ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ

ܡܚܫܒܢܐ (calculator)

ܡܕܒܪܢܐ (guide)

ܡܘܕܝ ܝܠܗ ܢܣܝܘܢܐ، ܘܩܐ ܡܘܕܝ ܠܬܩܢܐ ܠܗ

ܢܣܝܘܢܐ (experiment) ܝܠܗ "ܣܕܪܐ ܕܪ̈ܗܛܐ ܕܓܘܝܗܝ ܟܫܢܚ ܒܨܒܝܢܐ" ܡܫܬܚܠܦ̈ܢܐ (variables) ܕܟܥܐܠܝ ܠܓܘ ܥܒܕܐ، ܩܐ ܡܚܙܝܐ ܡܘܕܝ ܟܫܢܐ ܓܘ ܦܐܪܐ (outcome) ܘܩܐ ܡܘܕܝ (Montgomery, 2020, p. 1). ܠܐ ܐܝܟ ܒܘܨܝܐ ܕܐܣܘܪܐ ܕܚܕ̈ܕܐ (correlation)، ܒܨܘܝܬܐ ܟܦܣܩܐ ܕܪ̈ܓܐ – ܐܘܪܚܐ ܕܡܠܦܢܘܬܐ، ܟܡܝܘܬܐ ܕܣܡܐ، ܫܪܛܐ – ܘܟܦܠܓܐ ܠܗܘܢ ܠܡܫܬܘܬܦ̈ܢܐ (participants / respondents). ܡܛܠ ܗܕܐ ܢܣܝܘܢܐ ܡܨܐ ܕܦܢܐ ܥܠ ܫܘܐܠܐ ܕܥܠܬܢܘܬܐ (causality)، ܐܢ ܦܘܠܓܐ ܓܕܫܢܝܐ ܝܠܗ: ܚܘܫܒܐ ܕ-Fisher ܗܘܐ ܕܓܕܫܢܝܘܬܐ ܟܡܫܪܪܐ ܕ"ܚܕ ܥܠܬܐ ܕܠܐ ܝܠܗ ܬܚܘܬ ܐܝܕܐ ܠܐ ܡܨܠܐ" ܦܐܪܐ ܕܒܘܚܪܢܐ ܣܛܐܛܝܣܛܝܩܝܐ (Maxwell et al., 2018, p. 26).

ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ ܕܡܬܚܙܝܢ ܣܓܝ ܠܦܘܠܚܢܐ ܕܠܐ ܬܘܩܢܐ – "ܚܘܫܒܐ ܛܒܐ ܡܢ ܟܠܗ" (ܟܫܢܚ ܥܠܬܐ ܠܦܘܬ ܡܐ ܕܗܘܝܠܗ ܓܘ ܪܗܛܐ ܕܩܕܡ) ܘ"ܥܠܬܐ ܚܕܐ ܓܘ ܟܠ ܓܐܝܐ" – ܟܡܚܪܒܝ ܪ̈ܗܛܐ ܘܟܡܚܣܪܝ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ (interactions): ܡܐ ܕܟܦܠܚ ܓܘ ܕܪܓܐ ܚܕ ܕܥܠܬܐ ܚܕܐ ܠܐ ܐܝܟ ܣܢܝܩܘܬܐ ܟܦܠܚ ܓܘ ܐܚܪܢܐ (Montgomery, 2020, p. 4). ܡܪܚܩܢܘܬܐ ܝܠܗ ܠܬܩܢܐ ܡܢ ܩܕܡ، ܘܒܓܘܕܐ: ܠܟܬܒܐ ܥܘܩܐ، ܠܡܓܒܐ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable (DV))، ܥܠ̈ܠܬܐ، ܕܪ̈ܓܐ ܘܦܬܝ̈ܘܬܐ، ܘܒܬܪܟܢ ܬܘܩܢܐ – ܘܒܣ ܒܬܪܟܢ ܠܡܪܗܛ ܘܠܡܦܫܩ (ܬܡܢ، ܦܐܬܐ 13، ܠܘܚܐ 1.1). "ܐܘܪܚܐ ܕܦܘܫܩܐ ܬܠܝܐ ܝܠܗ ܬܪܝܨܐܝܬ ܒܬܘܩܢܐ" (ܬܡܢ، ܦܐܬܐ 11) – ܘܡܛܠ ܗܕܐ ܦܐܬܐ ܒܢܝܐ ܝܠܗ ܡܢ ܬܪܝܢ ܡܢܘ̈ܬܐ ܕܟܡܡܠܠܝ ܚܕܐ ܥܡ ܐܚܪܬܐ: ܦܐܬܐ ܕܪܗܛܐ ܕܟܡܚܬܚ ܓܘ ܬܘܩܢܐ ܛܥܝܢܐ ܝܠܗ ܬܘܒ ܓܘ ܦܘܫܩܐ.

ܬܠܬܐ ܫܪ̈ܫܐ: ܓܕܫܢܝܘܬܐ، ܬܢܝܬܐ، ܓܘ̈ܫܡܐ

ܓܕܫܢܝܘܬܐ ܝܠܗ "ܫܬܐܣܐ": ܐܦ ܦܘܠܓܐ ܕܡܫܬܘܬܦ̈ܢܐ ܠܫܪ̈ܛܐ ܘܐܦ ܛܟܣܐ ܕܪ̈ܗܛܐ ܦܣܝܩ̈ܐ ܝܢܐ ܒܓܕܫܐ. ܐܗܝ ܟܡܙܕܩܐ ܣܒܪܐ ܕܛܥܝ̈ܘܬܐ ܠܐ ܬܠܝ̈ܐ ܝܢܐ، ܘ"ܟܡܨܥܝܐ" ܥܠ̈ܠܬܐ ܢܘܟܪ̈ܝܬܐ – ܠܡܬܠܐ، ܐܢ ܟܠ ܡܫܬܘܬܦ̈ܢܐ ܡܠܝ̈ܐ ܫܩܠܠܗܘܢ ܒܓܕܫܐ ܗܘ ܫܪܛܐ (Montgomery, 2020, p. 11). ܬܢܝܬܐ (replication) – ܪܗܛܐ ܝܬܝܪܐ ܚܐܪܐ ܕܟܠ ܚܘܠܛܐ؛ ܐܗܝ ܟܡܨܝܐ ܠܡܫܥܪ ܛܥܝܘܬܐ ܕܢܣܝܘܢܐ ܘܟܡܚܬܬܐ ܡܨܥܝ̈ܬܐ. ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ (repeated measures) ܕܗܝ ܚܕܝܘܬܐ ܠܐ ܝܢܐ ܬܢܝ̈ܬܐ (ܬܡܢ، ܦܐܬܐ 12). ܓܘ̈ܫܡܐ (blocks) – ܓܘ̈ܕܐ ܕܚܕܝ̈ܘܬܐ ܕܡܝ̈ܬܐ (ܨܦܐ، ܝܘܡܐ، ܦܠܚܐ، batch)، ܕܓܘ ܟܠ ܚܕ ܡܢܝܗܝ ܟܦܚܡܚ treatments، ܩܐ ܡܦܠܛܐ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܐܢܝ ܡܢ ܛܥܝܘܬܐ (ܬܡܢ، ܦܐܬܐ 12-13). ܬܠܬܝܗܝ "ܡܢܬܐ ܕܟܠ ܢܣܝܘܢܐ" ܝܢܐ (ܬܡܢ، ܦܐܬܐ 13)، ܘܓܐ̈ܝܐ ܚܕ̈ܐ ܟܡܘܣܦܝ ܫܪܫܐ ܐܪܒܥܝܐ – ܬܘܩܢܐ ܦܩܛܘܪܝܐܠ (ܬܡܢ، ܦܐܬܐ 11).

ܐܝܢܐ ܬܘܩܢܐ ܦܐܫ؟

ܓܘ̈ܕܐ ܠܐ ܬܠܝ̈ܐ (ܦܘܠܓܐ ܓܕܫܢܝܐ ܠܓܘ̈ܕܐ ܓܡܝܪܐ) – ܥܠܬܐ ܚܕܐ، ܟܠ ܡܫܬܘܬܦܢܐ ܓܘ ܫܪܛܐ ܚܕ. ܦܫܝܩܐ ܡܢ ܟܠܗ؛ ܦܘܫܩܐ – ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܚܕ ܐܘܪܚܐ.

ܓܘ̈ܫܡܐ ܓܕܫܢ̈ܝܐ – ܐܡܬܝ ܕܐܝܬ nuisance factor ܝܕܝܥܬܐ ܕܡܨܝܐ ܠܡܚܕ ܠܗ: ܓܘ ܟܠ ܓܘܫܡܐ ܟܠ treatments، ܘܛܟܣܐ ܓܘ ܓܘܫܡܐ ܓܕܫܢܝܐ ܝܠܗ. ܓܘܫܡܐ ܝܠܗ "ܬܚܘܡܐ ܥܠ ܓܕܫܢܝܘܬܐ" (Montgomery, 2020, p. 117). ܬܠܬܐ ܫܥܘ̈ܬܐ ܕܝܘܠܦܢܐ ܓܘ ܟܠ ܚܕ ܡܢ ܐܪܒܥܐ ܨܦ̈ܐ، ܠܡܬܠܐ.

ܡܪܒܥܐ ܠܛܝܢܝܐ – ܬܪܬܝܢ nuisance factors، ܟܠ ܚܕܐ ܒ-p ܕܪ̈ܓܐ، ܘ-p treatments: ܟܠ ܐܘܪܚܐ ܚܕ ܓܐܝܐ ܓܘ ܟܠ ܣܕܪܐ ܘܓܘ ܟܠ ܥܡܘܕܐ (Montgomery, 2020, p. 133). ܓܘ ܝܕܥܬܐ ܕܢܦܫܐ – ܝܚܝܕܐܝܬ ܩܐ ܫܘܝܘܬܐ ܕܛܟܣܐ ܕܡܚܘܝܢܘܬܐ ܓܘ ܬܘܩܢܐ ܓܘ ܡܫܬܘܬܦ̈ܢܐ (within-subjects): ܣܕܪ̈ܐ ܡܫܬܘܬܦ̈ܢܐ ܝܢܐ، ܥܡܘ̈ܕܐ ܕܘܟܬܐ ܓܘ ܛܟܣܐ (Maxwell et al., 2018, pp. 645-648).

ܦܩܛܘܪܝܐܠ – ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܝܢ ܒܢܘܬ، ܟܠ ܚܘ̈ܠܛܐ. ܟܡܨܝܐ ܠܡܒܚܪ ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction)، ܘܕܠܐ ܥܒܕܐ ܕܚܕ̈ܕܐ – ܟܠ ܡܫܬܘܬܦܢܐ "ܟܦܠܚ" ܩܐ ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܐܟܚܕܐ، ܗܟܢܐ ܕܟܫܩܠܚ ܗܘ ܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ ܒܡܫܬܘܬܦ̈ܢܐ ܒܨܝܪ̈ܐ ܡܢ ܬܪܝܢ ܢܣܝ̈ܘܢܐ ܦܪ̈ܝܫܐ (Maxwell et al., 2018, pp. 355-356; Montgomery, 2020, pp. 155-156).

‎2^k‎ ܘܡܢܘ̈ܬܐ – ܥܠ̈ܠܬܐ ܣܓܝ̈ܐܐ، ܟܠ ܚܕܐ ܒܬܪܝܢ ܕܪ̈ܓܐ ("ܢܚܝܬܐ" ܘ"ܪܡܐ"). ܩܐ ܡܨܠܠܢܘܬܐ (screening): ܐܝܢܐ ܡܢ ܒܝܢܝ ܬܡܢܝܐ ܥܠ̈ܠܬܐ ܟܡܥܒܕܐ. ܐܡܬܝ ܕܠܐ ܡܨܝܐ ܠܡܪܗܛ ܟܠ ‎2^k‎ ܚܘ̈ܠܛܐ، ܟܡܪܗܛܚ ܦܠܓܐ ܝܢ ܪܘܒܥܐ ܡܢܝܗܝ (fraction) – ܒܛܝܡܐ ܕ-aliases ܒܝܢܝ ܡܥܒܕ̈ܢܘܝܬܐ (Montgomery, 2020, pp. 274-278).

ܬܘܩ̈ܢܐ ܥܡ ܥܠܬܐ ܓܘ ܡܫܬܘܬܦ̈ܢܐ (ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ) – ܒܘܚܪܢܐ t ܠܕܘ̈ܡܝܐ ܙܘܓܢ̈ܝܐ ܝܢ ANOVA ܠܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ؛ ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ، ܡܩܢܢܐ ܘܣܦܠܝܛ-ܦܠܘܛ – ܓܘ ܦܐܬܐ "ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ، ܡܩܢܢܐ ܘܣܦܠܝܛ-ܦܠܘܛ"؛ ܥܠ̈ܠܬܐ ܟܡ̈ܝܝܬܐ ܘܢܘܩܙܐ ܛܒܐ ܡܢ ܟܠܗ – ܓܘ ܦܐܬܐ "ܐܘܪܚܐ ܕܐܦ̈ܐ ܕܦܘܢܝܐ".

ܪܒܘܬܐ ܕܕܘܡܝܐ ܘܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ

ܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ (statistical power) ܝܠܗ ܡܣܬܒܪܢܘܬܐ (probability) ܠܡܣܠܝܐ ܗܘܦܘܬܣܝܣ ܕܣܦܝܩܘܬܐ (null hypothesis) ܐܡܬܝ ܕܫܪܝܪܐܝܬ ܦܘܫܝܬܐ ܝܠܗ. ܐܡܬܝ ܕܗܘܦܘܬܣܝܣ ܦܘܫܝܬܐ ܝܠܗ، ܡܚܘܝܢܐ F ܦܠܝܓܐ ܝܠܗ ܐܝܟ F ܠܐ ܡܨܥܝܬܐ (noncentral F)، ܥܡ ܦܪܡܝܛܪ ܕ-noncentrality λ ܕܟܚܠܛ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ، ܫܘܚܠܦܐ ܘܪܒܘܬܐ ܕܕܘܡܝܐ (Maxwell et al., 2018, pp. 144-145). ܦܐܬܐ ܟܚܫܒ ܚܝܠܐ ܬܪܝܨܐܝܬ ܡܢ ܐܗܐ ܦܘܠܓܐ – ܗܘ ܚܘܫܒܐ ܐܝܟ G*Power ܘ-pwr.anova.test ܓܘ R – ܘܟܒܨܐ n ܙܥܘܪܐ ܡܢ ܟܠܗ ܕܟܡܛܐ ܠܚܝܠܐ ܕܒܥܐܠܘܟܘܢ. "ܡܢܬܐ ܩܫܝܬܐ ܡܢ ܟܠܗ ܓܘ ܦܘܫܩܐ ܕܚܝܠܐ ܒܥܕܢܐ ܪܓܝܠܐ ܕܚܫܒܐ ܝܠܗ، ܠܐ ܕܚܘܫܒܢܐ": ܣܢܝܩ ܝܠܗ ܠܡܦܣܩ ܡܘܕܝ ܝܠܗ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ (ܬܡܢ، ܦܐܬܐ 145).

ܬܠܬܐ ܐܘܪ̈ܚܬܐ ܠܬܚܡܐ ܠܗ، ܓܘ ܦܐܬܐ ܘܓܘ ܟܬܒܐ: (1) f ܕ-Cohen – ܣܛܝܘܬܐ ܬܩܝܢܬܐ ܕܡܨܥܝ̈ܬܐ ܕܟܢܫܐ ܟܠܢܝܐ ܡܦܠܓܬܐ ܥܠ ܣܛܝܘܬܐ ܬܩܝܢܬܐ ܓܘ ܓܘܕܐ؛ ܙܥܘܪܐ .10، ܡܨܥܝܐ .25، ܪܒܐ .40 (Cohen, 1988, as cited in Maxwell et al., 2018, p. 146). (2) ܡܨܥܝ̈ܬܐ ܡܣܟ̈ܝܬܐ ܘܣܛܝܘܬܐ ܬܩܝܢܬܐ – ܡܢܝܗܝ ܚܫܝܒܐ ܝܠܗ f؛ ܓܘ ܡܬܠܐ ܕܟܬܒܐ، 400، 450 ܘ-500 ܥܡ σ = 100 ܟܝܗܒܝ f = .41 ܘ-21 ܡܫܬܘܬܦ̈ܢܐ ܩܐ ܟܠ ܓܘܕܐ ܩܐ ܚܝܠܐ ܕ-.80 (ܬܡܢ، ܦܐ̈ܬܐ 147-149). (3) ܦܘܪܫܢܐ ܒܝܢܝ ܬܪܬܝܢ ܡܨܥܝ̈ܬܐ ܕܣܦܪ̈ܐ، ܒܡܫܘ̈ܚܬܐ ܕܣܛܝܘܬܐ ܬܩܝܢܬܐ (δ)، ܐܡܬܝ ܕܟܠ ܐܚܪ̈ܢܐ ܓܘ ܦܠܓܐ ܝܢܐ – ܠܘܚܐ 3.10 ܓܘ ܟܬܒܐ (ܦܐܬܐ 148)؛ ܦܐܬܐ ܟܡܬܢܐ ܟܠ 96 ܩܠ̈ܝܬܐ ܕܝܠܗ.

ܓܘ ܥܠܬܐ ܝܚܝܕܝܬܐ ܚܝܠܐ ܬܠܝܐ ܝܠܗ ܒܣ ܒ-f، ܒ-n ܘܒܡܢܝܢܐ ܕܓܘ̈ܕܐ. ܓܘ ܬܘܩܢܐ ܦܩܛܘܪܝܐܠ ܩܐ ܟܠ ܡܥܒܕܢܘܬܐ ܐܝܬ ܚܝܠܐ ܕܝܠܗ، ܘܪܒܐ ܝܠܗ ܒܢܘܬ ܩܐ ܥܒܕܐ ܕܚܕ̈ܕܐ ܣܢܝܩܝ ܒܥܕܢܐ ܪܓܝܠܐ ܡܫܬܘܬܦ̈ܢܐ ܒܢܘܬ: ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܡܣܬܡܟܐ ܝܠܗ ܥܠ ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ ܕܡܫܬܘܬܦ̈ܢܐ ܣܓܝ̈ܐܐ، ܘܥܒܕܐ ܕܚܕ̈ܕܐ – ܥܠ ܡܨܥܝ̈ܬܐ ܕܩܠ̈ܝܬܐ (Maxwell et al., 2018, pp. 354-355). ܡܛܠ ܗܕܐ ܓܘ ܬܘܩ̈ܢܐ ܟܓܒܚ ܡܥܒܕܢܘܬܐ ܕܚܝܠܐ ܚܫܝܒܐ ܝܠܗ ܩܐܠܗ. ܓܘ ܓܘ̈ܫܡܐ، ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܩܐ ܛܥܝܘܬܐ ܙܥܘܪ̈ܐ ܝܢܐ – (a − 1)(b − 1) – ܘܐܦ ܐܗܐ ܓܘ ܚܘܫܒܐ ܝܠܗ. ܓܘ ܢܣܝܘܢܐ ‎2^k‎ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܝܠܗ ܦܘܪܫܢܐ ܒܝܢܝ ܕܪܓܐ ܪܡܐ ܘܢܚܝܬܐ ܒܡܫܘ̈ܚܬܐ ܕ-σ؛ ܐܝܟ ܓܘ ܡܬܠܐ ܕ-Montgomery – ܬܪܬܝܢ ܬܢܝ̈ܬܐ ܕ-‎2³‎ ܩܐ ܡܥܒܕܢܘܬܐ ܕ-2σ ܟܝܗܒܝ ܚܝܠܐ ܕ-.937 (Montgomery, 2020, pp. 209-210).

ܥܠ ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܡܢ ܢܣܝܘܢܐ ܩܕܡܝܐ (pilot) ܣܢܝܩ ܝܠܗ ܠܡܙܕܗܪ: ܐܗܝ ܡܣܠܝܬܐ ܝܠܗ ܠܥܠ ܘܠܐ ܚܬܝܬܬܐ، ܘܠܡܦܠܚ ܒܗ ܩܐ ܬܘܩܢܐ ܕܒܘܨܝܐ ܕܒܬܪܗ "ܡܚܬܡܠܐ ܚܘܫܒܐ ܦܪܥܐ ܝܠܗ" (Maxwell et al., 2015، ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Maxwell et al., 2018، ܦܐܬܐ 149). ܛܒܐ ܝܠܗ ܠܚܫܒܐ ܩܐ ܟܡܐ ܪ̈ܒܘܬܐ، ܝܢ ܠܦܘܬ ܬܚܘܡܐ ܬܚܬܝܐ ܕܬܚܘܡܐ ܕܬܘܟܠܢܐ (confidence interval (CI)) ܕܚܕ ܓܢܒܐ (ܬܡܢ، ܦܐ̈ܬܐ 150-151). Montgomery (2020، ܦܐ̈ܬܐ 94-95) ܟܡܩܪܒ ܐܦ ܐܘܪܚܐ ܐܚܪܬܐ: ܠܡܓܒܐ n ܗܟܢܐ ܕܬܚܘܡܐ ܕܬܘܟܠܢܐ ܩܐ ܦܘܪܫܢܐ ܒܝܢܝ ܬܪܬܝܢ ܡܨܥܝ̈ܬܐ ܗܘܐ ܒܦܬܝܘܬܐ ܕܟܒܥܝܚ.

ܦܘܠܓܐ ܓܕܫܢܝܐ ܠܓܘ̈ܕܐ ܒܥܒܕܐ

ܓܘ ܬܘܩܢܐ ܦܐܬܐ ܟܡܦܩ ܦܐܬܐ ܕܪܗܛܐ: ܩܐ ܟܠ ܪܗܛܐ (ܝܢ ܡܫܬܘܬܦܢܐ) – ܡܢܝܢܐ ܓܘ ܛܟܣܐ ܓܕܫܢܝܐ، ܡܢܝܢܐ ܓܘ ܛܟܣܐ ܬܩܝܢܐ، ܓܘܫܡܐ ܘܫܪܛܐ، ܘܥܡܘܕܐ ܣܦܝܩܐ ܩܐ ܦܐܪܐ. ܛܟܣܐ ܦܣܝܩܐ ܝܠܗ ܒܡܘܠܕܢܐ ܕܡܢܝ̈ܢܐ ܓܕܫܢ̈ܝܐ ܕ-R (Mersenne-Twister ܥܡ sample)، ܗܟܢܐ ܕ-set.seed(ܙܪܥܐ); sample(N) ܓܘ R ܟܝܗܒ ܚܬܝܬܐܝܬ ܗܘ ܛܟܣܐ؛ ܣܩܪܝܦܬ ܕܦܐܬܐ ܟܡܚܬܐ ܟܒܢܐ ܦܐܬܐ ܕܪܗܛܐ ܡܢ ܪܝܫ. ܛܒܐ ܝܠܗ ܠܡܢܛܪ ܙܪܥܐ ܥܡ ܝܗܝ̈ܒܬܐ.

ܓܕܫܢܝܘܬܐ ܓܡܝܪܬܐ – ܟܠ ܪ̈ܗܛܐ ܒܛܟܣܐ ܓܕܫܢܝܐ ܚܕ. ܓܘ ܓܘ̈ܫܡܐ – ܛܟܣܐ ܓܕܫܢܝܐ ܝܠܗ ܓܘ ܟܠ ܓܘܫܡܐ ܒܦܪܝܫܘܬܐ. ܓܘ ܡܪܒܥܐ ܠܛܝܢܝܐ – ܟܓܒܚ ܒܓܕܫܐ ܡܪܒܥܐ "ܬܩܝܢܐ" (ܕܣܕܪܐ ܘܥܡܘܕܐ ܩܕܡ̈ܝܐ ܕܝܠܗ ܒܛܟܣܐ ܕܐܠܦ-ܒܝܬ ܝܢܐ)، ܘܒܬܪܟܢ ܟܚܠܛܚ ܣܕܪ̈ܐ، ܥܡܘ̈ܕܐ ܘܫܡ̈ܗܐ ܕtreatments (Maxwell et al., 2018, pp. 647-648; Montgomery, 2020, p. 137). ܗܠ ‎6×6‎ ܦܐܬܐ ܟܓܒܐ ܡܢ ܟܠ ܡܪ̈ܒܥܐ ܬܩ̈ܝܢܐ (1، 4، 56 ܘ-9,408؛ ܬܡܢ، ܠܘܚܐ 4.13)؛ ܡܢ ‎7×7‎ – ܡܢ ܡܪܒܥܐ ܚܘܕܪܢܝܐ (cyclic). ܡܪܒܥܐ ܚܘܕܪܢܝܐ ܒܠܚܘܕ (A B C D, B C D A...) "ܠܐ ܝܠܗ ܬܘܩܢܐ ܛܒܐ" ܩܐ ܛܟܣܐ ܕܡܚܘܝܢܘܬܐ: B ܟܠ ܐܝܡܢ ܒܬܪ A (Maxwell et al., 2018, p. 646). ܐܡܬܝ ܕܟܙܕܥܚ ܡܢ ܡܥܒܕܢܘܬܐ ܕܫܪܟܐ (carryover) ܕܫܪܛܐ ܥܠ ܕܒܬܪܗ، ܟܓܒܚ ܡܪܒܥܐ ܫܘܝܐ ܠܛܟܣܐ (Williams): ܟܠ treatment ܟܐܬܐ ܒܥܓܠ ܒܬܪ ܟܠ ܐܚܪܢܐ ܚܬܝܬܐܝܬ ܚܕ ܓܐܝܐ؛ ܓܘ p ܠܐ ܙܘܓܝܐ ܣܢܝܩܝ ܬܪܝܢ ܡܪ̈ܒܥܐ (ܬܡܢ، ܦܐܬܐ 648). ܢܣܝܘܢܐ ܥܡ ܡܫܬܘܬܦ̈ܢܐ ܒܢܘܬ ܡܢ p – ܟܡܐ ܡܪ̈ܒܥܐ، ܟܠ ܚܕ ܓܒܝܐ ܒܓܕܫܐ ܒܦܪܝܫܘܬܐ (ܬܡܢ).

ܢܣܝܘܢܐ ܦܩܛܘܪܝܐܠ ܘܥܒܕܐ ܕܚܕ̈ܕܐ

ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction) ܦܘܫܩܗ ܝܠܗ ܕܡܥܒܕܢܘܬܐ ܕܥܠܬܐ ܚܕܐ ܬܠܝܐ ܝܠܗ ܒܕܪܓܐ ܕܐܚܪܬܐ – ܓܘ ܨܘܪܬܐ، ܣܪ̈ܛܐ ܕܠܐ ܝܢܐ ܒܓܢܒ ܚܕ̈ܕܐ. ܐܡܬܝ ܕܥܒܕܐ ܕܚܕ̈ܕܐ ܪܒܐ ܝܠܗ، ܠܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ "ܐܝܬ ܦܘܫܩܐ ܥܒܕܢܝܐ ܩܠܝܠܐ"، ܘܥܒܕܐ ܕܚܕ̈ܕܐ ܪܫܝܡܐ ܡܨܐ ܕܛܫܐ ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ (Montgomery, 2020, p. 155). ܗܝܕܝܟ ܟܒܚܪܚ ܡܥܒܕܢܘܬܐ ܕܥܠܬܐ ܚܕܐ ܓܘ ܟܠ ܕܪܓܐ ܕܐܚܪܬܐ (simple effects). ܓܘ ܥܒܕܐ ܕܚܕ̈ܕܐ ܕܬܠܬܐ ܪܫܝܡܐ "ܒܥܕܢܐ ܪܓܝܠܐ ܠܐ ܟܦܫܩܚ" ܐܦ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ ܘܐܦ ܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ (Maxwell et al., 2018, p. 422).

ܟܠ ܡܥܒܕܢܘܬܐ (ܪܫܝܬܐ، ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ، ܕܬܠܬܐ) ܝܠܗ "ܫܪܒܬܐ" ܦܪܝܫܬܐ، ܘܒܚܝܪܬܐ ܝܠܗ ܒܪܡܐ ܕܪܫܝܡܘܬܐ .05 ܕܠܐ ܬܘܪܨܐ ܒܝܢܝ ܫܪ̈ܒܬܐ (Maxwell et al., 2018, p. 329). ܓܘ ܟܠ ܩܠܝܬܐ ܣܢܝܩܝ ܬܪܬܝܢ ܚܙ̈ܝܬܐ ܠܟܡܝܘܬܐ ܕܗܘܐ ܫܘܥܪܐ ܩܐ ܛܥܝܘܬܐ؛ ܥܡ ܚܙܝܬܐ ܚܕܐ ܓܘ ܩܠܝܬܐ ܠܝܬ ܒܘܚܪܢܐ ܩܐ ܥܒܕܐ ܕܚܕ̈ܕܐ، ܘܟܣܒܪܚ ܕܠܝܬ – Tukey ܩܪܒܠܗ ܒܘܚܪܢܐ ܕܕܪܓܐ ܕܚܐܪܘܬܐ ܚܕ ܠܒܚܪܐ ܐܗܐ (Montgomery, 2020, pp. 171-173). ܦܐܬܐ ܟܡܪܗܛ ܠܗ ܡܢ ܓܢܒܗ ܓܘ ܠܘܚܐ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ ܥܡ ܚܙܝܬܐ ܚܕܐ ܓܘ ܩܠܝܬܐ، ܥܡ ܓܘ̈ܫܡܐ.

ܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ: ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܕ-Type III ܝܢ II

ܐܡܬܝ ܕܓܘ ܟܠ ܩܠ̈ܝܬܐ ܐܝܬ ܗܘ ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ، ܥܠ̈ܠܬܐ orthogonal ܝܢܐ، ܘܬܠܬܐ ܙܢ̈ܐ ܕܓܘܡܪܐ ܕܡܪ̈ܒܥܐ (sum of squares) ܫܘ̈ܝܐ ܝܢܐ. ܐܡܬܝ ܕܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ ܝܢܐ – ܡܫܬܘܬܦܢܐ ܕܦܠܛܠܗ، ܡܢܝܢܐ ܚܣܝܪܐ – ܐܢܝ ܟܒܚܪܝ ܗܘܦܘܬܣ̈ܝܣ ܦܪ̈ܝܫܐ (Maxwell et al., 2018, p. 370): Type I ܠܐ ܟܚܐܪ ܠܥܠܬܐ ܬܪܝܢܝܬܐ ܘܦܐܫ ܒܣ ܐܡܬܝ ܕܪܒܘܬܐ ܕܩܠ̈ܝܬܐ ܟܡܚܘܝܐ ܟܢܫܐ ܟܠܢܝܐ؛ Type II ܟܚܐܪ ܠܥܠܬܐ ܬܪܝܢܝܬܐ ܐܠܐ ܠܐ ܠܥܒܕܐ ܕܚܕ̈ܕܐ، ܘܚܝܠܢܐ ܝܠܗ ܒܢܘܬ ܐܢ ܠܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ ܓܘ ܟܢܫܐ ܟܠܢܝܐ؛ Type III ܟܒܚܪ ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ ܕܠܐ ܡܬܩܠܢܘܬܐ، ܘܦܫܝܩܐ ܝܠܗ ܠܡܦܫܩ ܠܗ ܐܦ ܐܡܬܝ ܕܐܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ. "ܡܪܚܩܢܘܬܐ ܟܠܢܝܬܐ ܕܝܠܢ" – Type III؛ "ܐܝܬ ܒܨܘ̈ܝܐ ܕܟܓܒܝ" Type II ܐܡܬܝ ܕܐܝܬ ܥܠܬܐ ܬܐܘܪܝܬܐ ܠܡܣܟܘܝܐ ܕܠܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ ܘܒܘܚܪܢܐ ܕܝܠܗ ܪܚܝܩܐ ܝܠܗ ܡܢ ܪܫܝܡܘܬܐ (ܬܡܢ، ܦܐܬܐ 371). ܐܗܐ ܝܠܗ "ܐܦ ܐܗܐ ܘܐܦ ܐܗܐ" – ܘܓܘ ܦܐܬܐ: Type III ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ (ܐܝܟ SPSS)، Type II ܓܘ ܬܘܩ̈ܢܐ. ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ ܓܘ ܠܘܚܐ – ܕܠܐ ܡܬܩܠܢܘܬܐ (ܐܝܟ emmeans ܓܘ R)، ܕܐܢܝ ܝܢܐ ܡܐ ܕ-Type III ܟܒܚܪ. ܒܚܕ ܓܕܫܐ ܠܐ ܟܫܩܠܚ ܡܫܬܘܬܦ̈ܢܐ ܩܐ ܡܫܘܝܐ ܩܠ̈ܝܬܐ (ܬܡܢ، ܦܐܬܐ 372).

ܓܘ̈ܫܡܐ ܘܡܪܒܥܐ ܠܛܝܢܝܐ

ܓܘܫܡܐ ܟܡܦܠܛ ܡܢ ܛܥܝܘܬܐ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܓܘ̈ܫܡܐ: ܓܘ ܡܬܠܐ ܕ-Montgomery ܗܢ ܝܗܝ̈ܒܬܐ، ܐܡܬܝ ܕܟܦܫܩܚ ܠܗܝܢ ܕܠܐ ܓܘ̈ܫܡܐ، ܟܡܥܦܦܝ ܡܨܥܝܬܐ ܕܡܪ̈ܒܥܐ ܕܛܥܝܘܬܐ (ܡܢ 7.33 ܠ-15.11)، ܘܒܘܚܪܢܐ ܟܡܚܠܫ (Montgomery, 2020, pp. 121-122). ܡܘܕܝܠ ܟܣܒܪ ܕܠܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܓܘ̈ܫܡܐ ܘtreatments (ܬܡܢ، ܦܐܬܐ 126). ܓܘ ܦܐܬܐ، ܥܠܬܐ ܕܪܫܝܡܬܐ ܝܠܗ "ܓܘܫܡܐ" ܟܥܐܠܐ ܠܡܘܕܝܠ ܐܝܟ ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܒܣ، ܘܥܠ̈ܠܬܐ ܕܢܣܝܘܢܐ – ܥܡ ܟܠ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܐܢܝ (ܐܝܟ ܓܘ ܢܣܝܘܢܐ ܦܩܛܘܪܝܐܠ ܓܘ ܓܘ̈ܫܡܐ؛ ܬܡܢ، ܦܐ̈ܬܐ 188-190). ܡܪܒܥܐ ܠܛܝܢܝܐ – ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܕܓܘܫܡܐ (ܣܕܪ̈ܐ ܘܥܡܘ̈ܕܐ)؛ ܡܪܒܥܐ ܝܘܢܝܐ-ܠܛܝܢܝܐ – ܬܠܬܐ (ܬܡܢ، ܦܐ̈ܬܐ 140-142)؛ ܘܐܦ ܢܣܝܘܢܐ ܦܩܛܘܪܝܐܠ ܕܓܘܗ ܟܠ ܚܘܠܛܐ ܐܬܘܬܐ ܝܠܗ ܓܘ ܡܪܒܥܐ (ܬܡܢ، ܦܐ̈ܬܐ 191-193).

ܒܘܚܪܢܐ ܩܐ ܓܘ̈ܫܡܐ. Montgomery (2020، ܦܐܬܐ 119) ܟܡܦܠܛ ܒܘܚܪܢܐ F ܩܐ ܓܘ̈ܫܡܐ ܡܢ ܠܘܚܐ ܕ-ANOVA: ܓܕܫܢܝܘܬܐ ܥܒܝܕܬܐ ܝܠܗ ܒܣ ܓܘ ܓܘ̈ܫܡܐ، ܘܡܛܠ ܗܕܐ ܐܗܐ ܠܐ ܝܠܗ ܒܘܚܪܢܐ ܚܬܝܬܐ ܩܐ ܦܘܪܫܢܐ ܒܝܢܝ ܓܘ̈ܫܡܐ – ܐܠܐ ܦܘܚܡܐ MS_ܓܘ̈ܫܡܐ/MS_ܛܥܝܘܬܐ ܟܡܚܘܐ ܐܢ ܓܘ̈ܫܡܐ ܥܕܪܠܗܘܢ. SPSS ܘ-R (ܘܐܦ JMP ܓܒܝܗ، ܦܐܬܐ 124) ܟܛܒܥܝ F ܘ-p. "ܐܦ ܐܗܐ ܘܐܦ ܐܗܐ": ܦܐܬܐ ܟܡܚܘܐ ܦܘܚܡܐ ܘ-p، ܥܡ ܡܥܝܪܢܘܬܐ.

ܚܣܝ̈ܪܐ. ܐܡܬܝ ܕܚܙܝܬܐ ܚܣܝܪܬܐ ܝܠܗ، ܓܘ̈ܫܡܐ ܘtreatments ܠܐ ܝܢܐ ܬܘܒ orthogonal. ܒܕܘܟܬܐ ܕܡܫܥܪܐ ܛܝܡܐ ܚܣܝܪܐ (ܦܘܫܩܐ ܩܪܝܒܐ ܕܟܡܢܦܚ ܪܫܝܡܘܬܐ)، ܦܐܬܐ ܟܥܒܕ ܦܘܫܩܐ ܚܬܝܬܐ – ܡܘܕܝܠ ܕܪܝܓܪܫܢ (Montgomery, 2020, pp. 129-132).

ܓܘ̈ܫܡܐ ܝܢ ANCOVA؟ ܐܡܬܝ ܕܓܘ̈ܫܡܐ ܒܢܝ̈ܐ ܝܢܐ ܡܢ ܦܘܠܓܐ ܕܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ (ܥܘܡܪܐ ܠܥܣܪܐ ܫܢ̈ܐ)، Maxwell ܘܚܒܪ̈ܘܗܝ ܟܓܒܝ ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܥܡ ܡܫܬܚܠܦܢܐ ܡܠܘܝܢܐ (ANCOVA) ܥܡ ܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ ܒܝܕܗ – ܐܗܘ ܟܡܦܠܚ ܒܟܠ ܝܕܥܬܐ ܘܒܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܒܨܝܪ̈ܐ؛ ܩܐ ܓܘ̈ܫܡܐ ܐܝܬ ܕܘܟܬܐ ܐܡܬܝ ܕnuisance factor ܕܓܢ̈ܣܐ ܝܠܗ ܡܢ ܟܝܢܗ (Maxwell et al., 2018, pp. 502-507). ܐܦ ܐܗܐ ܘܐܦ ܐܗܐ: ܓܘ ܬܘܩܢܐ – ܦܘܠܓܐ ܠܦܘܬ ܓܘ̈ܫܡܐ (ܕܦܐܬܐ ܟܒܢܐ) ܛܒܐ ܝܠܗ ܒܟܠ ܓܕܫܐ؛ ܓܘ ܦܘܫܩܐ – ܐܢ ܓܘܫܡܐ ܒܢܝܐ ܝܠܗ ܡܢ ܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ، ܚܫܘܒܘܢ ܪܝܓܪܫܢ ܥܡ ܡܫܬܚܠܦܢܐ ܡܬܡܫܚܢܐ.

ܡܪܒܥܐ ܠܛܝܢܝܐ ܩܐ ܛܟܣܐ ܕܡܚܘܝܢܘܬܐ. "ܦܘܫܩܐ ܣܢܝܩ ܝܠܗ ܕܦܐܫ ܠܬܘܩܢܐ" – ܠܦܫܩܐ ܡܪܒܥܐ ܠܛܝܢܝܐ ܐܝܟ ܕܗܘܐ ܬܘܩܢܐ ܕܠܐ ܛܟܣܐ ܟܚܪܒ ܡܐ ܕܬܘܩܢܐ ܝܗܒܠܗ (Maxwell et al., 2018, p. 648). ܦܐܬܐ ܟܦܫܩ ܡܫܬܘܬܦܢܐ ܘܕܘܟܬܐ ܐܝܟ ܥܠ̈ܠܬܐ ܕܓܘܫܡܐ – ܡܘܕܝܠ additive ܕ-Montgomery (2020، ܦܐܬܐ 134)، ܥܡ (p − 2)(p − 1) ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܩܐ ܛܥܝܘܬܐ ܓܘ ܡܪܒܥܐ ܚܕ. ܡܢ ܟܡܐ ܡܪ̈ܒܥܐ – ܟܠ ܡܫܬܘܬܦܢܐ ܥܡ ܡܢܝܢܐ ܡܦܪܫܢܐ ܕܝܠܗ (ܬܡܢ، ܦܐ̈ܬܐ 137-139).

‏2^k، ܡܢܘ̈ܬܐ ܘ-aliases

ܓܘ ܢܣܝܘܢܐ ܕܓܘܗ ܟܠ ܥܠܬܐ ܒܬܪܝܢ ܕܪ̈ܓܐ، ܡܥܒܕܢܘܬܐ ܕܥܠܬܐ ܝܠܗ ܦܘܪܫܢܐ ܒܝܢܝ ܡܨܥܝܬܐ ܓܘ ܕܪܓܐ ܪܡܐ ܘܡܨܥܝܬܐ ܓܘ ܕܪܓܐ ܢܚܝܬܐ؛ ܩܐ ܟܠ ܥܒܕܐ ܕܚܕ̈ܕܐ – ܡܥܒܕܢܘܬܐ ܒܗܝ ܐܘܪܚܐ، ܡܢ ܥܡܘܕܐ ܕܪ̈ܘܫܡܐ ܕܝܠܗ (Montgomery, 2020, p. 196). ܐܡܬܝ ܕܐܝܬ ܬܢܝ̈ܬܐ، ܟܒܚܪܚ ܟܠ ܡܥܒܕܢܘܬܐ ܒܒܘܚܪܢܐ F (ܕܪܓܐ ܕܚܐܪܘܬܐ ܚܕ). ܕܠܐ ܬܢܝ̈ܬܐ ܠܝܬ ܫܘܥܪܐ ܩܐ ܛܥܝܘܬܐ. ܗܝܕܝܟ ܟܣܡܟܚ ܥܠ ܫܪܫܐ ܕܩܠܝܠܘܬܐ (sparsity): ܣܓܝܐܐ ܡܢ ܛܘܦ̈ܣܐ ܡܬܕܒܪ̈ܢܐ ܝܢܐ ܒܝܕ ܟܡܐ ܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ ܘܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܢܚܝ̈ܬܐ (ܬܡܢ، ܦܐܬܐ 218). ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ، ܘܦܐܬܐ ܟܥܒܕ ܬܪܘܝܗܝ: ܨܘܪܬܐ half-normal ܕܛܝܡܐ ܫܪܝܪܐ ܕܡܥܒܕ̈ܢܘܝܬܐ – ܒܠܝܛ̈ܐ ܟܪܚܩܝ ܡܢ ܣܪܛܐ (ܬܡܢ، ܦܐܬܐ 224)؛ ܘܐܘܪܚܐ ܕ-Lenth: "ܛܥܝܘܬܐ ܬܩܝܢܬܐ ܕܓܠܐ" (PSE) ܡܢ ܡܥܒܕ̈ܢܘܝܬܐ ܙܥܘܪ̈ܐ، ܘܩܘܒܠܗ ܬܚܘܡܐ ܩܐ ܡܥܒܕܢܘܬܐ ܝܚܝܕܝܬܐ (ME) ܘܬܚܘܡܐ ܩܐ ܟܠ ܡܥܒܕ̈ܢܘܝܬܐ ܐܟܚܕܐ (SME) (ܬܡܢ، ܦܐܬܐ 226). Montgomery ܟܡܪܚܩ ܠܡܦܠܚ ܒ-Lenth "ܐܝܟ ܬܘܣܦܬܐ ܠܨܘܪܬܐ، ܠܐ ܒܕܘܟܬܗ" (ܬܡܢ، ܦܐܬܐ 227)، ܘܟܡܥܗܕ ܕܬܚܘ̈ܡܐ ܙܗܝܪ̈ܐ ܝܢܐ (ܬܡܢ، ܦܐܬܐ 226). ܒܬܪ ܕܓܒܚ ܡܥܒܕ̈ܢܘܝܬܐ، ܟܡܪܗܛܚ ܡܘܕܝܠ ܩܛܝܢܐ، ܘܡܥܒܕ̈ܢܘܝܬܐ ܐܚܪ̈ܢܐ ܟܗܘܝܐ ܛܥܝܘܬܐ (ܬܡܢ، ܦܐܬܐ 223) – ܓܘ ܬܘܩ̈ܢܐ "ܡܘܕܝܠ ܩܛܝܢܐ".

ܡܢܬܐ (fraction) ܡܢ ‎2^k‎ ܟܡܪܗܛ ܒܣ ܡܢܬܐ ܡܢ ܚܘ̈ܠܛܐ، ܕܓܒܝ̈ܐ ܝܢܐ ܠܦܘܬ "generator": ܠܡܬܠܐ ܓܘ ‎2⁴⁻¹‎، D = ABC – ܥܡܘܕܐ ܕ-D ܝܠܗ ܡܘܚܠܦܐ ܕ-A، B ܘ-C. ܛܝܡܐ: ܟܠ ܡܥܒܕܢܘܬܐ "alias" ܝܠܗ ܕܡܥܒܕܢܘܬܐ ܐܚܪܬܐ، ܘܠܐ ܡܨܝܐ ܠܡܦܪܫ ܒܝܢܝ ܐܢܝ – ܡܐ ܕܡܫܥܪܐ ܝܠܗ ܓܘܡܪܐ ܕܝܠܗܘܢ (Montgomery, 2020, pp. 275-276). Resolution ܟܟܢܫ ܐܗܐ: ܓܘ III ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ alias ܝܠܗ ܕܥܒܕܐ ܕܚܕ̈ܕܐ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ؛ ܓܘ IV – ܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ ܕܟܝ̈ܐ ܝܢܐ ܡܢ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ، ܐܠܐ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ aliases ܝܢܐ ܚܕ ܕܐܚܪܢܐ؛ ܓܘ V – ܐܦ ܕܬܪܝܢ ܐܘܪ̈ܚܬܐ ܕܟܝ̈ܐ ܝܢܐ (ܬܡܢ، ܦܐܬܐ 278). ܦܐܬܐ ܟܡܦܠܚ ܒ-generators ܕܠܘܚܐ 8.14 (ܬܡܢ، ܦܐܬܐ 299) – resolution ܪܡܐ ܡܢ ܟܠܗ، ܘܡܢ ܒܝܢܝ ܐܢܝ "minimum aberration" (ܬܡܢ، ܦܐ̈ܬܐ 297-298) – ܘܟܡܚܘܐ defining relation ܘܟܠ ܫܫܠ̈ܬܐ ܕ-aliases.

ܓܘ̈ܫܡܐ ܓܘ ‎2^k‎. ܐܡܬܝ ܕܟܠ ܪ̈ܗܛܐ ܠܐ ܟܥܐܠܝ ܠܓܘܫܡܐ ܚܕ، ܟܚܠܛܚ (confounding) ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܪ̈ܡܐ ܥܡ ܓܘ̈ܫܡܐ: ܓܘ ‎2⁴‎ ܒܬܪܝܢ ܓܘ̈ܫܡܐ – ABCD (Montgomery, 2020, pp. 259-265). ܦܐܬܐ ܟܒܢܐ ܓܘ̈ܫܡܐ ܠܦܘܬ ܠܘܚܐ 7.9 (ܬܡܢ، ܦܐܬܐ 270)، ܘܓܘ ܦܘܫܩܐ ܟܪܫܡ ܐܝܢܐ ܡܥܒܕܢܘܬܐ ܚܠܝܛܬܐ ܝܠܗ ܥܡ ܓܘ̈ܫܡܐ ܘܠܐ ܡܫܥܪܬܐ ܝܠܗ.

ܪ̈ܒܘܬܐ ܕܡܥܒܕܢܘܬܐ

ܓܘ ܬܘܩܢܐ ܕܥܠ̈ܠܬܐ ܣܓܝ̈ܐܐ ܣܢܝܩ ܝܠܗ ܠܡܦܣܩ ܡܘܕܝ ܚܬܝܬܐܝܬ ܟܡܦܠܓܚ ܥܠ ܡܘܕܝ (Maxwell et al., 2018, pp. 329-335). ܐܝܬܐ ܡܪܒܥܐ ܡܢܬܢܝܐ (partial eta squared) – SS_{ܡܥܒܕܢܘܬܐ}/(SS_{ܡܥܒܕܢܘܬܐ} + SS_ܛܥܝܘܬܐ) – ܦܐܫ ܐܡܬܝ ܕܥܠ̈ܠܬܐ ܐܚܪ̈ܢܐ "ܒܪ̈ܝܐ" ܝܢܐ ܠܫܘܐܠܐ (ܡܨܐ ܗܘܐ ܠܡܒܨܐ ܐܗܐ ܥܠܬܐ ܒܠܚܘܕ)؛ ܐܗܐ ܝܠܗ ܡܐ ܕ-SPSS ܟܛܒܥ. ܐܠܐ ܐܗܝ ܬܚܘܝܬܐ ܕܕܘܡܝܐ ܝܠܗ ܘܡܣܠܝܬܐ ܝܠܗ ܠܥܠ. ܐܘܡܝܓܐ ܡܪܒܥܐ ܡܢܬܢܝܐ – (SS − df·MS_ܛܥܝܘܬܐ)/(SS + (N − df)·MS_ܛܥܝܘܬܐ) – ܟܫܥܪܐ ܟܢܫܐ ܟܠܢܝܐ، ܘܚܝܘܒܝܬܐ ܝܠܗ ܒܣ ܐܡܬܝ ܕ-F ܪܒܐ ܝܠܗ ܡܢ 1 (ܬܡܢ، ܦܐܬܐ 334، ܫܘܝܘܬܐ 44)؛ ܐܡܬܝ ܕܒܠܐ ܝܠܗ ܟܡܚܘܚ 0 (ܬܡܢ، ܦܐܬܐ 421). ܓܘ ܠܘܚܐ 7.8 ܓܘ ܟܬܒܐ (ܦܐܬܐ 335) – .22 ܩܘܒܠ .17 ܩܐ ܗܝ ܡܥܒܕܢܘܬܐ. ܦܐܬܐ ܟܡܚܘܐ ܬܪܘܝܗܝ.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܓܘ ܬܘܩܢܐ: ܠܘܚܐ ܕܬܘܩܢܐ (ܡܢܝܢܐ ܕܪ̈ܗܛܐ، ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܩܐ ܛܥܝܘܬܐ)، ܠܘܚܐ ܕܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ (ܩܐ ܟܠ n – ܚܝܠܐ ܩܐ ܟܠ ܡܥܒܕܢܘܬܐ)، ܨܘܪܬܐ ܕܚܝܠܐ، ܘܦܐܬܐ ܕܪܗܛܐ. ܓܘ ܡܪܒܥܐ ܠܛܝܢܝܐ – ܡܪ̈ܒܥܐ ܒܝܕܗܘܢ؛ ܓܘ ‎2^k‎ – ܡܐܛܪܝܟܣ، defining relation، resolution ܘ-aliases. ܟܡܚܬܚ ܦܐܬܐ ܕܪܗܛܐ ܠܐܩܣܠ، ܟܡܡܠܚ ܥܡܘܕܐ ܕܦܐܪܐ (response)، ܘܟܛܥܢܚ ܠܗ ܓܘ "ܦܘܫܩܐ".

ܓܘ ܦܘܫܩܐ: ܠܘܚܐ ܕܡܨܥܝ̈ܬܐ ܓܘ ܟܠ ܩܠܝܬܐ؛ ܠܘܚܐ ܕ-ANOVA – ܩܐ ܟܠ ܡܥܒܕܢܘܬܐ SS، ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ، F، p ܘܬܪܝܢ ܪ̈ܒܘܬܐ ܕܡܥܒܕܢܘܬܐ؛ ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ ܥܡ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ؛ ܨܘܪܬܐ ܕܥܒܕܐ ܕܚܕ̈ܕܐ. ܓܘ ܢܣܝܘܢܐ ܕܬܪܝܢ ܕܪ̈ܓܐ – ܐܦ ܠܘܚܐ ܕܡܥܒܕ̈ܢܘܝܬܐ ܘܨܘܪܬܐ half-normal. "ܩܪܝܢܐ ܕܦܐܪܐ" ܟܡܚܘܐ ܡܐ ܕܚܫܝܚܐ ܝܠܗ ܠܒܚܪܐ ܓܘ ܓܕܫܐ ܕܝܘܟܘܢ – ܥܒܕܐ ܕܚܕ̈ܕܐ ܪܫܝܡܐ، ܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ، additivity ܓܘ ܓܘ̈ܫܡܐ.

ܫܪ̈ܛܐ

ܫܪ̈ܛܐ (assumptions) ܕܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance (ANOVA)): ܛܥܝ̈ܘܬܐ ܠܐ ܬܠܝ̈ܐ (ܕܓܕܫܢܝܘܬܐ ܟܡܙܕܩܐ ܠܗܘܢ)، ܦܠܝܓ̈ܐ ܢܘܪܡܠܐܝܬ، ܘܥܡ ܫܘܚܠܦܐ ܫܘܐ ܓܘ ܟܠ ܩܠ̈ܝܬܐ (Maxwell et al., 2018, pp. 133-141; Montgomery, 2020, pp. 68-76). ܒܘܚܪܢܐ F ܚܝܠܢܐ ܝܠܗ ܠܣܛܝܘܬܐ ܡܢ ܢܘܪܡܠܝܘܬܐ، ܘܠܫܘܚܠ̈ܦܐ ܠܐ ܫܘ̈ܝܐ – ܗܠ ܕܩܠ̈ܝܬܐ ܫܘ̈ܝܬܐ ܝܢܐ ܒܪܒܘܬܐ؛ ܓܘ ܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ ܫܘܚܠ̈ܦܐ ܦܪ̈ܝܫܐ "ܟܒܢܝ ܥܘ̈ܩܐ ܪ̈ܒܐ" (Maxwell et al., 2018, p. 379). ܦܐܬܐ ܟܡܪܗܛ ܒܘܚܪܢܐ ܕ-Levene ܩܐ ܫܘܝܘܬܐ ܕܫܘܚܠ̈ܦܐ (ܚܕܪ ܡܨܥܝܬܐ، ܐܝܟ SPSS) ܐܡܬܝ ܕܐܝܬ ܬܪܬܝܢ ܚܙ̈ܝܬܐ ܠܟܡܝܘܬܐ ܓܘ ܟܠ ܩܠܝܬܐ. ܓܘ ܬܘܩܢܐ ܥܡ ܓܘ̈ܫܡܐ ܟܣܒܪܚ ܐܦ additivity – ܕܠܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܓܘ̈ܫܡܐ ܘtreatments (Montgomery, 2020, p. 126).

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

ܦܘܫܩܐ: Analyze → General Linear Model → Univariate (ܦܘܩܕܢܐ UNIANOVA). Type III ܝܠܗ ܓܘܒܝܐ ܡܩܕܡܐ؛ ܡܘܕܝܠ ܥܡ ܓܘ̈ܫܡܐ – Model → Custom، ܥܡ ܓܘܫܡܐ ܐܝܟ ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܒܣ. ܦܐܬܐ ܟܡܚܬܐ ܣܝܢܬܐܟܣ ܥܡ /DESIGN ܦܐܝܫܐ، ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ (/EMMEANS)، ܐܝܬܐ ܡܪܒܥܐ ܡܢܬܢܝܐ ܘܒܘܚܪܢܐ ܕ-Levene. ܓܘ SPSS ܠܝܬ ܡܘܠܕܢܐ ܕܬܘܩ̈ܢܐ ܦܩܛܘܪܝܐܠ ܡܢܬܢ̈ܝܐ ܘܠܝܬ Lenth – ܐܢܝ ܐܝܬܝܗܘܢ ܓܘ ܦܐܬܐ ܘܓܘ ܣܩܪܝܦܬ R. ܣܝܢܬܐܟܣ ܒܚܝܪܐ ܝܠܗ ܓܘ PSPP (GLM).

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

Type III ܓܘ R ܟܒܥܐ contrasts ܕܓܘܡܪܐ-ܠܨܦܪܐ (sum-to-zero) – ܐܢ ܠܐ car::Anova(type = 3) ܟܝܗܒ ܡܢܝ̈ܢܐ ܦܘܫ̈ܝܐ. ܣܩܪܝܦܬ ܕܠܬܚܬ ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ (ܛܝ̈ܡܐ ܓܘ ܡܥܝܪ̈ܢܘܬܐ). ܩܐ ܬܘܩܢܐ – set.seed ܘ-sample ܟܝܗܒܝ ܗܘ ܛܟܣܐ ܐܝܟ ܦܐܬܐ، ܘܩܐ ܡܢܘ̈ܬܐ – ܚܒܝܫܘܬܐ FrF2.

# ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)     # ܩܪܝܢܐ ܕܐܩܣܠ
library(car)        # Anova ܕ-Type III/II
library(emmeans)    # ܡܨܥܝ̈ܬܐ ܕܣܦܪܐ
options(contrasts = c("contr.sum", "contr.poly"))   # ܣܢܝܩ ܩܐ Type III

# ܬܘܩܢܐ: ܚܝܠܐ ܘ-n (ܬܠܬܐ ܓܘ̈ܕܐ، f = .41) ܘܛܟܣܐ ܓܕܫܢܝܐ – ܗܘ ܛܟܣܐ ܐܝܟ ܓܘ ܦܐܬܐ ܥܡ ܗܘ ܙܪܥܐ
pwr::pwr.anova.test(k = 3, f = 0.4082, sig.level = .05, power = .80)   # n = 20.3 – ܝܥܢܝ 21 ܩܐ ܟܠ ܓܘܕܐ
set.seed(2026); sample(24)   # ܛܟܣܐ ܕܪܗܛܐ ܕ-24 ܪ̈ܗܛܐ: 1 6 13 15 12 4 ...

# ܦܘܫܩܐ: ܓܠܝܘܢܐ "Data"؛ ܬܪܝܢ ܡܢܝ̈ܢܐ ܚܣܝ̈ܪܐ
d <- read_excel("C:/Rdata/DATA_doe_example_eng.xlsx", sheet = "Data")
d <- d[!is.na(d$score), ]   # 46 ܡܢ ܒܝܢܝ 48
d$class <- factor(d$class); d$method <- factor(d$method); d$feedback <- factor(d$feedback)
m <- lm(score ~ class + method * feedback, data = d)   # ܨܦܐ – ܓܘܫܡܐ، ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܒܣ
Anova(m, type = 3)   # ܐܘܪܚܐ: F(2, 37) = 9.92, p < .001؛ feedback: F(1, 37) = 0.86, p = .360؛ ܥܒܕܐ ܕܚܕ̈ܕܐ: F(2, 37) = 1.75, p = .187
emmeans(m, ~ method)   # 66.8, 79.0, 71.1 (SE 1.92-2.00)

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܟܡܘܕܥܚ: ܬܘܩܢܐ (research design) (ܟܡܐ ܥܠ̈ܠܬܐ، ܕܪ̈ܓܐ، ܡܫܬܘܬܦ̈ܢܐ ܓܘ ܟܠ ܩܠܝܬܐ، ܓܘ̈ܫܡܐ)، ܐܝܟܢܐ ܥܒܝܕܐ ܝܠܗ ܦܘܠܓܐ ܓܕܫܢܝܐ ܠܓܘ̈ܕܐ، ܘܦܘܫܩܐ ܕܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ ܕܦܣܩܠܗ ܪܒܘܬܐ ܕܕܘܡܝܐ – ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܕܣܒܝܪܬܐ ܝܠܗ، α ܘܚܝܠܐ. ܓܘ ܡܫܟܚ̈ܢܘܬܐ – ܩܐ ܟܠ ܡܥܒܕܢܘܬܐ F(df₁, df₂)، p ܘܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ، ܘܙܢܐ ܕܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܐܡܬܝ ܕܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ ܝܢܐ؛ ܡܨܥܝ̈ܬܐ (ܘܠܐ ܒܣ ܒܘܚܪ̈ܢܐ) – ܓܘ ܠܘܚܐ ܝܢ ܓܘ ܨܘܪܬܐ. ܥܒܕܐ ܕܚܕ̈ܕܐ ܪܫܝܡܐ ܬܚܘܝܐ ܝܠܗ ܩܕܡ ܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ. ܠܐ ܟܡܘܕܥܚ "ܚܝܠܐ ܚܙܝܐ" (observed power) ܕܚܫܝܒܐ ܝܠܗ ܡܢ ܡܫܟܚ̈ܢܘܬܐ – ܐܗܘ ܒܣ ܬܘܪܓܡܐ ܕ-p ܝܠܗ (Maxwell et al., 2018, pp. 151-152).

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܠܐ ܠܦܣܐ ܒܓܕܫܐ – ܠܡܪܗܛ ܫܪ̈ܛܐ ܠܦܘܬ ܦܫܝܩܘܬܐ (ܟܠ ܫܪܛܐ ܐ ܒܨܦܪܐ) ܘܠܡܫܒܩ ܥܠܬܐ ܢܘܟܪܝܬܐ ܕܥܐܠܐ ܠܦܐܪܐ (Montgomery, 2020, p. 11).

ܠܦܫܩܐ ܓܘ̈ܫܡܐ ܐܝܟ ܕܠܐ ܗܘܘ – ܟܠ ܫܘܚܠܦܐ ܕܓܘ̈ܫܡܐ ܟܥܒܪ ܠܛܥܝܘܬܐ (Montgomery, 2020, pp. 121-122)؛ ܘܗܦܝܟܐܝܬ – ܠܦܫܩܐ ܡܪܒܥܐ ܠܛܝܢܝܐ ܕܠܐ ܣܕܪ̈ܐ ܘܥܡܘ̈ܕܐ (Maxwell et al., 2018, p. 648).

ܠܦܫܩܐ ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܐܡܬܝ ܕܐܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ – ܡܨܥܝܬܐ ܥܠ ܕܪ̈ܓܐ ܕܥܠܬܐ ܬܪܝܢܝܬܐ ܡܨܐ ܕܠܐ ܬܚܘܐ ܚܕ ܡܢܝܗܝ (Montgomery, 2020, p. 155).

Type I ܓܘ ܩܠ̈ܝܬܐ ܠܐ ܫܘ̈ܝܬܐ – ܓܘܒܝܐ ܡܩܕܡܐ ܕ-aov ܘ-anova ܓܘ R؛ ܦܐܪܐ ܬܠܝܐ ܝܠܗ ܒܛܟܣܐ ܕܥܠ̈ܠܬܐ (Maxwell et al., 2018, pp. 370-371).

"ܚܝܠܐ ܚܙܝܐ" ܐܝܟ ܣܗܕܘܬܐ ܕܠܝܬ ܡܥܒܕܢܘܬܐ – ܐܡܬܝ ܕܦܐܪܐ ܠܐ ܪܫܝܡܐ ܝܠܗ، ܐܗܘ ܟܠ ܐܝܡܢ ܬܚܘܬ .5 ܝܠܗ (Maxwell et al., 2018, p. 152).

ܠܚܫܒܐ ܪܒܘܬܐ ܕܕܘܡܝܐ ܒܣ ܩܐ ܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܐܡܬܝ ܕܫܘܐܠܐ ܥܒܕܐ ܕܚܕ̈ܕܐ ܝܠܗ (Maxwell et al., 2018, p. 355).

ܠܡܛܥܝܐ aliases ܓܘ ܡܢܬܐ – ܠܡܘܕܥ "ܥܒܕܐ ܕܚܕ̈ܕܐ AB ܪܫܝܡܐ ܝܠܗ" ܓܘ ‎2⁴⁻¹‎، ܐܡܬܝ ܕܫܪܝܪܐܝܬ ܡܫܥܪܐ ܝܠܗ AB + CD (Montgomery, 2020, p. 276).

ܡܫܘ̈ܚܬܐ ܡܬܢ̈ܝܬܐ ܒܕܘܟܬܐ ܕܬܢܝ̈ܬܐ – ܬܠܬܐ ܡܫܘ̈ܚܬܐ ܕܗܝ ܚܕܝܘܬܐ ܠܐ ܝܢܐ ܬܠܬܐ ܬܢܝ̈ܬܐ، ܘܠܐ ܟܫܥܪܝ ܛܥܝܘܬܐ ܕܢܣܝܘܢܐ (Montgomery, 2020, p. 12).

ܡܒܘ̈ܥܐ

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

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

Montgomery, D. C. (2020). Design and analysis of experiments (10th ed.). Wiley.

Cohen (1988)، Maxwell et al. (2015) ܘ-Harris ܘ-Quade (1992) – ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Maxwell ܘܚܒܪ̈ܘܗܝ (2018). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ (site rule)، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ (reference list) ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܓܘ ܢܘܣܚܐ ܩܕܡܝܐ؛ ܡܝܬܝ̈ܬܐ ܠܦܘܬ ܦܐܬܐ ܛܒܝܥܬܐ. ܒܢܝܐ ܝܠܗ ܡܢ ܬܪܝܢ ܟܬܒ̈ܐ ܕܓܘ folder "ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ" (Montgomery – ܡܢ ܢܘܣܚܐ ܬܪܝܨܐ ܕܨܘܪܬܐ ܕܣܟܐܢ، ܥܡ ܫܟܒܬܐ ܓܡܝܪܬܐ ܕܟܬܝܒܬܐ). ܥܝ̈ܕܐ ܕܣܐܝܛ: ܙܪܥܐ ܓܕܫܢܝܐ ܘܡܘܠܕܢܐ ܫܘܐ ܠܕ-R؛ Type III ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ؛ F ܘ-p ܐܦ ܩܐ ܓܘ̈ܫܡܐ، ܥܡ ܡܥܝܪܢܘܬܐ؛ ܬܪܝܢ ܡܢܝ̈ܢܐ ܒܬܪ ܢܘܩܙܐ ܓܘ ܠܘ̈ܚܐ. ܚܘܫܒܐ ܫܪܝܪܐ ܝܠܗ ܩܘܒܠ R 4.3.3 (car، emmeans، effectsize، pwr، FrF2، agricolae) ܘ-PSPP، ܘܩܘܒܠ ܡܢܝ̈ܢܐ ܛܒܝ̈ܥܐ ܓܘ ܬܪܝܢ ܟܬܒ̈ܐ: ܠܘܚܐ 3.10 ܘܡܬ̈ܠܐ ܕܚܝܠܐ ܓܘ Maxwell ܘܚܒܪ̈ܘܗܝ، ܠܘ̈ܚܐ 7.7، 7.8، 7.25، 8.14، 8.15 ܘ-9.12 ܬܡܢ؛ ܘܓܘ Montgomery – ܡܬ̈ܠܐ 4.1-4.3، 5.1-5.3، 5.6، 6.1، 6.2، 7.1، 7.2، 8.1، ܠܘ̈ܚܐ 4.13، 7.9، 8.13 ܘ-8.14، ܘܐܘܪܚܐ ܕ-Lenth (ܦܐܬܐ 226).