ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ (ANOVA)

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 ܕܒܥܝܚ ܕܝܕܥܚ ܐܢ ܡܨܥܝܬܐ (mean) ܕܡܫܬܚܠܦܢܐ ܟܡܝܝܐ (quantitative variable) ܦܪܝܫܐ ܝܠܗ ܒܝܢܝ ܬܠܬܐ ܓܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢ ܝܬܝܪ – ܬܠܬܐ ܕܪ̈ܓܐ ܕܝܘܠܦܢܐ، ܐܪܒܥܐ ܫܪ̈ܛܐ ܕܢܣܝܘܢܐ، ܚܡܫܐ ܐܬܪ̈ܘܬܐ. Pallant (2020، ܨܚܚܐ 18) ܟܣܝܡܐ ܛܘܦܣܐ: ܚܕ ܡܫܬܚܠܦܢܐ ܠܐ ܬܠܝܐ (independent variable) ܕܓܢ̈ܣܐ (categorical) ܥܡ ܬܠܬܐ ܓܘ̈ܕܐ ܘܠܥܠ ("ܕܚܕ ܐܘܪܚܐ" – ܚܕ ܡܫܬܚܠܦܢܐ ܠܐ ܬܠܝܐ؛ "ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ" – ܐܢܫ̈ܐ ܦܪ̈ܝܫܐ ܓܘ ܟܠ ܓܘܕܐ)، ܘܚܕ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܡܬܡܫܚܢܐ (continuous). Gravetter ܘ-Wallnau (2017، ܦܐܬܐ 366-370) ܟܡܦܫܩܝ ܩܡܘܕܝ ܠܐ ܣܕܪܐ ܕܒܘܚܪ̈ܢܐ t: ܟܠ ܒܘܚܪܢܐ ܛܥܝܢܐ ܝܠܗ 5% ܣܟܘܢܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I (Type I error)، ܘܬܠܬܐ ܓܘ̈ܕܐ ܬܠܬܐ ܒܘܚܪ̈ܢܐ ܝܢܐ – "ܪܡܐ ܕܐܠܦܐ ܕܢܣܝܘܢܐ" (experimentwise) ܟܡܬܟܢܫ؛ ANOVA ܟܒܚܪ ܟܠ ܦܘܪ̈ܫܢܐ ܒܚܕ ܒܘܚܪܢܐ. Field (2024، ܦܣܩܐ 12.3) ܟܡܚܘܐ ܕܐܗܐ ܝܠܗ ܡܘܕܝܠ (model) ܣܪܛܢܝܐ (linear) ܥܡ ܡܫܬܚܠܦ̈ܢܐ (dummy variables): ܗܘ F ܕܟܫܩܠܚ ܡܢ ܪܝܓܪܫܢ (regression) ܥܡ ܓܘ̈ܕܐ ܐܝܟ ܡܢܒ̈ܝܢܐ (predictors).

ܬܪܬܝܢ ܓܘ̈ܕܐ ܒܠܚܘܕ – ܦܐܬܐ ܟܚܫܒ، ܐܠܐ ܒܘܚܪܢܐ t ܠܕܘ̈ܡܝܐ ܠܐ ܬܠܝ̈ܐ ܟܝܗܒ ܗܘ ܦܘܣܩܢܐ ܓܢܒܗ (F = t²) ܘܝܬܝܪ ܡܢ ܗܕܐ d ܕ-Cohen ܘܬܚܘܡܐ ܕܬܘܟܠܢܐ (confidence interval) ܩܐ ܦܘܪܫܢܐ؛ ܗܢܝ ܡܫܬܘܬܦ̈ܢܐ (participants) ܓܘ ܟܡܐ ܡܫܘ̈ܚܬܐ – ܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance) ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ (repeated measures) (Pallant, Chapter 18; Gravetter & Wallnau, Chapter 13)، ܓܘ ܦܐܬܐ ANOVA ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ؛ ܦܘܚܡܐ ܒܝܢܝ ܓܘ̈ܕܐ ܥܡ ܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ ܐܚܪܢܐ ܕܡܫܝܚܐ ܝܠܗ ܩܕܡ (ܡܬܠܐ ܡܢܝܢܐ ܕ"ܩܕܡ") – ANCOVA، ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܥܡ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate)؛ ܬܪܝܢ ܡܫܬܚܠܦ̈ܢܐ ܠܐ ܬܠܝ̈ܐ ܝܢ ܝܬܝܪ، ܝܢ ܓܘ̈ܫܡܐ (blocks) – ANOVA ܕܥܠ̈ܠܬܐ ܣܓܝ̈ܐܐ، ܓܘ ܦܐܬܐ ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ (ܬܡܢ ܐܦ ܪܒܘܬܐ ܕܕܘܡܝܐ (sample size) ܘܦܐܬܐ ܕܪܗܛܐ ܓܕܫܢܝܐ)؛ ܡܫܬܚܠܦܢܐ ܬܠܝܐ ܕܛܟܣܐ (ordinal) ܝܢ ܬܒܝܪܘܬܐ ܩܫܝܬܐ ܕܫܪ̈ܛܐ (assumptions) ܓܘ ܕܘܡܝܐ ܙܥܘܪܐ – ܒܘܚܪܢܐ ܕ-Kruskal-Wallis (Pallant, Chapter 16; Gravetter & Wallnau, Appendix E). ܡܢܝܢܐ ܕܣܒܠܬܐ ܕܦܣܘ̈ܩܐ ܣܓܝ̈ܐܐ (multi-item scale score؛ ܦܐܬܐ ܐܠܦܐ ܕ-Cronbach ܘܡܢܝܢܐ ܕܣܒܠܬܐ) ܟܡܬܚܫܒ ܐܝܟ ܡܫܬܚܠܦܢܐ ܕܪܘ̈ܚܩܐ ܫܘ̈ܝܐ (interval variable) ܠܦܘܬ ܥܝܕܐ ܕܣܐܝܛ (site convention؛ ܦܐܬܐ ܡܫܬܚܠܦ̈ܢܐ ܘܕܪ̈ܓܐ ܕܡܫܘܚܬܐ)؛ ܦܣܘܩܐ ܕ-Likert (Likert item) ܝܚܝܕܝܐ – ܠܐ.

ܚܘܫܒܐ: ܫܘܚܠܦܐ ܒܝܢܝ ܓܘ̈ܕܐ ܦܠܝܓܐ ܥܠ ܫܘܚܠܦܐ ܓܘ ܓܘ̈ܕܐ

Gravetter ܘ-Wallnau (2017، ܦܐܬܐ 371-375) ܟܦܠܓܝ ܟܠܗ ܒܕܪܐ ܕܝܗܝ̈ܒܬܐ ܠܬܪܝܢ ܡܢܘ̈ܬܐ. ܫܘܚܠܦܐ ܒܝܢܝ ܓܘ̈ܕܐ (between-groups variance) – ܟܡܐ ܡܨܥܝ̈ܬܐ (means) ܕܓܘ̈ܕܐ ܪܚܝ̈ܩܐ ܝܢܐ ܚܕ ܡܢ ܐܚܪܢܐ؛ ܟܐܬܐ ܡܢ ܐܣܝܘܬܐ (ܐܢ ܐܝܬ ܠܗ̇ ܡܥܒܕܢܘܬܐ) ܘܐܦ ܡܢ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܐܢܫ̈ܐ ܘܡܢ ܛܥܝܘܬܐ ܕܓܕܫܐ. ܫܘܚܠܦܐ ܓܘ ܓܘ̈ܕܐ (within-groups variance) – ܟܡܐ ܡܫܬܘܬܦ̈ܢܐ (participants) ܓܘ ܗܝ ܓܘܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܚܕ ܡܢ ܐܚܪܢܐ؛ ܟܐܬܐ ܒܣ ܡܢ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܐܢܫ̈ܐ ܘܡܢ ܛܥܝܘܬܐ ܕܓܕܫܐ، ܡܛܠ ܕܓܘ ܓܘܕܐ ܟܠܗܘܢ ܩܒܝܠܗܘܢ ܗܝ ܐܣܝܘܬܐ. F ܝܠܗ ܦܘܚܡܐ ܒܝܢܝ ܬܪܘܬܝܗܝ: ܐܢ ܠܝܬ ܡܥܒܕܢܘܬܐ ܠܐܣܝܘܬܐ، ܬܪܝܢ ܡܒܘ̈ܥܐ ܟܡܫܚܝ ܚܕ ܡܕܡ ܘ-F ܩܪܝܒܐ ܝܠܗ ܠ-1؛ ܐܢ ܐܝܬ، ܡܢܝܢܐ ܥܠܝܐ ܟܪܒܐ ܘ-F ܟܣܠܩ. Field (2024، ܦܣ̈ܩܐ 12.3.4-12.3.8) ܟܡܬܢܐ ܗܘ ܚܘܫܒܐ ܒܠܫܢܐ ܕܡܘܕܝܠ (model): SS_T (ܟܠܗ ܒܕܪܐ ܚܕܪ ܡܨܥܝܬܐ ܟܠܢܝܬܐ) = SS_M (ܡܐ ܕܡܨܥܝ̈ܬܐ ܕܓܘ̈ܕܐ ܟܡܦܫܩܝ) + SS_R (ܡܐ ܕܦܝܫܐ ܝܠܗ)، ܘܟܦܠܓܚ ܟܠ ܚܕ ܥܠ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܕܝܠܗ ܩܐ ܕܫܩܠܚ ܡܨܥܝ̈ܬܐ ܕܡܪ̈ܒܥܐ (mean squares) – "ܡܨܥܝܬܐ" ܕܫܘܚܠܦܐ، ܡܛܠ ܕܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܟܪܒܐ ܥܡ ܡܢܝܢܐ ܕܚܙ̈ܝܬܐ (observations).

ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom): df_ܒܝܢܝ = k − 1 (ܡܢܝܢܐ ܕܓܘ̈ܕܐ ܒܨܝܪ ܚܕ)، df_ܓܘ = N − k (Gravetter & Wallnau, pp. 378-379). ܡܬܠܐ ܕ-Gravetter ܘ-Wallnau (ܡܬܠܐ 12.2، ܦܐܬܐ 385-388): ܬܠܬܐ ܐܘܪ̈ܚܬܐ ܕܝܘܠܦܢܐ، ܫܬܐ ܬܠܡܝ̈ܕܐ ܓܘ ܟܠ ܚܕܐ – SS_ܒܝܢܝ = 84، SS_ܓܘ = 88، MS ܕ-42 ܘ-5.87، F(2, 15) = 7.16، ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ (statistically significant)؛ ܡܬܠܐ ܕ-Field (ܦܣܩܐ 12.2، ܠܘܚܐ 12.1): ܐܣܝܘܬܐ ܒܓܘܪ̈ܝܐ ܕܟܠܒ̈ܐ ܒܬܠܬܐ ܡܫܘ̈ܚܬܐ ܕܣܡܐ، ܚܡܫܐ ܓܘ ܟܠ ܓܘܕܐ – F(2, 12) = 5.12, p = .025. ܬܪܘܬܝܗܝ ܡܚܘܫܒ̈ܐ ܝܢܐ ܓܘ ܦܐܬܐ ܘܟܝܗܒܝ ܚܬܝܬܐܝܬ ܡܢܝ̈ܢܐ ܕܐܝܬ ܓܘ ܟܬ̈ܒܐ.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܠܘܚܐ 1 – ܣܛܐܛܝܣܛܝܩܐ ܕܓܘ̈ܕܐ (Descriptives). Pallant (2020، ܨܚܚܐ 18): "ܟܠ ܙܒܢܐ ܟܒܚܪܚ ܐܗܐ ܠܘܚܐ ܩܕܡ – ܐܢ N ܕܟܠ ܓܘܕܐ ܬܪܝܨܐ ܝܠܗ؟" ܓܘܕܐ ܣܦܝܩܬܐ ܝܢ ܙܥܘܪܬܐ ܒܣܓܝ ܣܓܝ ܓܗ̈ܐ ܟܡܚܘܝܐ ܪܡܙܐ ܛܥܝܐ ܝܢ ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ (missing values) ܣܓܝ̈ܐܐ. ܣܕܪܐ ܕ"ܓܘܡܪܐ" – ܡܨܥܝܬܐ (mean) ܟܠܢܝܬܐ، ܕܚܕܪܗ̇ ܟܡܬܡܫܚ ܫܘܚܠܦܐ (variance) ܒܝܢܝ ܓܘ̈ܕܐ.

ܠܘܚܐ 2 – ܠܘܚܐ ܕ-ANOVA، ܘܓܘܗ ܐܦ ܣܕܪܐ ܕ-Welch (ܓܘ SPSS ܐܝܬܠܗ ܓܘ ܠܘܚܐ ܦܪܝܫܐ، Robust Tests of Equality of Means). ܐܝܢܐ ܣܕܪܐ ܟܩܪܚ؟ ܐܝܟ ܓܘ ܒܘܚܪܢܐ t، ܡܒܘ̈ܥܐ ܦܠܝܓ̈ܐ ܝܢܐ، ܘܦܐܬܐ ܟܡܚܘܐ ܬܪܝܢ ܪ̈ܥܝܢܐ:

ܟܘܬܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ" ܕܓܘ ܡܦܩܢܐ (output) ܟܐܡܪܐ ܐܢ ܬܪܝܢ ܣܕܪ̈ܐ ܡܫܠܡܝ. ܓܘ ܡܬܠܐ ܕܓܘܪ̈ܝܐ ܕ-Field ܦܪ̈ܝܫܐ ܝܢܐ: F ܫܚܝܡܐ p = .025، Welch p = .054 – ܘ-Field (ܦܣܩܐ 12.9.1) ܟܐܡܪ ܕܐܗܐ ܟܡܚܠܛ "ܒܣ ܐܢ ܝܗܒܚ ܠܬܚܘܡܐ .05 ܚܝ̈ܠܐ ܕܚܪܫܘܬܐ". F، df ܘ-p. p ܙܥܘܪܐ ܡܢ .05 – ܐܝܬ ܦܘܪܫܢܐ ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ "ܒܚܕ ܕܘܟܐ" ܒܝܢܝ ܓܘ̈ܕܐ؛ ܠܘܚܐ ܠܐ ܟܐܡܪ ܒܝܢܝ ܐܝܢܝ (Pallant, Chapter 18; Gravetter & Wallnau, p. 393). ܠ-F ܠܝܬ ܢܘܣܚܐ ܕܚܕ ܕܘܢܒܐ (one-tailed) – ܠܐ ܡܨܚ ܕܦܠܓܚ p ܥܠ ܬܪܝܢ (Field، ܡܠܟܐ 12.1).

ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ: ܒܝܢܝ ܐܝܢܝ ܓܘ̈ܕܐ؟

F ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ (statistically significant) ܟܐܡܪ ܕܠܟܡܐ ܚܕ ܦܘܪܫܢܐ ܪܫܝܡܐ ܝܠܗ؛ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ (post hoc) ܟܥܒܪܝ ܥܠ ܟܠ ܙܘܓ̈ܐ ܘܟܬܪܨܝ ܪܡܐ ܕܪܫܝܡܘܬܐ (significance level) ܗܕܟܐ ܕܣܟܘܢܐ ܟܠܢܝܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I (Type I error) ܥܠ ܟܠ ܙܘܓ̈ܐ ܒܚܕ ܓܗܐ ܦܐܫ 5% (Gravetter & Wallnau, 2017, pp. 393-394; Field, 2024, Section 12.6). SPSS ܟܡܩܪܒ ܩܪܝܒ ܠ-18 ܐܘܪ̈ܚܬܐ؛ Field (ܦܣܩܐ 12.6.3، ܒܬܪ Toothaker، 1993) ܟܟܢܫ: ܓܘ̈ܕܐ ܫܘ̈ܝܐ ܒܪܒܘܬܐ ܘܫܘܚܠ̈ܦܐ ܕܡܝ̈ܐ – Tukey (ܝܢ REGWQ)؛ "ܚܘܟܡܐ ܫܪܝܪܐ" ܥܠ ܛܥܝܘܬܐ ܕܙܢܐ I – Bonferroni؛ ܦܠܓܘܬ ܠܒܐ ܥܠ ܫܘܝܘܬܐ ܕܫܘܚܠ̈ܦܐ – Games-Howell، "ܕܐܢܐ ܟܡܩܪܒܢ ܕܡܪܗܛܝܬܘܢ ܠܗ ܥܠ ܟܠ ܐܘܪܚܐ ܐܚܪܬܐ". ܦܐܬܐ ܟܡܚܘܐ ܬܠܬܝܗܝ، ܘܠܘܚܐ 3 ܟܚܠܦ ܒܝܢܬܗܘܢ ܒܓܘܒܝܐ:

ܐܝܡܢ ܟܦܫܩܚ؟ Pallant (ܨܚܚܐ 18) ܘ-Gravetter ܘ-Wallnau (ܦܐܬܐ 393): ܒܣ ܒܬܪ F ܟܠܢܝܐ ܪܫܝܡܐ ܘܥܡ ܬܠܬܐ ܓܘ̈ܕܐ ܘܠܥܠ (ܓܘ ܬܪܬܝܢ ܓܘ̈ܕܐ ܠܝܬ ܡܐ ܕܫܐܠܚ). ܠܘܚܐ (table) ܡܚܘܝܐ ܝܠܗ ܟܠ ܙܒܢܐ، ܘܐܝܡܢ ܕ-F ܠܐ ܝܠܗ ܪܫܝܡܐ ܡܥܝܪܢܘܬܐ (note) ܟܐܡܪܐ ܕܠܐ ܟܡܬܢܚ ܦܘܪ̈ܫܢܐ ܡܢܗ. ܟܣܝܡܚ ܐܘܪܚܐ ܡܢ ܩܕܡ – ܠܐ ܟܓܒܚ ܒܬܪ ܒܘܚܪܢܐ ܗܝ ܕܟܝܗܒܐ ܟܘ̈ܟܒܐ ܝܬܝܪ؛ ܓܘܒܝܐ ܡܩܕܡܐ (default) ܕܦܐܬܐ (Tukey ܐܝܡܢ ܕ-Levene ܠܐ ܝܠܗ ܪܫܝܡܐ، Games-Howell ܐܝܡܢ ܕܐܝܠܗ) ܝܠܗ ܢܡܘܣܐ ܗܕܟܐ. ܬܘܩܢܐ ܡܢ ܩܕܡ ܕܦܘܚܡ̈ܐ ܝܚܝܕ̈ܝܐ (planned contrasts, Field, Section 12.5; Scheffé, Gravetter & Wallnau, pp. 395-396) – ܠܐ ܓܘ ܦܐܬܐ.

ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size)

ܪܫܝܡܘܬܐ (significance) ܠܐ ܝܠܗ ܪܒܘܬܐ: Pallant (2020، ܨܚܚܐ 18) ܟܡܚܘܝܐ ܥܠ N = 435 – F(2, 432) = 4.64, p = .01، ܐܠܐ ܡܨܥܝ̈ܬܐ (means) 21.36، 22.10 ܘ-22.96 ܘ-η² = .02: "ܥܡ ܕܘܡܝܐ (sample) ܪܒܐ ܕܡܣܦܩ ܦܘܪ̈ܫܢܐ ܙܥܘܪ̈ܐ ܟܦܠܛܝ ܪܫܝܡ̈ܐ، ܐܦ ܐܢ ܠܝܬ ܠܗܘܢ ܚܫܝܚܘܬܐ ܥܒܕܢܝܬܐ". ܠܘܚܐ ܟܛܒܥ ܬܪܝܢ ܡܚܘ̈ܝܢܐ (indices):

ܐܝܬܐ ܡܪܒܥܐ (eta squared؛ η²) – ܡܢܬܐ ܕܫܘܚܠܦܐ (variance) ܕܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܕܡܦܫܩܐ ܝܠܗ ܒܗܘܝܐ ܓܘ ܓܘܕܐ: SS_ܒܝܢܝ/SS_ܓܘܡܪܐ (Gravetter & Wallnau, 2017، ܦܐܬܐ 388-389، ܫܘܝܘܬܐ 12.14 – "ܡܢܬܐ ܒܡܐܐ ܕܫܘܚܠܦܐ ܡܦܫܩܐ"، η² = 0.488 ܓܘ ܡܬܠܐ 12.2؛ Pallant, 2020، ܨܚܚܐ 18 – ܡܚܘܫܒܐ ܒܐܝܕܐ ܡܢ ܠܘܚܐ ܕ-ANOVA؛ Field, 2024، ܫܘܝܘܬܐ 12.33 – ܐܗܐ ܝܠܗ R² ܕܡܘܕܝܠ (model) ܣܪܛܢܝܐ). ܬܚܘ̈ܡܐ ܕ-Cohen (1988، ܦܐܬܐ 284-287) ܕ-Pallant ܟܡܝܬܝܐ: .01 ܙܥܘܪܐ، .06 ܡܨܥܝܐ، .14 ܪܒܐ.

ܐܘܡܝܓܐ ܡܪܒܥܐ (omega squared؛ ω²) – Field (ܦܣܩܐ 12.7، ܫܘܝܘܬܐ 12.34): η² "ܡܨܠܝܐ ܝܠܗ ܩܠܝܠ" ܡܛܠ ܕܒܣ ܥܠ ܓܘܡܪ̈ܐ ܕܡܪ̈ܒܥܐ ܕܕܘܡܝܐ (sample) ܡܣܡܟܐ ܝܠܗ؛ ω² ܟܬܪܨ ܩܐ ܫܘܥܪܐ (estimate) ܓܘ ܟܢܫܐ ܟܠܢܝܐ (population): (SS_ܒܝܢܝ − df_ܒܝܢܝ·MS_ܓܘ) / (SS_ܓܘܡܪܐ + MS_ܓܘ) – ܓܘ ܡܬܠܐ ܕܓܘܪ̈ܝܐ η² = .46 ܐܠܐ ω² = .35، ܘ-Field ܟܡܬܢܐ ω² (ܦܣܩܐ 12.12) ܥܡ ܗܢܝ ܬܚܘ̈ܡܐ (Kirk، 1996). ܓܘ ܕܘܡܝܐ ܙܥܘܪܐ ܝܢ ܓܘ ܓܘ̈ܕܐ ܣܓܝ̈ܐܐ ܦܘܪܫܢܐ ܒܝܢܬܗܘܢ ܪܒܐ ܝܠܗ، ܘܓܘ ܕܘܡܝܐ ܪܒܐ ܩܪܝܒ ܚܕ ܝܢܐ؛ ܟܘܬܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ" ܟܡܚܘܝܐ ܬܪܘܬܝܗܝ. Field ܟܡܘܣܦ ܕܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size) ܟܠܢܝܬܐ ܒܨܝܪ ܚܫܝܚܬܐ ܝܠܗ ܡܢ ܪܒܘܬܐ ܕܦܘܪܫܢܐ ܒܝܢܝ ܙܘܓܐ ܕܓܘ̈ܕܐ – ܩܐ ܟܠ ܙܘܓܐ ܡܨܚ ܕܚܫܒܚ d ܕ-Cohen ܓܘ ܦܐܬܐ ܕܒܘܚܪܢܐ t. SPSS 27 ܘܠܥܠ ܟܛܒܥ ܬܪܘܬܝܗܝ (ANOVA Effect Sizes) ܥܡ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals)، ܕܠܐ ܝܢܐ ܓܘ ܦܐܬܐ.

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

ܬܠܬܐ ܫܪ̈ܛܐ (assumptions؛ Gravetter & Wallnau, 2017, pp. 391-392): ܚܙ̈ܝܬܐ (observations) ܓܘ ܟܠ ܓܘܕܐ ܠܐ ܬܠܝ̈ܐ ܝܢܐ؛ ܟܢܫ̈ܐ ܟܠܢ̈ܝܐ (populations) ܢܘܪܡܠ̈ܝܐ ܝܢܐ؛ ܘܠܟܠܗܘܢ ܐܝܬ ܚܕ ܫܘܚܠܦܐ (variance). ܠܐ ܬܠܝܘܬܐ ܝܠܗ ܫܪܛܐ ܚܫܝܚܐ ܒܝܬܝܪ: Field (ܦܣܩܐ 12.4.2) ܟܡܝܬܐ Scariano ܘ-Davenport – ܐܣܘܪܐ ܕܚܕ̈ܕܐ (correlation) ܕ-0.5 ܒܝܢܝ ܚܙ̈ܝܬܐ ܟܡܪܝܡ ܛܥܝܘܬܐ ܕܙܢܐ I (Type I error) ܡܢ 5% ܠ-74%. ܡܫܬܘܬܦܢܐ (participant) ܕܡܫܝܚܐ ܝܠܗ ܬܪܝܢ ܙܒ̈ܢܐ، ܬܠܡܝ̈ܕܐ ܡܢ ܗܢܝ ܨܦ̈ܐ، ܙܘܓ̈ܐ – ܟܬܒܪܝ ܠܗ، ܘܒܘܚܪܢܐ ܬܪܝܨܐ ܐܚܪܢܐ ܝܠܗ. ܢܘܪܡܠܝܘܬܐ (normality): Gravetter ܘ-Wallnau – "ܣܓܝ ܓܗ̈ܐ ܠܐ ܟܡܬܥܩܚ ܡܢܗ̇، ܒܝܬܝܪ ܓܘ ܕܘ̈ܡܝܐ ܪ̈ܒܐ"؛ Field (ܦܣܩܐ 12.4.2) ܠܐ ܟܡܫܠܡ ܥܡ ܬܫܥܝܬܐ ܕ"ANOVA ܚܣܝܢܬܐ ܝܠܗ": ܥܩܝܡܘܬܐ (skewness)، ܕܘܢ̈ܒܐ ܝܩܝܪ̈ܐ ܘܫܘܚܠ̈ܦܐ ܦܪ̈ܝܫܐ ܟܡܥܒܕܝ ܥܠ F ܐܦ ܓܘ ܓܘ̈ܕܐ ܫܘ̈ܝܐ، ܘܓܘ ܕܘܢ̈ܒܐ ܝܩܝܪ̈ܐ ܢܡܘܣܐ ܕܬܚܘܡܐ ܡܨܥܝܐ (central limit theorem) ܒܥܐ ܕܘ̈ܡܝܐ ܪ̈ܒܐ ܒܣܓܝ ܡܢ 30 – ܘܡܛܠ ܗܕܐ ܟܚܙܚ ܗܝܣܛܘܓܪܡ ܕܟܠ ܓܘܕܐ (ܦܐܬܐ ܣܓܝܐܘ̈ܬܐ ܘܣܛܐܛܝܣܛܝܩܐ ܡܚܘܝܢܝܬܐ) ܘܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers؛ ܦܐܬܐ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ ܘܫܪ̈ܛܐ)، ܘܐܢ ܐܝܬ ܦܠܓܘܬ ܠܒܐ ܟܡܪܗܛܚ ܐܦ ܒܘܚܪܢܐ ܚܣܝܢܐ (robust؛ Field, Section 12.10) ܝܢ Kruskal-Wallis.

ܗܘܡܘܓܢܝܘܬܐ ܕܫܘܚܠ̈ܦܐ (homogeneity of variances). Gravetter ܘ-Wallnau (ܦܐܬܐ 392، ܘܦܐܬܐ 314): "ܫܪܛܐ ܚܫܝܚܐ"، ܥܡ ܢܡܘܣܐ ܕܨܒܥܐ – ܫܘܚܠܦܐ ܚܕ ܪܒܐ ܬܠܬܐ-ܐܪܒܥܐ ܐܥܦ̈ܐ ܡܢ ܐܚܪܢܐ ܝܠܗ ܥܠܬܐ ܩܐ ܥܩܬܐ؛ Field (ܟܘܬܐ 12.2): ܐܝܡܢ ܕܓܘ̈ܕܐ ܠܐ ܫܘ̈ܝܐ ܝܢܐ ܒܪܒܘܬܐ ܘܪ̈ܒܬܐ ܒܕܝܪ̈ܐ ܝܢܐ ܝܬܝܪ، F ܫܚܝܡܐ ܢܛܘܪܐ ܝܠܗ، ܘܓܘ ܓܗܐ ܗܦܝܟܐ ܦܬܝܚܐ (liberal) – Welch ܘ-Brown-Forsythe ܟܬܪܨܝ. ܦܐܬܐ ܟܛܒܥ Levene (ܚܕܪ ܡܨܥܝܬܐ، ܐܝܟ SPSS) ܘܣܕܪܐ ܕ-Welch؛ ܓܘܒܝܐ – ܦܣܩܐ ܕ"ܩܪܝܢܐ ܕܡܦܩܢܐ".

ܓܘ̈ܕܐ ܠܐ ܫܘ̈ܝܐ ܒܪܒܘܬܐ. Gravetter ܘ-Wallnau (ܦܐܬܐ 390): ANOVA ܬܪܝܨܐ ܝܠܗ ܐܦ ܗܕܟܐ، "ܒܝܬܝܪ ܐܝܡܢ ܕܕܘ̈ܡܝܐ ܪ̈ܒܐ ܝܢܐ ܩܘܒܠ ܐܚܪ̈ܢܐ ܘܦܘܪܫܢܐ ܒܪ̈ܒܘܬܐ ܠܐ ܝܠܗ ܝܬܝܪܐ"، ܐܠܐ ܚܬܝܬܐ ܝܬܝܪ ܝܠܗ ܓܘ ܓܘ̈ܕܐ ܫܘ̈ܝܐ. Field (ܦܣܩܐ 12.6.2): ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ ܐܢܝ ܝܢܐ ܕܟܡܬܚܒܠܝ ܐܝܡܢ ܕܪ̈ܒܘܬܐ ܘܫܘܚܠ̈ܦܐ ܦܪ̈ܝܫܐ ܝܢܐ – Games-Howell ܥܒܝܕܐ ܝܠܗ ܩܐ ܐܗܐ ܓܗܐ. ܕܪܓܐ ܥܡ ܒܨܝܪ ܡܢ ܬܪܬܝܢ ܚܙ̈ܝܬܐ (observations) ܟܡܬܫܩܠ ܡܢ ܦܘܫܩܐ (analysis؛ ܘܦܐܬܐ ܟܡܘܕܥ).

ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ (missing values). ܡܫܬܘܬܦܢܐ (participant) ܕܚܣܝܪܐ ܝܠܗ ܛܝܡܐ ܓܘ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܝܢ ܓܘ ܡܫܬܚܠܦܢܐ ܕܓܘ̈ܕܐ ܠܐ ܝܠܗ ܥܠܝܠܐ ܓܘ ܦܘܫܩܐ ܕܗܘ ܡܫܬܚܠܦܢܐ ܬܠܝܐ ܒܠܚܘܕ – ܫܩܠܬܐ ܠܦܘܬ ܦܘܫܩܐ (/MISSING=ANALYSIS)، ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ (default) ܕ-SPSS ܘܕ-Pallant؛ ܡܛܠ ܗܕܐ N ܡܨܐ ܕܫܚܠܦ ܡܢ ܡܫܬܚܠܦܢܐ ܬܠܝܐ ܠܐܚܪܢܐ. ܡܘܕܝ ܥܒܕܚ ܥܡ ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ – ܦܐܬܐ ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ.

ܟܡܐ ܡܫܬܚܠܦ̈ܢܐ ܬܠܝ̈ܐ (dependent variables). Field (ܦܣܩܐ 12.8.1) ܠܐ ܟܡܩܒܠ ܓܒܝܬܐ ܕ-SPSS ܕܡܪܗܛܚ ܟܡܐ ܡܫܬܚܠܦ̈ܢܐ ܬܠܝ̈ܐ ܒܚܕ ܓܗܐ – "ܟܡܘܒܕܚ ܚܘܟܡܐ ܥܠ ܛܥܝܘܬܐ ܕܙܢܐ I" – ܘܟܡܩܪܒ MANOVA ܝܢ ܡܚܘܝܢܐ (index) ܡܚܠܛܐ؛ Pallant (ܨܚܚܐ 18) ܟܡܩܪܒܐ ܕܡܬܢܚ ܣܕܪܐ ܕܦܘܫܩ̈ܐ ܓܘ ܚܕ ܠܘܚܐ (table). ܦܐܬܐ ܟܡܦܣ ܟܡܐ، ܘܟܡܥܗܕ ܓܘ ܟܘܬܐ ܕܩܪܝܢܐ ܕܟܠ ܦܘܫܩܐ ܛܥܝܢܐ ܝܠܗ 5% ܣܟܘܢܐ.

ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. Brown-Forsythe، ܣܘܩܒ̈ܠܐ (contrasts) ܕܛܘܝܒ̈ܐ ܡܢ ܩܕܡ ܘܦܘܫܩܐ ܕܡܓܡܐ (trend analysis؛ Field, Section 12.5)، Scheffé ܘ-REGWQ، Dunnett، ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals) ܩܐ η² ܘ-ω² (SPSS 27 ܘܠܥܠ)، bootstrap ܘܒܘܚܪܢܐ ܚܣܝܢܐ (robust) ܥܠ ܡܨܥܝ̈ܬܐ ܦܣܝ̈ܩܬܐ (trimmed means؛ Field, Section 12.10)، ANOVA ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ ܘܕܥܠ̈ܠܬܐ ܣܓܝ̈ܐܐ (ܓܘ ܦܐܬܐ ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ) ܘܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ (ܓܘ ܦܐܬܐ ANOVA ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ)، Bayes factor.

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

ܡܥܝܪܢܘܬܐ ܕܢܘܣܚ̈ܐ. ܐܘܪܚܐ ܩܝܡܐ ܝܠܗ؛ ܡܢ ܢܘܣܚܐ 27 ܟܘܬܐ ܕܕܘܪܫܐ ܐܝܬ ܓܘܗ̇ Estimate effect size for overall tests ܘܟܛܒܥܐ ܠܘܚܐ ANOVA Effect Sizes ܥܡ η²، ε² ܘ-ω² ܘܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals؛ Field, 2024، ܦܣ̈ܩܐ 12.8.1 ܘ-12.9.1 – ܠܦܘܬ ܢܘܣܚܐ 29)، ܘPallant (2020، ܠܦܘܬ ܢܘܣܚܐ 26) ܟܟܬܒܐ ܕ-SPSS "ܠܐ ܟܡܦܩ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܩܐ ܐܗܐ ܦܘܫܩܐ" ܘܟܚܫܒܐ ܐܝܬܐ ܡܪܒܥܐ (eta squared) ܒܐܝܕܐ. ܓܘ ܢܘܣܚ̈ܐ ܥܬܝܩ̈ܐ – ܫܩܘܠܘܢ ܣܕܪܐ /ES=OVERALL ܡܢ ܣܝܢܬܐܟܣ (syntax).

ܐܘܪܚܐ (Pallant, 2020, Chapter 18; Field, 2024, Sections 12.8.1-12.8.4): Analyze › Compare Means › One-Way ANOVA… – ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable؛ ܝܢ ܟܡܐ) ܠܟܘܬܐ Dependent List، ܡܫܬܚܠܦܢܐ ܠܐ ܬܠܝܐ (independent variable) ܠ-Factor؛ Options… – Descriptive، Homogeneity of variance test، Welch (ܘ-Brown-Forsythe)، Means plot، ܘ-Exclude cases analysis by analysis؛ Post Hoc… – Tukey ܘ-Bonferroni ܬܚܘܬ "ܫܘܚܠ̈ܦܐ ܫܘ̈ܝܐ" (Equal Variances Assumed)، Games-Howell ܬܚܘܬ "ܠܐ ܫܘ̈ܝܐ" (Not Assumed). ܣܝܢܬܐܟܣ (syntax):

ONEWAY עמדות_ציון גיל BY השכלה
  /ES=OVERALL
  /STATISTICS=DESCRIPTIVES HOMOGENEITY WELCH
  /MISSING=ANALYSIS
  /POSTHOC=TUKEY GH BONFERRONI ALPHA(0.05).

ܩܪܝܢܐ ܕܡܦܩܢܐ. Descriptives = ܠܘܚܐ 1 (N, Mean, Std. Deviation, Std. Error، ܬܚܘܡܐ ܕܬܘܟܠܢܐ (confidence interval) ܩܐ ܡܨܥܝܬܐ، ܛܝܡܐ ܬܚܬܝܐ ܘܛܝܡܐ ܥܠܝܐ)؛ Test of Homogeneity of Variances = ܒܘܚܪܢܐ ܕ-Levene ܕܓܘ ܡܥܝܪܢܘܬܐ ܕܠܘܚܐ 2 – SPSS 26 ܘܠܥܠ ܟܛܒܥ ܐܪܒܥܐ ܢܘܣܚ̈ܐ، ܘܦܐܬܐ (ܘ-Pallant) ܟܩܪܝ Based on Mean؛ ANOVA = ܠܘܚܐ 2 (Between Groups, Within Groups, Total)؛ Robust Tests of Equality of Means = ܣܕܪܐ ܕ-Welch (Statistic, df1, df2, Sig.)؛ ANOVA Effect Sizes – Eta-squared ܘ-Omega-squared Fixed-effect ܕܓܘ ܡܥܝܪܢܘܬܐ ܕܠܘܚܐ؛ Multiple Comparisons = ܠܘܚܐ 3 – SPSS ܟܛܒܥ ܟܠ ܙܘܓܐ ܬܪܝܢ ܙܒ̈ܢܐ (I−J and J−I) ܓܘ ܬܠܬܐ ܓܘ̈ܫܡܐ (Tukey HSD، Games-Howell، Bonferroni)، ܥܡ ܟܘܟܒܐ ܓܢܒ ܦܘܪܫܢܐ ܪܫܝܡܐ. ܩܘܦܬܐ "ܡܚܬܬܐ ܕܣܝܢܬܐܟܣ SPSS" ܓܘ ܡܦܩܢܐ (output) ܟܥܒܕܐ ܛܥܢܐ (loading)، ܟܬܒܐ ܕܪ̈ܡܙܐ (data dictionary؛ ܐܢ ܐܝܬ ܗܘܐ ܡܦܫܩܢܐ) ܘܦܘܩܕܢܐ ONEWAY ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܕܓܒܝ̈ܐ ܝܢܐ.

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

ܓܘ R ܫܬܐܣܝܐ: aov() ܩܐ ܠܘܚܐ ܕ-ANOVA، oneway.test() ܩܐ Welch (ܓܘܒܝܐ ܡܩܕܡܐ (default) ܕܝܠܗ)، TukeyHSD() ܩܐ ܒܘܚܪ̈ܢܐ ܕ-Tukey (Tukey-Kramer، ܐܝܟ ܓܘ ܦܐܬܐ)، pairwise.t.test(p.adjust = "bonferroni") ܩܐ Bonferroni؛ ܒܘܚܪܢܐ ܕ-Levene ܚܕܪ ܡܨܥܝܬܐ – leveneTest(center = mean) ܡܢ ܚܙܡܐ car؛ Games-Howell – ܚܙܡܐ rstatix ܝܢ PMCMRplus (ܠܐ ܓܘ ܣܩܪܝܦܬ). η² ܘ-ω² ܡܚܘܫܒ̈ܐ ܝܢܐ ܒܐܝܕܐ ܡܢ ܠܘܚܐ ܕ-ANOVA. ܣܩܪܝܦܬ (script) ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ (sample data file) ܘܝܗܒܠܗ ܛܝ̈ܡܐ ܕܠܘ̈ܚܐ (ܓܘ ܡܥܝܪ̈ܢܘܬܐ). ܢܡܘܣܐ ܕܣܐܝܛ (site rule) ܩܐ ܣܘܪܝܬ ܒ-R: ܦܐܝܠ UTF-8، ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ، ܘܐܬܪܐ (locale) UTF-8 ܩܐ ܫܡ̈ܗܐ ܥܒܪ̈ܝܐ (ܓܘ Windows: R 4.2 ܘܠܥܠ، ܝܢ Sys.setlocale('LC_ALL', 'Hebrew')).

# ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)   # ܩܪܝܢܐ ܕܐܩܣܠ
library(car)      # ܒܘܚܪܢܐ ܕ-Levene: leveneTest

# ܛܥܢܐ: ܛܘܦܣܐ ܬܩܝܢܐ – ܣܕܪܐ = ܡܫܬܘܬܦܢܐ، ܥܡܘܕܐ = ܡܫܬܚܠܦܢܐ؛ ܓܠܝܘܢܐ "נתונים"
d <- read_excel("C:/Rdata/DATA_anova_example_heb.xlsx", sheet = "נתונים")
d <- d[complete.cases(d[["השכלה"]], d[["עמדות_ציון"]]), ]          # ܫܩܠܬܐ ܠܦܘܬ ܦܘܫܩܐ (117 ܡܢ 120)
g <- factor(d[["השכלה"]], levels = 1:4, labels = c("תיכונית", "על-תיכונית", "תואר ראשון", "תואר מתקדם"))
y <- d[["עמדות_ציון"]]

# ܠܘܚܐ 1 – ܣܛܐܛܝܣܛܝܩܐ ܕܓܘ̈ܕܐ
tapply(y, g, function(v) c(n = length(v), M = mean(v), SD = sd(v)))
# ܦܐܪܐ: 17 / 18 / 50 / 32 ܡܫܬܘܬܦ̈ܢܐ؛ M 3.51, 3.21, 3.06, 3.51; SD 0.67, 0.81, 0.66, 0.81

# ܒܘܚܪܢܐ ܕ-Levene ܚܕܪ ܡܨܥܝܬܐ – ܐܝܟ Based on Mean ܓܘ SPSS
leveneTest(y ~ g, center = mean)                         # F(3, 113) = 0.218, p = .884

# ܠܘܚܐ 2 – ܠܘܚܐ ܕ-ANOVA، ܘܣܕܪܐ ܕ-Welch
m <- aov(y ~ g); summary(m)                              # F(3, 113) = 3.21, p = .026; SSB 5.08, SSW 59.51
oneway.test(y ~ g)                                       # Welch F(3, 43.82) = 3.26, p = .030

# ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ: ܐܝܬܐ ܡܪܒܥܐ = SSB/SST، ܐܘܡܝܓܐ ܡܪܒܥܐ = (SSB − dfB·MSW)/(SST + MSW)
s <- summary(m)[[1]]; ssb <- s[["Sum Sq"]][1]; ssw <- s[["Sum Sq"]][2]; msw <- s[["Mean Sq"]][2]; dfb <- s[["Df"]][1]
ssb / (ssb + ssw)                                        # ܐܝܬܐ ܡܪܒܥܐ = .079
(ssb - dfb * msw) / (ssb + ssw + msw)                    # ܐܘܡܝܓܐ ܡܪܒܥܐ = .054

# ܠܘܚܐ 3 – ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ: Tukey HSD (Tukey-Kramer) ܘ-Bonferroni
TukeyHSD(m)                                              # ܕܪܓܐ ܡܬܩܕܡܐ–ܕܪܓܐ ܩܕܡܝܐ: ܦܘܪܫܢܐ 0.448, p = .037; ܙܘܓ̈ܐ ܐܚܪ̈ܢܐ ܠܐ ܪܫܝܡ̈ܐ
pairwise.t.test(y, g, p.adjust.method = "bonferroni")    # ܕܪܓܐ ܡܬܩܕܡܐ–ܕܪܓܐ ܩܕܡܝܐ: p = .044

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܟܡܬܢܚ ܡܫܟܚܢܘܬܐ ܘܒܬܪ ܣܛܐܛܝܣܛܝܩܐ: ܡܨܥܝܬܐ (mean) ܘܣܛܝܘܬܐ ܬܩܝܢܬܐ (standard deviation) ܕܟܠ ܓܘܕܐ (ܓܘ ܡ̈ܠܐ، ܓܘ ܠܘܚܐ ܝܢ ܓܘ ܨܘܪܬܐ)، F ܥܡ ܬܪܝܢ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܓܘ ܩܘ̈ܠܒܐ، p ܚܬܝܬܐ، ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size)، ܘܒܬܪ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ (post hoc) – ܐܝܢܝ ܙܘܓ̈ܐ ܦܪ̈ܝܫܐ ܗܘܘ (Gravetter & Wallnau, 2017, pp. 389-390; Field, 2024, Section 12.12; Pallant, 2020, Chapter 18; American Psychological Association [APA], 2020, Section 6.43). ܡܬܠܐ ܕ-Gravetter ܘ-Wallnau: "ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܓܠܐ ܦܘܪ̈ܫܢܐ ܪܫܝܡ̈ܐ ܒܝܢܝ ܬܠܬܐ ܐܘܪ̈ܚܬܐ ܕܝܘܠܦܢܐ، F(2, 15) = 7.16, p = .007, η² = 0.488" – ܘܐܝܡܢ ܕ-p ܚܬܝܬܐ ܐܝܬ ܡܢ ܚܘܫܒܢܐ ܟܡܬܢܚ ܠܗ، ܠܐ p < .05. ܡܬܠܐ ܕ-Field، ܥܡ Welch ܘ-ω²: "ܠܐ ܐܫܬܟܚ ܡܥܒܕܢܘܬܐ ܪܫܝܡܬܐ ܕܐܣܝܘܬܐ ܒܓܘܪ̈ܝܐ ܥܠ ܪܡܐ ܕܚܕܘܬܐ، F(2, 7.94) = 4.32, p = .054, ω² = .35 [−.15, .59]" – ܘܚܙܝܡܘܢ ܕܗܘ ܟܡܬܢܐ ܐܦ F ܕܠܐ ܪܫܝܡܐ ܘܐܦ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܕܡܬܚܙܝܐ. ܡܬܠܐ ܕ-Pallant: "ܐܫܬܟܚ ܦܘܪܫܢܐ ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ ܒܪܡܐ p < .05 ܓܘ ܡܢܝ̈ܢܐ ܕ-LOT ܒܝܢܝ ܬܠܬܐ ܓܘ̈ܕܐ ܕܥܘܡܪܐ: F(2, 432) = 4.64, p = .01. ܐܦ ܥܡ ܪܫܝܡܘܬܐ، ܦܘܪܫܢܐ ܒܫܪܪܐ ܒܝܢܝ ܡܨܥܝ̈ܬܐ ܙܥܘܪܐ ܗܘܐ؛ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ، ܐܝܬܐ ܡܪܒܥܐ، ܗܘܐ .02. ܦܘܚܡ̈ܐ ܕܒܬܪ ܗܕܐ ܒܒܘܚܪܢܐ ܕ-Tukey HSD ܓܠܝܠܗܘܢ ܕܡܨܥܝܬܐ ܕܓܘܕܐ 1 (M = 21.36, SD = 4.55) ܦܪܝܫܐ ܗܘܐ ܒܪܫܝܡܘܬܐ ܡܢ ܕܓܘܕܐ 3 (M = 22.96, SD = 4.49)؛ ܓܘܕܐ 2 (M = 22.10, SD = 4.15) ܠܐ ܗܘܐ ܦܪܝܫܐ ܒܪܫܝܡܘܬܐ ܡܢ ܚܕ ܡܢܝܗܝ." ܩܐ ܣܕܪܐ ܕܦܘܫܩ̈ܐ – ܠܘܚܐ ܕ-APA ܥܡ n, M, SD ܩܐ ܟܠ ܓܘܕܐ، F, p ܘ-η² (Pallant).

ܢܡܘ̈ܣܐ ܕܨܘܪܬܐ (APA, 2020, Section 6.36): F ܥܡ ܣܦܝܩܘܬܐ ܩܕܡ ܢܘܩܙܐ (ܡܨܐ ܕܗܘܐ ܝܬܝܪ ܡܢ 1)، p، η² ܘ-ω² ܕܠܐ (ܠܐ ܡܨܝ)؛ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܠܐ ܓܡܝܪ̈ܐ ܕ-Welch – ܒܬܪܝܢ ܡܢܝ̈ܢܐ. ܡܐܡܪ̈ܐ ܕܬܘܢܝܐ (report sentences) ܕܓܘ ܡܦܩܢܐ ܒܢܝ̈ܐ ܝܢܐ ܗܕܟܐ، ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ، ܥܡ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ ܕܐܘܪܚܐ ܕܓܒܝܬܐ ܝܠܗ ܒܓܘܒܝܐ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܣܕܪܐ ܕܒܘܚܪ̈ܢܐ t ܒܕܘܟܬܐ ܕ-ANOVA. ܐܪܒܥܐ ܓܘ̈ܕܐ ܫܬܐ ܒܘܚܪ̈ܢܐ t ܝܢܐ – ܘܣܟܘܢܐ ܟܠܢܝܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I (Type I error) ܟܣܠܩ ܒܣܓܝ ܠܥܠ ܡܢ 5% (Gravetter & Wallnau, 2017, pp. 369-370; Field, 2024, Section 2.10.8).

ܕܡܬܢܚ "ܐܝܬ ܦܘܪܫܢܐ" ܕܠܐ ܐܡܪܚ ܒܝܢܝ ܡܢܝ. F ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ (statistically significant) ܒܣ ܟܐܡܪ ܕܠܟܡܐ ܚܕ ܙܘܓܐ ܦܪܝܫܐ ܝܠܗ؛ ܕܠܐ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ (post hoc؛ ܝܢ ܣܘܩܒ̈ܠܐ ܕܛܘܝܒ̈ܐ ܡܢ ܩܕܡ) ܡܫܟܚܢܘܬܐ ܠܐ ܝܠܗ ܓܡܝܪܬܐ (Gravetter & Wallnau, p. 393).

ܕܦܫܩܚ ܒܘܚܪ̈ܢܐ ܕܒܬܪ ܗܕܐ ܐܝܡܢ ܕ-F ܠܐ ܝܠܗ ܪܫܝܡܐ – ܝܢ ܕܓܒܚ ܐܘܪܚܐ ܒܬܪ ܕܚܙܝܠܢ ܦܐܪ̈ܐ. ܟܣܝܡܚ ܡܢ ܩܕܡ (Pallant, 2020، ܨܚܚܐ 18؛ Field، ܦܣܩܐ 12.6 – "ܐܗܐ ܩܠܝܠ ܐܝܟ p-hacking ܝܠܗ، ܒܣ ܕܟܬܪܨܚ p").

ܕܠܐ ܚܙܚ ܣܕܪܐ ܕ-Welch. ܐܝܡܢ ܕ-Levene ܪܫܝܡܐ ܝܠܗ، ܝܢ ܐܝܡܢ ܕܓܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܒܣܓܝ ܒܪܒܘܬܐ ܘܒܒܕܪܐ ܕܝܠܗܘܢ، ܣܕܪܐ ܫܚܝܡܐ ܕ-F ܡܨܐ ܕܡܛܥܐ ܠܬܪܝܢ ܓܢܒ̈ܐ (Field, Box 12.2)؛ Field ܟܡܬܢܐ Welch ܟܠ ܙܒܢܐ.

ܕܡܬܢܚ p ܕܠܐ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size). ܓܘ ܕܘܡܝܐ (sample) ܪܒܐ ܦܘܪܫܢܐ ܙܥܘܪܐ ܒܣܓܝ ܟܦܠܛ ܪܫܝܡܐ – ܡܬܠܐ ܕ-Pallant ܥܡ η² = .02؛ Field ܟܡܬܢܐ ω² ܐܦ ܐܝܡܢ ܕ-F ܠܐ ܝܠܗ ܪܫܝܡܐ.

ANOVA ܕܚܕ ܐܘܪܚܐ ܥܠ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ (repeated measures). ܗܢܝ ܡܫܬܘܬܦ̈ܢܐ (participants) ܓܘ ܬܠܬܐ ܙܒ̈ܢܐ ܟܬܒܪܝ ܠܐ ܬܠܝܘܬܐ (Field, Section 12.4.2) – ANOVA ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ.

ܕܬܟܠܚ ܥܠ ܨܘܪܬܐ ܕܡܨܥܝ̈ܬܐ. ܣܪܛܐ ܕܠܐ ܟܫܪܐ ܡܢ ܣܦܝܩܘܬܐ ܟܡܪܒܐ ܦܘܪ̈ܫܢܐ ܙܥܘܪ̈ܐ ܒܣܓܝ (Pallant, Chapter 18)، ܘܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals) ܕܟܡܚܦܝ ܚܕ ܥܠ ܐܚܪܢܐ ܠܐ ܟܐܡܪܝ "ܠܐ ܪܫܝܡܐ" (Field, Section 12.9.1).

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.). Routledge.

Welch (1951)، Brown ܘ-Forsythe (1974)، Toothaker (1993)، Kirk (1996)، Scariano ܘ-Davenport (1987)، Wilcox (2017) ܘ-Weinstein، McDermott ܘ-Roediger (2010) – ܐܝܟ ܕܡܬܝ̈ܬܐ ܝܢܐ ܓܘ Field (2024) ܘܓܘ Gravetter ܘ-Wallnau (2017). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ (site rule)، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ (reference list) ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܓܘ ܫܪܫܐ ܕܝܗܝ. Field (2024) ܘ-Pallant (2020) ܒܢܘܣܚܐ ܕܝܓܝܛܠܝܐ ܕܪܕܝܐ (reflowable) – ܡܬܝܬܐ ܠܦܘܬ ܦܣܩܐ ܘܨܚܚܐ؛ Gravetter ܘ-Wallnau (2017)، Cohen (1988) ܘ-APA (2020) – ܠܦܘܬ ܦܐܬܐ ܘܦܣܩܐ ܕܡܦܩܬܐ ܛܒܝܥܬܐ.