ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ، ܡܩܢܢܐ ܘܣܦܠܝܛ-ܦܠܘܛ

ܩܝܡܬܐ ܝܢ ܓܕܫܢܝܬܐ؟

ܦܘܪܫܢܐ ܠܐ ܝܠܗ ܓܘ ܥܠܬܐ (factor) ܒܓܢܣܗ̇ ܐܠܐ ܓܘ ܫܘܐܠܐ. ܥܠܬܐ ܩܝܡܬܐ (fixed) – ܕܪ̈ܓܐ ܕܓܘ ܢܣܝܘܢܐ (experiment) ܚܬܝܬܐܝܬ ܐܢܝ ܝܢܐ ܕܟܒܥܝܚ، ܘ"ܦܬܝܘܬܐ ܕܦܠܛܬܐ ܝܠܗ̇ ܟܢܘܫܝܐ ܕܕܪ̈ܓܐ ܝܚܝܕ̈ܝܐ ܕܒܚܝܪ̈ܐ ܝܢܐ" (Montgomery, 2020, p. 493). ܥܠܬܐ ܓܕܫܢܝܬܐ (random) – ܕܪ̈ܓܐ "ܓܒܝ̈ܐ ܝܢܐ ܓܕܫܢܐܝܬ ܡܢ ܟܢܫܐ ܟܠܢܝܐ (population) ܪܒܐ ܝܬܝܪ ܕܕܪ̈ܓܐ، ܘܒܨܘܝܐ ܟܒܥܝܐ ܠܡܦܠܛܐ ܥܠ ܟܠܗ ܟܢܫܐ" (ܬܡܢ). Maxwell، Delaney ܘ-Kelley (2018) ܟܟܬܒܝ ܗܕܐ ܒܝܕ ܬܢܝܬܐ (replication): ܥܠܬܐ ܩܝܡܬܐ ܝܠܗ̇ ܗܝ ܕܗܢܝ ܕܪ̈ܓܐ ܒܕ ܗܘܘ ܥܠܝ̈ܠܐ ܐܦ ܓܘ ܬܢܝܬܐ ܕܒܘܨܝܐ؛ ܓܘ ܓܕܫܢܝܬܐ ܢܝܫܐ ܝܠܗ "ܠܡܓܘܢܐ ܠܐ ܠܡܥܒܕ̈ܢܘܝܬܐ ܕܟܡܐ ܕܪ̈ܓܐ ܝܚܝܕ̈ܝܐ، ܐܠܐ ܠܟܢܫܐ ܟܠܢܝܐ ܕܕܪ̈ܓܐ ܕܕܪ̈ܓܐ ܓܒܝ̈ܐ ܒܣ ܟܡܚܘܝ ܠܗ" (ܦܐܬܐ 548). ܥܠ̈ܠܬܐ ܐܝܟ ܐܢܝ "ܡܬܕܡܝ̈ܢܐ" ܝܢܐ ܝܬܝܪ ܡܢ ܕ"ܡܬܦܠܚ̈ܢܐ" (ܦܐܬܐ 549): ܡܕܪ̈ܫܝܬܐ، ܨܦ̈ܐ، ܡܐܣܝ̈ܢܐ (therapists)، ܦܠܚ̈ܐ ܕܡܟܝܢܐ (operators)، ܡܢܘ̈ܬܐ ܕܥܒܕܐ (batches)، ܝܘܡ̈ܢܐ – ܘܥܠܬܐ ܓܕܫܢܝܬܐ ܕܡܬܚܙܝܐ ܝܬܝܪ ܡܢ ܟܠܗܝܢ، ܡܫܬܘܬܦ̈ܢܐ (participants؛ ܦܐܬܐ 547).

ܐܦ ܐܝܡܢ ܕܡܕܪ̈ܫܝܬܐ ܠܐ ܝܢܐ ܕܘܡܝܐ (sample) ܓܕܫܢܝܐ ܫܪܝܪܐ – ܡܬܠܐ ܐܢܝ ܕܨܒܝܠܗܘܢ ܠܡܫܬܘܬܦܐ – ܫܘܐܠܐ ܝܠܗ ܐܢ ܦܠܛܬܐ ܣܢܝܩܬܐ ܝܠܗ̇ ܕܐܬܝܐ ܒܣ ܥܠܝܗܝ. ܐܢ ܐܝܢ، ܥܠܬܐ ܩܝܡܬܐ ܝܠܗ̇؛ ܐܢ ܟܒܥܝܚ ܠܡܡܠܠܐ ܥܠ "ܡܕܪ̈ܫܝܬܐ ܒܟܠܢܝܘܬܐ" – ܓܕܫܢܝܬܐ (Maxwell et al., 2018, p. 549). ܘܓܘ ܓܘ̈ܕܐ ܕܓܒܝ̈ܐ ܝܢܐ ܡܛܠ ܦܫܝܩܘܬܐ، Murray ܘ-Blitstein (2003، ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Maxwell et al., 2018، ܦܐܬܐ 607) ܟܟܬܒܝ ܕܠܐ ܒܣ ܫܠܝܛܐ ܝܠܗ ܐܠܐ "ܣܢܝܩܐ" ܝܠܗ ܠܡܚܕܕܐ ܓܘܕܐ ܐܝܟ ܡܥܒܕܢܘܬܐ ܓܕܫܢܝܬܐ (random effect). ܫܒܩܬܐ ܕܥܠܬܐ ܓܕܫܢܝܬܐ، ܝܢ ܚܘܕܕܐ ܕܝܠܗ̇ ܐܝܟ ܩܝܡܬܐ، "ܡܨܐ ܕܡܒܠܐ ܠܡܥܩܡܐ ܒܘܚܪ̈ܢܐ ܡܢܝܗܝ ܠܥܠ" (ܦܐܬܐ 551).

ܚܕ ܥܠܬܐ ܓܕܫܢܝܬܐ: ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ ܘܐܣܘܪܐ ܕܚܕ̈ܕܐ ܓܘ ܓܘܕܐ

ܓܘ ܡܘܕܝܠ (model) ܓܕܫܢܝܐ ܕܚܕ ܐܘܪܚܐ، y_ij = μ + τ_i + ε_ij، ܡܥܒܕ̈ܢܘܝܬܐ τ_i ܡܫܬܚܠܦܢܐ ܓܕܫܢܝܐ (random variable) ܝܢܐ ܥܡ ܫܘܚܠܦܐ (variance) σ²_τ، ܘܫܘܚܠܦܐ ܕܚܙܝܬܐ (observation) ܝܠܗ σ²_τ + σ² – "ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ" (variance components؛ Montgomery, 2020, p. 101). ܚܘܫܒܢܐ ܫܘܝܐ ܝܠܗ ܠܦܘܫܩܐ ܕܫܘܚܠܦܐ ܓܘܢܝܐ، "ܐܠܐ ܦܠ̈ܛܬܐ ܦܪ̈ܝܫܐ ܝܢܐ ܒܓܡܝܪܘܬܐ" (ܦܐܬܐ 103): ܗܘܦܘܬܣܝܣ (hypothesis) ܝܠܗ̇ H₀: σ²_τ = 0، ܥܠ ܟܠܗ ܟܢܫܐ ܕܕܪ̈ܓܐ. ܡܢ ܛܝ̈ܡܐ ܡܣܟ̈ܝܐ (expected values) E(MS_Tr) = σ² + nσ²_τ ܘ-E(MS_E) = σ² ܟܐܬܝ ܫܘܥܪ̈ܐ σ̂² = MS_E ܘ-σ̂²_τ = (MS_Tr − MS_E)/n (ܫܘܝ̈ܘܬܐ 3.48-3.51، ܦܐܬܐ 103).

ܩܐ σ²_τ ܒܓܢܣܗ ܠܝܬ ܬܚܘܡܐ ܕܬܘܟܠܢܐ (confidence interval) ܚܬܝܬܐ (ܦܐܬܐ 106)، ܐܠܐ ܩܐ ܦܘܚܡܐ σ²_τ/(σ²_τ + σ²) – ܐܣܘܪܐ ܕܚܕ̈ܕܐ ܓܘ ܓܘܕܐ (intraclass correlation, ICC) – ܐܝܬ (ܫܘܝ̈ܘܬܐ 3.56-3.57، ܦܐܬܐ 106)، ܘܦܐܬܐ ܟܚܫܒܐ ܠܗ. ICC "ܟܡܚܘܐ ܡܢܬܐ ܕܫܘܚܠܦܐ ܕܡܬܝܗܒܐ ܝܠܗ̇ ܠܥܠܬܐ ܓܕܫܢܝܬܐ" (Maxwell et al., 2018, p. 547)، ܘܕܡܝܐ ܝܠܗ ܠ-ω² ܕܡܥܒܕ̈ܢܘܝܬܐ ܩܝ̈ܡܬܐ (fixed effects؛ ܦܐܬܐ 565). ܓܘ ܡܬܠܐ ܕܓܪ̈ܕܝܐ (looms) ܕ-Montgomery (ܡܬܠܐ 3.10) – ICC ܒܝܢܝ .38 ܘ-.98 (ܦܐܬܐ 106). ܓܘ ܝܗܝ̈ܒܬܐ ܕܡܨܝܢܘܬܐ ܓܘ ܡܕܪ̈ܫܝܬܐ ICC ܓܘܢܝܐ ܝܠܗ .15-.25 (Hedges & Hedberg, 2007، ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Maxwell et al., 2018، ܦܐܬܐ 608).

ܡܨܥܝ̈ܬܐ ܡܣܟ̈ܝܬܐ ܕܡܪ̈ܒܥܐ: ܐܝܟܢܐ ܡܢܝܢܐ ܬܚܬܝܐ ܓܒܝܐ ܝܠܗ

ܓܘ ܦܘܫܩܐ ܕܫܘܚܠܦܐ (analysis of variance) ܓܘܢܝܐ ܟܠ ܒܘܚܪܢܐ ܩܘܒܠ ܫܪܟܐ (residual) ܝܠܗ. ܐܝܡܢ ܕܐܝܬ ܥܠܬܐ ܓܕܫܢܝܬܐ ܐܗܐ ܠܐ ܝܠܗ ܬܪܝܨܐ ܒܬܪ ܗܕܐ: "ܐܝܡܢ ܕܐܝܬ ܥܠܬܐ ܓܕܫܢܝܬܐ ܓܘ ܬܘܩܢܐ، ܟܠ ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction) ܕܝܠܗ̇ ܥܡ ܥܠܬܐ ܐܚܪܬܐ ܟܡܥܒܕ ܥܠ ܡܨܥܝܬܐ ܕܡܪ̈ܒܥܐ (mean square) ܕܡܥܒܕܢܘܬܐ ܪܫܝܬܐ ܕܥܠܬܐ ܐܚܪܬܐ" (Maxwell et al., 2018, p. 555). ܡܬܠܐ، ܓܘ ܬܘܩܢܐ ܕܝܘܠܦܢܐ (ܩܝܡܬܐ) × ܡܕܪ̈ܫܝܬܐ (ܓܕܫܢ̈ܝܬܐ): ܐܢ ܬܘܩܢܐ ܟܦܠܚ ܛܒ ܝܬܝܪ ܓܘ ܡܢܬܐ ܡܢ ܡܕܪ̈ܫܝܬܐ، ܦܘܪܫܢܐ ܒܝܢܝ ܬܘܩ̈ܢܐ ܓܘ ܕܘܡܝܐ ܬܠܝܐ ܝܠܗ ܒܐܝܢܝ ܡܕܪ̈ܫܝܬܐ ܡܬܕܡܝ̈ܐ ܝܢܐ – ܘܒܘܚܪܢܐ ܩܘܒܠ ܫܪܟܐ "ܟܝܗܒ ܛܝ̈ܡܐ ܕ-F ܪ̈ܒܐ ܒܣܓܝ، ܐܠܐ ܐܢ ܥܒܕܐ ܕܚܕ̈ܕܐ ܓܘ ܟܢܫܐ ܟܠܢܝܐ ܚܬܝܬܐܝܬ ܣܦܪ ܝܠܗ" (ܦܐܬܐ 558).

ܡܐܢܐ ܝܠܗ ܡܨܥܝܬܐ ܡܣܟܝܬܐ ܕܡܪ̈ܒܥܐ (expected mean square) E(MS) ܕܟܠ ܗܕܡܐ (term). Montgomery (2020، ܦܣܩܐ 13.4، ܦܐܬܐ 505-507) ܟܝܗܒ ܢܡܘ̈ܣܐ ܩܐ ܟܠ ܬܘܩܢܐ ܡܫܘܝܐ (balanced) – ܨܠܝܒܐ (crossed)، ܡܩܢܢܐ (nested) ܝܢ ܡܚܠܛܐ: ܛܝܡܐ ܡܣܟܝܐ ܕܗܕܡܐ ܝܠܗ σ²، ܘܥܡܗ ܡܢܬܐ ܕܝܠܗ ܒܓܢܣܗ، ܘܥܡܗ ܡܢܘ̈ܬܐ ܕܟܠ ܗܕ̈ܡܐ ܓܕܫܢ̈ܝܐ ܕܐܝܬ ܓܘܝܗܝ ܐܗܐ ܗܕܡܐ؛ "ܡܢܝܢܐ ܡܫܘܬܦܐ ܕܟܠ ܡܢܬܐ ܝܠܗ ܡܢܝܢܐ ܕܚܙ̈ܝܬܐ ܓܘ ܟܠ ܛܝܡܐ ܦܪܝܫܐ ܕܝܠܗ̇" (ܦܐܬܐ 506). F ܟܒܢܚ ܗܟܐ ܕܡܢܝܢܐ ܥܠܝܐ (numerator) ܘܡܢܝܢܐ ܬܚܬܝܐ (denominator) "ܦܪ̈ܝܫܐ ܝܢܐ ܒܣ ܒܡܢܬܐ ܝܢ ܒܥܠܬܐ ܩܝܡܬܐ ܕܟܒܥܝܚ" (ܦܐܬܐ 505). ܦܐܬܐ ܟܡܦܠܚܐ ܢܡܘ̈ܣܐ ܚܬܝܬܐܝܬ ܗܟܐ، ܘܟܡܚܘܝܐ ܛܝ̈ܡܐ ܡܣܟ̈ܝܐ ܓܘ ܠܘܚܐ 2 – ܘܡܨܝܚ ܠܡܦܚܡܐ ܐܢܝ ܥܡ ܠܘ̈ܚܐ 13.10، 14.1، 14.11 ܘ-14.17 ܓܘ ܟܬܒܐ. Maxwell et al. (2018) ܟܡܟܢܫܝ ܒܢܡܘܣܐ ܕܨܒܥܐ (rule of thumb): "ܠܒܪ ܡܢ ܡܥܒܕܢܘܬܐ ܒܓܢܣܗ̇، ܝܗܘ̈ܒܐ ܝܚܝܕ̈ܝܐ ܠܛܝܡܐ ܡܣܟܝܐ ܥܒܕ̈ܐ ܕܚܕ̈ܕܐ ܥܡ ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ ܝܢ ܡܥܒܕ̈ܢܘܝܬܐ ܓܕܫܢ̈ܝܬܐ ܡܩܢܢ̈ܬܐ ܝܢܐ" (ܦܐܬܐ 593).

ܡܘܕܝܠ ܡܚܠܛܐ: ܟܪܝܐ ܝܢ ܠܐ ܟܪܝܐ؟

ܐܝܡܢ ܕܥܠܬܐ ܩܝܡܬܐ A ܨܠܝܒܬܐ ܝܠܗ̇ ܥܡ ܥܠܬܐ ܓܕܫܢܝܬܐ B، ܐܝܬ ܬܪܝܢ ܢܘܣܚ̈ܐ ܩܐ ܥܒܕܐ ܕܚܕ̈ܕܐ (interaction). ܓܘ ܡܘܕܝܠ ܟܪܝܐ (restricted) ܓܘܡܪܐ ܕܡܥܒܕ̈ܢܘܝܬܐ ܕܥܒܕܐ ܕܚܕ̈ܕܐ ܥܠ ܕܪ̈ܓܐ ܕ-A ܣܦܪ ܝܠܗ ܓܘ ܟܠ ܕܪܓܐ ܕ-B (Montgomery, 2020, Equation 13.8, p. 500)؛ ܗܝܕܝܟ E(MS_B) = σ² + anσ²_β، ܘ-B ܒܚܝܪܐ ܝܠܗ̇ ܩܘܒܠ ܫܪܟܐ. ܓܘ ܡܘܕܝܠ ܠܐ ܟܪܝܐ (unrestricted) ܠܝܬ ܟܪܝܘܬܐ ܐܝܟ ܗܕܐ (ܦܐܬܐ 502)؛ ܗܝܕܝܟ E(MS_B) ܐܝܬ ܓܘܗ ܐܦ ܥܒܕܐ ܕܚܕ̈ܕܐ، ܘ-B ܒܚܝܪܐ ܝܠܗ̇ ܩܘܒܠ MS_AB (ܦܐܬܐ 503). ܒܘܚܪܢܐ ܕܥܠܬܐ ܩܝܡܬܐ ܫܘܝܐ ܝܠܗ ܓܘ ܬܪܘܝܗܝ.

Montgomery "ܟܓܒܐ ܡܘܕܝܠ ܟܪܝܐ" ܘܟܐܡܪ ܕܐܝܬܘܗܝ "ܒܫܪܪܐ ܩܠܝܠ ܓܘܢܝܐ ܝܬܝܪ" (ܦܐܬܐ 503)؛ Maxwell et al. (2018) "ܟܡܣܠܝ ܐܘܪܚܐ ܠܐ ܟܪܝܬܐ"، ܡܛܠ ܕܓܘܗ̇ ܗܘܦܘܬܣܝܣ ܕܣܦܝܩܘܬܐ (null hypothesis) ܥܠ ܥܠܬܐ ܓܕܫܢܝܬܐ ܠܐ ܟܐܡܪܐ ܒܬܪ ܗܕܐ ܕܡܨܥܝ̈ܬܐ ܕܣܦܪܐ (marginal means) ܫܘ̈ܝܐ ܝܢܐ (ܦܐܬܐ 606). ܡܢ ܓܢܒܐ ܐܚܪܢܐ، JMP ܘ-Minitab (ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ (default)) ܟܦܠܚܝ ܓܘ ܡܘܕܝܠ ܠܐ ܟܪܝܐ (Montgomery, 2020, p. 504)، ܘܗܟܐ ܐܦ lmer ܓܘ R؛ ܘܠܦܘܬ Maxwell et al. (2018) – ܐܦ SPSS ܘ-SAS (ܦܐܬܐ 605-606). ܡܛܠ ܗܕܐ ܓܘܒܝܐ ܡܩܕܡܐ ܓܘ ܦܐܬܐ ܝܠܗ ܡܘܕܝܠ ܟܪܝܐ، ܥܡ ܡܚܠܦܢܐ (switch) ܩܐ ܡܘܕܝܠ ܠܐ ܟܪܝܐ؛ "ܩܪܝܢܐ ܕܦܐܪܐ" ܟܡܚܘܐ ܐܝܢܝ ܒܘܚܪ̈ܢܐ ܟܡܫܬܚܠܦܝ ܓܘ ܫܘܚܠܦܐ. ܐܣܘܪܐ ܒܝܢܝ ܡܢܘ̈ܬܐ ܓܘ ܬܪܝܢ ܡܘܕܝ̈ܠܐ: σ²_β = σ²_γ + σ²_αγ/a – ܓܘ ܟܬܒܐ (ܦܐܬܐ 503) ܡܬܢܝܬܐ ܛܒܝܥܬܐ ܝܠܗ̇ ܗܦܝܟܐܝܬ (ܡܫܟܚܢܘܬܐ E3، ܚܙܝܡܘܢ ܠܬܚܬ)؛ ܓܘ ܡܬܠܐ 13.2: 10.2798 + (−0.1399)/3 = 10.2332.

ܐܝܡܢ ܕܠܝܬ ܒܘܚܪܢܐ ܚܬܝܬܐ: F ܩܪܝܒܐ

ܥܡ ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ ܝܢ ܝܬܝܪ، ܐܝܬ ܐܡܢ̈ܐ ܕܠܝܬ ܚܕ ܡܨܥܝܬܐ ܕܡܪ̈ܒܥܐ ܕܛܝܡܐ ܡܣܟܝܐ ܕܝܠܗ̇ ܦܐܐ. ܗܝܕܝܟ ܟܒܢܚ ܚܘܠܛܐ: MS' = MS_r + … + MS_s ܓܘ ܡܢܝܢܐ ܥܠܝܐ (numerator) ܘ-MS'' = MS_u + … + MS_v ܓܘ ܡܢܝܢܐ ܬܚܬܝܐ (denominator)، ܗܟܐ ܕܦܘܪܫܢܐ ܒܝܢܝ ܛܝ̈ܡܐ ܡܣܟ̈ܝܐ ܚܬܝܬܐܝܬ ܡܢܬܐ ܒܚܝܪܬܐ ܝܠܗ̇، ܘ-F = MS'/MS'' ܐܝܬܠܗ ܩܪܝܒܐܝܬ ܦܘܠܓܐ (distribution) ܕ-F ܥܡ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܕ-Satterthwaite (1946, as cited in Montgomery, 2020): p = (ΣMS)²/Σ(MS²/f) ܩܐ ܟܠ ܓܢܒܐ (ܫܘܝ̈ܘܬܐ 13.14-13.18، ܦܐܬܐ 509). ܡܨܝܚ ܐܦ ܠܡܒܨܪܐ ܡܨܥܝ̈ܬܐ ܕܡܪ̈ܒܥܐ، ܐܠܐ Montgomery ܟܓܒܐ ܚܘܠܛܐ ܕ"ܠܝܬ ܓܘܗ ܡܨܥܝ̈ܬܐ ܕܡܪ̈ܒܥܐ ܥܡ ܡܢܝܢܐ ܡܫܘܬܦܐ ܬܚܬ ܣܦܪ" (ܦܐܬܐ 512) – ܘܦܐܬܐ ܐܡܝܢ ܟܒܢܐ ܚܘܠܛܐ ܐܝܟ ܗܕܐ، ܘܟܓܒܐ ܩܛܝܢܐ ܝܬܝܪ ܡܢ ܟܠܗܘܢ. ܓܘ ܡܬܠܐ 13.5 (ܢܚܬܬܐ ܕܥܨܪܐ – pressure drop) ܒܘܚܪܢܐ ܕܚܡܝܡܘܬܐ ܝܠܗ (MS_A + MS_ABC)/(MS_AB + MS_AC) = 2.22 ܥܡ 2.11 ܘ-7.88 ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (ܦܐܬܐ 511-512)، ܘ-Minitab، ܕܟܒܨܪ، ܟܫܩܠ 2.30. ܐܗܐ ܒܘܚܪܢܐ "ܒܣ ܩܪܝܒܐܝܬ ܐܝܬܠܗ ܦܘܠܓܐ ܕ-F" (Maxwell et al., 2018, p. 560).

ܡܚܝܕܢܘܬܐ (pooling). ܚܠܘܦܐ ܝܠܗ ܠܡܦܠܛܐ ܥܒܕܐ ܕܚܕ̈ܕܐ ܙܥܘܪܐ ܛܒܐ (negligible) ܡܢ ܡܘܕܝܠ ܘܠܡܚܝܕܐ ܠܗ ܥܡ ܫܪܟܐ. ܐܠܐ ܦܣܩܐ ܥܠ ܥܒܕܐ ܕܚܕ̈ܕܐ ܡܢ ܓܘ ܝܗܝ̈ܒܬܐ "ܡܨܐ ܕܗܘܐ ܒܝܫܐ ܠܚܕ ܒܝܫܘܬܐ (dangerous)"، ܡܛܠ ܕܦܬܝܚܐ ܝܠܗ ܠܛܥܝ̈ܘܬܐ ܡܢ ܬܪܝܢ ܙܢ̈ܐ (Montgomery, 2020, p. 508). ܢܡܘܣܐ ܥܒܕܢܝܐ: "ܐܢ ܠܡܨܥܝܬܐ ܕܡܪ̈ܒܥܐ ܕܛܥܝܘܬܐ ܩܕܡܝܬܐ ܐܝܬ ܫܬܐ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܝܢ ܝܬܝܪ – ܠܐ ܠܡܚܝܕܐ"؛ ܘܐܢ ܠܐ، ܒܣ ܐܢ F ܕܗܕܡܐ ܠܐ ܝܠܗ ܪܫܝܡܐ ܒ-α ܪܡܐ، ܐܝܟ 0.25 (ܦܐܬܐ 509). Maxwell et al. (2018) ܟܡܘܣܦܝ ܕܐܘܪܚܐ ܢܛܘܪܬܐ (conservative) ܝܠܗ̇ ܠܡܦܠܚܐ ܒܥܒܕܐ ܕܚܕ̈ܕܐ ܐܝܟ ܡܢܝܢܐ ܬܚܬܝܐ ܕܠܐ ܚܝܪܐ ܠܦܐܪܐ ܕܒܘܚܪܢܐ ܩܕܡܝܐ (ܦܐܬܐ 562)، ܘܕܐܘܪܚܐ ܥܒܕܢܝܬܐ ܝܬܝܪ ܩܐ ܡܪܒܝܐ ܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ (statistical power) ܦܫܝܛܐܝܬ ܠܡܘܣܦܐ ܕܪ̈ܓܐ ܠܥܠܬܐ ܓܕܫܢܝܬܐ (ܦܐܬܐ 563). ܓܘ ܦܐܬܐ ܟܪܫܡܚ ܗܕ̈ܡܐ ܩܐ ܡܚܝܕܢܘܬܐ ܓܘ ܬܘܩ̈ܢܐ.

ܥܠ̈ܠܬܐ ܡܩܢܢ̈ܬܐ

"ܥܠܬܐ ܡܩܢܢܬܐ (nested) ܝܠܗ̇ ܓܘ ܥܠܬܐ ܐܚܪܬܐ ܐܢ ܟܠ ܕܪܓܐ ܕܝܠܗ̇ ܟܡܬܚܙܐ ܥܡ ܒܣ ܚܕ ܕܪܓܐ ܕܥܠܬܐ ܐܚܪܬܐ" (Maxwell et al., 2018, p. 573): ܨܦ̈ܐ ܓܘ ܡܕܪ̈ܫܝܬܐ، ܡܢܘ̈ܬܐ ܕܥܒܕܐ (batches) ܓܘ ܡܛܝܒ̈ܢܐ (suppliers)، ܡܐܣܝ̈ܢܐ ܓܘ ܐܘܪ̈ܚܬܐ ܕܐܣܝܘܬܐ. ܒܘܚܪܢܐ ܥܒܕܢܝܐ ܕ-Montgomery: "ܐܢ ܡܨܝܚ ܠܡܢܝܢܐ ܡܢ ܪܝܫ ܕܪ̈ܓܐ ܕܥܠܬܐ ܒܨܒܝܢܐ ܚܐܪܐ" – ܡܢܬܐ 1 ܕܡܛܝܒܢܐ ܐ ܘܡܢܬܐ 1 ܕܡܛܝܒܢܐ ܒ ܠܐ ܝܢܐ ܗܝ ܡܢܬܐ – "ܥܠܬܐ ܡܩܢܢܬܐ ܝܠܗ̇" (2020، ܦܐܬܐ 519). ܡܛܠ ܗܕܐ ܠܝܬ ܥܒܕܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܥܠܬܐ ܡܩܢܢܬܐ ܘܥܠܬܐ ܕܗܝ ܓܘܗ̇ (ܦܐܬܐ 519)، ܘܓܘܡܪܐ ܕܡܪ̈ܒܥܐ (sum of squares) ܕܝܠܗ̇ ܝܠܗ ܓܘܡܪܐ ܕܦܘܪ̈ܫܢܐ ܒܝܢܝ ܕܪ̈ܓܐ ܕܝܠܗ̇ ܓܘ ܟܠ ܐܒܐ (parent؛ ܦܐܬܐ 521). ܓܘ ܦܐܬܐ ܡܨܝܚ ܠܡܢܝܢܐ ܕܪ̈ܓܐ ܡܩܢܢ̈ܐ ܡܢ ܪܝܫ ܓܘ ܟܠ ܐܒܐ (1-4 ܓܘ ܟܠ ܡܛܝܒܢܐ) ܝܢ ܒܫܫܠܬܐ (1-12)؛ ܐܝܡܢ ܕܡܢܝܢܐ ܪܗܛ، ܦܐܬܐ ܟܡܦܪܫܐ ܡܩܢܢܘܬܐ ܒܓܢܣܗ̇.

ܥܠܬܐ ܡܩܢܢܬܐ ܣܓܝ ܓܗ̈ܝ ܓܕܫܢܝܬܐ ܝܠܗ̇ (Maxwell et al., 2018, p. 574)، ܘܗܝܕܝܟ ܝܠܗ̇ ܡܢܝܢܐ ܬܚܬܝܐ (denominator) ܕܥܠܬܐ ܕܠܥܠ ܡܢܗ̇: ܦܘܚܡܐ ܒܝܢܝ ܐܘܪ̈ܚܬܐ ܕܐܣܝܘܬܐ ܒܚܝܪܐ ܝܠܗ ܩܘܒܠ ܫܘܚܠܦܐ ܒܝܢܝ ܡܐܣܝ̈ܢܐ، ܠܐ ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ – "ܫܘܐܠܐ ܐܢ ܬܠܡܝ̈ܕܐ ܓܒܪ̈ܐ ܟܝܗܒܝ ܕܪ̈ܓܐ ܦܪ̈ܝܫܐ ܡܢ ܬܠܡܝ̈ܕܬܐ ܢܫ̈ܐ ܡܬܦܢܐ ܝܠܗ ܬܪܝܨܐܝܬ ܩܘܒܠ ܫܘܚܠܦܐ ܒܝܢܝ ܬܠܡܝ̈ܕܐ" (ܦܐܬܐ 583). ܗܟܐ ܐܦ ܓܘ ܒܘܨ̈ܝܐ ܕܓܘܝܗܝ ܨܦ̈ܐ ܓܡܝܪ̈ܐ ܡܬܦܠ̈ܓܐ ܝܢܐ ܠܚܠ̈ܝܬܐ (conditions؛ ܦܐܬܐ 575). Wampold ܘ-Serlin (2000، ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Maxwell et al., 2018، ܦܐܬܐ 590) ܟܡܚܘܝ ܕܫܒܩܬܐ ܕܥܠܬܐ ܡܩܢܢܬܐ ܓܕܫܢܝܬܐ ܥܡ ICC ܕ-.3 ܡܨܝܐ ܕܡܢܦܚܐ ܡܢܬܐ ܕܛܥܝܘܬܐ ܕܙܢܐ I (Type I error) ܬܠܬܐ ܓܗ̈ܝ ܘܝܬܝܪ. ܦܘܫܩܐ ܕܬܘܩܢܐ ܡܩܢܢܐ ܐܝܟ ܕܨܠܝܒܐ ܗܘܐ ܟܒܢܐ "ܥܒܕܐ ܕܚܕ̈ܕܐ" ܕܠܐ ܣܘܟܠܐ؛ SS_B + SS_A×B ܕܦܘܫܩܐ ܨܠܝܒܐ ܚܬܝܬܐܝܬ ܝܠܗ SS_B(A) (Montgomery, 2020, pp. 522-524). ܬܘܩ̈ܢܐ ܡܚܠܛ̈ܐ – ܡܢܬܐ ܡܢ ܥܠ̈ܠܬܐ ܨܠܝ̈ܒܬܐ ܘܡܢܬܐ ܡܩܢܢ̈ܬܐ، ܐܝܟ ܓܘ ܡܬܠܐ 14.2 (fixtures × layouts، operators ܡܩܢܢ̈ܐ ܓܘ layout) – ܡܬܦܫܩ̈ܢܐ ܝܢܐ ܒܗܢܝ ܢܡܘ̈ܣܐ (ܦܐܬܐ 530-534).

ܣܦܠܝܛ-ܦܠܘܛ

ܐܝܬ ܐܡܢ̈ܐ ܕ"ܠܐ ܡܨܝܚ ܠܡܥܒܕܐ ܪ̈ܗܛܐ ܒܛܟܣܐ ܓܕܫܢܝܐ ܓܡܝܪܐ" (Montgomery, 2020, p. 534). ܓܘ ܡܬܠܐ ܕܚܝܠܐ ܕܡܬܝܚܘܬܐ ܕܟܪܛܝܣܐ (tensile strength؛ ܦܣܩܐ 14.4) ܬܠܬܐ ܐܘܪ̈ܚܬܐ ܕܡܛܝܒܘܬܐ ܕܥܓܝܢܐ ܕܟܪܛܝܣܐ (pulp) ܒܚܝܪ̈ܐ ܝܢܐ ܒܐܪܒܥܐ ܚܡܝܡܘ̈ܬܐ: ܟܡܛܝܒܚ ܚܕ ܡܢܬܐ ܕܥܒܕܐ (batch) ܒܚܕ ܐܘܪܚܐ، ܟܦܠܓܚ ܠܗ̇ ܠܐܪܒܥܐ ܡܢܘ̈ܬܐ ܘܟܐܦܚ ܟܠ ܚܕ ܒܚܡܝܡܘܬܐ ܐܚܪܬܐ – "ܬܫܥܐ ܡܢܘ̈ܬܐ ܒܟܠܗ" ܒܕܘܟܬܐ ܕ-36 (ܦܐܬܐ 535). ܡܢܬܐ ܕܥܒܕܐ ܝܠܗ̇ ܚܩܠܐ ܪܫܝܐ (whole plot)، ܐܘܪܚܐ – ܥܠܬܐ ܕܚܩܠܐ ܪܫܝܐ، ܘܟܠ ܡܢܬܐ – ܚܩܠܐ ܙܥܘܪܐ (subplot؛ ܦܐܬܐ 535). ܫܪܫܐ: "ܥܠ̈ܠܬܐ ܕܥܣܩܐ ܝܢܐ ܠܫܘܚܠܦܐ ܟܒܢܝ ܚܩ̈ܠܐ ܪ̈ܫܝܐ، ܘܦܫܝ̈ܩܐ – ܚܩ̈ܠܐ ܙܥܘ̈ܪܐ" (ܦܐܬܐ 539). ܠܬܘܩܢܐ ܐܝܬ ܬܪܬܝܢ ܛܥܝ̈ܘܬܐ: ܥܠܬܐ ܕܚܩܠܐ ܪܫܝܐ ܒܚܝܪܬܐ ܝܠܗ̇ ܩܘܒܠ ܬܢܝܬܐ × ܥܠܬܐ ܕܚܩܠܐ ܪܫܝܐ (ܛܥܝܘܬܐ ܕܚܩܠܐ ܪܫܝܐ)، ܘܥܠܬܐ ܕܚܩܠܐ ܙܥܘܪܐ ܘܥܒܕܐ ܕܚܕ̈ܕܐ – ܩܘܒܠ ܛܥܝ̈ܘܬܐ ܕܚܩܠܐ ܙܥܘܪܐ، ܕܣܓܝ ܓܗ̈ܝ ܙܥܘܪ̈ܐ ܝܢܐ ܝܬܝܪ، "ܡܛܠ ܕܚܩ̈ܠܐ ܙܥܘ̈ܪܐ ܫܘ̈ܝܐ ܝܢܐ ܝܬܝܪ ܒܓܢܣܗܘܢ (homogeneous)" (ܦܐܬܐ 537). ܡܛܠ ܗܕܐ "ܛܒ ܝܠܗ ܠܡܬܘܐ ܓܘ ܚܩ̈ܠܐ ܙܥܘ̈ܪܐ ܥܠܬܐ ܕܟܒܥܝܚ ܝܬܝܪ" (ܦܐܬܐ 535).

ܓܘ ܡܘܕܝܠ ܓܡܝܪܐ (ܫܘܝܘܬܐ 14.15) ܬܢܝ̈ܬܐ ܓܕܫܢ̈ܝܬܐ ܝܢܐ، ܘܩܐ ܬܢܝ̈ܬܐ ܘܩܐ ܬܢܝܬܐ × ܚܩܠܐ ܙܥܘܪܐ ܠܝܬ ܒܘܚܪܢܐ (ܦܐܬܐ 536)؛ ܐܢ ܟܣܒܪܚ ܕܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܥܡ ܬܢܝ̈ܬܐ ܓܘ ܚܩܠܐ ܙܥܘܪܐ ܙܥܘܪ̈ܐ ܛܒܐ ܝܢܐ، ܟܡܚܝܕܚ ܐܢܝ ܘܟܫܩܠܚ ܡܘܕܝܠ ܦܫܝܛܐ ܝܬܝܪ (ܫܘܝܘܬܐ 14.16، ܦܐܬܐ 537) – ܐܗܐ ܝܠܗ ܡܘܕܝܠ ܕ-JMP ܟܡܦܠܚ ܒܗ (ܬܡܢ)، ܘܐܦ aov(y ~ A*B + Error(rep/A)) ܓܘ R. ܓܘ ܦܐܬܐ: ܓܘ ܠܫܢܝܬܐ (tab) "ܬܘܩܢܐ ܕܣܦܠܝܛ-ܦܠܘܛ" – ܛܟܣܐ ܕܪܗܛܐ (run order) ܘܠܘܚܐ ܡܣܟܝܐ؛ ܓܘ ܦܘܫܩܐ – ܬܢܝܬܐ ܓܕܫܢܝܬܐ، ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ، ܘܩܐ ܡܚܝܕܢܘܬܐ ܟܪܫܡܚ ܬܢܝܬܐ × ܚܩܠܐ ܙܥܘܪܐ ܘܬܢܝܬܐ × ܥܒܕܐ ܕܚܕ̈ܕܐ. ܓܘ ܝܕ̈ܥܬܐ ܕܬܗܦܘܟܬܐ (behavioral sciences) ܗܘ ܒܢܝܢܐ ܟܡܬܩܪܐ ܐܦ "ܬܘܩܢܐ ܡܚܠܛܐ" (mixed design): ܥܠܬܐ ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ (between-subjects) ܘܥܠܬܐ ܓܘ ܡܫܬܘܬܦ̈ܢܐ (within-subjects)، ܐܝܡܢ ܕܡܫܬܘܬܦ̈ܢܐ ܡܩܢܢ̈ܐ ܝܢܐ ܓܘ ܥܠܬܐ ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ (Maxwell et al., 2018, pp. 689-690) – ܓܘ ܦܐܬܐ: ܡܫܬܘܬܦܢܐ ܓܕܫܢܝܐ ܘܡܩܢܢܐ ܓܘ ܓܘܕܐ، ܘܨܠܝܒܐ ܥܡ ܥܠܬܐ ܓܘ ܡܫܬܘܬܦ̈ܢܐ.

ܣܦܠܝܛ-ܦܠܘܛ "ܕܠܐ ܨܒܝܢܐ". ܐܝܡܢ ܕܟܪܗܛܚ ܢܣܝܘܢܐ ܦܩܛܘܪܝܐܠ (factorial) ܒܛܟܣܐ ܕܟܡܙܥܪ ܡܢܝܢܐ ܕܫܘ̈ܚܠܦܐ ܕܥܠܬܐ ܕܥܣܩܐ ܝܠܗ̇ ܠܫܘܚܠܦܐ، ܘܒܬܪ ܗܕܐ ܟܦܫܩܚ ܐܝܟ ܕܛܟܣܐ ܓܕܫܢܝܐ ܗܘܐ، "ܟܡܝܠܚ ܠܡܥܒܕܐ ܛܥܝ̈ܘܬܐ ܕܙܢܐ I ܝܬܝܪ ܡܢ ܡܫܘܚܬܐ ܓܘ ܥܠ̈ܠܬܐ ܕܚܩܠܐ ܪܫܝܐ، ܘܛܥܝ̈ܘܬܐ ܕܙܢܐ II ܝܬܝܪ ܡܢ ܡܫܘܚܬܐ ܓܘ ܥܠ̈ܠܬܐ ܕܚܩܠܐ ܙܥܘܪܐ" (Montgomery, 2020, p. 538). ܘܢܡܘܣܐ ܕܨܒܥܐ: "ܐܢ ܟܠ ܡܥܒܕ̈ܢܘܝܬܐ ܘܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܪ̈ܫܝܡܐ ܝܢܐ، ܛܒ ܝܠܗ ܠܡܒܚܪܐ ܛܒ ܐܝܟܢܐ ܢܣܝܘܢܐ ܥܒܝܕܐ ܝܠܗ ܒܫܪܪܐ" (ܦܐܬܐ 540).

ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ: ܫܘܥܪ̈ܐ، ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ ܘܛܝ̈ܡܐ ܬܚܬ ܣܦܪ

ܦܐܬܐ ܟܡܫܘܥܪܐ ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ (variance components) ܒܐܘܪܚܐ ܕ-ANOVA: ܟܫܪܚ ܫܘܝ̈ܘܬܐ ܕܛܝ̈ܡܐ ܡܣܟ̈ܝܐ ܕܗܕ̈ܡܐ ܓܕܫܢ̈ܝܐ "ܒܬܪ ܕܫܩܠܚ ܡܨܥܝ̈ܬܐ ܕܡܪ̈ܒܥܐ ܕܐܝܬ ܓܘܝܗܝ ܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ" (Montgomery, 2020, p. 501). ܐܗܐ ܝܠܗ ܐܘܪܚܐ ܕ-moments، ܕ"ܠܐ ܣܢܝܩܐ ܝܠܗ̇ ܠܣܒܪܐ ܕܢܘܪܡܠܝܘܬܐ (normality)" (ܦܐܬܐ 103). ܛܥܝܘܬܐ ܬܩܝܢܬܐ (standard error) – √(Σ2c_i²MS_i²/f_i)، ܐܝܟ ܓܘ ܡܦܩܢܐ ܕ-JMP ܓܘ ܟܬܒܐ (ܠܘ̈ܚܐ 3.20 ܘ-14.7، ܦܐܬܐ 109، 527). ܩܐ ܫܘܚܠܦܐ ܕܛܥܝܘܬܐ ܐܝܬ ܬܚܘܡܐ ܕܬܘܟܠܢܐ ܚܬܝܬܐ ܡܢ ܟܝ ܡܪܒܥܐ (chi-square؛ ܫܘܝܘܬܐ 13.19، ܦܐܬܐ 512)؛ ܩܐ ܐܚܪ̈ܢܐ – ܩܪܝܒܘܬܐ ܕ-Satterthwaite (ܫܘܝ̈ܘܬܐ 13.20-13.23، ܦܐܬܐ 514-515)، ܕ"ܡܨܝܚ ܠܡܦܠܚܐ ܒܗ̇ ܒܣ ܐܝܡܢ ܕܫܘܥܪܐ ܥܠ ܡܢ ܣܦܪ ܝܠܗ" (ܦܐܬܐ 515)، ܘܟܦܠܚܐ ܛܒ ܐܝܡܢ ܕܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܪ̈ܒܐ ܝܢܐ ܘܡܢܝ̈ܢܐ ܡܫܘܬܦ̈ܐ ܥܠ ܡܢ ܣܦܪ (ܦܐܬܐ 516). ܓܘ ܡܬܠܐ 13.6: σ̂²_τβ = 31.36، r = 5.19، ܘܬܚܘܡܐ 12.39 ܗܠ 179.83 (ܓܘ ܟܬܒܐ 180.84، ܡܛܠ ܓܪܓܠܬܐ ܕ-quantile – ܡܫܟܚܢܘܬܐ E25).

ܫܘܥܪܐ ܬܚܬ ܣܦܪ. ܦܘܪܫܢܐ ܒܝܢܝ ܬܪܝܢ ܡܨܥܝ̈ܬܐ ܕܡܪ̈ܒܥܐ ܡܨܐ ܕܦܠܛ ܬܚܬ ܣܦܪ. Montgomery ܟܡܢܐ ܬܠܬܐ ܐܘܪ̈ܚܬܐ: ܠܡܩܒܠܐ ܠܗ ܐܝܟ ܐܬܐ ܕܡܢܬܐ ܣܦܪ ܝܠܗ̇ – ܐܠܐ "ܣܦܪ ܒܕܘܟܬܐ ܕܫܘܥܪܐ ܬܚܬ ܣܦܪ ܡܨܐ ܕܡܚܒܠ ܕܝܠܝ̈ܘܬܐ ܕܫܘܥܪ̈ܐ ܐܚܪ̈ܢܐ"؛ ܠܡܫܘܥܪܐ ܡܢ ܪܝܫ ܒܐܘܪܚܐ ܕܐܡܝܢ ܟܝܗܒܐ ܛܝ̈ܡܐ ܠܐ ܬܚܬ ܣܦܪ؛ ܝܢ ܠܡܚܙܝܐ ܠܗ ܐܝܟ ܐܬܐ ܕܡܘܕܝܠ ܠܐ ܝܠܗ ܬܪܝܨܐ (ܦܐܬܐ 104). ܐܘܪܚܐ ܥܒܕܢܝܬܐ ܐܚܪܬܐ: ܐܝܡܢ ܕ-p ܕܥܒܕܐ ܕܚܕ̈ܕܐ ܪܒܐ ܝܠܗ، ܠܡܦܠܛܐ ܠܗ ܘܠܡܦܫܩܐ ܡܢ ܪܝܫ – "ܐܘܪܚܐ ܦܫܝܩܬܐ ܝܬܝܪ ܕܣܓܝ ܓܗ̈ܝ ܟܦܠܚܐ ܩܪܝܒ ܐܝܟ ܐܘܪ̈ܚܬܐ ܚܟܝܡ̈ܬܐ ܝܬܝܪ" (ܦܐܬܐ 497). Maxwell et al. (2018) ܟܐܡܪܝ ܕܥܝܕܐ ܝܠܗ ܠܡܬܘܐ ܫܘܥܪܐ ܣܦܪ (ܦܐܬܐ 566)، ܘܕܣܓܝ ܓܗ̈ܝ ܐܗܐ ܐܬܐ ܝܠܗ̇ ܕܕܘܡܝܐ ܙܥܘܪܐ ܝܢ ܕܡܢܬܐ ܩܪܝܒܬܐ ܠܣܦܪ (ܦܐܬܐ 964). ܦܐܬܐ ܟܡܚܘܝܐ ܫܘܥܪܐ ܬܚܬ ܣܦܪ ܐܝܟ ܕܐܝܬܘܗܝ (ܒܬܘܦܚܢܝܐ)، ܘܟܚܫܒܐ ܡܢܬܐ ܡܢ ܫܘܚܠܦܐ ܟܠܢܝܐ ܐܝܟ ܕܣܦܪ ܗܘܐ.

ANOVA ܝܢ REML؟ REML ܝܠܗ̇ "ܢܘܣܚܐ ܡܩܒܠܐ ܕܡܬܚܙܝܢܘܬܐ ܥܠܝܬܐ (maximum likelihood) ܩܐ ܫܘܥܪܐ ܕܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ" (Montgomery, 2020, p. 108)، ܘܝܘܬܪܢܐ ܕܝܠܗ̇ ܝܠܗ ܕ"ܟܒܢܝܐ ܛܥܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ، ܘܡܛܠ ܗܕܐ ܦܫܝܩܐ ܝܠܗ ܠܡܫܟܚܐ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ" (ܦܐܬܐ 513). ܓܘ ܬܘܩܢܐ ܡܫܘܝܐ (balanced) "ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ ܫܠܡ̈ܐ ܝܢܐ" (ܦܐܬܐ 109، 499) – ܐܠܐ ܒܣ ܐܝܡܢ ܕ-REML ܠܐ ܝܠܗ ܟܪܝܐ ܠܛܝ̈ܡܐ ܠܐ ܬܚܬ ܣܦܪ (ܐܝܟ ܓܘ JMP). lmer ܓܘ R ܟܟܪܐ، ܘܡܛܠ ܗܕܐ ܐܝܡܢ ܕܚܕ ܡܢ ܫܘܥܪ̈ܐ ܕ-ANOVA ܬܚܬ ܣܦܪ ܝܠܗ، ܟܡܬܘܐ ܠܗ ܣܦܪ ܘܟܡܫܚܠܦ ܐܦ ܐܚܪ̈ܢܐ: ܓܘ ܡܬܠܐ 13.1 – ܡܢܘ̈ܬܐ (parts) 10.2513 ܒܕܘܟܬܐ ܕ-10.2798 (ܡܫܟܚܢܘܬܐ E9). ܓܘ ܡܬܠܐ ܕܦܐܬܐ – ܐܘܪܚܐ × ܡܕܪܫܬܐ: −0.089 ܓܘ ANOVA، 0 ܓܘ REML. ܓܘ ܬܘܩܢܐ ܠܐ ܡܫܘܝܐ ܟܡܦܠܚܚ ܒܡܘܕܝܠ ܡܚܠܛܐ (mixed model) – "ܐܘܪܚܐ ܓܒܝܬܐ ܐܝܡܢ ܕܪ̈ܒܘܬܐ ܕܕܘ̈ܡܝܐ ܠܐ ܫܘ̈ܝܐ ܝܢܐ" (Maxwell et al., 2018, p. 954)؛ ܦܐܬܐ ܟܦܫܩܐ ܒܣ ܬܘܩ̈ܢܐ ܡܫܘ̈ܝܐ، ܘܓܘ ܬܘܩܢܐ ܠܐ ܡܫܘܝܐ ܟܡܩܪܒܐ ܣܩܪܝܦܬ (script) R ܥܡ lmer.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܠܘܚܐ 1 – ܒܢܝܢܐ ܕܬܘܩܢܐ ܐܝܟ ܕܦܐܬܐ ܦܗܡܠܗ̇: ܙܢܐ ܕܟܠ ܥܠܬܐ، ܡܢܝܢܐ ܕܕܪ̈ܓܐ، ܘܓܘ ܡܘܕܝ ܡܩܢܢܬܐ ܝܠܗ̇. ܛܒ ܝܠܗ ܠܡܒܚܪܐ ܠܗ ܩܕܡ ܟܠ – ܣܓܝܐܐ ܡܢ ܛܥܝ̈ܘܬܐ ܓܘ ܦܘܫܩܐ ܐܝܟ ܗܕܐ ܛܥܝ̈ܘܬܐ ܓܘ ܒܢܝܢܐ ܝܢܐ.

ܠܘܚܐ 2 – ܦܘܫܩܐ ܕܫܘܚܠܦܐ: SS، df ܘ-MS ܐܝܟ ܥܝܕܐ؛ ܥܡܘܕܐ E(MS) – ܡܘܕܝ ܟܡܫܘܥܪܐ ܟܠ ܡܨܥܝܬܐ ܕܡܪ̈ܒܥܐ (σ²_X – ܡܢܬܐ ܕܫܘܚܠܦܐ ܕܗܕܡܐ ܓܕܫܢܝܐ، θ²_X – ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܕܗܕܡܐ ܩܝܡܐ، ܐܝܟ ܓܘ Maxwell et al., 2018، ܦܐܬܐ 557)؛ "ܡܢܝܢܐ ܬܚܬܝܐ ܕܒܘܚܪܢܐ" – ܩܘܒܠ ܡܘܕܝ ܒܚܝܪܐ ܝܠܗ ܟܠ ܗܕܡܐ. F ܥܡ ^a – F ܩܪܝܒܐ، ܥܡ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܠܐ ܓܡܝܪ̈ܐ. "ܠܝܬ ܒܘܚܪܢܐ" – ܠܝܬ ܚܘܠܛܐ ܦܐܝܐ؛ "ܡܬܦܠܚܐ ܐܝܟ ܫܪܟܐ" – ܗܕܡܐ ܥܠܝܐ ܐܝܡܢ ܕܠܝܬ ܬܢܝ̈ܬܐ ܓܘ ܩܠ̈ܝܬܐ.

ܠܘܚܐ 3 ܘܨܘܪܬܐ 1 – ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ: ܫܘܥܪܐ، ܛܥܝܘܬܐ ܬܩܝܢܬܐ، ܡܢܬܐ ܡܢ ܫܘܚܠܦܐ ܟܠܢܝܐ (ICC ܕܟܠ ܡܢܬܐ)، ܘܬܚܘܡܐ ܕܬܘܟܠܢܐ. ܥܡܘܕܐ "ܫܘܥܪܐ" ܟܡܚܘܐ ܡܬܢܝܬܐ – ܡܨܝܚ ܠܡܬܢܐ ܠܗ̇ ܒܐܝܕܐ ܡܢ ܠܘܚܐ 2. ICC (ܓܘ ܬܘܩܢܐ ܕܚܕ ܐܘܪܚܐ) – ܥܡ ܬܚܘܡܐ ܕܬܘܟܠܢܐ ܚܬܝܬܐ. ܡܨܥܝ̈ܬܐ – ܩܐ ܟܠ ܥܠܬܐ ܩܝܡܬܐ، ܥܡ ܛܥܝܘܬܐ ܬܩܝܢܬܐ ܡܢ ܡܢܝܢܐ ܬܚܬܝܐ ܕܒܘܚܪܢܐ ܕܝܠܗ̇ (Montgomery, 2020, p. 501).

ܓܘ ܡܬܠܐ (ܬܠܬܐ ܐܘܪ̈ܚܬܐ ܕܡܠܦܢܘܬܐ × ܐܪܒܥܐ ܡܕܪ̈ܫܝܬܐ، ܬܪܬܝܢ ܨܦ̈ܐ ܓܘ ܟܠ ܐܘܪܚܐ ܓܘ ܟܠ ܡܕܪܫܬܐ، ܬܠܬܐ ܬܠܡܝ̈ܕܐ ܓܘ ܨܦܐ): ܐܘܪܚܐ ܒܚܝܪܬܐ ܝܠܗ̇ ܩܘܒܠ ܐܘܪܚܐ × ܡܕܪܫܬܐ، F(2, 6) = 5.18, p = .049 – ܪܫܝܡܐ ܒܥܣܩܘܬܐ، ܥܡ ܒܣ ܫܬܐ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ؛ ܡܕܪܫܬܐ، ܓܘ ܡܘܕܝܠ ܟܪܝܐ، ܒܚܝܪܬܐ ܝܠܗ̇ ܩܘܒܠ ܨܦ̈ܐ، F(3, 12) = 2.14, p = .148؛ ܦܘܪ̈ܫܢܐ ܒܝܢܝ ܨܦ̈ܐ ܪ̈ܫܝܡܐ ܝܢܐ، F(12, 48) = 2.48, p = .013، ܘܨܦܐ ܟܡܦܫܩ ܩܪܝܒ 30% ܡܢ ܫܘܚܠܦܐ. ܥܒܕܐ ܕܚܕ̈ܕܐ ܐܘܪܚܐ × ܡܕܪܫܬܐ – ܫܘܥܪܐ ܬܚܬ ܣܦܪ ܙܥܘܪܐ.

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

ܫܘܝܘܬܐ (balance). ܢܡܘ̈ܣܐ ܕܛܝ̈ܡܐ ܡܣܟ̈ܝܐ ܥܠ ܬܘܩ̈ܢܐ ܡܫܘ̈ܝܐ (balanced) ܒܠܚܘܕ ܝܢܐ – ܗܘ ܡܢܝܢܐ ܕܕܪ̈ܓܐ ܓܘ ܟܠ ܐܒܐ ܘܗܘ ܡܢܝܢܐ ܕܚܙ̈ܝܬܐ ܓܘ ܟܠ ܩܠܝܬܐ (Montgomery, 2020, p. 505). ܓܘ ܥܠܬܐ ܡܩܢܢܬܐ ܓܕܫܢܝܬܐ ܥܡ ܪ̈ܒܘܬܐ ܠܐ ܫܘ̈ܝܐ "ܠܝܬ ܒܘܚܪܢܐ ܚܬܝܬܐ" (Maxwell et al., 2018, p. 586).

ܦܘܠܓܐ (distribution). ܡܥܒܕ̈ܢܘܝܬܐ ܓܕܫܢ̈ܝܬܐ ܘܛܥܝ̈ܘܬܐ ܐܝܬܠܗܝܢ ܦܘܠܓܐ ܢܘܪܡܠܝܐ (normal distribution)، ܠܐ ܬܠܝ̈ܬܐ، ܥܡ ܫܘܚܠܦܐ ܫܘܝܐ – ܐܦ ܓܘ ܟܠ ܕܪܓܐ ܕܥܠܬܐ ܩܝܡܬܐ (Maxwell et al., 2018, p. 585). F ܩܪܝܒܐ ܘܬܚܘ̈ܡܐ ܕ-Satterthwaite – ܩܪܝܒ̈ܘܬܐ ܒܠܚܘܕ ܝܢܐ.

ܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ (statistical power). "ܚܝܠܐ ܬܠܝܐ ܝܠܗ ܐܦ ܒܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܘܐܦ ܒܡܢܝܢܐ ܕܕܪ̈ܓܐ ܕܥܠܬܐ ܓܕܫܢܝܬܐ – ܘܡܢܝܢܐ ܕܕܪ̈ܓܐ ܣܓܝ ܓܗ̈ܝ ܚܫܒܢܐ ܐܝܩܪܢܐ ܝܬܝܪ ܝܠܗ" (Maxwell et al., 2018, p. 572). ܥܡ ܐܪܒܥܐ ܡܕܪ̈ܫܝܬܐ، ܠܒܘܚܪܢܐ ܕܐܘܪܚܐ ܐܝܬ ܒܣ ܫܬܐ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ، ܐܦܢ ܓܘ ܟܠ ܡܕܪܫܬܐ ܐܝܬ ܡܐܐ̈ ܬܠܡܝ̈ܕܐ.

ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ ܨܠܝ̈ܒܬܐ ܣܓܝ̈ܐܐ. ܐܝܡܢ ܕܐܝܬ ܝܬܝܪ ܡܢ ܚܕ ܥܠܬܐ ܓܕܫܢܝܬܐ ܠܒܪ ܡܢ ܐܢܝ ܕܓܘ ܗܕܡܐ، ܣܓܝ ܓܗ̈ܝ ܠܝܬ ܒܘܚܪܢܐ ܚܬܝܬܐ (Maxwell et al., 2018, p. 559)، ܘܒܘܚܪܢܐ ܩܪܝܒܐ ܣܢܝܩ ܝܠܗ ܠܕܪ̈ܓܐ ܣܓܝ̈ܐܐ.

ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. ܬܘܩ̈ܢܐ ܠܐ ܡܫܘ̈ܝܐ (ܩܐ ܐܢܝ – ܣܩܪܝܦܬ lmer)، ܬܘܩ̈ܢܐ ܡܩܢܢ̈ܐ ܕܕܪ̈ܓܐ (staggered; Montgomery, 2020, Section 14.1.4)، ܣܦܠܝܛ-ܦܠܘܛ ܡܢܬܢܝܐ ܕܠܐ ܬܢܝ̈ܬܐ ܕܡܬܦܫܩ ܒܨܘܪ̈ܬܐ half-normal (ܦܣܩܐ 14.5.1)، ܬܚܘ̈ܡܐ MLS (ܫܘܝ̈ܘܬܐ 13.24-13.27)، ܡܨܠܝ̈ܢܘܬܐ ܓܕܫܢ̈ܝܬܐ (random slopes) ܘܡܢܒ̈ܝܢܐ ܒܕܪܓܐ ܕܓܘܕܐ (Maxwell et al., 2018, Chapter 16)، ܘ-ICC ܩܐ ܫܪܝܪܘܬܐ (reliability) ܕܕܝ̈ܢܐ (ܨܚܚܐ 11). ܣܦܠܝܛ-ܣܦܠܝܛ-ܦܠܘܛ ܘܬܘܩܢܐ ܕܥܪ̈ܩܐ (strip-plot؛ ܦܣ̈ܩܐ 14.5.2-14.5.3) ܡܬܦܫܩ̈ܢܐ ܝܢܐ ܓܘ ܦܐܬܐ ܐܝܟ ܬܘܩܢܐ ܥܡ ܬܢܝܬܐ ܓܕܫܢܝܬܐ ܘܬܠܬܐ ܝܢ ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ، ܥܡ ܡܚܝܕܢܘܬܐ ܕܗܕ̈ܡܐ ܦܐܝ̈ܐ. ܓܘ ܦܐܬܐ "ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ" – ܢܣܝ̈ܘܢܐ ܦܩܛܘܪܝܐܠ ܥܡ ܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ، 2^k ܘܡܢܘ̈ܬܐ (fractions)؛ ܘܓܘ ܦܐܬܐ "ܐܘܪܚܐ ܕܐܦ̈ܐ ܕܦܘܢܝܐ" – ܥܠ̈ܠܬܐ ܟܡܝ̈ܬܐ ܘܢܘܩܙܐ ܛܒܐ ܝܬܝܪ.

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

ܓܘ Analyze › General Linear Model › Univariate ܟܡܥܠܚ ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ ܠ-Random Factor(s)؛ ܗܕܡܐ ܡܩܢܢܐ ܟܟܬܒܚ ܓܘ ܣܝܢܬܐܟܣ (syntax)، /DESIGN = A B(A). SPSS ܟܚܫܒ ܡܨܥܝ̈ܬܐ ܡܣܟ̈ܝܬܐ ܕܡܪ̈ܒܥܐ ܘܟܒܢܐ ܒܘܚܪ̈ܢܐ، ܥܡ ܩܪܝܒ̈ܐ – ܓܘ ܡܘܕܝܠ ܠܐ ܟܪܝܐ (Maxwell et al., 2018, pp. 605-606)، ܘܡܛܠ ܗܕܐ ܩܐ ܦܘܚܡܐ ܟܓܒܚ ܓܘ ܦܐܬܐ "ܠܐ ܟܪܝܐ". ܒܘܚܪ̈ܢܐ ܕܡܘܕܝܠ ܟܪܝܐ ܟܡܘܣܦܚ ܒ-/TEST = method VS method*school (ܦܐܬܐ 568)؛ ܣܝܢܬܐܟܣ ܕܦܐܬܐ ܟܟܬܒ ܐܢܝ ܒܓܢܣܗ. ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ – ܓܘ VARCOMP ܥܡ /METHOD=SSTYPE(3) (ܫܘܥܪ̈ܐ ܕ-ANOVA، ܠܐ ܟܪܝܐ). ܫܒ̈ܝܠܐ ܕܬܦܪܝܛܐ (menu paths) ܝܕܥܬܐ ܓܘܢܝܬܐ ܝܢܐ ܥܠ ܬܘܪܓܡܐ ܕܡܚܫܒܢܐ (software)، ܘܣܝܢܬܐܟܣ ܠܐ ܪܗܛܐ ܝܠܗ ܓܘ SPSS. ܓܘ PSPP، ܬܘܪܓܡܐ ܚܐܪܐ ܕܟܩܪܐ ܣܝܢܬܐܟܣ ܕ-SPSS، ܠܝܬ UNIANOVA ܘ-VARCOMP، ܘܦܘܩܕܢܐ GLM ܕܝܠܗ ܠܐ ܟܡܩܒܠ ܗܕ̈ܡܐ ܡܩܢܢ̈ܐ؛ ܓܘܡܪ̈ܐ ܕܡܪ̈ܒܥܐ ܕܬܘܩܢܐ ܨܠܝܒܐ ܒܚܪܠܢ ܓܘܗ، ܘܫܘ̈ܝܐ ܝܢܐ ܠܦܐܬܐ.

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

ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ. ܥܬܝܩܬܐ – lm ܩܐ ܓܘܡܪ̈ܐ ܕܡܪ̈ܒܥܐ، ܘܒܬܪ ܗܕܐ ܒܘܚܪ̈ܢܐ ܘܡܢܘ̈ܬܐ ܠܦܘܬ ܛܝ̈ܡܐ ܡܣܟ̈ܝܐ، ܒܐܝܕܐ؛ ܐܗܐ ܝܠܗ ܡܐ ܕܣܩܪܝܦܬ ܕܦܐܬܐ ܟܥܒܕ، ܥܡ ܡܢܝ̈ܢܐ ܬܚܬ̈ܝܐ ܕܦܐܬܐ ܓܒܝܠܗܘܢ. aov ܥܡ Error() ܟܝܗܒ ܒܘܚܪ̈ܢܐ ܕܡܘܕܝܠ ܠܐ ܟܪܝܐ (ܡܬܠܐ aov(y ~ A*B + Error(rep/A)) ܩܐ ܣܦܠܝܛ-ܦܠܘܛ). ܚܕܬܬܐ – lmer ܡܢ ܚܒܝܫܘܬܐ lme4: (1 | school) ܩܐ ܥܠܬܐ ܓܕܫܢܝܬܐ، (1 | method:school) ܩܐ ܥܒܕܐ ܕܚܕ̈ܕܐ ܕܝܠܗ̇، ܘ-(1 | school:class) ܩܐ ܥܠܬܐ ܡܩܢܢܬܐ؛ lmerTest ܟܡܘܣܦܐ ܒܘܚܪ̈ܢܐ F. lmer ܝܠܗ ܡܘܕܝܠ ܠܐ ܟܪܝܐ ܥܡ ܡܢܘ̈ܬܐ ܠܐ ܬܚܬ ܣܦܪ – ܚܙܝܡܘܢ "ANOVA ܝܢ REML". ܣܩܪܝܦܬ ܕܠܬܚܬ ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ ܐܢܓܠܝܙܝܐ، ܘܛܝ̈ܡܐ ܓܘ ܡܥܝܪ̈ܢܘܬܐ ܫܘ̈ܝܐ ܝܢܐ ܠܦܐܬܐ. ܢܡܘܣܐ ܕܣܐܝܛ ܩܐ ܣܘܪܝܬ ܓܘ R: ܦܐܝܠ UTF-8، ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ، ܘܫܡ̈ܗܐ ܕܡܫܬܚܠܦ̈ܢܐ ܒܐܢܓܠܝܙܝܐ.

# ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ ܘܡܩܢܢ̈ܬܐ – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)   # ܩܪܝܢܐ ܕܐܩܣܠ
library(lme4)     # ܡܘܕܝ̈ܠܐ ܡܚܠܛ̈ܐ (install.packages("lme4"))

# ܛܥܢܐ: ܒܢܝܢܐ ܐܪܝܟܐ – ܣܕܪܐ = ܬܠܡܝܕܐ؛ ܓܠܝܘܢܐ "Data"
d <- as.data.frame(read_excel("C:/Rdata/DATA_mix_example_eng.xlsx", sheet = "Data"))
for (v in c("method", "school", "class")) d[[v]] <- factor(d[[v]])

# ܓܘܡܪ̈ܐ ܕܡܪ̈ܒܥܐ: ܐܘܪܚܐ، ܡܕܪܫܬܐ، ܥܒܕܐ ܕܚܕ̈ܕܐ، ܘܨܦܐ ܡܩܢܢܐ ܓܘ ܐܘܪܚܐ × ܡܕܪܫܬܐ
options(contrasts = c("contr.sum", "contr.poly"))
a <- anova(lm(score ~ method + school + method:school + method:school:class, data = d))
MS <- setNames(a[["Mean Sq"]], c("method", "school", "method:school", "class", "Residuals")); round(MS, 3)
# ܦܐܪܐ: 511.097 212.759 98.690 99.222 40.056

# ܒܘܚܪ̈ܢܐ ܓܘ ܡܘܕܝܠ ܟܪܝܐ (ܓܘܒܝܐ ܡܩܕܡܐ ܓܘ ܦܐܬܐ): ܡܕܪܫܬܐ ܒܚܝܪܬܐ ܝܠܗ̇ ܩܘܒܠ ܨܦܐ، ܠܐ ܩܘܒܠ ܐܘܪܚܐ × ܡܕܪܫܬܐ
c(method = MS[["method"]] / MS[["method:school"]], school = MS[["school"]] / MS[["class"]],
  method.school = MS[["method:school"]] / MS[["class"]], class = MS[["class"]] / MS[["Residuals"]])
# ܦܐܪܐ: 5.179 (2, 6), 2.144 (3, 12), 0.995 (6, 12), 2.477 (12, 48)

# ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ ܒܐܘܪܚܐ ܕ-ANOVA، ܓܘ ܡܘܕܝܠ ܟܪܝܐ
c(school = (MS[["school"]] - MS[["class"]]) / 18, method.school = (MS[["method:school"]] - MS[["class"]]) / 6,
  class = (MS[["class"]] - MS[["Residuals"]]) / 3, residual = MS[["Residuals"]])
# ܦܐܪܐ: 6.308, -0.089, 19.722, 40.056

# ܩܐ ܦܘܚܡܐ: REML – ܡܘܕܝܠ ܠܐ ܟܪܝܐ، ܘܡܢܘ̈ܬܐ ܠܐ ܬܚܬ ܣܦܪ (ܐܘܪܚܐ × ܡܕܪܫܬܐ ܟܗܘܝܐ ܣܦܪ)
m <- lmer(score ~ method + (1 | school) + (1 | method:school) + (1 | method:school:class), data = d)
print(VarCorr(m), comp = "Variance")
# ܦܐܪܐ: ܡܕܪܫܬܐ 6.318، ܐܘܪܚܐ × ܡܕܪܫܬܐ 0.000، ܨܦܐ 19.663، ܫܪܟܐ 40.056

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܬܘܢܝܐ ܣܢܝܩ ܝܠܗ ܕܡܦܣܐ ܠܡܬܢܐ ܬܘܩܢܐ ܘܦܘܫܩܐ: ܐܝܢܝ ܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ (fixed) ܘܐܝܢܝ ܓܕܫܢ̈ܝܬܐ (random) ܘܩܐ ܡܘܕܝ، ܡܘܕܝ ܡܩܢܢܐ ܝܠܗ ܓܘ ܡܘܕܝ، ܟܡܐ ܕܪ̈ܓܐ ܘܟܡܐ ܚܙ̈ܝܬܐ (observations) ܓܘ ܟܠ ܩܠܝܬܐ (cell)؛ ܐܝܢܐ ܡܘܕܝܠ (ܟܪܝܐ ܝܢ ܠܐ ܟܪܝܐ) ܘܐܝܢܝ ܗܕ̈ܡܐ ܡܚܝ̈ܕܐ ܝܢܐ؛ ܩܐ ܟܠ ܒܘܚܪܢܐ – F ܥܡ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܕܡܢܝܢܐ ܬܚܬܝܐ ܫܪܝܪܐ (ܠܐ ܕܫܪܟܐ، ܐܝܡܢ ܕܡܢܝܢܐ ܬܚܬܝܐ ܐܚܪܢܐ ܝܠܗ) ܘ-p؛ ܘܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ (variance components) ܝܢ ICC، ܥܡ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ (confidence intervals). APA (2020، ܦܣܩܐ 6.43) ܟܒܥܐ "ܝܕܥܬܐ ܣܦܩܬܐ ܕܩܪܘܝܐ ܡܨܐ ܕܡܫܪܪ ܦܘܫܩ̈ܐ". ܢܡܘ̈ܣܐ ܕܨܘܪܬܐ (ܦܣܩܐ 6.36): p ܘ-ICC ܕܠܐ ܣܦܪ ܩܕܡ ܢܘܩܙܐ؛ F ܘܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ – ܥܡ. ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ (results paragraph) ܓܘ ܡܦܩܢܐ ܒܢܝܐ ܝܠܗ ܗܟܐ، ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܠܡܒܚܪܐ ܟܠ ܡܕܡ ܩܘܒܠ ܫܪܟܐ. ܐܝܡܢ ܕܐܝܬ ܥܠܬܐ ܓܕܫܢܝܬܐ (random factor)، ܐܗܐ ܟܡܢܦܚ F ܕܥܠ̈ܠܬܐ ܩܝ̈ܡܬܐ ܕܨܠܝ̈ܒܬܐ ܝܢܐ ܥܡܗ̇ ܝܢ ܕܠܥܠ ܡܢܗ̇ ܝܢܐ (Maxwell et al., 2018, pp. 558, 590).

ܠܡܦܫܩܐ ܬܘܩܢܐ ܡܩܢܢܐ (nested) ܐܝܟ ܕܨܠܝܒܐ ܗܘܐ. ܟܦܠܛ "ܥܒܕܐ ܕܚܕ̈ܕܐ" (interaction) ܕܠܝܬ ܠܗ ܦܘܫܩܐ (Montgomery, 2020, pp. 522-524). ܟܒܥܝܚ: ܐܝܢ ܡܢܬܐ ܕܥܒܕܐ 1 ܓܘ ܡܛܝܒܢܐ ܐ ܗܝ ܡܢܬܐ ܝܠܗ̇ ܐܝܟ ܡܢܬܐ ܕܥܒܕܐ 1 ܓܘ ܡܛܝܒܢܐ ܒ؟

ܠܫܒܩܐ ܚܩ̈ܠܐ ܪ̈ܫܝܐ. ܣܦܠܝܛ-ܦܠܘܛ (split-plot) ܕܡܬܦܫܩ ܐܝܟ ܬܘܩܢܐ ܓܕܫܢܝܐ ܓܡܝܪܐ – ܛܥܝ̈ܘܬܐ ܕܙܢܐ I ܝܬܝܪ ܡܢ ܡܫܘܚܬܐ ܓܘ ܥܠܬܐ ܕܚܩܠܐ ܪܫܝܐ ܘܕܙܢܐ II ܓܘ ܚܩ̈ܠܐ ܙܥܘ̈ܪܐ (ܦܐܬܐ 538).

ܕܪ̈ܓܐ ܩܠܝ̈ܠܐ ܩܐ ܥܠܬܐ ܓܕܫܢܝܬܐ. ܬܠܬܐ ܝܢ ܐܪܒܥܐ ܡܕܪ̈ܫܝܬܐ – ܘܒܘܚܪܢܐ ܕܐܘܪܚܐ ܩܪܝܒ ܕܠܐ ܚܝܠܐ ܣܛܐܛܝܣܛܝܩܝܐ (statistical power) ܝܠܗ، ܠܐ ܚܝܪܐ ܟܡܐ ܬܠܡܝ̈ܕܐ (Maxwell et al., 2018, p. 572).

ܠܡܦܚܡܐ ܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ ܒܝܢܝ ܬܘܪ̈ܓܡܐ ܕܡܚܫܒܢܐ ܕܠܐ ܒܘܚܪܢܐ ܕܡܘܕܝܠ. ܟܪܝܐ ܝܢ ܠܐ ܟܪܝܐ، ANOVA ܝܢ REML، ܥܡ ܝܢ ܕܠܐ ܡܬܘܝܬܐ ܕܣܦܪ – ܟܠ ܚܕ ܟܝܗܒ ܡܢܝܢܐ ܐܚܪܢܐ ܩܐ ܗܝ ܡܢܬܐ. ܓܘ ܡܬܠܐ ܕܚܝܠܐ ܕܟܪܛܝܣܐ، JMP ܟܡܘܕܥ 2.4757 ܩܐ ܬܢܝ̈ܬܐ (ܠܘܚܐ 14.19، ܦܐܬܐ 538 – ܠܐ ܟܪܝܐ، ܡܚܝܕܐ)، ܘܡܘܕܝܠ ܟܪܝܐ ܟܝܗܒ 2.90.

ܠܡܚܝܕܐ ܥܒܕܐ ܕܚܕ̈ܕܐ ܒܣ ܡܛܠ ܕܠܐ ܝܠܗ ܪܫܝܡܐ. ܥܡ ܫܬܐ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܝܢ ܝܬܝܪ ܩܐ ܛܥܝܘܬܐ – ܠܐ ܟܡܚܝܕܚ؛ ܘܐܢ ܠܐ – ܒܣ ܐܢ p ܒܢܘܬ ܡܢ 0.25 ܝܠܗ (Montgomery, 2020, p. 509).

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.

Satterthwaite (1946)، Murray ܘ-Blitstein (2003)، Hedges ܘ-Hedberg (2007) ܘ-Wampold ܘ-Serlin (2000) – ܐܝܟ ܕܡܝܬܝ̈ܐ ܝܢܐ ܓܘ Montgomery (2020) ܘܓܘ Maxwell et al. (2018). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ (site rule)، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ (reference list) ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܓܘ ܫܪܫܐ ܕܝܗܝ؛ ܡܬ̈ܝܬܐ ܠܦܘܬ ܦܐܬܐ ܛܒܝܥܬܐ. ܩܪܝ̈ܐ ܝܢܐ ܦܣܩܐ 3.9 ܘܨܚ̈ܚܐ 13-14 ܕ-Montgomery، ܘܨܚܚܐ 10، ܦܣ̈ܩܐ ܡܢ ܨܚ̈ܚܐ 11-12 ܘܨܚܚܐ 16 ܕ-Maxwell et al. ܚܘܫܒܢܐ ܓܘ ܦܐܬܐ ܡܫܪܪܐ ܝܠܗ ܩܘܒܠ R 4.3.3 (lm، aov ܥܡ Error، lme4، lmerTest)، ܘܩܘܒܠ ܟܠ ܡܢܝ̈ܢܐ ܛܒܝ̈ܥܐ ܓܘ ܡܬܠ̈ܐ 3.10، 13.1-13.6، 14.1-14.2 ܘܓܘ ܡܬܠܐ ܕܚܝܠܐ ܕܟܪܛܝܣܐ (ܦܣܩܐ 14.4) ܕ-Montgomery ܘܓܘ ܠܘ̈ܚܐ 10.5، 10.9 ܘ-12.19 ܕ-Maxwell et al.