ܐܘܪܚܐ ܕܐܦ̈ܐ ܕܦܘܢܝܐ

ܡܘܕܝ ܝܠܗ̇ ܐܦܐ ܕܦܘܢܝܐ، ܘܐܝܡܢ ܡܦܠܚܝܚ ܒܗ̇

ܐܘܪܚܐ ܕܐܦ̈ܐ ܕܦܘܢܝܐ (Response Surface Methodology, RSM) ܝܠܗ̇ "ܟܢܘܫܝܐ ܕܛܟܢܝܩ̈ܐ ܡܬܡܛܝܩ̈ܝܐ ܘܣܛܐܛܝܣܛܝܩ̈ܝܐ" ܩܐ ܒܢܝܢܐ ܕܡܘܕܝܠ ܕܦܘܢܝܐ ܕܬܠܝܐ ܝܠܗ ܒܟܡܐ ܡܫܬܚܠܦ̈ܢܐ، ܐܝܡܢ ܕ"ܢܝܫܐ ܝܠܗ ܛܘܒܐ ܕܦܘܢܝܐ" (Montgomery, 2020, p. 408). ܦܘܢܝܐ ܡܨܥܝܐ ܦܘܢܟܫܢ ܝܠܗ ܕܥܠ̈ܠܬܐ، η = f(x₁, x₂)، ܘܨܘܪܬܐ ܕܝܠܗ ܝܠܗ̇ "ܐܦܐ ܕܦܘܢܝܐ"؛ ܟܡܚܘܚ ܠܗ̇ ܒܨܘܪܬܐ ܕܣܪ̈ܛܐ ܕܦܘܢܝܐ ܫܘܝܐ (contour plot) – "ܣܪ̈ܛܐ ܕܦܘܢܝܐ ܩܝܡܐ" ܓܘ ܫܛܝܚܘܬܐ ܕܬܪܬܝܢ ܥܠ̈ܠܬܐ (ܦܐ̈ܬܐ 408-409).

ܦܘܪܫܢܐ ܡܢ ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܥܡܝܐ: ܬܡܢ ܥܠ̈ܠܬܐ ܕܓܢ̈ܣܐ ܝܢܐ، ܘܟܫܐܠܚ ܐܢ ܡܨܥܝ̈ܬܐ ܦܪ̈ܝܫܐ ܝܢܐ. ܗܪܟܐ ܥܠ̈ܠܬܐ ܟܡܝ̈ܝܐ ܘܡܬܡܫܚ̈ܢܐ ܝܢܐ – ܐܘܪܟܐ ܕܫܥܘܬܐ، ܡܢܝܢܐ ܕܬܡܪ̈ܢܐ، ܡܫܘܚܬܐ ܕܣܡܐ، ܚܡܝܡܘܬܐ – ܘܟܫܐܠܚ ܐܝܟܐ ܓܘ ܬܚܘܡܐ ܕܝܠܗܘܢ ܦܘܢܝܐ ܛܒܐ ܝܠܗ ܡܢ ܟܠ، ܘܒܐܝܢܐ ܐܘܪܚܐ ܛܒ ܝܠܗ ܕܐܙܠܚ. f ܫܪܝܪܬܐ ܠܐ ܝܕܥܚ، ܘܡܛܠ ܗܕܐ ܟܡܩܪܒܚ ܠܗ̇ ܒܦܘܠܝܢܘܡܝܐܠ: ܕܛܟܣܐ ܩܕܡܝܐ (ܫܘܝܘܬܐ 11.1) ܝܢ ܕܛܟܣܐ ܬܪܝܢܐ (11.2؛ ܦܐܬܐ 409). ܐܗܐ ܩܪܝܒܘܬܐ ܕܘܟܬܢܝܬܐ ܝܠܗ̇ ܒܣ – "ܠܐ ܝܠܗ ܡܩܒܠܐ ܕܦܘܠܝܢܘܡܝܐܠ ܗܘܐ ܩܪܝܒܘܬܐ ܡܩܒܠܬܐ ܥܠ ܟܠܗ ܦܘܬܝܐ"، ܐܠܐ "ܓܘ ܐܬܪܐ ܙܥܘܪܐ ܒܩܪܝܒܘܬܐ ܒܥܡܝܘܬܐ ܛܒ ܟܦܠܚ" (ܦܐܬܐ 409).

ܡܬ̈ܠܐ ܕ-Montgomery ܟܐܬܝ ܡܢ ܗܢܕܣܐ ܟܝܡܝܐܝܬܐ – ܙܒܢܐ ܕܥܒܕܐ ܟܝܡܝܐ ܘܚܡܝܡܘܬܐ ܩܘܒܠ ܡܦܩܬܐ (yield). ܗܘ ܚܘܫܒܐ ܦܐܝܐ ܝܠܗ ܠܟܠ ܒܘܨܝܐ ܕܓܘܗ ܒܨܘܝܬܐ ܟܦܣܩܐ ܕܪ̈ܓܐ ܕܡܫܬܚܠܦ̈ܢܐ ܡܬܡܫܚ̈ܢܐ: ܐܘܪܟܐ ܘܣܓܝܐܘܬܐ ܕܡܥܠܬܐ (intervention)، ܡܫܘܚܬܐ ܕܣܡܐ، ܡܢܝܢܐ ܕܦܣܘ̈ܩܐ، ܙܒܢܐ ܕܓܠܝܘܬܐ (exposure). ܦܐܝܠ ܕܡܬܠܐ ܓܘ ܦܐܬܐ – ܝܗܝ̈ܒܬܐ ܓܒܝ̈ܠܬܐ ܥܠ ܫܥܘܬܐ ܕܬܡܪܢܐ – ܒܢܝܐ ܝܠܗ ܗܟܢܐ.

ܥܒܕܐ ܒܕܪ̈ܓܐ

"RSM ܐܘܪܚܐ ܡܬܢܝܬܐ ܝܠܗ̇" (Montgomery, 2020, p. 410). ܐܝܡܢ ܕܐܝܬܚ ܪܚܘܩ ܡܢ ܢܘܩܙܐ ܛܒܬܐ، ܠܐܦܐ ܩܪܝܒ ܠܝܬ ܟܦܝܦܘܬܐ (curvature) ܘܡܘܕܝܠ ܕܛܟܣܐ ܩܕܡܝܐ ܣܦܩ ܝܠܗ؛ ܥܡܗ ܟܣܠܩܚ ܠܦܘܬ ܢܘܩܙܐ ܛܒܬܐ. ܩܪܝܒ ܠܢܘܩܙܐ ܛܒܬܐ ܟܦܝܦܘܬܐ ܟܡܬܚܙܝܐ، ܘܟܡܦܐܚ ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ ܕܡܫܟܚܚ ܠܗ̇ – ܐܝܟ "ܣܠܩܐ ܥܠ ܬܠܐ" (ܦܐܬܐ 410). ܡܚܫܒܢܐ ܒܢܝܐ ܝܠܗ ܩܐ ܬܪܘܝܗܝ ܕܪ̈ܓܐ: ܡܘܕܝܠ ܣܪܛܢܝܐ ܥܡ ܐܘܪܚܐ ܕܣܠܩܐ ܚܪܝܦܐ ܡܢ ܟܠ، ܘܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ ܥܡ ܢܘܩܙܐ ܩܝܡܬܐ.

ܓܘ ܕܪܓܐ ܩܕܡܝܐ ܣܦܩ ܝܠܗ ܢܣܝܘܢܐ 2^k (ܝܢ ܡܢܬܐ ܕܝܠܗ) ܥܡ ܟܡܐ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ. ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܟܡܨܝܝ ܬܠܬܐ ܡܕ̈ܡܐ: ܫܘܥܪܐ ܕܛܥܝܘܬܐ، ܒܘܚܪܢܐ ܕܥܒ̈ܕܐ ܕܚܕ̈ܕܐ، ܘܒܘܚܪܢܐ ܕܟܦܝܦܘܬܐ (ܦܐܬܐ 412). ܒܘܚܪܢܐ ܕܟܦܝܦܘܬܐ ܝܠܗ ܒܚܕ ܕܪܓܐ ܕܚܐܪܘܬܐ: ܐܢ ܐܦܐ ܫܛܝܚܬܐ ܝܠܗ̇، ܡܨܥܝܬܐ ܕܢܘܩ̈ܙܐ ܦܩܛܘܪܝܐܠ ȳ_F ܫܘܝܐ ܝܠܗ̇ ܠܡܨܥܝܬܐ ܕܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ȳ_C؛ ܦܘܪܫܢܐ ܟܫܥܪ Σβ_jj، ܘ-SS = n_Fn_C(ȳ_F − ȳ_C)²/(n_F + n_C) ܒܚܝܪܐ ܝܠܗ ܩܘܒܠ ܛܥܝܘܬܐ ܕܟܝܬܐ (ܦܐܬܐ 413). ܦܐܬܐ ܟܚܫܒܐ ܠܗ ܐܝܡܢ ܕܝܗܝ̈ܒܬܐ ܚܬܝܬܐܝܬ 2^k ܝܢܐ ܥܡ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ، ܘܟܦܪܩܐ ܫܪ̈ܟܐ ܕܡܘܕܝܠ ܣܪܛܢܝܐ ܠܥܒ̈ܕܐ ܕܚܕ̈ܕܐ، ܠܡܪ̈ܒܥܐ ܕܟܝ̈ܐ ܘܠܛܥܝܘܬܐ ܕܟܝܬܐ – ܐܝܟ ܠܘ̈ܚܐ 11.2 ܘ-11.5 (ܦܐ̈ܬܐ 413، 415).

ܐܘܪܚܐ ܕܣܠܩܐ ܚܪܝܦܐ ܡܢ ܟܠ

"ܐܘܪܚܐ ܕܣܠܩܐ ܚܪܝܦܐ ܡܢ ܟܠ ܝܠܗ̇ ܐܘܪܚܐ ܕܙܘܥܐ ܡܬܢܝܐ ܒܐܘܪܚܐ ܕܣܠܩܐ ܪܒܐ ܡܢ ܟܠ ܓܘ ܦܘܢܝܐ"؛ ܩܐ ܙܥܘܪܐ – ܢܚܬܐ ܚܪܝܦܐ ܡܢ ܟܠ (Montgomery, 2020, p. 411). ܐܘܪܚܐ ܙܩܝܦܐ ܝܠܗ̇ ܥܠ ܣܪ̈ܛܐ ܕܦܘܢܝܐ ܫܘܝܐ ܕܡܘܕܝܠ ܣܪܛܢܝܐ، ܘ"ܓܫ̈ܩܐ ܒܐܘܪܚܐ ܡܫܘܚ̈ܝܐ ܝܢܐ ܠܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܕܪܝܓܪܫܢ" (ܦܐܬܐ 411). ܐܠܓܘܪܝܬܡ (ܦܐܬܐ 415): ܟܓܒܚ ܪܒܘܬܐ ܕܓܫܩܐ Δx_j ܓܘ ܚܕ ܥܠܬܐ – "ܒܥܡܝܘܬܐ ܥܠܬܐ ܕܟܝܕܥܚ ܥܠܗ̇ ܝܬܝܪ ܡܢ ܟܠ، ܝܢ ܐܗܝ ܕܐܝܬ ܠܗ̇ ܡܢܝܢܐ ܡܫܘܬܦܐ ܪܒܐ ܡܢ ܟܠ ܒܛܝܡܐ ܫܪܝܪܐ (absolute value)"؛ ܓܘ ܟܠ ܥܠܬܐ ܐܚܪܬܐ Δx_i = b_i/(b_j/Δx_j)؛ ܘܟܡܫܚܠܦܚ ܠܚܕ̈ܝܘܬܐ ܟܝܢܝ̈ܬܐ. ܓܘ ܡܬܠܐ 11.1 ܓܫܩܐ ܝܠܗ 5 ܩܛܝ̈ܢܬܐ ܓܘ ܙܒܢܐ ܘ-2 ܕܪ̈ܓܐ ܓܘ ܚܡܝܡܘܬܐ (ܦܐܬܐ 416).

"ܪܒܘܬܐ ܕܓܫܩܐ ܒܥܒܕܐ ܟܦܣܩ ܡܢܣܝܢܐ، ܠܦܘܬ ܝܕܥܬܐ ܥܠ ܥܒܕܐ" (ܦܐܬܐ 411). ܟܪܗܛܚ ܢܣܝ̈ܘܢܐ ܒܐܘܪܚܐ "ܥܕܡܐ ܕܠܐ ܟܡܬܚܙܐ ܬܘܒ ܣܠܩܐ ܓܘ ܦܘܢܝܐ"، ܘܬܡܢ ܟܡܦܐܚ ܡܘܕܝܠ ܣܪܛܢܝܐ ܚܕܬܐ؛ ܡܛܝܬܐ ܠܩܘܪܒܐ ܕܢܘܩܙܐ ܛܒܬܐ "ܒܥܡܝܘܬܐ ܪܫܝܡܬܐ ܝܠܗ̇ ܒܠܐ ܦܐܝܘܬܐ ܕܡܘܕܝܠ ܕܛܟܣܐ ܩܕܡܝܐ" (ܦܐܬܐ 411). ܓܘ ܡܬܠܐ 11.1 ܦܘܢܝܐ ܣܠܩܠܗ ܥܕܡܐ ܓܫܩܐ ܥܣܝܪܝܐ ܘܢܚܬܠܗ ܒܬܪܗ (ܦܐ̈ܬܐ 413-414)، ܘܡܘܕܝܠ ܚܕܬܐ ܚܕܪ ܓܫܩܐ ܥܣܝܪܝܐ ܡܚܘܝܠܗ ܟܦܝܦܘܬܐ ܪܫܝܡܬܐ – "ܡܘܕܝܠ ܕܛܟܣܐ ܩܕܡܝܐ ܠܐ ܝܠܗ ܩܪܝܒܘܬܐ ܦܐܝܬܐ" (ܦܐܬܐ 414). ܓܘ ܦܐܬܐ – ܟܓܒܚ ܥܠܬܐ ܕܫܬܐܣܐ، ܪܒܘܬܐ ܕܓܫܩܐ ܘܡܢܝܢܐ ܕܓܫ̈ܩܐ، ܘܠܘܚܐ ܟܝܗܒ ܐܘܪܚܐ ܒܚܕ̈ܝܘܬܐ ܪܡܝܙ̈ܬܐ ܘܟܝܢܝ̈ܬܐ، ܥܡ ܡܢܒܝܢܘܬܐ ܒܣ ܓܘ ܦܬܝܘܬܐ ܕܝܗܝ̈ܒܬܐ.

ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ: ܢܘܩܙܐ ܩܝܡܬܐ ܘܦܘܫܩܐ ܩܢܘܢܝܐ

ܒܟܬܝܒܬܐ ܕܡܐܛܪܝܟ̈ܣܐ، ŷ = b₀ + x'b + x'Bx (ܫܘܝܘܬܐ 11.5): b – ܘܩܛܘܪ ܕܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܣܪܛܢ̈ܝܐ، ܘ-B – ܡܐܛܪܝܟܣ ܫܘܝܐ ܠܬܪܝܢ ܓܢ̈ܒܐ (symmetric) ܥܡ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܡܪܒܥ̈ܝܐ ܥܠ ܣܪܛܐ ܡܨܠܝܐ (diagonal) ܘܦܠܓܐ ܕܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܕܥܒܕܐ ܕܚܕ̈ܕܐ ܠܒܪ ܡܢܗ (Montgomery, 2020, pp. 417-418). ܢܘܩܙܐ ܩܝܡܬܐ – ܕܓܘܗ̇ ܟܠ ܡܫܬܩ̈ܠܢܘܝܬܐ (derivatives) ܨܦܪܐ ܝܢܐ – ܝܠܗ̇ x_s = −½B⁻¹b، ܘܦܘܢܝܐ ܓܘܗ̇ ŷ_s = b₀ + ½x_s'b (ܫܘܝ̈ܘܬܐ 11.7-11.8، ܦܐܬܐ 418). ܗܝ "ܡܨܝܐ ܕܗܘܝܐ ܢܘܩܙܐ ܕܛܝܡܐ ܥܠܝܐ (maximum)، ܬܚܬܝܐ (minimum) ܝܢ ܣܪܓܐ (saddle)" (ܦܐܬܐ 416).

ܕܝܕܥܚ ܡܘܕܝ ܝܠܗ̇، ܟܥܒܪܚ ܠܐܣܟܝܡܐ ܩܢܘܢܝܐ: ܟܡܫܢܚ ܫܘܪܝܐ ܠܢܘܩܙܐ ܩܝܡܬܐ ܘܟܡܚܕܪܚ ܣܪ̈ܛܐ ܠܣܪ̈ܛܐ ܪ̈ܫܝܐ ܕܐܦܐ، ܘܗܟܢܐ ŷ = ŷ_s + λ₁w₁² + … + λ_kw_k² (ܫܘܝܘܬܐ 11.9)، ܐܝܡܢ ܕ-λ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ (eigenvalues) ܝܢܐ ܕ-B (ܦܐܬܐ 418). ܟܠܗܘܢ ܒܨܘܪܐ ܡܢ ܨܦܪܐ – ܛܝܡܐ ܥܠܝܐ؛ ܟܠܗܘܢ ܥܠ ܡܢ ܨܦܪܐ – ܛܝܡܐ ܬܚܬܝܐ؛ ܒܪ̈ܘܫܡܐ ܚܠܝ̈ܛܐ – ܣܪܓܐ؛ ܘ"ܐܦܐ ܚܪܝܦܬܐ ܝܠܗ̇ ܝܬܝܪ ܡܢ ܟܠ ܒܐܘܪܚܐ ܕ-w_i ܕܓܘܗ̇ |λ_i| ܪܒܐ ܝܠܗ ܡܢ ܟܠ" (ܦܐܬܐ 418). ܣܪ̈ܛܐ ܚܕ̈ܬܐ ܝܢܐ w = M'(x − x_s)، ܐܝܡܢ ܕ-M – ܘܩܛܘܪ̈ܐ ܕܝܠܢ̈ܝܐ (eigenvectors) ܢܘܪܡܠ̈ܝܐ (ܫܘܝܘܬܐ 11.10، ܦܐܬܐ 423). ܓܘ ܡܬܠܐ 11.2 ܕ-Montgomery: x_s = (0.389, 0.306)، ܗܢܘ 87 ܩܛܝ̈ܢܬܐ ܘ-176.5 ܕܪ̈ܓܐ، ŷ_s = 80.21، ܘܬܪܝܢ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܒܨܘܪܐ ܡܢ ܨܦܪܐ ܝܢܐ (−0.963، −1.414) – ܛܝܡܐ ܥܠܝܐ (ܦܐܬܐ 420)؛ ܦܐܬܐ ܟܥܒܕܐ ܬܘܒ ܟܠ ܗܢܝ ܡܢܝ̈ܢܐ.

ܚܨ̈ܐ (ridges). ܐܝܡܢ ܕܚܕ ܡܢ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܩܪܝܒܐ ܝܠܗ ܠܨܦܪܐ، ܦܘܢܝܐ "ܩܪܝܒ ܠܐ ܝܠܗ ܪܓܝܫܐ" ܠܙܘܥܐ ܒܐܘܪܚܐ ܕܣܪܛܐ ܕܝܠܗ (ܦܐܬܐ 424). ܐܢ ܢܘܩܙܐ ܩܝܡܬܐ ܓܘ ܐܬܪܐ ܕܢܣܝܘܢܐ ܝܠܗ̇ – ܐܗܐ ܝܠܗ "ܚܨܐ ܩܝܡܐ" (stationary ridge): ܣܪܛܐ ܕܢܘܩ̈ܙܐ ܩܪܝܒ ܫܘ̈ܝܐ، ܘܫܘܠܛܢܐ ܕܓܒܚ ܒܝܢܬܗܘܢ ܠܦܘܬ ܛܝܡܐ ܝܢ ܦܫܝܩܘܬܐ. ܐܢ ܪܚܝܩܬܐ ܝܠܗ̇ ܠܒܪ ܡܢ ܐܬܪܐ – ܚܨܐ ܕܣܠܩ (ܝܢ ܕܢܚܬ): "ܠܐ ܡܨܝܚ ܕܡܦܩܚ ܫܘܠܡܐ ܥܠ ܐܦܐ ܫܪܝܪܬܐ ܝܢ ܥܠ ܢܘܩܙܐ ܩܝܡܬܐ"، ܐܠܐ ܛܒ ܝܠܗ ܕܒܨܚ ܝܬܝܪ ܒܐܘܪܚܐ ܕܣܪܛܐ ܫܛܝܚܐ (ܦܐ̈ܬܐ 424-425). ܦܐܬܐ ܟܪܫܡܐ ܛܝܡܐ ܕܝܠܢܝܐ ܙܥܘܪܐ ܡܢ ܚܕ ܡܢ ܥܣܪܐ ܕܪܒܐ ܒܛܝܡܐ ܫܪܝܪܐ – ܬܚܘܡܐ ܕ-canonical ܓܘ ܨܪܪܐ rsm (ܥܝܕܐ ܕܣܐܝܛ) – ܘܟܡܘܕܥ ܐܝܡܢ ܕܢܘܩܙܐ ܠܒܪ ܡܢ ܦܬܝܘܬܐ ܕܝܗܝ̈ܒܬܐ ܝܠܗ̇.

ܟܡܐ ܦܘܢ̈ܝܐ. ܐܝܡܢ ܕܐܝܬ ܟܡܐ ܦܘܢ̈ܝܐ، ܟܡܦܐܚ ܡܘܕܝܠ ܩܐ ܟܠ ܚܕ ܘܟܒܥܚ ܚܘܠܛܐ ܕܡܣܦܩ ܟܠܗܘܢ – ܒܚܠܛܘܬܐ ܕܨܘܪ̈ܬܐ ܕܣܪ̈ܛܐ ܕܦܘܢܝܐ ܫܘܝܐ، ܒܛܘܒܐ ܥܡ ܐܘܠܨ̈ܢܐ (constraints)، ܝܢ ܒܦܘܢܟܫܢ ܕܨܒܝܢܘܬܐ (desirability) ܕܟܚܝܕܐ ܟܠ ܦܘܢ̈ܝܐ ܠܚܕ ܡܚܘܝܢܐ (ܦܐ̈ܬܐ 425-429). ܦܐܬܐ ܟܦܫܩܐ ܚܕ ܦܘܢܝܐ ܓܘ ܟܠ ܪܗܛܐ؛ ܩܐ ܟܡܐ ܦܘܢ̈ܝܐ – ܟܪܗܛܚ ܩܐ ܟܠ ܚܕ ܘܟܦܚܡܚ ܨܘܪ̈ܬܐ.

ܐܝܢܐ ܬܘܩܢܐ؟

ܩܐ ܡܘܕܝܠ ܕܛܟܣܐ ܩܕܡܝܐ – 2^k ܝܢ ܡܢܬܐ ܕܓܘܗ̇ ܡܥܒܕ̈ܢܘܝܬܐ ܪ̈ܫܝܬܐ ܠܐ ܝܢܐ aliases ܚܕ ܕܐܚܪܢܐ؛ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܠܐ ܟܡܫܚܠܦܝ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܣܪܛܢ̈ܝܐ ܘܐܘܪܬܘܓܘܢܠܝܘܬܐ، ܘܟܡܘܣܦܝ ܫܘܥܪܐ ܩܐ ܛܥܝܘܬܐ (Montgomery, 2020, p. 430). ܓܘ ܦܐܬܐ "ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ" – 2^k ܘܡܢ̈ܘܬܐ.

ܬܘܩܢܐ ܡܨܥܝܐ ܡܪܟܒܐ (CCD) – "ܫܪܒܬܐ ܕܡܦܪܣܬܐ ܝܬܝܪ ܡܢ ܟܠ" ܩܐ ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ (ܦܐܬܐ 430): 2^k (ܝܢ ܡܢܬܐ ܒ-resolution V) ܥܡ n_F ܪ̈ܗܛܐ، 2k ܢܘܩ̈ܙܐ ܕܣܪ̈ܛܐ (axial points) ܓܘ ±α ܥܠ ܟܠ ܣܪܛܐ، ܘ-n_C ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ (ܦܐܬܐ 431). ܗܘ ܦܐܝܐ ܝܠܗ ܩܐ ܒܢܝܢܐ ܡܬܢܝܐ: ܟܪܗܛܚ ܩܕܡܝܐ 2^k ܥܡ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ، ܘܐܢ ܐܝܬ ܠܐ ܦܐܝܘܬܐ – ܟܡܘܣܦܚ ܢܘܩ̈ܙܐ ܕܣܪ̈ܛܐ (ܦܐܬܐ 431). ܬܪܝܢ ܦܘܣ̈ܩܢܐ: α ܘܡܢܝܢܐ ܕܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ.

• ܡܬܚܕܪܢܐ (rotatable) – α = n_F^1/4 (1.414 ܩܐ k = 2، 1.682 ܩܐ k = 3): ܫܘܚܠܦܐ ܕܡܢܒܝܢܘܬܐ ܫܘܝܐ ܝܠܗ ܓܘ ܟܠ ܢܘܩ̈ܙܐ ܕܐܝܬܝܢ ܒܗܘ ܪܘܚܩܐ ܡܢ ܡܨܥܬܐ – "ܫܘܚܠܦܐ ܕܦܘܢܝܐ ܡܢܒܝܐ ܩܝܡܐ ܝܠܗ ܥܠ ܐܣܦܝܪ̈ܐ" (ܦܐ̈ܬܐ 431-432).
• ܐܣܦܝܪܝܐ (spherical) – α = √k: ܟܠ ܢܘܩ̈ܙܐ ܥܠ ܚܕ ܐܣܦܝܪ؛ ܩܐ ܐܬܪܐ ܐܣܦܝܪܝܐ ܐܗܐ ܝܠܗ̇ ܓܘܒܝܐ ܛܒܐ ܡܢ ܚܘܪܐ ܕܫܘܚܠܦܐ ܕܡܢܒܝܢܘܬܐ، ܘ"ܠܝܬ ܣܢܝܩܘܬܐ ܠܡܬܚܕܪܢܘܬܐ ܚܬܝܬܬܐ ܕܬܘܩܢܐ ܗܘܐ ܛܒܐ" (ܦܐܬܐ 432). ܡܘܥ̈ܕܐ ܝܢܐ ܡܢ ܬܠܬܐ ܥܕܡܐ ܚܡܫܐ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ (ܦܐܬܐ 432).
• ܡܨܥܝܐ ܥܠ ܐܦ̈ܐ (face-centered) – α = 1: ܒܣ ܬܠܬܐ ܕܪ̈ܓܐ، ܩܐ ܐܬܪܐ ܕܩܘܒܐ (cuboidal)، ܐܝܡܢ ܕܠܐ ܡܨܝܚ ܕܦܠܛܚ ܡܢ ܦܬܝܘܬܐ؛ ܠܐ ܝܠܗ ܡܬܚܕܪܢܐ، ܘ"ܬܪܝܢ ܝܢ ܬܠܬܐ" ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܣܦܩܝ (ܦܐܬܐ 433).
• ܓܘܫܡܢܘܬܐ ܐܘܪܬܘܓܘܢܠܝܬܐ (orthogonal blocking) – ܐܝܡܢ ܕܡܢܬܐ ܦܩܛܘܪܝܐܠ ܘܕܣܪ̈ܛܐ ܟܪܗܛܝ ܓܘ ܙܒ̈ܢܐ ܦܪ̈ܝܫܐ، ܐܢܝ ܬܪܝܢ ܓܘ̈ܫܡܐ ܝܢܐ. ܬܘܩܢܐ "ܓܘܫܡܢܐ ܝܠܗ ܐܘܪܬܘܓܘܢܠܐܝܬ" ܐܝܡܢ ܕܡܥܒܕܢܘܬܐ ܕܓܘ̈ܫܡܐ ܠܐ ܟܡܥܒܕܐ ܥܠ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ (ܦܐܬܐ 437)؛ ܓܘ CCD ܐܗܐ ܟܗܘܐ ܐܝܡܢ ܕ-α = [n_F(n_A + n_CA)/(2(n_F + n_CF))]^1/2 (ܫܘܝܘܬܐ 11.18، ܦܐܬܐ 439). ܡܨܝܚ "ܕܫܒܩܚ ܩܠܝܠ ܡܢ ܡܬܚܕܪܢܘܬܐ ܝܢ ܡܢ ܓܘܫܡܢܘܬܐ ܐܘܪܬܘܓܘܢܠܝܬܐ ܕܠܐ ܡܘܒܕܢܐ ܪܒܐ ܕܡܘܕܥܢܘܬܐ" (ܦܐܬܐ 439). ܒܣ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܕܪܗܛܝ ܓܘ ܗܘ ܓܘܫܡܐ ܬܢܝ̈ܬܐ (replicates) ܝܢܐ (ܦܐܬܐ 440).

Box-Behnken – ܬܘܩܢܐ ܒܬܠܬܐ ܕܪ̈ܓܐ، "ܚܘܠܛܐ ܕ-2^k ܥܡ ܬܘܩ̈ܢܐ ܕܓܘ̈ܫܡܐ ܠܐ ܓܡܝܪ̈ܐ": ܟܠ ܙܘܓܐ ܕܥܠ̈ܠܬܐ ܓܘ ±1 ܘܐܚܪ̈ܢܐ ܓܘ ܡܨܥܬܐ، ܘܙܘܕܐ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ. ܗܘ "ܡܥܕܪܢܐ ܝܠܗ ܣܓܝ ܒܡܢܝܢܐ ܕܪ̈ܗܛܐ"، "ܡܬܚܕܪܢܐ ܝܢ ܩܪܝܒ ܡܬܚܕܪܢܐ"، ܟܠ ܢܘܩ̈ܙܐ ܕܝܠܗ ܥܠ ܐܣܦܝܪ ܒܦܠܓܘܬ ܩܘܛܪܐ (radius) √2، ܘܠܝܬ ܓܘܗ ܩܪ̈ܢܬܐ – ܝܬܝܪܘܬܐ ܐܝܡܢ ܕܚܘܠܛܐ ܚܪܝܦܐ ܕܟܠ ܥܠ̈ܠܬܐ ܝܩܝܪܐ ܝܢ ܠܐ ܡܨܝܢܐ ܝܠܗ (ܦܐܬܐ 432؛ ܠܘܚܐ 11.8 ܩܐ k = 3، ܦܐܬܐ 433). ܓܘ ܦܐܬܐ – k = 3 ܥܕܡܐ 5، ܒܗܘ ܒܢܝܢܐ.

ܓܘ ܬܘܩܢܐ ܦܐܬܐ ܟܡܚܘܐ ܫܘܚܠܦܐ ܡܬܩܢܐ ܕܡܢܒܝܢܘܬܐ N·x'(X'X)⁻¹x (ܫܘܝܘܬܐ 11.16، ܦܐܬܐ 436) ܓܘ ܡܨܥܬܐ، ܓܘ ܢܘܩܙܐ ܦܩܛܘܪܝܐܠ ܘܓܘ ܢܘܩܙܐ ܕܣܪܛܐ – ܓܘ ܬܘܩܢܐ ܡܬܚܕܪܢܐ ܬܪܬܝܢ ܐܚܪ̈ܝܬܐ ܫܘ̈ܝܐ ܝܢܐ ܐܝܡܢ ܕܐܝܬܝܢ ܒܗܘ ܪܘܚܩܐ. ܬܘܩ̈ܢܐ ܛܒ̈ܐ ܡܢ ܟܠ (optimal؛ D, I)، ܬܘܩ̈ܢܐ ܕܚܘܠ̈ܛܐ (mixture designs) ܘܬܘܩ̈ܢܐ ܩܐ ܢܣܝ̈ܘܢܐ ܕܡܚܫܒܢܐ (ܦܐ̈ܬܐ 440-472) – ܠܝܬ ܓܘ ܦܐܬܐ.

ܐܝܬܘ ܡܘܕܝܠ ܦܐܝܐ؟

ܠܐ ܦܐܝܘܬܐ (lack of fit) ܩܘܒܠ ܛܥܝܘܬܐ ܕܟܝܬܐ (pure error). ܫܪ̈ܟܐ (residuals) ܕܡܘܕܝܠ ܟܡܬܦܪܩܝ ܠܬܪܝܢ: ܛܥܝܘܬܐ ܕܟܝܬܐ – ܫܘܚܠܦܐ ܒܝܢܝ ܪ̈ܗܛܐ ܡܬܢ̈ܝܐ ܓܘ ܗܝ ܢܘܩܙܐ، ܕܠܐ ܬܠܝܐ ܝܠܗ̇ ܒܡܘܕܝܠ؛ ܘܠܐ ܦܐܝܘܬܐ – ܡܐ ܕܦܝܫ. ܐܢ ܠܐ ܦܐܝܘܬܐ ܪܒܬܐ ܝܠܗ̇ ܡܢ ܛܥܝܘܬܐ ܕܟܝܬܐ، ܡܘܕܝܠ ܟܡܛܥܐ ܡܕܡ (Montgomery, 2020, Tables 11.2, 11.5 and 11.7). ܡܛܠ ܗܕܐ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܣܢܝܩ̈ܐ ܝܢܐ: ܕܠܐ ܬܢܝܬܐ ܠܝܬ ܛܥܝܘܬܐ ܕܟܝܬܐ ܘܠܝܬ ܐܗܐ ܒܘܚܪܢܐ.

ܦܘܚܡܐ ܕܡܘܕ̈ܝܠܐ. ܠܘܚܐ 1 ܓܘ ܡܦܩܢܐ ܟܦܚܡܐ ܬܠܬܐ ܡܘܕ̈ܝܠܐ – ܣܪܛܢܝܐ، ܥܡ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ، ܘܕܛܟܣܐ ܬܪܝܢܐ – ܐܝܟ "Model Summary Statistics" ܘ"Lack of Fit Tests" ܓܘ Montgomery (ܠܘܚܐ 11.7، ܦܐܬܐ 421): ܣܛܝܘܬܐ ܬܩܝܢܬܐ ܕܫܪ̈ܟܐ، R²، R² ܡܬܘܪܨܐ، R² ܩܐ ܡܢܒܝܢܘܬܐ ܘ-PRESS، ܘܠܐ ܦܐܝܘܬܐ. ܓܘ ܡܬܠܐ 11.2 ܣܪܛܢܝܐ ܘܗܘ ܕܥܡ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܟܦܠܝ (R² ܩܐ ܡܢܒܝܢܘܬܐ ܒܨܘܪܐ ܡܢ ܨܦܪܐ، ܠܐ ܦܐܝܘܬܐ ܪܫܝܡܬܐ)، ܘܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ ܦܐܝܐ ܝܠܗ (R² ܩܐ ܡܢܒܝܢܘܬܐ .918، ܠܐ ܦܐܝܘܬܐ ܠܐ ܪܫܝܡܬܐ، p = .290). R² ܩܐ ܡܢܒܝܢܘܬܐ ܣܡܝܟܐ ܝܠܗ ܥܠ PRESS – ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܕܫܪ̈ܟܐ ܐܝܡܢ ܕܟܠ ܪܗܛܐ ܡܢܒܝܐ ܝܠܗ ܕܠܐ ܓܢܒܗ.

ܒܘܚܪܢܐ ܕܟܘܪܗܢܐ (diagnostics). ܐܝܟ ܓܘ ܟܠ ܪܝܓܪܫܢ – ܫܪ̈ܟܐ، ܡܪܝܡܢܘܬܐ (leverage) ܘܡܥܒܕܢܘܬܐ (influence) (ܠܘܚܐ 11.7، ܦܐܬܐ 422). ܓܘ ܦܐܬܐ – ܠܘܚܐ ܕܒܘܚܪܢܐ ܕܟܘܪܗܢܐ ܕܟܦܝܠܐ ܝܠܗ ܓܘ ܫܘܠܡܐ ܕܡܦܩܢܐ.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܓܘ ܬܘܩܢܐ: ܠܘܚܐ 1 – ܬܘܩܢܐ ܘܕܝܠܝ̈ܘܬܐ ܕܝܠܗ: ܡܢ̈ܘܬܐ، α، ܡܢܝܢܐ ܕܪ̈ܗܛܐ ܘܕܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ، ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ ܩܐ ܠܐ ܦܐܝܘܬܐ ܘܩܐ ܛܥܝܘܬܐ ܕܟܝܬܐ، ܓܘܫܡܢܘܬܐ، ܘܫܘܚܠܦܐ ܕܡܢܒܝܢܘܬܐ. ܠܘܚܐ 2 – ܦܐܬܐ ܕܪܗܛܐ ܒܛܟܣܐ ܓܕܫܢܝܐ، ܥܡ ܛܝ̈ܡܐ ܪܡܝܙ̈ܐ ܘܟܝܢܝ̈ܐ. ܨܘܪܬܐ 1 – ܢܘܩ̈ܙܐ ܕܬܘܩܢܐ. ܡܚܬܬܐ ܠܐܩܣܠ ܐܝܬ ܓܘܗ̇ ܡܦܫܩܢܐ (legend / מקרא) ܥܡ ܪܡܙܢܘܬܐ، ܘܦܐܬܐ ܟܩܪܐ ܠܗ ܬܘܒ ܓܘ ܦܘܫܩܐ.

ܓܘ ܦܘܫܩܐ: ܩܕܡ ܪܗܛܐ – ܠܘܚܐ ܕܪܡܙܢܘܬܐ: ܩܐ ܟܠ ܥܠܬܐ ܡܨܥܬܐ ܘܦܠܓܐ ܕܦܬܝܘܬܐ (x = (ξ − ܡܨܥܬܐ)/ܦܠܓܐ-ܕܦܬܝܘܬܐ)، ܡܢ ܡܦܫܩܢܐ ܝܢ ܠܦܘܬ ܛܝ̈ܡܐ. ܠܘܚܐ 1 – ܦܘܚܡܐ ܕܡܘܕ̈ܝܠܐ. ܠܘܚܐ 2 – ܦܘܫܩܐ ܕܫܘܚܠܦܐ ܕܡܘܕܝܠ ܕܓܒܝܐ ܝܠܗ: ܟܠ ܗܕܡܐ ܥܡ ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܡܢܬܢܝܐ، ܘܫܪ̈ܟܐ ܕܟܡܬܦܪܩܝ ܠܠܐ ܦܐܝܘܬܐ ܘܛܥܝܘܬܐ ܕܟܝܬܐ. ܠܘܚܐ 3 – ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܒܚܕ̈ܝܘܬܐ ܪܡܝܙ̈ܬܐ، ܥܡ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ ܘ-VIF، ܘܠܬܚܬܗ ܫܘܝܘܬܐ ܒܚܕ̈ܝܘܬܐ ܫܪܫܝ̈ܬܐ. ܓܘ ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ – ܠܘܚܐ 4، ܢܘܩܙܐ ܩܝܡܬܐ، ܘܠܘܚܐ 5، ܦܘܫܩܐ ܩܢܘܢܝܐ؛ ܓܘ ܡܘܕܝܠ ܣܪܛܢܝܐ – ܐܘܪܚܐ ܕܣܠܩܐ ܚܪܝܦܐ ܡܢ ܟܠ. ܨܘܪܬܐ – ܨܘܪܬܐ ܕܣܪ̈ܛܐ ܕܦܘܢܝܐ ܫܘܝܐ ܩܐ ܟܠ ܙܘܓܐ ܕܥܠ̈ܠܬܐ، ܐܝܡܢ ܕܐܚܪ̈ܢܐ ܓܘ ܡܨܥܬܐ ܝܢ ܓܘ ܢܘܩܙܐ ܩܝܡܬܐ. ܒܬܪܗ̇ – ܡܢܒܝܢܘܬܐ ܓܘ ܢܘܩܙܐ ܥܡ ܬܚܘܡܐ ܕܬܘܟܠܢܐ ܘܬܚܘܡܐ ܕܡܢܒܝܢܘܬܐ (prediction interval)، ܠܘܚܐ ܕܒܘܚܪܢܐ ܕܟܘܪܗܢܐ، ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ ܘܩܘܛܝܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ".

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

ܩܪܝܒܘܬܐ ܕܘܟܬܢܝܬܐ ܝܠܗ̇. ܦܘܠܝܢܘܡܝܐܠ ܦܐܝܐ ܝܠܗ ܠܐܬܪܐ ܕܒܚܝܪܐ ܝܠܗ، ܠܐ ܠܒܪ ܡܢܗ (Montgomery, 2020, p. 409). ܟܠ ܡܢܒܝܢܘܬܐ ܠܒܪ ܡܢ ܢܘܩ̈ܙܐ ܕܬܘܩܢܐ extrapolation ܝܠܗ̇؛ ܦܐܬܐ ܠܐ ܟܚܫܒܐ ܡܢܒܝܢܘ̈ܬܐ ܓܘ ܐܘܪܚܐ ܠܒܪ ܡܢ ܦܬܝܘܬܐ، ܘܟܡܙܗܪܐ ܐܝܡܢ ܕܢܘܩܙܐ ܩܝܡܬܐ ܠܒܪ ܡܢܗ̇ ܝܠܗ̇. ܦܘܫܩܐ ܕܢܘܩܙܐ ܩܝܡܬܐ ܟܣܡܟ ܕܗܝ "ܓܘ ܐܬܪܐ ܕܒܨܝܐ" ܝܠܗ̇ (ܦܐܬܐ 418).

ܓܕܫܢܝܘܬܐ (randomization). ܐܝܟ ܓܘ ܟܠ ܢܣܝܘܢܐ، ܛܟܣܐ ܕܪܗܛܐ ܓܕܫܢܝܐ ܝܠܗ – ܦܐܬܐ ܕܪܗܛܐ ܕܦܐܬܐ ܟܒܢܐ. ܫܪ̈ܛܐ ܥܠ ܛܥܝܘܬܐ ܕܪܝܓܪܫܢ ܥܡܝܐ ܝܢܐ: ܠܐ ܬܠܝ̈ܐ، ܫܘܚܠܦܐ ܫܘܝܐ، ܦܘܠܓܐ ܩܪܝܒܐ ܠܢܘܪܡܠܝܐ – ܟܒܚܪܚ ܓܘ ܫܪ̈ܟܐ.

ܪܡܙܢܘܬܐ (coding). ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ، ܢܘܩܙܐ ܩܝܡܬܐ ܘܦܘܫܩܐ ܩܢܘܢܝܐ – ܒܚܕ̈ܝܘܬܐ ܪܡܝܙ̈ܬܐ، ܐܝܟ ܓܘ Montgomery. ܫܘܝܘܬܐ ܒܚܕ̈ܝܘܬܐ ܫܪܫܝ̈ܬܐ ܫܘܝܐ ܝܠܗ̇، ܐܠܐ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܕܝܠܗ̇ ܬܠܝ̈ܐ ܝܢܐ ܒܚܕ̈ܝܘܬܐ ܘܠܐ ܐܝܬ ܠܢ ܕܦܚܡܚ ܒܝܢܬܗܘܢ (ܦܐܬܐ 422). ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܬܠܝ̈ܐ ܝܢܐ ܒܪܡܙܢܘܬܐ؛ ܪܡܙܢܘܬܐ ܕܓܘܗ̇ ±1 ܕܪ̈ܓܐ ܦܩܛܘܪܝܐܠ ܝܢܐ ܟܢܛܪܐ ܣܘܟܠܐ ܕܝܠܗܘܢ.

ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. ܬܘܩ̈ܢܐ ܛܒ̈ܐ ܡܢ ܟܠ (optimal؛ D, I, G)، ܬܘܩ̈ܢܐ ܕܚܘܠ̈ܛܐ (mixture designs)، ܢܣܝ̈ܘܢܐ ܕܡܚܫܒܢܐ، EVOP (Montgomery, 2020, Sections 11.4.4-11.7)، ܦܘܢܟܫܢ ܕܨܒܝܢܘܬܐ (desirability) ܩܐ ܟܡܐ ܦܘܢ̈ܝܐ، ܢܣܝ̈ܘܢܐ ܕ-3^k (ܨܚܚܐ 9) ܘܬܘܩ̈ܢܐ ܚܣܝܢ̈ܐ (robust designs؛ ܨܚܚܐ 12). ܓܘ ܦܐܬܐ "ܬܘܩܢܐ ܕܢܣܝ̈ܘܢܐ" – ܥܠ̈ܠܬܐ ܕܓܢ̈ܣܐ، 2^k ܘܡܢ̈ܘܬܐ؛ ܘܓܘ ܦܐܬܐ "ܥܠ̈ܠܬܐ ܓܕܫܢ̈ܝܬܐ، ܡܩܢܢ̈ܐ ܘܣܦܠܝܛ-ܦܠܘܛ" – ܬܘܩ̈ܢܐ ܥܡ ܟܪܝܘܬܐ ܥܠ ܓܕܫܢܝܘܬܐ.

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

ܓܘ SPSS Statistics ܠܝܬ ܦܪܘܣܝܕܘܪ ܩܐ ܐܦ̈ܐ ܕܦܘܢܝܐ. ܟܚܫܒܚ ܡܫܬܚܠܦ̈ܢܐ ܪܡܝܙ̈ܐ، ܥܘܦ̈ܦܐ ܕܝܠܗܘܢ ܘܡܪ̈ܒܥܐ ܕܝܠܗܘܢ ܓܘ Transform › Compute Variable، ܘܟܪܗܛܚ Analyze › General Linear Model › Univariate ܥܡ ܟܠܗܘܢ ܐܝܟ Covariates، ܥܡ ܓܘܡܪ̈ܐ ܕܡܪ̈ܒܥܐ ܕ-Type III – ܐܢܝ ܝܢܐ SS ܡܢܬܢ̈ܝܐ ܕܠܘܚܐ 2 – ܘܥܡ Lack of fit ܓܘ Options. ܣܝܢܬܐܟܣ ܕܦܐܬܐ ܟܒܢܐ ܟܥܒܕ ܚܬܝܬܐܝܬ ܐܗܐ (COMPUTE ܘܗܝܕܝܟ UNIANOVA … WITH … /PRINT=PARAMETER LOF). ܢܘܩܙܐ ܩܝܡܬܐ، ܦܘܫܩܐ ܩܢܘܢܝܐ ܘܨܘܪ̈ܬܐ SPSS ܠܐ ܟܚܫܒ – ܩܐ ܐܢܝ ܐܝܬ R ܝܢ ܦܐܬܐ. ܕܠܐ ܓܘ̈ܫܡܐ، ܣܝܢܬܐܟܣ ܟܡܘܣܦ ܨܐ REGRESSION ܥܡ ܗܢܝ ܗܕ̈ܡܐ – ܗܢܝ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ. ܐܘܪ̈ܚܬܐ ܓܘ ܪ̈ܫܝܡܬܐ ܡܘܕܥܢܘܬܐ ܟܠܢܝܬܐ ܝܢܐ ܥܠ ܬܘܟܢܝܬܐ، ܘܣܝܢܬܐܟܣ ܠܐ ܪܗܛܐ ܝܠܗ ܓܘ SPSS. ܓܘ PSPP، ܬܘܟܢܝܬܐ ܚܐܪܬܐ ܕܟܩܪܝܐ ܣܝܢܬܐܟܣ ܕ-SPSS، ܠܝܬ UNIANOVA؛ ܡܢܬܐ ܕ-REGRESSION ܪܗܛܠܢ ܓܘܗ̇ ܥܠ ܦܐܝܠ ܕܡܬܠܐ، ܘܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܫܘ̈ܝܐ ܝܢܐ ܠܦܐܬܐ.

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

ܨܪܪܐ rsm: coded.data ܩܐ ܪܡܙܢܘܬܐ، rsm ܥܡ FO، TWI، PQ ܝܢ SO ܩܐ ܡܘܕܝܠ، ܘ-summary ܟܝܗܒ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܘܦܘܫܩܐ ܕܫܘܚܠܦܐ ܠܦܘܬ ܓܘ̈ܕܐ ܥܡ ܠܐ ܦܐܝܘܬܐ ܘܛܥܝܘܬܐ ܕܟܝܬܐ؛ canonical – ܢܘܩܙܐ ܩܝܡܬܐ ܘܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ؛ contour – ܨܘܪ̈ܬܐ؛ ccd ܘ-bbd – ܬܘܩ̈ܢܐ. canonical ܟܥܒܕ ܨܦܪܐ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܙܥܘܪ̈ܐ ܡܢ ܚܕ ܡܢ ܥܣܪܐ ܕܪܒܐ ܩܕܡ ܚܘܫܒܢܐ ܕܢܘܩܙܐ، ܘܡܛܠ ܗܕܐ ܓܘ ܐܦܐ ܥܡ ܚܨܐ (ridge) ܢܘܩܙܐ ܕܝܠܗ ܦܪܝܫܐ ܝܠܗ̇ ܡܢ −½B⁻¹b ܕ-Montgomery ܕܓܘ ܦܐܬܐ. ܐܘܪܚܐ ܕܣܠܩܐ ܕ-steepest ܡܫܝܚܐ ܝܠܗ̇ ܒܪܘܚܩܐ ܫܘܝܐ ܘܠܐ ܠܦܘܬ ܓܫܩܐ ܓܘ ܚܕ ܥܠܬܐ. ܣܩܪܝܦܬ (script) ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ ܐܢܓܠܝܙܝܐ، ܘܛܝ̈ܡܐ ܓܘ ܡܥܝܪ̈ܢܘܬܐ ܫܘ̈ܝܐ ܝܢܐ ܠܦܐܬܐ. ܢܡܘܣܐ ܕܣܐܝܛ (site rule) ܩܐ ܣܘܪܝܬ ܒ-R: ܦܐܝܠ UTF-8، ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ، ܘܫܡ̈ܗܐ ܕܡܫܬܚܠܦ̈ܢܐ ܒܐܢܓܠܝܙܝܐ.

# ܐܦ̈ܐ ܕܦܘܢܝܐ – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)   # ܩܪܝܢܐ ܕܐܩܣܠ
library(rsm)      # ܐܦ̈ܐ ܕܦܘܢܝܐ (install.packages("rsm"))

# ܛܥܢܐ: ܛܘܦܣܐ ܬܩܝܢܐ – ܣܕܪܐ = ܪܗܛܐ؛ ܓܠܝܘܢܐ "Data"
d <- as.data.frame(read_excel("C:/Rdata/DATA_rsm_example_eng.xlsx", sheet = "Data"))

# ܪܡܙܢܘܬܐ: x = (ξ − ܡܨܥܬܐ)/ܦܠܓܐ-ܕܦܬܝܘܬܐ – ܕܪ̈ܓܐ ܦܩܛܘܪܝܐܠ ܟܗܘܝ 1− ܘ-1+
cd <- coded.data(d, x1 ~ (lesson_length - 45) / 10, x2 ~ (practice_items - 10) / 3, x3 ~ (review_weeks - 3) / 1)

# ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ: ܠܘܚܐ ܕ-rsm – FO, TWI, PQ، ܫܪ̈ܟܐ، ܠܐ ܦܐܝܘܬܐ ܘܛܥܝܘܬܐ ܕܟܝܬܐ
fit <- rsm(retention ~ SO(x1, x2, x3), data = cd)
summary(fit)       # ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ: b0 71.74; x1 4.26, x2 −2.68, x3 1.69; x1:x2 1.10, x1:x3 0.23, x2:x3 −1.68; x1² −3.26, x2² −2.57, x3² −1.68; ܠܐ ܦܐܝܘܬܐ: F(5, 5) = 0.89, p = .551

# SS ܡܢܬܢܝܐ ܩܐ ܟܠ ܗܕܡܐ (ܠܘܚܐ 2 ܓܘ ܦܐܬܐ): ܥܡܘ̈ܕܐ ܓܠܝ̈ܐ، ܘܗܝܕܝܟ drop1 – ܟܠ ܗܕܡܐ ܒܬܪ ܟܠ ܐܚܪ̈ܢܐ
e <- transform(as.data.frame(cd), x12 = x1 * x2, x13 = x1 * x3, x23 = x2 * x3, x1q = x1^2, x2q = x2^2, x3q = x3^2)
drop1(lm(retention ~ x1 + x2 + x3 + x12 + x13 + x23 + x1q + x2q + x3q, data = e), test = "F")   # ܡܢ x1 F = 123.97 … ܥܕܡܐ x3q F = 21.07

# ܢܘܩܙܐ ܩܝܡܬܐ ܘܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ (ܠܘܚܐ 4 ܓܘ ܦܐܬܐ)
canonical(fit)     # xs = 0.568, −0.687, 0.883؛ ܒܚܕ̈ܝܘܬܐ ܫܪܫܝ̈ܬܐ 50.7 ܩܛܝ̈ܢܬܐ، 7.9 ܬܡܪ̈ܢܐ، 3.9 ܫܒ̈ܘܥܐ؛ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ −1.16, −2.64, −3.71 – ܛܝܡܐ ܥܠܝܐ

# ܨܘܪܬܐ ܕܣܪ̈ܛܐ ܕܦܘܢܝܐ ܫܘܝܐ ܩܐ ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܩܕܡܝ̈ܬܐ، ܐܝܡܢ ܕܬܠܝܬܝܬܐ ܓܘ ܢܘܩܙܐ ܩܝܡܬܐ
contour(fit, ~ x1 + x2, image = TRUE, at = canonical(fit)$xs)

# ܦܘܚܡܐ ܕܡܘܕ̈ܝܠܐ (ܠܘܚܐ 1 ܓܘ ܦܐܬܐ): R² ܡܬܘܪܨܐ ܘ-PRESS
for (f in list(retention ~ FO(x1, x2, x3), retention ~ FO(x1, x2, x3) + TWI(x1, x2, x3), retention ~ SO(x1, x2, x3))) {
  m <- rsm(f, data = cd)
  cat(sprintf("%-45s adjR2 = %.3f  PRESS = %.2f\n", deparse(f), summary(m)$adj.r.squared, sum((resid(m) / (1 - hatvalues(m)))^2)))
}   # R² ܡܬܘܪܨܐ .482 / .432 / .944؛ PRESS 423.72 / 625.07 / 88.27

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܬܘܢܝܐ ܐܝܬ ܠܗ ܕܡܨܐ ܠܡܥܒܕܐ ܬܘܒ ܬܘܩܢܐ ܘܦܘܫܩܐ: ܙܢܐ ܕܬܘܩܢܐ (CCD، ܘ-α ܕܝܠܗ؛ Box-Behnken)، ܥܠ̈ܠܬܐ ܘܦܬܝ̈ܘܬܐ ܕܝܠܗܝܢ، ܪܡܙܢܘܬܐ، ܡܢܝܢܐ ܕܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܘܕܓܘ̈ܫܡܐ، ܘܡܢܝܢܐ ܕܪ̈ܗܛܐ؛ ܡܘܕܝܠ ܕܓܒܝܐ ܝܠܗ ܘܩܡܘܕܝ (ܠܐ ܦܐܝܘܬܐ، R² ܡܬܘܪܨܐ ܘ-R² ܩܐ ܡܢܒܝܢܘܬܐ)؛ ܠܘܚܐ ܕܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ؛ ܘܐܢ ܐܝܬ – ܢܘܩܙܐ ܩܝܡܬܐ ܒܚܕ̈ܝܘܬܐ ܫܪܫܝ̈ܬܐ، ܦܘܢܝܐ ܡܢܒܝܐ ܓܘܗ̇ ܘܟܝܢܐ ܕܝܠܗ̇ (ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ). APA (2020، ܦܣܩܐ 6.43) ܟܒܥܐ "ܡܘܕܥܢܘܬܐ ܣܦܩܬܐ ܕܩܪܘܝܐ ܡܨܐ ܕܡܫܪܪ ܦܘܫܩ̈ܐ". ܢܡܘ̈ܣܐ ܕܐܣܟܝܡܐ (ܦܣܩܐ 6.36): R² ܘ-p ܕܠܐ ܨܦܪܐ ܩܕܡ ܢܘܩܙܐ؛ ܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ، ܡܨܥܝ̈ܬܐ ܘܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ – ܥܡ ܨܦܪܐ. ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ ܓܘ ܡܦܩܢܐ ܒܢܝܐ ܝܠܗ̇ ܗܟܢܐ، ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ؛ ܘܩܕܡ ܕܩܒܠܚ ܢܘܩܙܐ – ܟܪܗܛܚ ܓܘܗ̇ ܪ̈ܗܛܐ ܕܫܘܪܪܐ (confirmation runs).

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

"ܚܕ ܥܠܬܐ ܓܘ ܟܠ ܓܐܝܐ". ܟܡܙܝܥܚ ܟܠ ܥܠܬܐ ܒܦܪܝܫܘܬܐ ܘܟܡܛܥܝܚ ܥܒ̈ܕܐ ܕܚܕ̈ܕܐ ܘܐܘܪܚܐ ܫܪܝܪܬܐ ܕܛܘܒܐ – ܐܘܪܚܐ ܚܪܝܦܬܐ ܡܢ ܟܠ ܟܡܫܚܠܦܐ ܟܠ ܥܠ̈ܠܬܐ ܚܕ̈ܕܐ، ܡܫܘܚܝܐܝܬ ܠܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ (Montgomery, 2020, p. 415).

ܡܘܕܝܠ ܕܛܟܣܐ ܬܪܝܢܐ ܡܢ ܬܘܩܢܐ ܕܠܐ ܟܡܥܕܪ ܠܗ. ܓܘ 2^k ܥܡ ܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ ܟܠ ܗܕ̈ܡܐ ܡܪܒܥ̈ܝܐ ܫܘ̈ܝܐ ܝܢܐ، ܘܠܐ ܡܨܝܚ ܕܫܥܪܚ ܠܗܘܢ ܒܦܪܝܫܘܬܐ – ܒܣ ܟܦܝܦܘܬܐ ܟܠܢܝܬܐ (ܦܐܬܐ 413). ܟܡܘܣܦܚ ܢܘܩ̈ܙܐ ܕܣܪ̈ܛܐ.

ܦܘܫܩܐ ܕܢܘܩܙܐ ܩܝܡܬܐ ܠܒܪ ܡܢ ܐܬܪܐ ܕܢܣܝܘܢܐ ܐܝܟ ܕܐܢ ܗܝ ܢܘܩܙܐ ܛܒܬܐ ܝܠܗ̇. ܠܒܪ ܡܢ ܐܬܪܐ ܡܘܕܝܠ ܠܐ ܝܠܗ ܬܪܝܨܐ (ܦܐܬܐ 425).

ܓܘܒܝܐ ܕܡܘܕܝܠ ܠܦܘܬ R² ܒܠܚܘܕ. R² ܒܟܠ ܙܒܢ ܟܣܠܩ ܥܡ ܗܕ̈ܡܐ ܙܘܕܐ؛ R² ܡܬܘܪܨܐ، R² ܩܐ ܡܢܒܝܢܘܬܐ ܘܠܐ ܦܐܝܘܬܐ ܟܐܡܪܝ ܐܢ ܙܘܕܐ ܫܘܝܐ ܝܠܗ (ܠܘܚܐ 11.7، ܦܐܬܐ 421).

ܦܘܚܡܐ ܕܡܢܝ̈ܢܐ ܡܫܘܬ̈ܦܐ ܓܘ ܫܘܝܘܬܐ ܟܝܢܝܬܐ. ܐܢܝ ܬܠܝ̈ܐ ܝܢܐ ܒܚܕ̈ܝܘܬܐ؛ ܟܦܚܡܚ ܓܘ ܪܡܝܙܬܐ (ܦܐܬܐ 422).

ܫܒܩܐ ܕܢܘܩ̈ܙܐ ܡܨܥ̈ܝܐ. ܕܠܐ ܐܢܝ ܠܝܬ ܛܥܝܘܬܐ ܕܟܝܬܐ، ܠܝܬ ܒܘܚܪܢܐ ܕܠܐ ܦܐܝܘܬܐ، ܘܠܝܬ ܒܘܚܪܢܐ ܕܟܦܝܦܘܬܐ (ܦܐ̈ܬܐ 412-413).

ܡܥܓܠܐ ܕ-α. ܓܘ ܡܬܠܐ 11.2 ܠܘܚܐ ܟܡܚܘܐ α = 1.414، ܘܡܦܩܢܐ ܚܫܝܒܐ ܝܠܗ ܥܡ √2 – ܘܦܐܪ̈ܐ ܦܪ̈ܝܫܐ ܝܢܐ ܓܘ ܡܢܝܢܐ ܬܠܝܬܝܐ (ܡܫܟܚܢܘܬܐ ܓܘ ܫܘܪܪܐ ܕܦܐܬܐ). ܓܘ ܬܘܩܢܐ ܟܣܝܡܚ α ܚܬܝܬܐ؛ ܐܢ ܕܪܓܐ ܒܥܒܕܐ ܡܥܓܠܐ ܝܠܗ، ܟܟܬܒܚ ܛܝܡܐ ܕܪܗܛܐ ܝܠܗ.

ܡܒܘ̈ܥܐ

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

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

Box ܘ-Hunter (1957) ܘ-Box ܘ-Behnken (1960) – ܐܝܟ ܕܡܬܝ̈ܬܐ ܝܢܐ ܓܘ Montgomery (2020). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ (site rule)، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ (reference list) ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܓܘ ܫܪܫܐ ܕܝܗܝ؛ ܡܬܝ̈ܬܐ ܠܦܘܬ ܦܐܬܐ ܛܒܝܥܬܐ. ܨܚܚܐ 11 ܕ-Montgomery ܩܪܝܐ ܝܠܗ ܒܟܠܗ ܡܢ ܨܘܪ̈ܬܐ ܕܦܐ̈ܬܐ (ܫܬܐܣܐ ܕܟܬܒܐ ܕܣܪܝܩܘܬܐ (scan) ܟܫܒܩܐ ܣܓܝܐܘܬܐ ܕܫܘܝ̈ܘܬܐ). ܚܘܫܒܢܐ ܓܘ ܦܐܬܐ ܡܫܪܪܐ ܝܠܗ ܩܘܒܠ R 4.3.3 (lm, drop1, predict, hatvalues, rstudent, cooks.distance) ܘܨܪܪܐ rsm (rsm, canonical, ccd, bbd)، ܘܩܘܒܠ ܟܠ ܡܢܝ̈ܢܐ ܛܒܝ̈ܥܐ ܓܘ ܡܬ̈ܠܐ 11.1 ܘ-11.2 ܘܓܘ ܠܘ̈ܚܐ 11.8 ܘ-11.11.