Whitney U (Wilcoxon rank‑sum)

• Purpose: non‑parametric alternative to the independent two‑sample t‑test; compares distributions/medians of two independent samples.
• Typical sample size guideline (lecture): each sample ideally between about 5 and 20 when using table‑based critical values; for larger samples use normal approximation.
• Null hypothesis: the two populations have equal medians (or identical distributions).
• Common formulation for U:
  • U1 = n1·n2 + n1(n1+1)/2 − R1
    • n1, n2 = sample sizes; R1 = sum of ranks for sample 1.
  • Compute U2 similarly (swap indices). The smaller U is used for decision.
• Decision rule (table approach): if U ≤ U_critical, reject H0. For large samples, use normal approximation.

Wilcoxon signed‑rank test

• Purpose: paired non‑parametric alternative to the paired t‑test; compares median differences for paired data.
• Use when data are paired and assumptions for paired t‑test are not met.

Kruskal–Wallis test

• Purpose: non‑parametric alternative to one‑way ANOVA; compares more than two independent groups using ranks.
• Null hypothesis: population medians (or distributions) are equal across groups.
• Test statistic:
  • H = (12 / [N(N+1)]) × Σ (Rj^2 / nj) − 3(N+1)
    • N = total sample size, Rj = sum of ranks for group j, nj = size of group j.
• Decision: if H ≥ H_critical (or p < α), reject H0.

Friedman test

• Purpose: non‑parametric alternative to repeated‑measures/two‑way ANOVA; used for comparing more than two related groups (blocks/subjects).
• Null hypothesis: treatment medians are equal across repeated measures/blocks.
• Common test statistic (approximate χ²):
  • χ^2_F = (12 / [n k (k+1)]) × Σ R_j^2 − 3 n (k+1)
    • n = number of blocks (subjects), k = number of treatments, R_j = sum of ranks for treatment j.
• Decision: if χ^2_F ≥ χ^2_critical (or p < α), reject H0.

Classification / memory aids

• Parametric tests: focus on means and standard deviations (examples: t‑tests, ANOVA).
• Non‑parametric classification (lecture examples):
  • One‑sample tests: sign test, chi‑square (as applicable).
  • Two‑sample tests: Mann–Whitney.
  • k‑sample tests: Kruskal–Wallis, median test.
  • Repeated measures: Friedman.
• Lecturer emphasized common exam points: definitions, assumptions, when to apply each test, formulas for test statistics, and decision criteria.

Practical / testing tips (emphasized)

• When describing parametric tests, explicitly list assumptions (normality, independence, homogeneity).
• For t‑tests mention sample size guidance (t commonly used when n < 30), unknown population SD, and random independent sampling.
• For rank tests, state that the approach uses ranks/medians and note sample size considerations (e.g., minimum size for Mann–Whitney when using tables).
• Exam structure recommendation: present H0/H1, test statistic formula, decision rule, and conclusion (reject / fail to reject H0).
• After a significant ANOVA, perform a post‑hoc test (LSD or other) to identify which pairs of means differ.

Corrections / clarifications (subtitle errors)

• Auto‑captioning garbled several test names; standard names and equivalents:
  • “Mann‑Whitney U test” = Wilcoxon rank‑sum / Mann–Whitney.
  • Garbled “Wilcox‑on‑Rungsum” likely refers to Wilcoxon rank‑sum or Wilcoxon signed‑rank.
  • Garbled “Krause‑Kul‑Wallis” = Kruskal–Wallis.
  • Garbled “Fried‑Mann” = Friedman test.
• Where lecture formulas were partially garbled, standard textbook formulas are given above.

Speakers / sources referenced

• Lecturer / narrator (Depth of Biology instructor/video author).
• William Sealy Gosset (developer of Student’s t‑test; pen name “Student”).
• Ronald A. Fisher (developer of ANOVA).
• Milton Friedman (developer of the Friedman test).
• Resource/platform mentioned: Depth of Biology (dpbiology.com / Play Store).

Category

Educational

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