ܒܘܚܪܢܐ ܕ-Kruskal-Wallis

ܛܥܢܐ ܕܝܗܝ̈ܒܬܐ (loading the data)

ܟܡܕܒܩܚ ܥܡܘ̈ܕܐ ܬܪܝܨܐܝܬ ܡܢ ܓܠܝܘܢܐ (worksheet)، ܥܡ ܣܕܪܐ ܕܪ̈ܫܐ (header row) – ܟܪܫܡܚ ܓܘ ܐܩܣܠ، ܟܢܣܚܚ، ܘܟܡܕܒܩܚ ܠܐܟܐ:

ܓܘܒܝܐ ܕܡܫܬܚܠܦ̈ܢܐ (Variable selection)

ܐܝܡܢ ܡܦܠܚܝܚ (When to use)؟

ܐܝܡܢ ܕܐܝܬ ܬܠܬܐ ܓܘ̈ܕܐ ܝܢ ܝܬܝܪ ܕܡܫܬܘܬܦ̈ܢܐ ܦܪ̈ܝܫܐ ܘܚܕ ܡܫܬܚܠܦܢܐ ܩܐ ܦܘܚܡܐ ܒܝܢܝ ܐܢܝ – ܘܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance) ܕܚܕ ܐܘܪܚܐ ܠܐ ܦܐܐ ܝܠܗ. Gravetter ܘ-Wallnau (2017, Appendix E.4) ܟܡܚܘܝܝ Kruskal-Wallis ܐܝܟ ܚܘܠܦܐ ܕܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ ܕܨܚܚܐ 12: "ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܣܢܝܩ ܝܠܗ ܠܡܢܝ̈ܢܐ ܡܢܝܢܝ̈ܐ ܕܡܢܝܗܝ ܟܚܫܒܚ ܡܨܥܝ̈ܬܐ ܘܫܘܚܠ̈ܦܐ؛ Kruskal-Wallis ܣܢܝܩ ܝܠܗ ܒܣ ܕܡܨܝܐ ܠܡܛܟܣ ܡܫܬܘܬܦ̈ܢܐ" – ܘܐܝܠܗ ܡܬܚܬܐ ܕ-Mann-Whitney ܡܢ ܬܪܬܝܢ ܓܘ̈ܕܐ ܠܬܠܬܐ ܘܝܬܝܪ. ܓܘ̈ܕܐ ܡܨܝܐ ܕܗܘܝܐ ܡܘܨ̈ܒܐ ܕ-treatment (ܝܠܘ̈ܕܐ ܕܟܨܝܪܝ ܒܠܚܘܕܝܗܘܢ، ܓܘ ܓܘܕܐ، ܝܢ ܥܡ ܢܛܘܪܘܬܐ – ܘܟܡܛܟܣܚ ܟܠ ܨܘ̈ܪܬܐ ܐܟܚܕܐ ܠܦܘܬ ܒܪܝܬܢܘܬܐ) ܝܢ ܟܢܫ̈ܐ ܕܐܝܬ (ܒܢܝ̈ 5، 6 ܘ-7 ܫܢ̈ܐ). Field (2024، ܦܣܩܐ 7.6) ܟܡܚܘܐ ܠܗ ܐܝܟ ܒܘܚܪܢܐ ܩܐ "ܓܘ̈ܕܐ ܠܐ ܬܠܝ̈ܬܐ ܣܓܝ̈ܐܐ" ܒܓܢܒ ܒܘܚܪܢܐ ܕ-Friedman ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ، ܓܘ ܡܬܠܐ ܕܐܟܠ̈ܬܐ ܕܣܘܝܐ (soy) ܘܡܢܝܢܐ ܕܙܪܥܐ: ܐܪܒܥܐ ܓܘ̈ܕܐ ܕ-20 ܓܒܪ̈ܐ، ܘܒܘܚܪܢܐ ܕܝܗܝ̈ܒܬܐ (ܦܣܩܐ 7.6.3) ܟܡܚܘܐ ܓܘ ܟܠ ܓܘܕܐ ܨܘܪܬܐ Q-Q ܕܣܛܝܐ ܡܢ ܣܪܛܐ ܡܨܠܝܐ، Shapiro-Wilk ܪܫܝܡܐ (ܠܒܪ ܡܢ ܚܕܐ ܓܘܕܐ)، ܘ-Levene ܪܫܝܡܐ – "ܬܫܥܝܬܐ ܥܨܝܒܬܐ: ܒܠܟܝ ܠܐ ܢܘܪܡܠܝܐ ܘܒܠܟܝ ܫܘܚܠ̈ܦܐ ܠܐ ܫܘ̈ܝܐ". Pallant (2020، ܨܚܚܐ 16): "ܚܘܠܦܐ ܠܐ ܦܪܡܝܛܪܝܐ ܕܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ… ܡܢܝ̈ܢܐ ܡܫܘܚܠܦ̈ܐ ܝܢܐ ܠܕܪ̈ܓܐ ܕܛܟܣܐ، ܘܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ ܕܟܠ ܓܘܕܐ ܡܬܦܚܡܐ ܝܠܗ" – "ܐܢ ܕܪܓܐ ܕܣܒܪܐ ܛܒܐ (optimism) ܦܪܝܫܐ ܝܠܗ ܓܘ ܬܠܬܐ ܓܘ̈ܕܐ ܕܥܘܡܪܐ؟".

ܓܘ ܐܗܐ ܣܐܝܛ ܕܘܟܐ ܟܝܢܝܐ ܝܠܗ ܩܐ ܬܪܝܢ ܡܘܨ̈ܒܐ: ܦܣܘܩܐ ܕܕܪܓܐ ܝܚܝܕܝܐ (Likert 1-5)، ܕܠܦܘܬ ܥܝܕܐ ܕܣܐܝܛ ܡܫܬܚܠܦܢܐ ܕܛܟܣܐ ܝܠܗ ܘܠܝܬ ܠܗ ܡܨܥܝܬܐ (ܡܫܬܚܠܦ̈ܢܐ ܘܕܪ̈ܓܐ ܕܡܫܘܚܬܐ) – ܦܘܚܡܐ ܒܝܢܝ ܟܡܐ ܓܘ̈ܕܐ ܥܒܝܕܐ ܝܠܗ ܗܪܟܐ، ܝܢ ܓܘ ܣܓܝܐܘ̈ܬܐ ܘܒܟܝ ܡܪܒܥܐ؛ ܘܡܢܝܢܐ ܕܣܒܠܬܐ ܝܢ ܡܫܬܚܠܦܢܐ ܟܡܝܝܐ ܓܘ ܓܘ̈ܕܐ ܙܥܘܪ̈ܐ ܝܢ ܥܡ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ، ܐܝܡܢ ܕܒܘܚܪܢܐ ܕܢܘܪܡܠܝܘܬܐ (ܒܘܚܪܢܐ ܕܢܘܪܡܠܝܘܬܐ، ܠܦܘܬ ܓܘܕܐ) ܝܢ boxplot ܟܡܚܘܝܝ ܦܘܠܓܐ ܕܠܐ ܦܐܐ ܠܦܘܫܩܐ ܕܫܘܚܠܦܐ. ܐܝܡܢ ܕܟܠ ܓܘ̈ܕܐ ܪ̈ܒܐ ܝܢܐ (30 ܘܠܥܠ) ܘܝܗܝ̈ܒܬܐ ܟܡܝ̈ܝܬܐ ܝܢܐ، ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ ܝܠܗ ܓܘܒܝܐ ܡܩܕܡܐ – Gravetter ܘ-Wallnau: "ܐܢ ܝܗܝ̈ܒܬܐ ܦܐܝ̈ܐ ܝܢܐ ܠ-ANOVA ܝܢ ܠܚܕ ܡܢ ܒܘܚܪ̈ܢܐ t، ܒܘܚܪܢܐ ܬܩܝܢܐ ܛܒ ܝܠܗ ܡܢ ܚܘܠܦܐ ܩܐ ܝܗܝ̈ܒܬܐ ܕܛܟܣܐ" (Appendix E.1) – ܘ-Kruskal-Wallis ܟܡܚܝܠ ܠܗ ܝܢ ܟܫܚܠܦ ܠܗ ܠܦܘܬ ܪܥܝܢܐ ܕܒܘܨܝܐ؛ ܩܘܛܝܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ" ܟܐܡܪܐ ܓܘ ܐܝܢܝ ܡܘܨܒܐ ܝܬܘܢ. ܬܪܬܝܢ ܓܘ̈ܕܐ ܒܠܚܘܕ – ܒܘܚܪܢܐ ܕ-Mann-Whitney (ܗܘ p، ܘܝܬܝܪ U، p ܚܬܝܬܐ ܘܬܪܝܢ ܡܚܘ̈ܝܢܐ ܕܡܥܒܕܢܘܬܐ)؛ ܗܢܝ ܡܫܬܘܬܦ̈ܢܐ ܓܘ ܟܡܐ ܡܫܘ̈ܚܬܐ – ܒܘܚܪܢܐ ܕ-Friedman (Field, Section 7.7)، ܠܐ ܓܘ ܐܗܐ ܦܐܬܐ.

ܗܘܢܐ: ܕܪ̈ܓܐ ܕܛܟܣܐ ܘ-H

ܐܝܟ ܓܘ Mann-Whitney، ܟܡܛܟܣܚ ܟܠ ܡܢܝ̈ܢܐ ܐܟܚܕܐ ܡܢ ܢܚܝܬܐ (ܕܪܓܐ ܕܛܟܣܐ 1) ܠܪܡܐ "ܕܠܐ ܚܫܒܢܐ ܕܓܘܕܐ ܕܡܢܝܢܐ ܡܢܗ ܝܠܗ" (Field, 2024, Section 7.6.1)، ܘܛܝ̈ܡܐ ܫܘ̈ܝܐ ܟܫܩܠܝ ܡܨܥܝܬܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ ܕܝܠܗܘܢ (Section 7.4.1; Gravetter & Wallnau, 2017, Appendix E.1). ܒܬܪ ܗܕܐ "ܟܡܟܢܫܚ ܡܢܝ̈ܢܐ ܬܘܒ ܠܓܘ̈ܕܐ" ܘܟܓܡܪܚ ܕܪ̈ܓܐ ܕܛܟܣܐ ܓܘ ܟܠ ܓܘܕܐ – R_i ܓܘ Field، T ܓܘ Gravetter ܘ-Wallnau. ܐܢ ܠܝܬ ܦܘܪܫܢܐ، ܕܪ̈ܓܐ ܪ̈ܡܐ ܘܢܚܝ̈ܬܐ ܟܡܬܒܕܪܝ ܒܝܢܝ ܓܘ̈ܕܐ ܘܕܪ̈ܓܐ ܕܛܟܣܐ ܡܨܥܝ̈ܐ ܕܡܝ̈ܐ ܝܢܐ ܠܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ ܓܘܢܝܐ (N + 1)/2؛ ܐܢ ܚܕܐ ܓܘܕܐ ܪܡܬܐ ܝܠܗ، ܕܪ̈ܓܐ ܪ̈ܡܐ ܟܡܬܟܢܫܝ ܓܘܗ. ܡܚܘܝܢܐ ܣܛܐܛܝܣܛܝܩܝܐ (statistic) ܟܡܫܚܐ ܒܕܪܐ ܕܓܘ̈ܡܪܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ: H = 12/(N(N + 1)) · Σ(R_i²/n_i) − 3(N + 1) (ܫܘܝܘܬܐ 7.26؛ ܬܘܣܦܬܐ E.4 – ܗܝ ܫܘܝܘܬܐ ܥܡ T). ܓܘ ܡܬܠܐ ܕܣܘܝܐ ܓܘ̈ܡܪܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ ܝܢܐ 927، 883، 883 ܘ-547 (ܓܘܕܐ ܕܐܟܠܠܗ ܣܘܝܐ ܫܒܥܐ ܙܒ̈ܢܐ ܒܫܒܬܐ – ܢܚܝܬܐ)، ܘ-H = 251.66 − 243 = 8.659 (ܫܘܝܘܬܐ 7.27)؛ ܓܘ ܡܬܠܐ E.2 ܕ-Gravetter ܘ-Wallnau (ܬܠܬܐ ܓܘ̈ܕܐ ܕܚܡܫܐ، T = 39.5, 26.5, 54) – H = 3.785. ܝܗܝ̈ܒܬܐ ܕܬܪܝܢ ܡܬܠ̈ܐ ܐܝܬ ܓܘ ܒܘܚܪ̈ܢܐ ܕܦܐܬܐ.

ܪܫܝܡܘܬܐ. H ܡܬܦܠܓܐ ܝܠܗ ܩܪܝܒܐܝܬ ܐܝܟ ܟܝ ܡܪܒܥܐ ܥܡ df = k − 1 – ܡܢܝܢܐ ܕܓܘ̈ܕܐ ܒܨܪ ܚܕܐ (Field, Section 7.6.1; Gravetter & Wallnau, Appendix E.4). ܩܐ 4 ܓܘ̈ܕܐ df = 3 ܘ-H = 8.659 ܟܝܗܒ p = .034؛ ܩܐ 3 ܓܘ̈ܕܐ ܛܝܡܐ ܚܪܝܦܐ (critical value) ܓܘ .05 ܝܠܗ 5.99، ܘ-3.785 ܠܐ ܥܒܪ ܠܗ – "ܝܗܝ̈ܒܬܐ ܠܐ ܟܝܗܒܝ ܣܗܕܘܬܐ ܣܦܩܬܐ ܩܐ ܦܘܪܫܢܐ ܒܝܢܝ ܬܠܬܐ treatments". ܩܘܪܒܐ ܣܢܝܩ ܝܠܗ ܠܐ ܒܨܪ ܡܢ ܚܡܫܐ ܚܙ̈ܝܬܐ ܓܘ ܟܠ ܓܘܕܐ (Gravetter & Wallnau). SPSS ܟܬܪܨ H ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ (ties) – ܡܥܝܪܢܘܬܐ "The test statistic is adjusted for ties" ܓܘ ܡܦܩܢܐ ܕ-Pallant – ܒܦܘܠܓܐ ܥܠ C = 1 − Σ(t³ − t)/(N³ − N)؛ ܫܘܝܘܬܐ ܕܟܬܒ̈ܐ ܠܐ ܟܬܪܨܐ، ܘܦܘܪܫܢܐ ܓܘ ܡܬܠܐ E.2 (ܬܪܝܢ ܛܝ̈ܡܐ ܡܬܢ̈ܝܐ) ܝܠܗ 3.785 ܩܘܒܠ 3.819. ܦܐܬܐ ܟܛܒܥܐ ܡܬܘܪܨܐ، ܘܓܘ ܡܥܝܪܢܘܬܐ ܕܠܘܚܐ ܐܦ ܠܐ ܡܬܘܪܨܐ.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܠܘܚܐ 1 – ܕܪ̈ܓܐ ܕܛܟܣܐ ܘܛܝ̈ܡܐ ܡܨܥܝ̈ܐ. ܠܘܚܐ Ranks ܕ-SPSS (N, Mean Rank) ܥܡ ܓܘܡܪܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ، ܛܝܡܐ ܡܨܥܝܐ، ܪ̈ܘܒܥܐ ܘܦܬܝܘܬܐ ܕܟܠ ܓܘܕܐ. ܡܕܡ ܩܕܡܝܐ ܕܟܩܪܝܚ: ܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ – Pallant (2020، ܨܚܚܐ 16): "ܟܒܚܪܚ ܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ ܕܓܘ̈ܕܐ… ܐܗܐ ܟܐܡܪ ܐܝܢܝ ܓܘܕܐ ܐܝܬ ܠܗ ܛܟܣܐ ܓܘܢܝܐ ܪܡܐ ܡܢ ܟܠܗ، ܕܦܐܐ ܠܡܢܝܢܐ ܪܡܐ ܡܢ ܟܠܗ"؛ ܓܘ ܡܬܠܐ ܕܝܠܗ ܓܘܕܐ ܕܥܘܡܪܐ ܣܒܐ (45+) ܥܡ ܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ ܪܡܐ – "ܣܒܪܐ ܛܒܐ ܪܡܐ ܡܢ ܟܠܗ"، ܘܥܠܝܡܬܐ – ܢܚܝܬܐ. ܓܘ ܡܬܠܐ ܕܣܘܝܐ: 46.35، 44.15، 44.15 ܘ-27.35 – ܬܠܬܐ ܓܘ̈ܕܐ ܕܡܝ̈ܐ ܘܚܕܐ ܢܚܝܬܐ. ܛܝܡܐ ܡܨܥܝܐ ܝܠܗ ܡܐ ܕܟܡܘܬܢܚ ܒܕܘܟܬܐ ܕܡܨܥܝܬܐ (Field, 2024، ܦܣܩܐ 7.6.8؛ Pallant ܟܡܦܩܐ ܠܗ ܒܓܫܩܐ ܦܪܝܫܐ ܓܘ Means: 22، 22 ܘ-23).

ܠܘܚܐ 2 – ܒܘܚܪܢܐ. ܓܘ ܡܦܩܢܐ ܚܕܬܐ ܕ-SPSS: Total N، Test Statistic (ܐܗܐ ܝܠܗ H؛ Pallant: "ܟܝ ܡܪܒܥܐ، ܕܡܚܘܝܐ ܝܠܗ ܐܝܟ Test Statistic")، Degree of Freedom ܘ-Asymptotic Sig. (2-sided test)؛ ܓܘ ܡܦܩܢܐ ܥܬܝܩܐ – Chi-Square, df, Asymp. Sig.. ܢܡܘܣܐ (Field، ܥܝܬ̈ܐ ܕ-Cramming Sam ܩܐ ܨܚܚܐ 7): "ܟܚܙܚ ܓܘ ܣܕܪܐ Asymptotic Sig. – ܛܝܡܐ ܕ-.05 ܘܠܬܚܬ ܒܪܘܒܐ ܡܬܚܫܒܐ ܝܠܗ ܦܘܪܫܢܐ ܪܫܝܡܐ ܒܝܢܝ ܓܘ̈ܕܐ". ܓܘ ܡܬܠܐ ܕܣܘܝܐ H(3) = 8.66, p = .034 – "ܒܢܝ̈ܢܫܐ ܒܪܘܒܐ ܒܕ ܦܠܛܝ ܦܘܣܩܢܐ ܕܟܡܝܘܬܐ ܕܐܟܠ̈ܬܐ ܕܣܘܝܐ ܟܡܥܒܕܐ ܪܫܝܡܐܝܬ ܥܠ ܡܢܝܢܐ ܕܙܪܥܐ"؛ ܓܘ Pallant χ²(2, n = 435) = 8.57, p = .014. ܐܠܐ ܒܘܚܪܢܐ ܓܘܢܝܐ "ܟܐܡܪ ܕܡܢܝ̈ܢܐ ܕܙܪܥܐ ܦܪ̈ܝܫܐ ܗܘܘ ܓܘ ܡܢܬܐ ܡܢ ܓܘ̈ܕܐ، ܐܠܐ ܠܐ ܐܝܢܝ ܓܘ̈ܕܐ" (Field, Section 7.6.5) – ܘܩܐ ܗܕܐ ܠܘܚܐ ܕܒܬܪܗ.

Boxplot ܝܠܗ ܡܐ ܕܡܦܩܢܐ ܚܕܬܐ ܕ-SPSS ܟܡܙܝܕ ܠܒܘܚܪܢܐ (ܡܦܩܢܐ 7.6): Field ܟܩܪܐ ܡܢܗ ܕܩܘܛܝܐ ܕܓܘܕܐ ܕܫܒܥܐ ܐܟܠ̈ܬܐ ܙܥܘܪܐ ܝܠܗ ܝܬܝܪ ("ܡܢܝ̈ܢܐ ܚܒܝܫ̈ܐ ܝܢܐ ܝܬܝܪ") ܘܕܛܝܡܐ ܡܨܥܝܐ ܕܝܠܗ "ܟܡܬܚܙܐ ܩܠܝܠ ܢܚܝܬܐ ܡܢ ܐܚܪ̈ܢܐ" – "ܒܠܟܝ ܦܘܪܫܢܐ ܝܬܝܒܐ ܝܠܗ ܒܝܢܝ ܓܘܕܐ ܕܫܒܥܐ ܐܟܠ̈ܬܐ ܘܐܚܪ̈ܢܐ؟" – ܘܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers؛ ܚܘ̈ܕܪܐ ܠܥܠ ܡܢ whiskers) ܝܢܐ "ܟܡܐ ܓܒܪ̈ܐ ܕܥܒܕܠܗܘܢ ܟܡܝܘܬܐ ܫܢܝܬܐ ܣܓܝ".

ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ – ܠܘܚܐ 3

"ܐܘܪܚܐ ܦܫܝܛܬܐ ܡܢ ܟܠܗ ܩܐ ܦܪܩܐ ܡܥܒܕܢܘܬܐ ܓܘܢܝܬܐ ܝܠܗ ܦܘܚܡܐ ܕܟܠ ܙܘ̈ܓܐ ܕܓܘ̈ܕܐ" (Field, 2024, Section 7.6.2) – ܓܘ 4 ܓܘ̈ܕܐ ܫܬܐ ܙܘ̈ܓܐ. ܡܨܝܐ ܗܘܐ ܠܡܪܗܛ ܫܬܐ ܒܘܚܪ̈ܢܐ ܕ-Mann-Whitney، ܐܠܐ "ܐܝܡܢ ܕܥܒܕܚ ܣܓܝ ܒܘܚܪ̈ܢܐ ܥܠ ܗܢܝ ܝܗܝ̈ܒܬܐ ܟܡܢܦܚܚ ܡܢܬܐ ܕܛܥܝܘܬܐ ܕܫܪܒܬܐ (familywise)" (ܦܣܩܐ 2.10.8) – ܘܡܛܠ ܗܕܐ p ܕܟܠ ܦܘܚܡܐ ܡܬܘܪܨܐ ܝܠܗ ܗܟܢܐ ܕܓܘ ܟܠ ܒܘܚܪ̈ܢܐ ܐܟܚܕܐ ܣܘܟܢܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I ܦܐܫ 5%. ܠܘܚܐ ܕܦܐܬܐ ܟܛܒܥܐ ܝܠܗ ܠܘܚܐ Pairwise Comparisons ܕ-SPSS (ܡܦܩܢܐ 7.7؛ ܓܘ Pallant – Pairwise Comparisons of age 3 groups): ܩܐ ܟܠ ܙܘܓܐ ܡܚܘܝܢܐ ܣܛܐܛܝܣܛܝܩܝܐ ܝܠܗ ܦܘܪܫܢܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ ܡܨܥܝ̈ܐ – "ܩܐ 7 ܩܘܒܠ 1 ܐܟܠ̈ܬܐ 44.15 − 27.35 = 16.80" – ܛܥܝܘܬܐ ܬܩܝܢܬܐ ܕܝܠܗ، z (Std. Test Statistic: "ܦܘܪ̈ܫܢܐ ܡܫܘܚܠܦ̈ܐ ܝܢܐ ܠܡܢܝ̈ܢܐ z ܒܦܘܠܓܐ ܥܠ ܛܥܝܘܬܐ ܬܩܝܢܬܐ، ܘܡܢܝܗܝ p"؛ ܩܐ 7 ܩܘܒܠ 1: z = 2.286, p = .022)، ܘܬܪܝܢ ܛܝ̈ܡܐ ܕ-p: Sig. ܘ-Adj. Sig. – p ܡܟܘܦܠܐ ܒܡܢܝܢܐ ܕܦܘܚܡ̈ܐ (ܬܘܪܨܐ ܕ-Bonferroni، ܡܥܝܪܢܘܬܐ ܓܘ ܠܘܚܐ)، ܥܕܡܐ 1. Field: "ܥܡܘܕܐ Adj. Sig. ܐܝܬ ܒܓܘܗ p ܡܬܘܪ̈ܨܐ، ܘܠܐܢܝ ܟܡܦܫܩܚ (ܠܐ ܦܐܫ ܟܡܐ ܕܡܓܪܓܐ ܝܠܗ ܠܡܦܫܩ ܥܡܘܕܐ Sig.)". ܓܘ ܡܬܠܐ ܕܣܘܝܐ ܚܕ ܛܝܡܐ ܡܬܘܪܨܐ ܠܐ ܝܠܗ ܬܚܘܬ .05 – "ܐܦܢ ܕܦܘܚܡܐ ܒܝܢܝ ܫܒܥܐ ܐܟܠ̈ܬܐ ܘܣܦܝܩܘܬܐ ܟܩܪܒ، p = .058، ܘܐܗܐ ܟܡܥܗܕ ܠܢ ܕܒܘܚܪ̈ܢܐ ܕܪܫܝܡܘܬܐ ܟܡܓܪܓܝ ܚܘܫܒܐ ܕܐܘܟܡܐ-ܚܘܪܐ، ܘܡܚܘ̈ܝܢܐ ܕܡܥܒܕܢܘܬܐ ܡܨܝܐ ܕܥܕܪܝ": "ܥܡ ܕܐܝܬ ܡܥܒܕܢܘܬܐ ܓܘܢܝܬܐ ܪܫܝܡܬܐ، ܚܕ ܦܘܚܡܐ ܦܪܝܫܐ ܠܐ ܟܡܚܘܐ ܦܘܪܫܢܐ ܪܫܝܡܐ". Pallant، ܒܚܘܠܦܐ، ܟܒܨܝܐ ܥܡܘܕܐ Sig. ("ܟܒܥܝܚ ܛܝ̈ܡܐ ܬܚܘܬ .05") ܘܟܡܫܟܚܐ ܚܕ ܦܘܚܡܐ ܪܫܝܡܐ، ܥܠܝܡ̈ܐ ܩܘܒܠ ܣܒ̈ܐ – ܐܦܢ ܕܓܘ ܠܘܚܐ ܕܝܠܗ Sig. = .004 ܘ-Adj. Sig. = .011 (ܬܪܘܝܗܝ ܪ̈ܫܝܡܐ؛ ܓܘ ܟܬܒܐ ܕܝܠܗ ܛܒܝܥܐ .002). ܥܝܕܐ ܕܣܐܝܛ: ܟܩܪܝܚ p ܡܬܘܪܨܐ، ܐܝܟ Field ܘܐܝܟ SPSS؛ ܠܘܚܐ ܟܛܒܥ ܐܦ ܠܐ ܡܬܘܪܨܐ. ܐܢܝ ܦܘܚܡ̈ܐ ܝܢܐ ܒܘܚܪܢܐ ܕܝܕܝܥܐ ܝܠܗ ܐܝܟ ܒܘܚܪܢܐ ܕ-Dunn – ܗܟܢܐ ܡܬܩܪܐ ܝܠܗ ܓܘ R (dunn_test)، ܘ-SPSS ܠܐ ܟܡܥܗܕ ܫܡܗ.

SPSS ܟܡܚܝܪ ܐܦ ܦܘܠܚܢܐ ܕܓܫ̈ܩܐ (Stepwise step-down، ܡܦܩܢܐ 7.8): ܓܘ̈ܕܐ ܡܛܟܣ̈ܐ ܝܢܐ ܠܦܘܬ ܓܘܡܪܐ ܕܕܪ̈ܓܐ ܕܛܟܣܐ، ܘܓܘ ܟܠ ܓܫܩܐ ܟܒܚܪܚ ܐܢ ܓܘܕܐ ܕܒܬܪܗ "ܫܘܝܐ ܝܠܗ" ܠܩܕܡ̈ܝܬܐ؛ ܦܐܪܐ ܝܠܗ ܓܘ̈ܕܐ ܙܥܘܪ̈ܐ ܗܘܡܘܓܢ̈ܝܐ، ܥܡ ܬܘܪܨܐ ܕܠܐ ܩܫܝܐ ܝܬܝܪ – ܓܘ ܡܬܠܐ ܕܣܘܝܐ ܓܘܕܐ ܕܫܒܥܐ ܐܟܠ̈ܬܐ ܒܠܚܘܕܗ، ܘܬܠܬܐ ܐܚܪ̈ܢܐ ܐܟܚܕܐ (p = .943). Field: "ܐܢܝ ܦܘܠܚ̈ܢܐ ܡܣܘܒ̈ܟܐ ܝܢܐ ܒܝܫ، ܠܐ ܡܨܛܥܪܝܬܘܢ ܐܢ ܠܐ ܦܗܡܝܬܘܢ ܠܗܘܢ ܒܓܡܝܪܘܬܐ". ܦܐܬܐ ܠܐ ܟܚܫܒܐ ܠܗ (ܚܙܝܡܘܢ "ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ").

ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ – r_rb ܩܐ ܟܠ ܙܘܓܐ

"ܠܝܬ ܐܘܪܚܐ ܦܫܝܩܬܐ ܩܐ ܫܘܚܠܦܐ ܕܡܚܘܝܢܐ ܕ-Kruskal-Wallis ܥܡ ܝܬܝܪ ܡܢ ܚܕ ܕܪܓܐ ܕܚܐܪܘܬܐ ܠܡܚܘܝܢܐ ܕܡܥܒܕܢܘܬܐ r. ܐܠܐ ܡܚܘܝܢܐ ܕܡܥܒܕܢܘܬܐ ܐܝܟ ܐܗܐ ܒܟܠ ܐܘܪܚܐ ܠܐ ܝܠܗ ܡܘܬܪܢܐ ܣܓܝ (ܡܛܠ ܕܟܢܫܐ ܡܥܒܕܢܘܬܐ ܓܘܢܝܬܐ). ܓܘ ܪܘܒܐ ܕܡܘܨ̈ܒܐ ܚܫܝܚ ܝܬܝܪ ܠܡܝܕܥ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܕܦܘܚܡܐ ܡܢܝܫܐ" (Field, 2024, Section 7.6.7) – ܘܡܛܠ ܗܕܐ Field ܟܚܫܒ ܡܚܘܝܢܐ ܕܡܥܒܕܢܘܬܐ ܩܐ ܟܠ ܙܘܓܐ: U ܕܒܘܚܪܢܐ ܕ-Mann-Whitney ܥܠ ܬܪܬܝܢ ܓܘ̈ܕܐ، ܘܡܢ r_rb = 1 − 2U/(n₁n₂) (ܫܘܝܘܬܐ 7.11؛ ܠܘܚܐ 7.4). ܓܘ ܡܬܠܐ ܕܣܘܝܐ ܡܥܒ̈ܕܢܘܝܬܐ "ܪ̈ܒܐ ܝܢܐ (ܘܕܡܝ̈ܐ ܒܪܒܘܬܐ) ܩܐ ܫܒܥܐ ܐܟܠ̈ܬܐ ܩܘܒܠ ܟܠ ܓܘ̈ܕܐ ܐܚܪ̈ܢܐ، ܩܪܝܒ .5، ܐܠܐ ܙܥܘܪ̈ܐ ܛܒܐ (r_rb < .1) ܩܐ ܦܘܚܡ̈ܐ ܒܝܢܝ ܓܘ̈ܕܐ ܐܚܪ̈ܢܐ" – .48، .39 ܘ-.44 ܩܘܒܠ .04، .06 ܘ-.02. ܩܐ ܕܫܩܠ ܫܬܐ ܛܝ̈ܡܐ ܕ-U ܓܘ SPSS، Field ܟܡܪܗܛ ܫܬܐ NPAR TESTS /M-W ܓܘ ܣܝܢܬܐܟܣ ("ܥܕܡܐ ܕܡܝܬܚ ܡܢ ܡܥܩܬܐ – ܒܛܝܒܘܬܐ ܣܝܢܬܐܟܣ ܚܒܪܐ ܕܝܘܟܘܢ ܝܠܗ")؛ ܦܐܬܐ ܟܚܫܒܐ ܠܗܘܢ ܡܚܕܐ ܘܟܛܒܥܐ ܠܗܘܢ ܓܘ ܠܘܚܐ 3. ܪܘܫܡܐ ܟܐܡܪ ܐܘܪܚܐ: ܚܝܘܒܝܐ – ܓܘܕܐ ܩܕܡܝܬܐ ܓܘ ܙܘܓܐ ܪܡܬܐ ܝܠܗ (ܓܘ ܠܘܚܐ 7.4 ܕ-Field ܛܝ̈ܡܐ ܛܒܝ̈ܥܐ ܝܢܐ ܕܠܐ ܪܘܫܡܐ)؛ ܘܩܘܛܝܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ" ܟܡܬܪܓܡܐ ܠܗ ܠܡܢܬܐ ܒܡܐܐ ܕܙܘ̈ܓܐ ܕܟܣܡܟܝ (Kerby, as cited in Field) ܘܠܡܠܬܐ ܓܘ ܣܪܓܠܐ ܕ-Cohen (1988، ܦܐ̈ܬܐ 79-83: .1 ܙܥܘܪܐ، .3 ܡܨܥܝܐ، .5 ܪܒܐ – ܐܝܟ ܓܘ ܦܐܬܐ ܕ-Mann-Whitney). Pallant ܠܐ ܟܡܝܬܝܐ ܡܚܘܝܢܐ ܕܡܥܒܕܢܘܬܐ ܩܐ ܐܗܐ ܒܘܚܪܢܐ – ܓܘ ܬܘܢܝܐ ܕܝܠܗ ܛܝ̈ܡܐ ܡܨܥܝ̈ܐ ܝܢܐ ܡܚܘܝܢܐ.

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

ܬܪܝܢ ܫܪ̈ܛܐ ܓܘܢܝ̈ܐ (Pallant, 2020, Chapter 16): ܓܘܒܝܐ ܓܕܫܢܝܐ ܕܕܘܡܝܐ، ܘܚܙ̈ܝܬܐ ܠܐ ܬܠܝ̈ܬܐ – "ܦܘܫܩܐ ܒܝܢܝ ܓܘ̈ܕܐ، ܘܡܛܠ ܗܕܐ ܓܘ ܟܠ ܓܘܕܐ ܒܢܝ̈ܢܫܐ ܦܪ̈ܝܫܐ"؛ ܗܢܝ ܡܫܬܘܬܦ̈ܢܐ ܓܘ ܟܡܐ ܡܘܨ̈ܒܐ – ܒܘܚܪܢܐ ܕ-Friedman. ܚܡܫܐ ܚܙ̈ܝܬܐ ܓܘ ܓܘܕܐ. "ܫܪܛܐ ܚܕ… ܩܐ ܬܪܨܐ ܡܦܠܚܢܘܬܐ ܕܦܘܠܓܐ ܕܟܝ ܡܪܒܥܐ: ܟܠ ܡܘܨܒܐ ܣܢܝܩ ܝܠܗ ܕܐܚܕ ܠܐ ܒܨܪ ܡܢ ܚܡܫܐ ܡܢܝ̈ܢܐ" (Gravetter & Wallnau, 2017, Appendix E.4)؛ ܦܐܬܐ ܟܡܙܗܪܐ، ܘ-SPSS ܟܡܚܝܪ p ܚܬܝܬܐ ܓܘ ܡܘܕܘܠ ܕܒܘܚܪ̈ܢܐ ܚܬܝ̈ܬܐ (Exact Tests). ܚܝܠܐ. Field (ܩܘܛܝܐ 7.1): ܕܪ̈ܓܐ ܕܛܟܣܐ ܟܡܘܒܕܝ ܝܕܥܬܐ ܥܠ ܪܒܘܬܐ ܕܦܘܪ̈ܫܢܐ، ܘܡܛܠ ܗܕܐ ܐܝܡܢ ܕܫܪ̈ܛܐ ܕܒܘܚܪܢܐ ܦܪܡܝܛܪܝܐ ܡܫܬܡܠܝܐ ܝܢܐ ܐܗܘ ܚܝܠܢܐ ܝܠܗ ܝܬܝܪ؛ ܘܐܝܡܢ ܕܠܐ ܡܫܬܡܠܝܐ ܝܢܐ، ܡܐܡܪܐ "ܩܐ ܒܘܚܪ̈ܢܐ ܠܐ ܦܪܡܝܛܪ̈ܝܐ ܒܨܪ ܚܝܠܐ ܐܝܬ" ܠܐ ܝܠܗ ܡܬܩܢܐ. ܒܘܚܪܢܐ ܩܕܡܝܐ. Field (ܦܣܩܐ 7.6.3) ܟܒܚܪ ܩܕܡ ܒܘܚܪܢܐ ܢܘܪܡܠܝܘܬܐ ܠܦܘܬ ܓܘܕܐ (ܨܘ̈ܪܬܐ Q-Q، ܘܓܘ ܓܘ̈ܕܐ ܙܥܘܪ̈ܐ ܐܦ Shapiro-Wilk – "ܐܢ ܒܘܚܪܢܐ ܚܙܐܠܗ ܣܛܝܘܬܐ ܓܘ ܕܘܡܝܐ ܙܥܘܪܐ ܗܟܢܐ، ܒܠܟܝ ܪܒܬܐ ܝܠܗ") – ܦܐܬܐ ܒܘܚܪܢܐ ܕܢܘܪܡܠܝܘܬܐ ܠܦܘܬ ܓܘܕܐ.

ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ (missing values). ܡܫܬܘܬܦܢܐ ܥܠܝܠܐ ܝܠܗ ܐܝܡܢ ܕܐܝܬ ܠܗ ܛܝܡܐ ܬܪܝܨܐ ܓܘ ܡܫܬܚܠܦܢܐ ܬܠܝܐ ܘܓܘ ܡܫܬܚܠܦܢܐ ܕܓܘ̈ܕܐ – ܫܩܠܬܐ ܠܦܘܬ ܦܘܫܩܐ (/MISSING ANALYSIS). ܒܘܚܪ̈ܢܐ ܣܓܝ̈ܐܐ. ܟܡܐ ܡܫܬܚܠܦ̈ܢܐ ܒܚܕ ܛܪܦܐ ܟܡܐ ܒܘܚܪ̈ܢܐ ܓܘܢ̈ܝܐ ܝܢܐ، ܟܠ ܚܕ ܥܡ 5% ܣܘܟܢܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I؛ ܬܘܪܨܐ ܓܘ ܠܘܚܐ 3 ܓܘ ܟܠ ܒܘܚܪܢܐ ܒܠܚܘܕ ܝܠܗ. ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. ܦܘܠܚܢܐ ܕܓܫ̈ܩܐ (Stepwise step-down)، ܒܘܚܪܢܐ ܕܛܝܡܐ ܡܨܥܝܐ (ܥܝܬܐ 7.3 ܕ-Field)، ܒܘܚܪܢܐ ܕܡܨܠܝܢܘܬܐ (trend) Jonckheere-Terpstra (ܦܣܩܐ 7.6.6: "ܐܝܡܢ ܕܡܣܟܝܚ ܕܛܝ̈ܡܐ ܡܨܥܝ̈ܐ ܣܠܩܝ ܝܢ ܢܚܬܝ ܒܛܟܣܐ ܕܓܘ̈ܕܐ" – ܓܘ ܡܬܠܐ ܕܣܘܝܐ J = 912, z = −2.48, p = .013)، p ܚܬܝܬܐ ܘ-Monte Carlo، ܡܚܘܝܢܐ ܕܡܥܒܕܢܘܬܐ ܓܘܢܝܐ (ܐܝܬܐ ܡܪܒܥܐ ܕ-H – ܠܝܬ ܓܘ ܚܕ ܡܢ ܡܒܘ̈ܥܐ)، ܘܒܘܚܪܢܐ ܕ-Friedman ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ.

ܘܕܟܝ ܟܥܒܕܚ ܐܗܐ ܒ-SPSS؟ (And how do you do it in SPSS?)

ܐܘܪܚܐ ܚܕܬܐ (Pallant, 2020, Chapter 16; Field, 2024, Section 7.6.4): Analyze › Nonparametric Tests › Independent Samples… – ܓܘ ܠܫܢܝܬܐ (tab) Objective ܟܓܒܚ Customize analysis؛ ܓܘ Fields – ܡܫܬܚܠܦܢܐ ܬܠܝܐ ܠ-Test Fields ܘܡܫܬܚܠܦܢܐ ܕܓܢ̈ܣܐ ܠ-Groups؛ ܓܘ Settings – Customize tests، Kruskal-Wallis 1-way ANOVA، ܘܓܘ Multiple comparisons – All pairwise (ܝܢ Stepwise step-down)؛ ܩܕܡ ܗܕܐ، Pallant: ܓܘ Edit › Options › Output ܠܡܓܒܐ Pivot tables and charts. ܡܦܩܢܐ: Hypothesis Test Summary (ܒܣ ܦܘܣܩܢܐ)، ܘܬܪܝܢ ܛܪ̈ܦܐ ܟܦܬܚܝ Independent-Samples Kruskal-Wallis Test Summary ܥܡ boxplot ܘܠܘܚܐ ܕ-Pairwise Comparisons. ܛܝ̈ܡܐ ܡܨܥܝ̈ܐ ܠܝܬ ܓܘ ܡܦܩܢܐ – Pallant ܟܡܦܩܐ ܠܗܘܢ ܓܘ Analyze › Compare Means › Means ܥܡ Median ܓܘ Cell Statistics.

ܣܝܢܬܐܟܣ ܕܦܐܬܐ ܟܒܪܝܐ ܐܚܝܕ ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ: ܥܬܝܩܬܐ (Analyze › Nonparametric Tests › Legacy Dialogs › K Independent Samples…، ܥܡ Define Range ܩܐ ܪ̈ܡܙܐ)، ܕܟܛܒܥܐ Ranks ܘ-Test Statistics ܐܝܟ ܓܘ ܠܘ̈ܚܐ 1-2 ܘܟܪܗܛܐ ܐܦ ܓܘ PSPP؛ ܘܚܕܬܐ ܕ-Pallant، ܕܟܡܙܝܕܐ ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ (ܠܘܚܐ 3):

NPAR TESTS
  /K-W=עמדות_ציון BY השכלה(1 4)
  /MISSING ANALYSIS.

NPTESTS
  /INDEPENDENT TEST (עמדות_ציון) GROUP (השכלה) KRUSKAL_WALLIS(COMPARE=PAIRWISE)
  /MISSING SCOPE=ANALYSIS USERMISSING=EXCLUDE
  /CRITERIA ALPHA=0.05 CILEVEL=95.

MEANS TABLES=עמדות_ציון BY השכלה
  /CELLS=COUNT MEDIAN.

ܩܪܝܢܐ ܕܡܦܩܢܐ (output). Ranks = ܠܘܚܐ 1 (N, Mean Rank)؛ Test Statistics = ܠܘܚܐ 2 (Chi-Square = H ܡܬܘܪܨܐ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ، df, Asymp. Sig.)؛ Pairwise Comparisons = ܠܘܚܐ 3 (Test Statistic, Std. Error, Std. Test Statistic, Sig., Adj. Sig. – ܚܙܝܡܘܢ ܕ-SPSS ܟܡܛܟܣ ܙܘ̈ܓܐ ܠܦܘܬ ܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ، ܘܪܘܫܡܐ ܕܡܚܘܝܢܐ ܬܠܝܐ ܝܠܗ ܒܢܘܣܚܐ: ܓܘ ܡܦܩܢܐ ܕ-Field ܚܝܘܒܝܐ، ܓܘ ܡܦܩܢܐ ܕ-Pallant ܣܠܒܝܐ؛ ܦܐܬܐ ܟܛܒܥܐ ܩܕܡܝܬܐ ܒܨܪ ܬܪܝܢܝܬܐ ܒܛܟܣܐ ܕܕܪ̈ܓܐ)؛ MEANS – ܛܝ̈ܡܐ ܡܨܥܝ̈ܐ. ܣܝܢܬܐܟܣ ܒܚܝܪܐ ܝܠܗ ܩܘܒܠ PSPP (ܡܫܘܡܠܝܢܘܬܐ ܚܐܪܬܐ): ܕܪ̈ܓܐ ܕܛܟܣܐ ܡܨܥܝ̈ܐ، H، df ܘ-p ܚܕ ܝܢܐ ܥܡ ܠܘ̈ܚܐ 1-2؛ NPTESTS ܠܝܬ ܓܘ PSPP.

ܘܕܟܝ ܟܥܒܕܚ ܐܗܐ ܒ-R؟ (And how do you do it in R?)

ܓܘ R ܫܬܐܣܝܐ kruskal.test(y, g) ܟܝܗܒ H ("Kruskal-Wallis chi-squared"، ܡܬܘܪܨܐ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ ܐܝܟ SPSS)، df ܘ-p. ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ ܕ-SPSS ܟܝܗܒ dunn_test ܡܢ ܚܙܡܐ (package) rstatix ܥܡ p.adjust.method = "bonferroni" – ܗܢܝ z ܘ-p (ܪܘܫܡܐ ܕ-z ܬܡܢ ܝܠܗ ܬܪܝܢܝܬܐ ܒܨܪ ܩܕܡܝܬܐ)؛ ܘ-r_rb ܟܚܫܒܚ ܡܢ wilcox.test ܩܐ ܟܠ ܙܘܓܐ (W ܕ-R ܝܠܗ n₁n₂ − U₁). ܣܩܪܝܦܬ ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ ܘܝܗܒܠܗ ܛܝ̈ܡܐ ܕܠܘ̈ܚܐ (ܓܘ ܡܥܝܪ̈ܢܘܬܐ). ܢܡܘܣܐ ܕܣܐܝܛ ܩܐ ܣܘܪܝܬ ܓܘ R: ܦܐܝܠ UTF-8، ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ، ܘܐܬܪܐ UTF-8 ܩܐ ܫܡܗ̈ܐ ܥܒܪ̈ܝܐ.

# ܒܘܚܪܢܐ ܕ-Kruskal-Wallis – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)    # ܩܪܝܢܐ ܕܐܩܣܠ
library(rstatix)   # ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ ܕ-Dunn ܥܡ ܬܘܪܨܐ ܕ-Bonferroni (ܐܝܟ NPTESTS ܓܘ SPSS)

# ܛܥܢܐ: ܛܘܦܣܐ ܬܩܝܢܐ – ܣܕܪܐ = ܡܫܬܘܬܦܢܐ، ܥܡܘܕܐ = ܡܫܬܚܠܦܢܐ؛ ܓܠܝܘܢܐ "נתונים"
d <- read_excel("C:/Rdata/DATA_kw_example_heb.xlsx", sheet = "נתונים")
ok <- !is.na(d[["עמדות_ציון"]]) & !is.na(d[["השכלה"]])
y <- d[["עמדות_ציון"]][ok]; g <- factor(d[["השכלה"]][ok])

# ܛܝܡܐ ܡܨܥܝܐ ܘܪ̈ܘܒܥܐ (type = 6 = SPSS) ܘ-n ܩܐ ܟܠ ܓܘܕܐ – ܠܘܚܐ 1
do.call(rbind, tapply(y, g, function(v) c(n = length(v), Mdn = median(v), quantile(v, c(.25, .75), type = 6))))
                                                      # n 17/18/50/32; Mdn 3.38, 3.31, 3.06, 3.50

# ܕܪ̈ܓܐ ܕܛܟܣܐ: ܟܠ ܛܝ̈ܡܐ ܡܛܟܣ̈ܐ ܝܢܐ ܐܟܚܕܐ (ܕܪܓܐ ܡܨܥܝܐ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ)؛ ܓܘܡܪܐ ܘܕܪܓܐ ܕܛܟܣܐ ܡܨܥܝܐ ܩܐ ܟܠ ܓܘܕܐ
r <- rank(y); cbind(sumRank = tapply(r, g, sum), meanRank = tapply(r, g, mean))   # 68.94, 57.94, 49.16, 69.69

# ܒܘܚܪܢܐ: H ܥܡ ܬܘܪܨܐ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ (ܐܝܟ SPSS)، df = k − 1، p ܡܢ ܟܝ ܡܪܒܥܐ – ܠܘܚܐ 2
kruskal.test(y, g)                                   # chi-squared = 8.884, df = 3, p = .031

# ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ: z، p ܘ-p ܡܬܘܪܨܐ ܕ-Bonferroni – ܠܘܚܐ 3
dunn_test(data.frame(y = y, g = g), y ~ g, p.adjust.method = "bonferroni")   # 3 ܩܘܒܠ 4: z = 2.68, p = .007، ܡܬܘܪܨܐ .045

# ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܩܐ ܟܠ ܙܘܓܐ – r_rb = 1 − 2U1/(n1n2) (Field, 2024، ܠܘܚܐ 7.4)؛ W ܕ-wilcox.test = n1n2 − U1
a <- y[g == "3"]; b <- y[g == "4"]
U1 <- length(a) * length(b) - wilcox.test(a, b, exact = FALSE)$statistic
1 - 2 * U1 / (length(a) * length(b))                 # −.345 (ܓܘܕܐ 4 ܪܡܬܐ)

# ܡܬܠܐ ܕ-Field (2024، ܠܘܚܐ 7.3): ܐܪܒܥܐ ܓܘ̈ܕܐ ܕ-20 – H(3) = 8.66, p = .034؛ 7 ܩܘܒܠ 1 ܐܟܠ̈ܬܐ: z = 2.29, p = .022، ܡܬܘܪܨܐ .133
# ܡܬܠܐ E.2 ܕ-Gravetter ܘ-Wallnau: H = 3.785 ܕܠܐ ܬܘܪܨܐ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ (ܟܬܒܐ)؛ kruskal.test ܟܬܪܨ ܩܐ ܛܝ̈ܡܐ ܫܘ̈ܝܐ – 3.819
kruskal.test(list(c(14, 3, 21, 5, 16), c(2, 14, 9, 12, 5), c(26, 8, 14, 19, 20)))

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

Field (2024، ܦܣܩܐ 7.6.8): ܟܡܘܬܢܚ ܡܚܘܝܢܐ ܣܛܐܛܝܣܛܝܩܝܐ H، ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܘܪܫܝܡܘܬܐ – "ܡܢܝ̈ܢܐ ܕܙܪܥܐ ܡܬܥܒܕ̈ܐ ܗܘܘ ܪܫܝܡܐܝܬ ܡܢ ܐܟܠܬܐ ܕܐܟܠ̈ܬܐ ܕܣܘܝܐ، H(3) = 8.66, p = .034" – ܘܒܬܪܗܘܢ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ (post hoc) ܥܡ ܡܚܘ̈ܝܢܐ ܕܡܥܒܕܢܘܬܐ: "ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ ܥܡ p ܡܬܘܪ̈ܨܐ ܚܘܝܠܗܘܢ ܕܠܝܬ ܦܘܪ̈ܫܢܐ ܪ̈ܫܝܡܐ… ܒܝܢܝ ܫܒܥܐ ܐܟܠ̈ܬܐ ܘܐܪܒܥܐ (p = .133, r_rb = .44)، ܚܕܐ (p = .133, r_rb = .39) ܝܢ ܣܦܝܩܘܬܐ (p = .058, r_rb = .48)…". Pallant (2020، ܨܚܚܐ 16) ܟܡܙܝܕܐ n ܩܐ ܟܠ ܓܘܕܐ ܘܛܝ̈ܡܐ ܡܨܥܝ̈ܐ: "ܒܘܚܪܢܐ ܕ-Kruskal-Wallis ܚܘܝܠܗ ܦܘܪܫܢܐ ܪܫܝܡܐ ܓܘ ܕܪ̈ܓܐ ܕܣܒܪܐ ܛܒܐ ܒܝܢܝ ܬܠܬܐ ܓܘ̈ܕܐ ܕܥܘܡܪܐ (ܓܘܕܐ 1، n = 147: 18-29…)، χ²(2, n = 435) = 8.57, p = .014. ܓܘܕܐ ܕܥܘܡܪܐ ܣܒܐ (45+) ܚܘܝܠܗ ܛܝܡܐ ܡܨܥܝܐ ܪܡܐ ܪܫܝܡܐܝܬ (Md = 23) ܡܢ ܥܠܝܡܬܐ… (Md = 22)". Gravetter ܘ-Wallnau (Appendix E.4): "ܠܝܬ ܛܘܦܣܐ ܬܩܝܢܐ… ܡܘܕܥܢܘܬܐ ܣܢܝܩܬܐ ܝܠܗ ܕܐܚܕܐ ܟܢܘܫܝܐ ܕܝܗܝ̈ܒܬܐ، ܛܝܡܐ ܕܟܝ ܡܪܒܥܐ، df، N ܘ-p" – "H = 3.785 (2, N = 15), p > .05". ܡܐܡܪ̈ܐ ܕܬܘܢܝܐ ܕܓܘ ܡܦܩܢܐ ܟܡܚܠܛܝ ܬܠܬܝܗܘܢ ("ܐܦ ܐܗܐ ܐܦ ܐܗܐ"): n ܘܛܝܡܐ ܡܨܥܝܐ ܩܐ ܟܠ ܓܘܕܐ، H(df, N) = …, p، ܘܒܬܪ ܒܘܚܪܢܐ ܓܘܢܝܐ ܪܫܝܡܐ – ܙܘ̈ܓܐ ܪ̈ܫܝܡܐ ܥܡ z، p ܡܬܘܪܨܐ ܘ-r_rb. ܢܡܘ̈ܣܐ ܕܨܘܪܬܐ (American Psychological Association [APA], 2020, Section 6.36): p ܘ-r_rb ܕܠܐ ܣܦܝܩܘܬܐ ܩܕܡ ܢܘܩܙܐ؛ H ܘ-z ܥܡ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܟܠܝܐ ܓܘ ܒܘܚܪܢܐ ܓܘܢܝܐ. H ܪܫܝܡܐ ܟܐܡܪ ܕܐܝܬ ܦܘܪܫܢܐ، ܠܐ ܒܝܢܝ ܐܝܢܝ ܓܘ̈ܕܐ (Field, 2024, Section 7.6.2) – ܟܡܘܬܢܚ ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ.

ܩܪܝܢܐ ܕ-Sig. ܒܕܘܟܬܐ ܕ-Adj. Sig. ܫܬܐ ܒܘܚܪ̈ܢܐ ܥܠ ܗܢܝ ܝܗܝ̈ܒܬܐ – ܕܠܐ ܬܘܪܨܐ، "ܠܐ ܒܨܪ ܡܢ ܚܕܐ ܛܥܝܘܬܐ ܕܙܢܐ I ܓܘ ܝܬܝܪ ܡܢ 5% ܡܢ ܡܘܨ̈ܒܐ" (Field; Section 2.10.8). Pallant ܟܩܪܝܐ Sig.؛ ܥܝܕܐ ܕܣܐܝܛ – ܡܬܘܪܨܐ.

ܦܘܫܩܐ ܕ"ܚܕ ܙܘܓܐ ܠܐ ܪܫܝܡܐ" ܐܝܟ "ܠܝܬ ܡܥܒܕܢܘܬܐ". ܓܘ ܡܬܠܐ ܕܣܘܝܐ ܒܘܚܪܢܐ ܓܘܢܝܐ ܪܫܝܡܐ ܝܠܗ، ܚܕ ܙܘܓܐ ܠܐ، ܘ-r_rb ≈ .5 ܩܐ ܫܒܥܐ ܐܟܠ̈ܬܐ ܩܘܒܠ ܟܠ ܐܚܪ̈ܢܐ (Field, Section 7.6.7) – ܟܡܘܬܢܚ ܡܥܒܕܢܘܬܐ.

ܬܘܢܝܐ ܕܡܨܥܝ̈ܬܐ. ܒܘܚܪܢܐ ܥܠ ܕܪ̈ܓܐ ܕܛܟܣܐ ܝܠܗ؛ ܟܡܘܬܢܚ ܛܝ̈ܡܐ ܡܨܥܝ̈ܐ (Field, Section 7.6.8; Pallant, 2020, Chapter 16).

ܦܘܚܡܐ ܕ-z ܕܦܐܬܐ ܥܡ ܕ-R ܝܢ ܕ-SPSS ܕܠܐ ܒܘܚܪܢܐ ܕܪܘܫܡܐ. ܗܘ ܛܝܡܐ ܫܪܝܪܐ؛ ܪܘܫܡܐ ܬܠܝܐ ܝܠܗ ܒܛܟܣܐ ܕܙܘܓܐ ܘܒܬܘܟܢܝܬܐ.

ܓܘ̈ܕܐ ܕܬܪܝܢ-ܬܠܬܐ ܡܫܬܘܬܦ̈ܢܐ. ܩܘܪܒܐ ܠܟܝ ܡܪܒܥܐ ܣܢܝܩ ܝܠܗ ܠܚܡܫܐ ܓܘ ܓܘܕܐ (Gravetter & Wallnau, 2017, Appendix E.4).

ܥܪܩܐ ܠ-Kruskal-Wallis ܟܠ ܐܝܡܢ ܕܒܘܚܪܢܐ ܕܢܘܪܡܠܝܘܬܐ ܪܫܝܡܐ ܝܠܗ. ܓܘ ܓܘ̈ܕܐ ܪ̈ܒܐ ܢܡܘܣܐ ܕܬܚܘܡܐ ܡܨܥܝܐ ܟܡܓܢܐ ܥܠ ܦܘܫܩܐ ܕܫܘܚܠܦܐ (Field, 2024، ܦܣܩܐ 6.2.7؛ ܦܐܬܐ ܒܘܚܪܢܐ ܕܢܘܪܡܠܝܘܬܐ) – ܘܕܪ̈ܓܐ ܕܛܟܣܐ ܟܡܘܒܕܝ ܪܒܘܬܐ ܕܦܘܪ̈ܫܢܐ (ܩܘܛܝܐ 7.1).

ܡܒܘ̈ܥܐ

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.

Field, A. (2024). Discovering statistics using IBM SPSS statistics (6th ed.). SAGE.

Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the behavioral sciences (10th ed.). Cengage Learning.

Pallant, J. (2020). SPSS survival manual: A step by step guide to data analysis using IBM SPSS (7th ed.). Open University Press.

Kruskal ܘ-Wallis (1952)، Jonckheere (1954)، Terpstra (1952) ܘ-Kerby (2014) – ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Field (2024). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܡܢ ܥܩܪܐ؛ Gravetter ܘ-Wallnau (2017, Appendix E, pp. 687-697)، Cohen (1988، ܦܐ̈ܬܐ 79-83 – ܥܝܕ̈ܐ ܕ-r)، Pallant (2020) ܘ-APA (2020) – ܠܦܘܬ ܦܐܬܐ، ܨܚܚܐ ܘܦܣܩܐ ܕܢܘܣܚܐ ܛܒܝܥܐ؛ Field (2024) ܓܘ ܢܘܣܚܐ ܕܝܓܝܛܠܝܐ ܪܕܝܐ (reflowable) – ܠܦܘܬ ܦܣܩܐ، ܫܘܝܘܬܐ، ܠܘܚܐ ܘܡܦܩܢܐ (ܠܘܚܐ 7.3 ܕܝܠܗ، 80 ܡܫܬܘܬܦ̈ܢܐ، ܝܠܗ ܦܐܝܠ ܕܒܘܚܪܢܐ ܕܦܐܬܐ). ܣܝܢܬܐܟܣ – ܒܚܝܪܐ ܩܘܒܠ PSPP؛ R ܪܗܛܐ ܝܠܗ ܓܘ 4.3.3 ܥܡ rstatix 0.7.2.