ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܥܡ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (ANCOVA)

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 ܕܟܦܚܡܚ ܡܨܥܝ̈ܬܐ (means) ܕܓܘ̈ܕܐ (ܐܝܟ ܓܘ ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ)، ܘܐܝܬ ܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ (continuous variable) ܐܚܪܢܐ ܕܐܣܝܪܐ ܝܠܗ ܥܡ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) – ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate؛ ܡܢ ܩܕܡ ܒܐܢܓܠܝܙܝܐ ܡܬܩܪܐ ܗܘܐ concomitant variable، "ܡܫܬܚܠܦܢܐ ܕܐܬܐ ܥܡ" ܥܩܪܐ ܕܒܘܨܝܐ – Maxwell et al., 2018، ܦܐܬܐ 452؛ ܡܬܩܪܐ ܐܦ "ܡܫܬܚܠܦܢܐ ܕܦܘܚܡܐ"). ܓܗܐ ܕܡܬܚܙܐ ܣܓܝ: ܢܣܝܘܢܐ (experiment) ܥܡ ܡܫܘܚܬܐ ܩܕܡ ܘܒܬܪ – ܦܘܚܡܐ ܓܘ ܡܢܝܢܐ ܕ"ܒܬܪ" ܝܠܗ، ܘܡܢܝܢܐ ܕ"ܩܕܡ" ܝܠܗ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (Pallant, 2020, Chapter 22; Field, 2024, Section 13.5.1). ANCOVA ܝܠܗ ܡܘܕܝܠ (model) ܣܪܛܢܝܐ (linear) ܕܐܝܬ ܓܘܗ ܐܦ ܥܠܬܐ (factor) ܕܓܢ̈ܣܐ (categorical) ܘܐܦ ܡܢܒܝܢܐ (predictor) ܡܬܡܫܚܢܐ (Field, 2024, Section 13.3). ܐܝܡܢ ܕܐܝܬ ܟܡܐ ܡܫܘ̈ܚܬܐ ܩܐ ܗܢܝ ܡܫܬܘܬܦ̈ܢܐ (participants) ܘܫܘܐܠܐ ܝܠܗ ܫܘܢܝܐ ܓܢܒܗ – ANOVA ܩܐ ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ.

ܚܘܫܒܐ: ܣܪ̈ܛܐ ܡܩܒܠ̈ܐ

ܡܘܕܝܠ (model) ܟܡܦܐܐ ܩܐ ܟܠ ܓܘܕܐ ܣܪܛܐ ܕܪܝܓܪܫܢ (regression line) ܕܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܥܠ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) – ܒܚܕܐ ܡܨܠܝܢܘܬܐ (slope) ܩܐ ܟܠ ܓܘ̈ܕܐ، ܡܨܠܝܢܘܬܐ ܡܫܘܬܦܬܐ ܓܘ ܓܘ̈ܕܐ (b_W) – ܘܟܒܚܪ ܐܢ ܐܢܝ ܣܪ̈ܛܐ ܡܩܒܠ̈ܐ ܐܝܬܠܗܘܢ ܪ̈ܘܡܐ ܦܪ̈ܝܫܐ (Maxwell et al., 2018, pp. 463-466). ܛܥܝܘܬܐ ܝܠܗ ܣܛܝܘܬܐ ܕܟܠ ܡܫܬܘܬܦܢܐ (participant) ܡܢ ܣܪܛܐ ܕܓܘܕܐ ܕܝܠܗ، ܘܡܛܠ ܗܕܐ ܐܝܡܢ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܟܡܢܒܐ ܛܒ، ܛܥܝܘܬܐ ܙܥܘܪܬܐ ܝܠܗ ܒܣܓܝ ܡܢ ܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance) ܫܚܝܡܐ. ܓܘ ܡܬܠܐ ܙܥܘܪܐ ܕ-Maxwell ܘܚܒܪ̈ܘܗܝ، ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ (sum of squares) ܕܛܥܝܘܬܐ ܟܢܚܬ ܡܢ 28 ܠ-12 (ܬܡܢ، ܦܐܬܐ 467-468). ܓܘ ܨܘܪܬܐ ܕܒܕܪܐ (scatter plot) ܕܦܐܬܐ ܟܨܐܪ ܟܚܙܚ ܐܗܐ ܒܐܘܪܚܐ ܬܪܝܨܬܐ.

ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ

ܡܨܥܝܬܐ ܡܬܘܪܨܬܐ (adjusted mean) ܕܓܘܕܐ ܝܠܗ ܪܘܡܐ ܕܣܪܛܐ ܕܝܠܗ ܓܘ ܡܨܥܝܬܐ (mean) ܟܠܢܝܬܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate): Ȳ′_j = Ȳ_j − b_W(X̄_j − X̄) (Maxwell et al., 2018, Equation 9.27, p. 469). ܓܘܕܐ ܕܫܪܝܠܗ ܪܡܐ ܓܘ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ "ܟܢܚܬ" ܥܠ ܣܪܛܐ، ܘܓܘܕܐ ܕܫܪܝܠܗ ܢܡܘܟܐ "ܟܣܠܩ" – ܬܘܪܨܐ ܝܠܗ "ܫܘܓܫܐ ܕܡܨܥܝܬܐ ܥܠ ܣܪܛܐ ܕܪܝܓܪܫܢ" (ܬܡܢ، ܦܐܬܐ 471). ܒܘܚܪܢܐ ܕ-F ܕܥܠܬܐ (factor) ܟܒܚܪ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ، ܘܡܛܠ ܗܕܐ ܟܦܫܩܚ ܠܗܝܢ ܘܠܐ ܠܡܨܥܝ̈ܬܐ ܓܘܠܡ̈ܝܬܐ (Field, 2024, Section 13.7.2). ܓܘ ܡܬܠܐ ܕ-Maxwell ܘܚܒܪ̈ܘܗܝ: ܡܨܥܝ̈ܬܐ ܕܥܩܬܐ ܕܪܥܝܢܐ (depression) ܒܬܪ ܐܣܝܘܬܐ 7.3، 12.2 ܘ-14.0، ܘܡܬܘܪ̈ܨܬܐ 7.54، 11.98 ܘ-13.98 (2018، ܦܐܬܐ 483-484) – ܦܐܬܐ ܟܝܗܒ ܗܢܝ ܡܢܝ̈ܢܐ ܓܢܒܗܘܢ.

ܥܡ ܦܘܠܓܐ ܓܕܫܢܝܐ ܘܕܠܐ ܐܗܐ

ܐܝܡܢ ܕܓܘ̈ܕܐ ܦܠܝܓ̈ܐ ܝܢܐ ܒܓܕܫܐ، ܫܘ̈ܝܐ ܝܢܐ ܒܡܨܥܝܬܐ (mean) ܓܘ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate)، ܘܥܘܕܪܢܐ ܪܝܫܝܐ ܕ-ANCOVA ܝܠܗ ܙܥܘܪܐ ܕܛܥܝܘܬܐ – ܒܘܚܪܢܐ ܚܝܠܬܢܐ ܝܬܝܪ ܘܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals) ܥܝܩ̈ܐ ܝܬܝܪ، ܐܝܡܢ ܕܡܬܘܪ̈ܨܬܐ ܘܓܘܠܡ̈ܝܬܐ ܟܫܥܪܝ ܠܗܘ ܦܘܪܫܢܐ ܓܢܒܗ (Maxwell et al., 2018, pp. 454-455, 471). ܐܦ ܐܢ ܐܬܐ ܕܓܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܩܠܝܠ ܓܘ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܒܓܕܫܐ، ANCOVA ܛܒܐ ܝܬܝܪ ܝܠܗ (ܬܡܢ، ܦܐܬܐ 476).

ܐܝܡܢ ܕܓܘ̈ܕܐ ܐܝܬܠܗܘܢ ܡܢ ܩܕܡ – ܨܦ̈ܐ، ܒ̈ܬܐ ܕܣܦܪܐ، ܐܢܝ ܕܓܒܝܠܗܘܢ ܕܡܫܬܘܬܦܝ – ܬܘܪܨܐ ܡܨܐ ܕܡܫܚܠܦ ܦܐܪܐ (outcome) ܘܐܦ ܕܗܦܟ ܠܗ. ANCOVA "ܠܐ ܝܠܗ ܚܠܦܐ ܕܦܘܠܓܐ ܓܕܫܢܝܐ" (Maxwell et al., 2018, p. 454): ܟܦܢܐ ܠܫܘܐܠܐ "ܐܢ ܓܘ̈ܕܐ ܦܪ̈ܝܫܐ ܗܘܘ ܐܝܡܢ ܕܫܘ̈ܝܐ ܗܘܘ ܓܘ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ"، ܘܠܝܬ ܐܘܪܚܐ ܕܡܫܪܪܚ ܕܟܬܪܨܐ ܒܬܪܝܨܘܬܐ ܦܘܪ̈ܫܢܐ ܕܠܐ ܗܘܘ ܡܫܝ̈ܚܐ (ibid., p. 455; Field, 2024, Section 13.5.1). ܘܐܦ، ܐܝܡܢ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܡܫܝܚܐ ܝܠܗ ܥܡ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ، ܬܘܪܨܐ ܚܣܝܪܐ ܝܠܗ (Maxwell et al., 2018, p. 482). ܦܐܬܐ ܟܒܚܪ ܐܢ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܦܪܝܫܐ ܝܠܗ ܒܝܢܝ ܓܘ̈ܕܐ ("ܫܘܝܘܬܐ"). ܐܗܐ ܠܐ ܝܠܗ ܣܢܝܩܘܬܐ ܣܛܐܛܝܣܛܝܩܝܬܐ، ܐܠܐ ܐܗܝ ܟܦܣܩܐ ܕܟܝ ܟܦܫܩܚ (ibid., pp. 475-476; Field, 2024, Section 13.5.1).

ܓܘܒܝܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ

ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܓܪܓ ܕܗܘܐ ܡܫܝܚܐ ܩܕܡ ܐܣܝܘܬܐ – ܡܫܬܚܠܦܢܐ ܕܐܣܝܘܬܐ ܡܨܐ ܗܘܐ ܕܡܥܒܕ ܥܠܘܗܝ "ܟܡܦܩ" ܡܢܬܐ ܡܢ ܡܥܒܕܢܘܬܐ (effect) ܓܢܒܗ (Pallant, 2020, Chapter 22; Maxwell et al., 2018, p. 478). ܟܓܒܚ ܠܗ ܡܢ ܩܕܡ، ܠܦܘܬ ܒܘܨ̈ܝܐ ܩܕܡܝ̈ܐ ܕܟܡܚܘܝܝ ܕܐܝܠܗ ܟܡܢܒܐ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable)، ܘܠܐ ܠܦܘܬ ܝܗܝ̈ܒܬܐ ܕܒܘܨܝܐ ܗܫܝܐ (ܬܡܢ، ܦܐܬܐ 494). ܡܫܘܚܬܐ ܕ"ܩܕܡ" ܕܡܫܬܚܠܦܢܐ ܬܠܝܐ ܝܠܗ ܓܒܝܐ ܛܒܐ (ܬܡܢ، ܦܐܬܐ 451). ܓܪܓ ܕܗܘܐ ܫܪܝܪܐ – Pallant ܟܡܩܪܒܐ ܐܠܦܐ ܕ-Cronbach (Cronbach's alpha) ܠܥܠ ܡܢ .7، ܛܒܐ ܝܬܝܪ ܠܥܠ ܡܢ .8 (ܨܚܚܐ 22؛ ܚܙܝܡܘܢ ܐܠܦܐ ܕ-Cronbach). ܟܡܐ ܡܫܬܚܠܦ̈ܢܐ ܡܠܘܝܢ̈ܐ؟ ܓܘ ܒܘܨܝܐ ܙܥܘܪܐ ܠܐ ܝܬܝܪ ܡܢ ܬܪܝܢ ܝܢ ܬܠܬܐ (Maxwell et al., 2018, p. 493)، ܘܠܐ ܐܢܝ ܕܐܣܝܪ̈ܐ ܝܢܐ ܒܣܓܝ ܚܕ ܥܡ ܐܚܪܢܐ – ܩܪܝܒ ܠ-r = .80 ܛܒܐ ܝܠܗ ܕܫܩܠܚ ܚܕ (Pallant, 2020, Chapter 22).

ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ

ܫܪܛܐ (assumption) ܪܝܫܝܐ: ܐܣܘܪܐ ܒܝܢܝ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܘܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܚܕ ܝܠܗ ܓܘ ܟܠ ܓܘ̈ܕܐ – ܣܪ̈ܛܐ ܒܫܪܪܐ ܡܩܒܠ̈ܐ ܝܢܐ. ܟܒܚܪܚ ܠܗ ܒܡܘܕܝܠ (model) ܕܟܡܘܣܦ ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction) ܥܠܬܐ × ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ: ܐܢ ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ (statistically significant) ܝܠܗ، ܡܨܠܝܢ̈ܘܬܐ ܦܪ̈ܝܫܐ ܝܢܐ (Pallant, 2020, Chapter 22; Field, 2024, Section 13.8; Maxwell et al., 2018, p. 520). ܦܐܬܐ ܟܡܪܗܛ ܒܘܚܪܢܐ ܐܗܐ ܟܠ ܙܒܢܐ، ܘܟܡܚܘܐ ܐܦ ܡܨܠܝܢܘܬܐ (slope) ܓܘ ܟܠ ܓܘܕܐ. ܡܘܕܝ ܟܥܒܕܚ ܐܝܡܢ ܕܐܝܠܗ ܬܒܝܪܐ؟ Pallant (2020، ܨܚܚܐ 22) ܟܟܬܒܐ ܕܗܝܓܝ ܠܐ ܛܒܐ ܝܠܗ ܕܡܪܗܛܚ ANCOVA؛ Field (2024، ܦܣܩܐ 13.8) ܟܡܩܪܒ ܕܢܛܪܚ ܡܘܕܝܠ ܥܡ ܥܒܕܐ ܕܚܕ̈ܕܐ ܘܦܫܩܚ ܠܗ؛ Maxwell ܘܚܒܪ̈ܘܗܝ (2018، ܦܐܬܐ 536-537) ܟܡܩܪܒܝ ܡܘܕܝܠ ܥܡ ܡܨܠܝܢ̈ܘܬܐ ܦܪ̈ܝܫܬܐ – ܡܢ ܐܝܡܢ ܕܒܘܚܪܢܐ ܕܗܘܡܘܓܢܝܘܬܐ ܩܪܝܒܐ ܝܠܗ ܠܪܫܝܡܘܬܐ (significance)، ܡܬܠܐ p < .2 – ܘܐܘܪܚܐ ܕ-Johnson-Neyman، ܕܟܡܚܘܝܐ ܒܐܝܢܝ ܛܝ̈ܡܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܦܘܪܫܢܐ ܒܝܢܝ ܓܘ̈ܕܐ ܪܫܝܡܐ ܝܠܗ (ܬܡܢ، ܦܐܬܐ 531-535). ܦܐܬܐ ܠܐ ܟܡܪܗܛ Johnson-Neyman؛ ܠ-SPSS ܐܝܬ macro ܕܟܡܪܗܛ ܠܗ (ܬܡܢ، ܦܐܬܐ 535-536).

ANCOVA ܝܢ ܡܢܝ̈ܢܐ ܕܫܘܢܝܐ؟

ܐܘܪܚܐ ܐܚܪܬܐ ܕܡܬܚܙܝܐ ܣܓܝ: ܚܫܒܚ ܩܐ ܟܠ ܡܫܬܘܬܦܢܐ (participant) "ܒܬܪ ܒܨܝܪ ܩܕܡ" ܘܦܚܡܚ ܦܘܪ̈ܫܢܐ ܒܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance). ܓܘ ܢܣܝܘܢܐ (experiment) ܥܡ ܦܘܠܓܐ ܓܕܫܢܝܐ ܠܓܘ̈ܕܐ ܬܪܘܬܝܗܝ ܐܘܪ̈ܚܬܐ ܟܒܚܪܝ ܚܕܐ ܗܘܦܘܬܣܝܣ (hypothesis)، ܐܠܐ ܠ-ANCOVA ܐܝܬ ܛܥܝܘܬܐ ܙܥܘܪܬܐ ܝܬܝܪ، ܘܡܛܠ ܗܕܐ "ܩܪܝܒ ܟܠ ܙܒܢܐ ܛܒܐ ܝܬܝܪ ܝܠܗ" (Maxwell et al., 2018, pp. 499-501). ܡܢܝܢܐ ܕܫܘܢܝܐ (change score) ܫܘܝܐ ܝܠܗ ܠ-ANCOVA ܥܡ ܡܨܠܝܢܘܬܐ (slope) ܕܣܝܡܬܐ ܝܠܗ ܡܢ ܩܕܡ ܠ-1 (ܬܡܢ، ܦܐܬܐ 499). ܐܝܡܢ ܕܓܘ̈ܕܐ ܠܐ ܝܢܐ ܓܕܫܢ̈ܝܐ، ܬܪܘܬܝܗܝ ܐܘܪ̈ܚܬܐ ܟܫܐܠܝ ܫܘܐ̈ܠܐ ܦܪ̈ܝܫܐ ܘܡܨܝܢ ܕܝܗܒܝ ܦܘܢ̈ܝܐ ܦܪ̈ܝܫܐ – "ܦܪܕܘܟܣܐ ܕ-Lord" (ܬܡܢ، ܦܐܬܐ 501-502). ܐܘܪܚܐ ܬܠܝܬܝܬܐ – ܚܫܒܚ ܫܪ̈ܟܐ (residuals) ܡܢ ܪܝܓܪܫܢ (regression) ܥܠ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܘܦܫܩܚ ܠܗܘܢ – ܠܐ ܝܠܗ ܫܘܝܐ ܠ-ANCOVA، ܘܛܒܐ ܝܠܗ ܕܡܪܚܩܚ ܡܢܗ̇ (ܬܡܢ، ܦܐܬܐ 498).

ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ

ܐܝܡܢ ܕܐܝܬ ܝܬܝܪ ܡܢ ܬܪܬܝܢ ܓܘ̈ܕܐ، ܟܦܚܡܚ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ (adjusted means) ܒܙܘܓ̈ܐ. ܛܥܝܘܬܐ ܬܩܝܢܬܐ (standard error) ܕܟܠ ܦܘܪܫܢܐ ܐܝܬ ܓܘܗ̇ ܦܘܪܫܢܐ ܓܘ ܡܨܥܝ̈ܬܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) (Maxwell et al., 2018, Equation 9.37, p. 490). ܦܐܬܐ ܟܬܪܨ ܠܦܘܬ Bonferroni – ܡܩܪܒܬܐ ܕ-Field (2024، ܦܣܩܐ 13.6.6)، ܘܐܦ ܐܘܪܚܐ ܕ-SPSS، ܕܓܘܗ ܒܘܚܪ̈ܢܐ Post Hoc ܫܚܝܡ̈ܐ ܐܚܝܕ̈ܐ ܝܢܐ ܐܝܡܢ ܕܐܝܬ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (ܬܡܢ، ܦܣܩܐ 13.6.5). Maxwell ܘܚܒܪ̈ܘܗܝ ܟܡܩܪܒܝ ܩܐ ܟܠ ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ (pairwise comparisons) Tukey ܐܝܡܢ ܕܛܝ̈ܡܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܣܝܡ̈ܐ ܝܢܐ ܡܢ ܩܕܡ، ܘ-Bryant-Paulson ܐܝܡܢ ܕܡܫܝ̈ܚܐ ܝܢܐ (2018، ܦܐܬܐ 490-492)؛ ܡܛܠ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܣܓܝ ܓܗ̈ܐ ܡܫܝܚܐ ܝܠܗ ܐܝܟ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable)، ܐܝܬ ܐܢܫ̈ܐ ܕܟܡܩܪܒܝ ܕܚܙܚ ܠܗ ܐܝܟ ܡܫܬܚܠܦܢܐ ܓܕܫܢܝܐ (ܬܡܢ، ܦܐܬܐ 475).

ܪ̈ܒܘܬܐ ܕܡܥܒܕܢܘܬܐ

SPSS ܟܛܒܥ ܐܝܬܐ ܡܪܒܥܐ ܡܢܬܢܝܐ (partial eta squared) – ܫܘܚܠܦܐ (variance) ܕܥܠܬܐ (factor) ܟܡܦܫܩܐ ܡܢ ܡܐ ܕܡܫܬܚܠܦ̈ܢܐ ܐܚܪ̈ܢܐ ܠܐ ܦܫܩܠܗܘܢ (Field, 2024, Section 13.4). Maxwell ܘܚܒܪ̈ܘܗܝ (2018، ܦܐܬܐ 486-489) ܟܡܩܪܒܝ ܕܚܫܒܚ ܥܠ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܐܝܟ "ܡܢܬܐ ܥܩܪܢܝܬܐ" ܕܬܘܩܢܐ – ܐܝܬܠܗ ܓܘ ܟܢܫܐ ܟܠܢܝܐ (population) – ܘܡܛܠ ܗܕܐ ܠܐ ܡܦܩܚ ܠܗ ܡܢ ܫܘܚܠܦܐ ܟܠܢܝܐ: ܐܘܡܝܓܐ ܡܪܒܥܐ (omega squared) ܠܦܘܬ ܫܘܝܘܬܐ 9.31 (ܓܘ ܡܬܠܐ ܕܝܠܗܘܢ .12؛ ܩܐ ܦܘܚܡܐ، R² ܕܥܠܬܐ ܝܠܗ .18 ܡܢ ܟܠܗ ܫܘܚܠܦܐ ܘ-.22 ܐܝܡܢ ܕܟܡܦܠܛܚ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ – ܦܐܬܐ 487-488)، ܘ-d ܩܐ ܦܘܚܡܐ ܙܘܓܢܝܐ – ܦܘܪܫܢܐ ܒܝܢܝ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ (adjusted means) ܦܠܝܓܐ ܥܠ ܣܛܝܘܬܐ ܬܩܝܢܬܐ (standard deviation) ܡܢ ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܠܐ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (ܫܘܝܘܬܐ 9.32). ܦܐܬܐ ܟܡܚܘܐ ܬܠܬܝܗܝ (ܐܘܡܝܓܐ ܡܪܒܥܐ – ܐܝܡܢ ܕܐܝܬ ܚܕ ܥܠܬܐ).

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܠܦܘܬ ܛܟܣܐ: ܡܨܥܝ̈ܬܐ ܓܘܠܡ̈ܝܬܐ (raw means؛ ܐܦ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ)؛ ܒܘܚܪ̈ܢܐ ܩܕܡܝ̈ܐ – ܫܘܝܘܬܐ، ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ (homogeneity of regression slopes) ܘ-Levene؛ ܠܘܚܐ (table) ܕ-ANCOVA – ܣܕܪܐ ܩܐ ܟܠ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܘܩܐ ܟܠ ܥܠܬܐ (factor)؛ ܡܨܠܝܢܘܬܐ (slope)؛ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ (adjusted means) ܩܘܒܠ ܓܘܠܡ̈ܝܬܐ؛ ܦܘܚܡ̈ܐ ܙܘܓܢ̈ܝܐ (pairwise comparisons)؛ ܠܘܚܐ ܕܟܦܚܡ ܥܡ ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܠܐ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ – ܟܡܐ ܛܥܝܘܬܐ ܢܚܬܠܗ ܘܐܢ ܦܘܪܫܢܐ ܕܫܘܠܡܐ ܫܚܠܦܠܗ؛ ܘܨܘܪ̈ܬܐ. "ܩܪܝܢܐ ܕܦܐܪܐ" (Interpreting the result) ܟܡܚܘܐ ܠܡܐ ܕܐܝܠܗ ܚܫܝܚܐ ܓܘ ܓܗܐ ܕܝܘܟܘܢ.

ܫܪ̈ܛܐ

ܫܪ̈ܛܐ (assumptions) ܕܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance) – ܛܥܝ̈ܘܬܐ ܠܐ ܬܠܝ̈ܐ، ܢܘܪܡܠܝܘܬܐ (normality) ܘܫܘܚܠܦܐ ܫܘܝܐ – ܘܬܪܝܢ ܐܚܪ̈ܢܐ: ܐܣܘܪܐ ܣܪܛܢܝܐ (linear) ܒܝܢܝ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܘܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܓܘ ܟܠ ܓܘܕܐ، ܘܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ (homogeneity of regression slopes) (Maxwell et al., 2018, pp. 473-475; Pallant, 2020, Chapter 22). ܣܪܛܢܝܘܬܐ ܟܒܚܪܚ ܓܘ ܨܘܪܬܐ ܕܒܕܪܐ (scatter plot)؛ ܐܣܘܪܐ ܟܦܝܦܐ ܡܨܚ ܕܡܥܠܚ ܥܡ ܗܕܡܐ ܡܪܒܥܐ (Maxwell et al., 2018, p. 493). ܦܐܬܐ ܟܡܪܗܛ Levene ܥܠ ܛܝܡܐ ܫܪܝܪܐ (absolute value) ܕܫܪ̈ܟܐ (residuals) ܡܢ ܡܘܕܝܠ (model). ܐܝܡܢ ܕܫܪ̈ܛܐ ܠܐ ܟܡܬܡܠܝ – Field (2024، ܦܣ̈ܩܐ 13.5.3، 13.9) ܟܡܬܢܐ ܒܘܛܣܛܪܐܦ (bootstrap) ܘ-ANCOVA ܚܣܝܢܬܐ (robust).

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

Analyze → General Linear Model → Univariate: ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable)، ܥܠܬܐ (factor) ܓܘ Fixed Factor(s) ܘܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܓܘ Covariate(s) (Field, 2024, Section 13.6; Pallant, 2020, Chapter 22). ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ (homogeneity of regression slopes) ܟܒܚܪܚ ܒܪܗܛܐ ܦܪܝܫܐ ܥܡ Model → Build terms ܕܐܝܬ ܓܘܗ ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction) (Field, 2024, Section 13.8). ܓܘ EM Means – ܦܘܚܡ̈ܐ ܥܡ Bonferroni؛ ܓܘ Options – ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size)، ܒܘܚܪ̈ܢܐ ܕܗܘܡܘܓܢܝܘܬܐ ܘ-Parameter estimates. ܦܐܬܐ ܟܡܚܬ ܣܝܢܬܐܟܣ (syntax) ܥܡ ܬܪܝܢ ܪ̈ܗܛܐ (UNIANOVA ... WITH). PSPP ܠܐ ܟܡܪܗܛ GLM ܥܡ ܡܫܬܚܠܦ̈ܢܐ ܡܠܘܝܢ̈ܐ، ܘܡܛܠ ܗܕܐ ܣܝܢܬܐܟܣ ܠܐ ܝܠܗ ܒܚܝܪܐ ܓܘܗ.

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

ܡܘܕܝܠ (model) ܣܪܛܢܝܐ (linear) (lm) ܥܡ ܣܘܩܒ̈ܠܐ (contrasts) ܕܓܘܡܪܐ-ܠܣܦܝܩܘܬܐ (sum-to-zero)، car::Anova(type = 3) ܩܐ ܠܘܚܐ (table)، anova(m, ms) ܩܐ ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ (homogeneity of regression slopes)، ܘ-emmeans ܩܐ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ (adjusted means) ܘܩܐ ܦܘܚܡ̈ܐ (ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܓܘ ܡܨܥܝܬܐ ܕܝܠܗ – ܓܘܒܝܐ ܡܩܕܡܐ (default)). ܣܩܪܝܦܬ (script) ܕܠܬܚܬ ܪܗܝܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܠܗ ܥܠ ܡܬܠܐ ܕ-Maxwell ܘܚܒܪ̈ܘܗܝ (2018، ܠܘܚܐ 9.7، ܦܐܬܐ 483)، ܘܛܝ̈ܡܐ ܓܘ ܡܥܝܪ̈ܢܘܬܐ (notes).

# ANCOVA ܒ-R – ܡܬܠܐ ܕ-Maxwell ܘܚܒܪ̈ܘܗܝ (2018، ܠܘܚܐ 9.7): ܥܩܬܐ ܕܪܥܝܢܐ ܩܕܡ ܘܒܬܪ، ܬܠܬܐ ܓܘ̈ܕܐ ܕ-10
library(car)       # ܐܘܪܚܐ Anova ܕ-Type III
library(emmeans)   # ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ
options(contrasts = c("contr.sum", "contr.poly"))   # ܣܢܝܩܐ ܝܠܗ ܩܐ Type III

d <- data.frame(group = factor(rep(c("SSRI", "Placebo", "Wait"), each = 10), levels = c("SSRI", "Placebo", "Wait")),
                pre  = c(18, 16, 16, 15, 14, 20, 14, 21, 25, 11, 18, 16, 15, 14, 20, 25, 11, 25, 11, 22, 15, 19, 10, 29, 24, 15, 9, 18, 22, 13),
                post = c(12, 0, 10, 9, 0, 11, 2, 4, 15, 10, 11, 4, 19, 15, 3, 14, 10, 16, 10, 20, 17, 25, 10, 22, 23, 10, 2, 10, 14, 7))
m  <- lm(post ~ pre + group, data = d)
ms <- lm(post ~ pre + group + group:pre, data = d)
anova(m, ms)                          # ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ: F(2, 24) = 1.06, p = .36
Anova(m, type = 3)                    # ܓܘܕܐ: F(2, 26) = 3.73, p = .038; ܩܕܡ: F(1, 26) = 10.77
coef(m)["pre"]                        # ܡܨܠܝܢܘܬܐ ܡܫܘܬܦܬܐ b = 0.645
emmeans(m, pairwise ~ group, adjust = "bonferroni")
# ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ: 7.54, 11.98, 13.98; SSRI ܩܘܒܠ ܣܘܟܝܐ: -6.44, p = .039

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܟܡܬܢܚ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܘܒܘܚܪ̈ܢܐ ܩܕܡܝ̈ܐ، ܒܝܬܝܪ ܗܘܡܘܓܢܝܘܬܐ ܕܡܨܠܝܢ̈ܘܬܐ (homogeneity of regression slopes) (Pallant, 2020, Chapter 22)، ܘܒܬܪ F ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ ܘܕܥܠܬܐ (factor)، ܥܡ p ܘܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (effect size) (Field, 2024, Section 13.11; Pallant, ibid.). ܓܘ ܠܘܚܐ (table) – ܡܨܥܝ̈ܬܐ ܓܘܠܡ̈ܝܬܐ ܥܡ ܣܛܝܘܬܐ ܬܩܝܢܬܐ (standard deviation) ܘܡܬܘܪ̈ܨܬܐ ܥܡ ܛܥܝܘܬܐ ܬܩܝܢܬܐ (standard error)، ܘ-n ܓܘ ܟܠ ܓܘܕܐ (ܬܡܢ). ܦܐܬܐ ܟܟܬܒ ܦܣܩܐ ܗܕܟܐ ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

"ܕܡܚܟܡܚ" ܥܠ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܓܘ̈ܕܐ ܕܐܝܬܠܗܘܢ ܡܢ ܩܕܡ ܘܕܦܠܛܚ ܥܠܬܢܘܬܐ (causality) – ANCOVA ܠܐ ܝܠܗ ܚܠܦܐ ܕܦܘܠܓܐ ܓܕܫܢܝܐ ܠܓܘ̈ܕܐ (random assignment) (Maxwell et al., 2018, p. 454; Field, 2024, Section 13.5.1).

ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) ܕܡܫܝܚܐ ܝܠܗ ܒܬܪ ܐܣܝܘܬܐ – ܡܨܐ ܕܡܦܩ ܡܢܬܐ ܡܢ ܡܥܒܕܢܘܬܐ (effect) (Maxwell et al., 2018, p. 478).

ܕܦܫܩܚ ܡܨܥܝ̈ܬܐ ܓܘܠܡ̈ܝܬܐ (raw means) ܒܕܘܟܬܐ ܕܡܬܘܪ̈ܨܬܐ (Field, 2024, Section 13.7.2).

ܕܫܒܩܚ ܒܘܚܪܢܐ ܕܡܨܠܝܢ̈ܘܬܐ (slopes) – ܐܝܡܢ ܕܦܪ̈ܝܫܐ ܝܢܐ، ܦܘܪܫܢܐ ܒܝܢܝ ܓܘ̈ܕܐ ܬܠܝܐ ܝܠܗ ܒܛܝܡܐ ܕܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (covariate) (Field, 2024, Section 13.8).

ܕܓܒܚ ܡܫܬܚܠܦ̈ܢܐ ܡܠܘܝܢ̈ܐ (covariates) ܠܦܘܬ ܝܗܝ̈ܒܬܐ – ܡܬܠܐ، ܗܘ ܕܐܣܝܪܐ ܝܠܗ ܒܚܝܠܐ ܝܬܝܪ ܥܡ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable) ܓܘ ܕܘܡܝܐ (sample) ܗܫܝܐ (Maxwell et al., 2018, p. 494).

ܦܘܫܩܐ ܕܫܪ̈ܟܐ (residuals) ܒܕܘܟܬܐ ܕ-ANCOVA (Maxwell et al., 2018, p. 498).

ܡܒܘ̈ܥܐ

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

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

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2018). Designing experiments and analyzing data: A model comparison perspective (3rd ed.). Routledge.

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

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؛ Maxwell ܘܚܒܪ̈ܘܗܝ (2018) – ܠܦܘܬ ܦܐܬܐ ܛܒܝܥܬܐ. ܥܝ̈ܕܐ ܕܣܐܝܛ (site conventions): ܫܩܠܬܐ ܕܣܕܪܐ ܟܠܗ (listwise deletion)؛ Type III؛ ܡܨܥܝ̈ܬܐ ܡܬܘܪ̈ܨܬܐ (adjusted means) ܓܘ ܡܨܥܝܬܐ ܟܠܢܝܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܡܠܘܝܢ̈ܐ، ܕܠܐ ܡܬܩܠܢܘܬܐ (unweighted) ܥܠ ܥܠܬܐ ܬܪܝܢܐ؛ Levene ܥܠ ܛܝܡܐ ܫܪܝܪܐ (absolute value) ܕܫܪ̈ܟܐ (residuals) ܡܢ ܡܘܕܝܠ (model)؛ ܬܪܝܢ ܡܢܝ̈ܢܐ ܒܬܪ ܢܘܩܙܐ ܓܘ ܠܘ̈ܚܐ. "ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ" – ܬܘܪܓܡܐ ܕ-covariate ܓܘ ܣܐܝܛ (ܒܬܪ concomitant variable)، ܩܐ ܕܠܐ ܡܚܠܛܚ ܥܡ ܡܫܬܚܠܦܢܐ ܡܨܥܝܐ (mediator). ܚܘܫܒܢܐ ܡܫܪܪܐ ܝܠܗ ܩܘܒܠ R 4.3.3 (car, emmeans) ܘܩܘܒܠ ܡܢܝ̈ܢܐ ܛܒܝ̈ܥܐ ܓܘ Maxwell ܘܚܒܪ̈ܘܗܝ (2018): ܠܘ̈ܚܐ 9.1، 9.7-9.10 ܘ-9.12، ܬܘܪܓܡܐ (exercise) 4.