ܡܨܥܝܘܬܐ ܒܝܢܝ ܥܠ̈ܠܬܐ ܟܣ̈ܝܬܐ (bootstrap)

ܛܥܢܐ ܕܝܗܝ̈ܒܬܐ (loading the data)

ܟܡܕܒܩܚ ܥܡܘ̈ܕܐ ܬܪܝܨܐܝܬ ܡܢ ܓܠܝܘܢܐ (worksheet)، ܥܡ ܣܕܪܐ ܕܪ̈ܫܐ (header row) – ܟܪܫܡܚ ܓܘ ܐܩܣܠ، ܟܢܣܚܚ، ܘܟܡܕܒܩܚ ܠܐܟܐ:

ܬܘܩܢܐ ܕܡܘܕܝܠ (model)

ܐܝܡܢ ܡܦܠܚܝܚ (When to use)؟

ܐܡܬܝ ܕܗܘܦܘܬܣܝܣ (hypothesis) ܥܠ ܡܟܢܘܬܐ ܝܠܗ: ܥܠܬܐ ܚܕܐ ܟܡܥܒܕܐ ܥܠ ܦܐܪܐ ܒܐܘܪܚܐ ܕܥܠܬܐ ܬܠܝܬܝܬܐ. Kline (2023، ܦܐܬܐ 349) ܟܡܬܚܘܐ ܦܘܫܩܐ ܕܡܨܥܝܘܬܐ ܐܝܟ ܫܘܐܠܐ "ܕܟܝ ܫܘܢ̈ܝܐ ܟܥܒܪܝ ܡܢ ܥܠܬܐ ܕܥܠܬܢܘܬܐ ܒܐܘܪܚܐ ܕܡܫܬܚܠܦܢܐ ܡܨܥܝܐ ܚܕ ܝܢ ܝܬܝܪ، ܕܐܢܝ̈ ܒܬܪ ܗܕܐ ܟܡܝܬܝ ܫܘܢ̈ܝܐ ܓܘ ܦܐܪܐ". ܓܘ ܦܐܝܠ ܕܡܬܠܐ: ܬܘܟܠܢܐ ܥܠ ܡܕܒܪ̈ܢܘܝܬܐ ← ܪܓܫܐ ܕܕܝܠܢܝܘܬܐ ܠܥܕܬܐ ← ܫܘܬܦܘܬܐ، ܘܓܢܒܗ ܫܒܝܠܐ ܬܪܝܨܐ ܡܢ ܬܘܟܠܢܐ ܠܫܘܬܦܘܬܐ.

ܩܐ ܡܘܕܝ ܥܡ ܥܠ̈ܠܬܐ ܟܣ̈ܝܬܐ (latent factors) ܘܠܐ ܥܡ ܡܢܝ̈ܢܐ ܕܣܒܠܬܐ (scale scores)؟ Kline (2023، ܦܐܬܐ 350) ܟܡܝܬܐ ܢܣܝ̈ܘܢܐ ܕܕܡܘܬܐ (simulations) ܕ-Iacobucci ܘܐܚܪ̈ܢܐ (2007، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘܗ) ܕܠܦܘܬܗܘܢ ܦܘܫܩ̈ܐ ܕܡܨܥܝܘܬܐ ܥܡ ܟܡܐ ܡܚܘ̈ܝܢܐ ܩܐ ܟܠ ܡܫܬܚܠܦܢܐ، ܕܟܡܫܠܛܝ ܥܠ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ، "ܒܥܕܢܐ ܕܣܓܝ ܛܒ̈ܐ ܝܢܐ" ܡܢ ܦܘܫܩ̈ܐ ܥܡ ܡܚܘܝܢܐ ܝܚܝܕܝܐ (ܪܝܓܪܫܢ (regression) ܥܕܢܝܐ). ܥܡ ܡܢܝ̈ܢܐ ܕܣܒܠܬܐ – ܦܐܬܐ ܦܘܫܩܐ ܕܫܒ̈ܝܠܐ ܘܡܨܥܝܘܬܐ (ܬܡܢ ܐܝܬ ܐܦ bootstrap)، ܝܢ ܡܪܟܟܢܘܬܐ ܘܡܨܥܝܘܬܐ (ܐܝܟ PROCESS، ܐܦ ܡܨܥܝܘܬܐ ܡܪܟܟܬܐ)؛ ܩܐ ܡܘܕܝܠ ܓܡܝܪܐ ܕܠܐ ܦܘܪܩܐ ܕܡܨܥܝܘܬܐ – ܡܘܕܝܠ ܕܫܘܝ̈ܘܬܐ ܒܢܝܢܝ̈ܬܐ؛ ܩܐ ܒܘܚܪܢܐ ܕܡܘܕܝܠ ܕܡܫܘܚܬܐ ܒܠܚܘܕܘܗܝ – ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܡܫܪܪܢܐ.

ܡܠܬܐ ܕܗܘܢܐ: product ܕܫܒ̈ܝܠܐ

ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ (indirect effect) ܕ-X ܥܠ Y ܒܐܘܪܚܐ ܕ-M product ܝܠܗ ܕܬܪܝܢ ܡܢܝ̈ܢܐ ܡܫܘܬܦ̈ܐ (coefficients) – a (X ← M) ܥܦܝܦܐ ܒ-b (M ← Y)، ܐܡܬܝ ܕ-b ܡܢ ܩܕܡ ܟܫܠܛ ܥܠ X؛ ܡܥܒܕܢܘܬܐ ܟܠܢܝܬܐ (total effect) ab + c′ ܝܠܗ، ܥܡ c′ ܬܪܝܨܬܐ. Kline (2023، ܦܐܬܐ 125) ܟܡܦܫܩ: ab ܫܘܢܝܐ ܡܣܟܝܐ ܝܠܗ ܓܘ Y ܐܡܬܝ ܕ-X ܟܦܝܫ ܩܝܡܐ ܘ-M ܟܫܚܠܦ ܠܛܝܡܐ ܕܡܛܐ ܗܘܐ ܠܗ ܒܫܘܢܝܐ ܕܚܕ ܢܘܩܙܐ ܓܘ X (Pearl, 2009، ܐܝܟ ܕܡܬܡܠܠ ܓܘܗ). ܥܡ ܟܡܐ ܡܨܥ̈ܝܐ – ܩܐ ܟܠ ܐܘܪܚܐ product ܕܝܠܗ (ܒܓܢܒ ܚܕ̈ܕܐ: X ← M1 ← Y ܘ-X ← M2 ← Y؛ ܒܣܕܪܐ: X ← M1 ← M2 ← Y)، ܘܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܟܠܢܝܬܐ ܓܘܡܪܐ ܝܠܗ (Kline, pp. 351-352, Figure 20.2).

ܛܥܝܘܬܐ ܬܩܝܢܬܐ (standard error). ܩܐ ܫܒܝܠܐ ܓܢܒܗ – ܛܥܝܘܬܐ ܬܩܝܢܬܐ ܥܕܢܝܬܐ. ܩܐ product ܐܝܬ ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ. ܐܘܪܚܐ ܕ-delta (Sobel, 1982، ܐܝܟ ܕܡܬܡܠܠ ܓܘ Kline, 2023، ܦܐܬܐ 125، ܫܘܝܘܬܐ 8.2؛ ܐܗܐ ܝܠܗ ܐܦ ܓܘܒܝܐ ܡܩܕܡܐ ܕ-:= ܓܘ lavaan – Rosseel, 2012، ܦܣܩܐ 7.4) – ܐܠܐ "ܒܘܚܪܢܐ ܕ-Sobel ܟܣܐܡ ܢܘܪܡܠܝܘܬܐ (normality)، ܘܦܘܠ̈ܓܐ ܕܫܘ̈ܥܪܐ ܕ-product ܒܥܕܢܐ ܕܣܓܝ ܠܐ ܝܢܐ ܢܘܪܡܠ̈ܝܐ؛ ܒܪܘܒܐ ܠܐ ܫܘ̈ܝܐ ܝܢܐ ܠܬܪܝܢ ܓܢ̈ܒܐ ܥܡ ܓܒܢܘܬܐ (kurtosis) ܪܡܬܐ"، ܗܘ ܒܥܐ ܕܘܡܝܐ ܪܒܐ، ܘܛܝ̈ܡܐ p ܕܝܠܗ ܓܘ ܕܘܡܝܐ ܙܥܘܪܐ "ܡܨܝ ܕܗܘܝ ܣܓܝ ܠܐ ܚܬܝ̈ܬܐ" (Kline, p. 128). bootstrap – "ܕܠܐ ܟܣܐܡ ܢܘܪܡܠܝܘܬܐ" ܘܦܐܝܐ ܝܠܗ ܐܦ ܩܐ ܐܘܪ̈ܚܬܐ ܕܬܠܬܐ ܫܒ̈ܝܠܐ ܘܝܬܝܪ (ܬܡܢ). ܦܐܬܐ ܟܛܒܥܐ ܬܪܘܝܗܝ ܚܕܐ ܓܢܒ ܚܕܐ.

bootstrap ܘܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ

Byrne (2016، ܦܐܬܐ 367-368) ܟܡܬܚܘܝܐ ܪܥܝܢܐ: ܕܘܡܝܐ ܫܪܫܝܐ ܟܦܠܚ ܐܝܟ "ܟܢܫܐ ܟܠܢܝܐ" (population)، ܘܡܢ ܓܘܗ ܟܓܒܝ "ܥܡ ܕܥܪܬܐ" ܕܘܡ̈ܝܐ ܣܓܝ̈ܐܐ ܒܗܝ ܪܒܘܬܐ؛ ܓܘ ܟܠ ܕܘܡܝܐ ܟܫܥܪܚ ܡܘܕܝܠ ܡܢ ܪܝܫ، ܘܦܘܠܓܐ ܕܫܘ̈ܥܪܐ ܦܘܠܓܐ ܕܓܘܒܝܐ ܕܕܘܡܝܐ (sampling distribution) "ܓܫܝܡܐ" ܝܠܗ، ܕܠܐ ܬܠܝܐ ܝܠܗ ܒܣܐܡܬܐ ܕܢܘܪܡܠܝܘܬܐ. ܦܐܬܐ ܟܓܒܝܐ ܣܕ̈ܪܐ ܓܡܝܪ̈ܐ (ܡܫܬܘܬܦ̈ܢܐ) ܡܢ ܓܘ ܡܫܬܘܬܦ̈ܢܐ ܥܡ ܛܝܡܐ ܓܘ ܟܠ ܦܣܘ̈ܩܐ، ܟܫܥܪܐ ܡܘܕܝܠ ܓܘ ܟܠ ܕܘܡܝܐ ܡܢ ܫܘ̈ܥܪܐ ܕܕܘܡܝܐ ܓܡܝܪܐ، ܘܟܢܛܪܐ ܟܠ ܡܥܒܕ̈ܢܘܝܬܐ. ܕܘܡܝܐ ܕܓܘܗ ܫܘܥܪܐ ܠܐ ܡܬܟܢܫ ܗܘܐ – ܟܦܐܩ ܡܢ ܚܘܫܒܢܐ؛ ܕܘܡܝܐ ܥܡ ܦܘܫܪܐ ܠܐ ܡܩܒܠܐ – ܟܦܐܫ ܘܬܢܝܐ ܝܠܗ، ܐܝܟ ܓܘ lavaan. ܓܘ ܨܘܪܬܐ ܕܡܦܩܢܐ (output) ܟܚܙܝܚ ܦܘܠܓܐ ܕܐܬܐ، ܘܒܥܕܢܐ ܕܣܓܝ ܐܦ ܠܐ ܫܘܝܘܬܐ ܕܝܠܗ (skewness).

ܬܪܝܢ ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ. ܕܪ̈ܓܐ ܡܢ ܡܐܐ (percentiles): ܬܚܘ̈ܡܐ ܕܪ̈ܓܐ ܡܢ ܡܐܐ 2.5 ܘ-97.5 ܝܢܐ ܕܦܘܠܓܐ ܕ-bootstrap (ܦܐܬܐ ܟܡܦܠܚܐ ܒ-interpolation ܕ-lavaan، boot.ci.type = "perc"). ܡܬܘܪܨܐ ܡܢ ܡܣܒ ܒܐܦ̈ܐ (bias-corrected): Kline (2023، ܦܐܬܐ 129) ܟܡܬܚܘܐ ܠܗ ܐܝܟ ܐܘܪܚܐ ܕ"ܟܡܦܐܝܐ ܓܢܗ ܠܠܐ ܫܘܝܘܬܐ ܕܡܨܝܐ ܝܠܗ ܓܘ ܦܘܠܓܐ ܕܓܘܒܝܐ ܕܕܘܡܝܐ ܕܢܣܝܢܘܬܐ، ܒܝܕ ܦܪܫܬܐ ܕܡܢܬܐ ܕܫܘ̈ܥܪܐ ܕ-bootstrap ܕܟܢܦܠܝ ܬܚܝܬ ܦܐܪܐ ܕܚܙܝܐ ܗܘܐ" (Efron, 1987، ܐܝܟ ܕܡܬܡܠܠ ܓܘܗ) – ܐܗܐ ܝܠܗ "bca.simple" ܕ-lavaan، ܘܐܗܐ ܝܠܗ ܬܚܘܡܐ ܕ-AMOS ܟܛܒܥ (Byrne, 2016، ܦܐܬܐ 374، ܓܘܒܝܐ ܡܩܕܡܐ ܬܡܢ – 90%). ܦܐܬܐ ܟܛܒܥܐ ܬܪܘܝܗܘܢ ("ܐܦ ܐܦ"): ܐܡܬܝ ܕܫܠܡܝ ܝܢܐ – ܠܝܬ ܫܘܐܠܐ؛ ܐܡܬܝ ܕܚܕ ܟܡܚܘܐ ܨܦܪܐ ܘܐܚܪܢܐ ܠܐ – ܦܐܪܐ ܥܠ ܬܚܘܡܐ ܝܠܗ. ܙܪܥܐ: Kline (ܦܐܬܐ 129) ܟܡܥܗܕ ܕܬܚܘ̈ܡܐ ܕܬܚܘܡܐ ܕ-bootstrap "ܠܐ ܝܢܐ ܡܢ ܐܠܨܘܬܐ ܝܚܝ̈ܕܝܐ ܐܠܐ ܐܢ ܒܨܘܝܐ ܦܪܫ ܙܪܥܐ" – ܥܡ ܙܪܥܐ ܐܚܪܢܐ ܘܓܘ ܕܘܡܝܐ ܐܚܪܢܐ ܫܘܐܠܐ ܐܢ ܨܦܪܐ ܓܘܗ ܝܠܗ ܡܨܝܐ ܕܗܦܟ. ܦܐܬܐ ܟܥܒܕܐ ܡܘܠܕܢܐ ܕܡܢܝ̈ܢܐ ܕ-R (Mersenne-Twister ܘ-sample.int)، ܘܡܛܠ ܗܕܐ ܙܪܥܐ 2026 ܠܟܐ ܘ-iseed = 2026 ܓܘ lavaan ܟܝܗܒܝ ܗܢܝ̈ ܕܘܡ̈ܝܐ ܚܬܝܬܐܝܬ – ܘܗܢܝ̈ ܬܚܘ̈ܡܐ.

ܟܡܐ ܕܘܡ̈ܝܐ؟ 1,000 – ܓܘܒܝܐ ܡܩܕܡܐ ܓܘ ܚܙ̈ܡܐ ܕ-Kline (ܦܐܬܐ 129) ܪܗܛܠܗܘܢ، ܘܐܦ ܓܘ ܦܐܬܐ؛ Byrne (ܦܐܬܐ 374) ܚܘܝܠܗ ܥܡ 500. ܐܝܟ ܕܐܝܬ ܝܬܝܪ ܕܘܡ̈ܝܐ، ܬܚܘ̈ܡܐ ܒܨܝܪ ܬܠܝ̈ܐ ܝܢܐ ܒܙܪܥܐ؛ 5,000 ܓܘ ܒܪܐܘܙܪ ܟܫܩܠܝ ܩܪܝܒ ܦܠܓܐ ܕܩܛܝܢܬܐ.

ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ

Kline (2023، ܦܐܬܐ 353-355) ܟܡܚܘܐ ܕܠܝܬ "ܪܒܘܬܐ ܕܡܥܒܕܢܘܬܐ ܕܡܨܥܝܘܬܐ ܟܠܢܝܬܐ ܘܓܡܝܪܬܐ" ܘܕܩܐ ܟܠ ܡܚܘ̈ܝܢܐ ܐܝܬ ܬܚܘ̈ܡܐ. ܠܘܚܐ 5 ܓܘ ܡܦܩܢܐ ܟܡܝܬܐ ܬܠܬܐ: ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܡܬܩܢܬܐ – ܒܣܛܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ ܕܦܐܪܐ، ܩܐ ܫܘܢܝܐ ܕܣܛܝܘܬܐ ܬܩܝܢܬܐ ܚܕܐ ܓܘ ܡܢܒܝܢܐ (ܦܐܬܐ 353)، ܥܡ ܬܚܘܡܐ ܕܛܝ̈ܡܐ ܡܬܩ̈ܢܐ ܠܐ ܡܨܝ ܕܡܬܦܚܡܝ ܒܝܢܝ ܕܘܡ̈ܝܐ ܥܡ ܫܘܚܠ̈ܦܐ ܦܪ̈ܝܫܐ. ܦܘܚܡܐ ܕܡܨܥܝܘܬܐ P_M = ab/(ab + c′) (ܫܘܝܘܬܐ 20.1) – "ܒܠܟܝ ܬܢܝܐ ܝܬܝܪ ܡܢ ܟܠ"، ܐܠܐ "ܠܐ ܝܠܗ ܦܘܚܡܐ ܦܫܝܛܐ": ܗܘ ܡܨܐ ܕܣܠܩ ܠܥܠ ܡܢ 1 ܝܢ ܕܗܘܐ ܬܚܝܬ ܨܦܪܐ، ܠܐ ܝܠܗ ܟܪܝܐ ܐܡܬܝ ܕܟܠܢܝܬܐ ܩܪܝܒܬܐ ܝܠܗ ܠܨܦܪܐ، ܘܦܘܚܡܐ ܕܠܐ ܬܪܝܨܬܐ ܣܓܝ ܙܥܘܪܬܐ ܠܟܠܢܝܬܐ ܕܪܒܬܐ ܝܠܗ ܡܢܗ ܒܩܠܝܠ ܡܨܐ ܕܡܬܚܙܐ "ܪܡܐ" (Preacher ܘ-Kelley, 2011، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Kline، ܦܐܬܐ 354)؛ ܘܛܝ̈ܡܐ ܕܝܠܗ "ܡܨܝ ܕܗܘܝ ܠܐ ܩܝ̈ܡܐ ܐܡܬܝ ܕ-N < 500". ܓܘ ܦܐܝܠ ܕܡܬܠܐ P_M = 1.05 – ܬܪܝܨܬܐ ܩܪܝܒ ܨܦܪܐ ܘܬܚܝܬ ܨܦܪܐ – ܘܬܚܘܡܐ ܕ-bootstrap ܕܝܠܗ ܣܓܝ ܦܬܝܐ ܝܠܗ: ܐܗܐ ܚܬܝܬܐܝܬ ܝܠܗ ܥܠܬܐ. û = (a_sb_s)² (Lachowicz ܘܐܚܪ̈ܢܐ, 2018، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Kline، ܫܘܝܘܬܐ 20.3) – ܡܪܒܥܐ ܕܠܐ ܬܪܝܨܬܐ ܡܬܩܢܬܐ، ܕܫܘܝܐ ܝܠܗ ܓܘ ܟܢܫܐ ܟܠܢܝܐ ܠܡܢܬܐ ܕܫܘܚܠܦܐ ܕܦܐܪܐ ܕܡܢܒܝܢܐ ܘܡܨܥܝܐ ܟܡܦܫܩܝ ܚܕ ܓܗܐ ܒܬܪ ܬܘܪܨܐ ܩܐ ܐܣܘܪ̈ܐ ܕܓܠܐ؛ ܡܣܒ ܒܐܦ̈ܐ ܠܥܠ، ܪܫܝܬ ܓܘ ܕܘܡܝܐ ܙܥܘܪܐ. ܦܐܬܐ ܟܛܒܥܐ ܠܗ ܒܣ ܩܐ ܐܘܪܚܐ ܝܚܝܕܝܐ ܕܬܪܝܢ ܫܒ̈ܝܠܐ، ܐܝܟ ܓܘ ܦܪܫܬܐ.

ܦܘܫܩܐ: ܡܘܕܝ ܟܐܡܪܐ ܡܨܥܝܘܬܐ – ܘܡܘܕܝ ܠܐ

ܓܡܝܪܬܐ، ܡܢܬܢܝܬܐ، ܠܐ ܫܠܡܬܐ. ܫܒܝܠܐ ܬܪܝܨܐ ܕܠܐ ܝܠܗ ܦܪܝܫܐ ܡܢ ܨܦܪܐ ܓܢܒ ܠܐ ܬܪܝܨܬܐ ܕܦܪܝܫܬܐ ܝܠܗ – "ܡܨܥܝܘܬܐ ܓܡܝܪܬܐ"؛ ܬܪܝܨܬܐ ܘܠܐ ܬܪܝܨܬܐ ܥܡ ܗܘ ܪܘܫܡܐ – "ܡܢܬܢܝܬܐ" (Kline, 2023, p. 127). ܐܡܬܝ ܕܩܐ ܠܐ ܬܪܝܨܬܐ ܘܩܐ ܬܪܝܨܬܐ ܐܝܬ ܪ̈ܘܫܡܐ ܗܦܝ̈ܟܐ – "ܡܨܥܝܘܬܐ ܠܐ ܫܠܡܬܐ" (inconsistent mediation): ܟܠܢܝܬܐ ܡܨܝܐ ܕܗܘܝܐ ܩܪܝܒܬܐ ܠܨܦܪܐ ܐܦ ܐܡܬܝ ܕܠܐ ܬܪܝܨܬܐ ܪܒܬܐ ܝܠܗ، ܘܡܛܠ ܗܕܐ ܒܥܬܐ ܥܬܝܩܬܐ ܕ"ܐܣܘܪܐ ܪܫܝܡܐ ܒܝܢܝ ܡܢܒܝܢܐ ܘܦܐܪܐ" ܐܝܟ ܫܪܛܐ ܩܕܡܝܐ ܩܐ ܡܨܥܝܘܬܐ ܟܡܣܠܝܐ ܠܗ ܒܛܥܝܘܬܐ؛ ܐܡܬܝ ܕܫܘܠܛܢܐ ܥܠ ܡܨܥܝܐ ܟܡܚܝܠ ܐܣܘܪܐ – "ܡܛܫܝܢܘܬܐ" (suppression) (ܬܡܢ، ܦܐܬܐ 127-128). ܩܪܝܢܐ ܕܦܐܪܐ ܓܘ ܡܦܩܢܐ ܟܡܦܪܫ ܬܠܬܐ ܓܕ̈ܫܐ.

ܬܫܥܝܬܐ ܕܡܨܥܝܘܬܐ. Kline (2023، ܦܐܬܐ 349) ܟܫܪܐ ܨܚܚܐ ܕܡܨܥܝܘܬܐ ܒ"ܬܫܥܝܬܐ" (myth) – ܗܝܡܢܘܬܐ ܕܒܘܨܝܐ ܕܡܨܥܝܘܬܐ ܥܕܢܝܐ ܫܪܝܪܐܝܬ ܟܫܥܪܐ ܡܨܥܝܘܬܐ، ܐܡܬܝ ܕܬܘܩܢܐ ܕܚܕ ܙܒܢܐ ܝܠܗ (cross-sectional)، ܫܪ̈ܛܐ ܠܐ ܝܢܐ ܡܬܡܠܠ̈ܐ، ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܪܫܝܡܬܐ ܚܫܝܒܬܐ ܝܠܗ "ܚܘܝܐ" ܘܡܘ̈ܕܝܠܐ ܫܘ̈ܝܐ ܠܐ ܝܢܐ ܡܥܗ̈ܕܐ. ܬܠܬܐ ܡܬܩ̈ܠܐ ܓܘ ܒܘܨܝܐ ܕܚܕ ܙܒܢܐ (ܦܐܬܐ 350): (1) "ܠܝܬܘܬܐ ܕܩܕܡܘܬܐ ܓܘ ܙܒܢܐ" – ܡܨܥܝܘܬܐ ܦܪܝܫܬܐ ܝܠܗ ܐܝܟ ܫܘܢܝܐ ܕܟܥܒܪ ܡܢ ܥܠܬܐ ܠܥܠܬܐ، ܘܓܘ ܡܫܘܚܬܐ ܚܕܐ ܠܐ ܡܨܝܐ ܝܠܗ ܠܡܫܥܪܐ ܫܘܢܝܐ؛ (2) ܕܠܐ ܛܟܣܐ ܕܪ̈ܥܝܢܐ ܚܝܠܬܢܐ ܓܘ ܙܒܢܐ ܠܐ ܡܨܝܐ ܝܠܗ ܠܓܒܝܐ ܡܘܕܝܠ ܚܕ ܥܠ ܡܘ̈ܕܝܠܐ ܫܘ̈ܝܐ – ܘܓܘ ܡܘܕܝܠ ܕܬܠܬܐ ܡܫܬܚܠܦ̈ܢܐ ܐܝܬ 18 ܐܝܟ ܗܢܝ̈، ܡܢܝܗܘܢ ܕܠܐ ܡܨܥܝܘܬܐ ܟܠ ܥܩܪ (ܨܘܪܬܐ 20.1)؛ (3) ܡܣܒ ܒܐܦ̈ܐ ܡܢ ܥܠܬܐ ܡܫܘܬܦܬܐ ܕܠܐ ܡܫܝܚܬܐ ܝܠܗ. Thoemmes (2015, as cited in Kline, pp. 352-353) ܚܘܝܠܗ ܕܗܘܦܟܐ ܕܓܐܪ̈ܐ ܩܐ ܓܒܝܐ ܡܘܕܝܠ ܥܡ ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܪܫܝܡܬܐ ܠܐ ܝܠܗ ܬܪܝܨܐ – ܡܘܕܝܠ ܛܥܝܐ ܣܓܝ ܓܗ̈ܐ ܟܝܗܒ ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܒܪܒܘܬܐ ܕܡܝܬܐ، ܘܡܘܕܝܠ ܬܪܝܨܐ ܠܐ ܝܠܗ ܡܢ ܐܠܨܘܬܐ ܗܘ ܕܐܝܬ ܠܗ ܡܥܒܕܢܘܬܐ ܪܒܬܐ ܝܬܝܪ. ܡܐ ܕܟܡܥܕܪ: ܬܘܩܢܐ ܕܐܘܪܟܐ ܕܙܒܢܐ (longitudinal) (ܦܠܓܐ-ܐܘܪܟܐ ܝܢ ܓܡܝܪܐ – ܦܐܬܐ 356-357)، ܡܫܬܚܠܦܢܐ instrumental، ܡܚܘ̈ܝܢܐ ܛܒ̈ܐ ܘܐܘܪ̈ܚܬܐ ܦܪ̈ܝܫܐ ܕܡܫܘܚܬܐ ܩܐ ܡܨܥܝܐ ܘܩܐ ܦܐܪܐ (ܦܐܬܐ 353).

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

bootstrap ܠܐ ܝܠܗ ܚܪܫܐ. Kline (2023، ܦܐܬܐ 129): ܫܘ̈ܥܪܐ ܕ-bootstrap "ܡܨܝ ܕܗܘܝ ܣܓܝ ܡܣܒܝ ܒܐܦ̈ܐ ܓܘ ܕܘܡ̈ܝܐ ܙܥܘܪ̈ܐ، ܪܫܝܬ ܐܢ ܦܘܠ̈ܓܐ ܕܕܘܡܝܐ ܠܐ ܟܡܚܘܝ ܦܘܠ̈ܓܐ ܕܟܢܫܐ ܟܠܢܝܐ". Byrne (2016، ܦܐܬܐ 369-370) ܟܡܢܝܐ ܬܚܘ̈ܡܐ: ܕܘܡܝܐ ܫܪܫܝܐ ܣܢܝܩܐ ܝܠܗ ܕܡܚܘܐ ܟܢܫܐ ܟܠܢܝܐ؛ ܚܙ̈ܝܬܐ (observations) ܣܢܝܩ̈ܐ ܝܢܐ ܕܗܘܝ ܠܐ ܬܠܝ̈ܐ ܘܥܡ ܗܘ ܦܘܠܓܐ (Yung ܘ-Bentler, 1996، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘܗ)؛ ܓܘ ܝܗܝ̈ܒܬܐ ܢܘܪܡܠ̈ܝܬܐ ܛܥܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ ܕ-bootstrap ܝܬܝܪ ܡܣܒ ܒܐܦ̈ܐ ܝܢܐ ܡܢ ܕ-ML (Ichikawa ܘ-Konishi, 1995، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘܗ)؛ ܘ-bootstrap "ܫܪܝܪܐܝܬ ܠܐ ܝܠܗ ܣܡܐ ܕܟܠ ܡܕܡ ܩܐ ܕܘܡ̈ܝܐ ܙܥܘܪ̈ܐ". ܩܘܝܡܐ ܕܣܒܠܬܐ: Byrne (ܦܐܬܐ 370) ܟܡܝܬܝܐ Arbuckle (2007، ܐܝܟ ܕܡܬܡܠܠ ܓܘܗ): ܓܘ bootstrap ܟܩܝܡܚ ܛܥܢܐ ܚܕ ܩܐ ܟܠ ܥܠܬܐ، ܠܐ ܫܘܚܠܦܐ ܕܥܠܬܐ – ܩܘܝܡܐ ܕܫܘܚܠ̈ܦܐ ܥܠ 1 "ܟܡܝܬܐ ܛܥܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ ܕ-bootstrap ܣܓܝ ܢܦܝ̈ܚܬܐ" (Hancock ܘ-Nevitt, 1999، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘܗ). ܡܛܠ ܗܕܐ ܓܘ ܦܐܬܐ – ܒܣ marker؛ ܡܥܒܕ̈ܢܘܝܬܐ ܡܬܩ̈ܢܬܐ ܠܐ ܝܢܐ ܬܠܝ̈ܐ ܒܐܗܐ ܓܘܒܝܐ.

ܪܒܘܬܐ ܕܕܘܡܝܐ ܘܫܪ̈ܛܐ ܕ-SEM – ܐܝܟ ܓܘ ܦܐܬܐ ܡܘܕܝܠ ܕܫܘܝ̈ܘܬܐ ܒܢܝܢܝ̈ܬܐ: ܬܚܝܬ 200 ܡܫܬܘܬܦ̈ܢܐ، ܝܢ ܒܨܝܪ ܡܢ 10 ܩܐ ܟܠ ܦܪܡܝܛܪ، ܦܐܬܐ ܟܡܙܗܪܐ (Kline, 2023, p. 16)؛ ܦܣܘ̈ܩܐ ܥܡ ܒܨܝܪ ܡܢ ܚܡܫܐ ܓܢ̈ܣܐ (Rhemtulla et al., 2012, p. 354)؛ ܦܘܫܪܐ ܠܐ ܡܩܒܠܐ (Kline, p. 421). Kline (ܦܐܬܐ 354) ܟܡܘܣܦ ܕܦܘܚܡܐ ܕܡܨܥܝܘܬܐ P_M ܡܨܐ ܕܗܘܐ ܠܐ ܩܝܡܐ ܬܚܝܬ 500 ܡܫܬܘܬܦ̈ܢܐ. ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ – ܫܩܠܬܐ ܕܣܕܪܐ ܟܠܗ ܩܕܡ bootstrap (ܐܝܟ lavaan).

ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. ܡܨܥܝܘܬܐ ܬܠܝܬܐ (ܡܪܟܟܢܘܬܐ ܕܫܒܝܠܐ – "conditional process analysis"، Kline، ܦܐܬܐ 358-361)، ܡܘ̈ܕܝܠܐ ܕܐܘܪܟܐ ܕܙܒܢܐ ܥܡ ܡܥܒܕ̈ܢܘܝܬܐ ܡܐܘܚܪ̈ܬܐ، ܡܨܥܝܘܬܐ ܕܥܠܬܢܘܬܐ ܐܝܟ ܣܘܟܠܐ ܕ-Pearl (ܡܥܒܕ̈ܢܘܝܬܐ ܟܝܢܝ̈ܬܐ، ܦܐܬܐ 360)، ܬܚܘܡܐ BCa ܓܡܝܪܐ (ܥܡ ܡܢܝܢܐ ܡܫܘܬܦܐ ܕܡܣܪܗܒܢܘܬܐ – ܒܣ bca.simple)، bootstrap ܕ-Bollen-Stine ܩܐ ܒܘܚܪܢܐ ܕܦܐܝܘܬܐ، ܡܘ̈ܕܝܠܐ ܠܐ ܪܩܘܪܣܝܒ̈ܝܐ ܘܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܕܫܘܓܫ̈ܐ (disturbances)، ܦܘܚܡܐ ܕܓܘ̈ܕܐ ܘܠܐ ܡܫܬܚܠܦܢܘܬܐ ܕܡܫܘܚܬܐ.

ܘܕܟܝ ܟܥܒܕܚ ܐܗܐ ܒ-SPSS ܘ-AMOS؟

ܓܘ SPSS Statistics ܠܝܬ ܡܘ̈ܕܝܠܐ ܕܫܘܝ̈ܘܬܐ ܒܢܝܢܝ̈ܬܐ (Pallant, 2020, Chapter 10)؛ ܩܐ ܡܨܥܝܘܬܐ ܒܝܢܝ ܡܢܝ̈ܢܐ ܕܣܒܠܬܐ ܐܝܬ ܡܐܩܪܘ (macro) PROCESS ܕ-Hayes (2022, as cited in Kline, 2023, pp. 129, 358)، ܕܟܚܫܒ ܬܚܘ̈ܡܐ ܕ-bootstrap ܩܐ ܡܫܬܚܠܦ̈ܢܐ ܚܙ̈ܝܐ ܘܟܣܐܡ "ܨܦܪܐ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ" (Kline, p. 358). ܓܘ AMOS ܟܨܝܪܚ ܡܘܕܝܠ ܐܝܟ ܓܘ ܦܐܬܐ ܕ-SEM، ܘܓܘ ܟܘܬܐ Analysis Properties، ܓܘ ܠܫܢܐ Bootstrap، ܟܪܫܡܚ Perform bootstrap، ܡܢܝܢܐ ܕܕܘܡ̈ܝܐ ܘ-Bias-corrected confidence intervals – ܓܘܒܝܐ ܡܩܕܡܐ 90% (Byrne, 2016, pp. 373-374). AMOS ܟܬܢܐ ܓܘ ܟܢܘܫܝܐ ܟܡܐ ܕܘܡ̈ܝܐ "ܠܐ ܦܠܝܚ̈ܐ ܝܢܐ" ܡܛܠ ܕܠܐ ܐܫܬܟܚ ܦܘܫܪܐ (ܬܡܢ، ܦܐܬܐ 382) – ܐܝܟ ܣܕܪܐ ܕ-bootstrap ܓܘ ܪܝܫܐ ܕܡܦܩܢܐ ܠܟܐ. ܦܘܪ̈ܫܢܐ ܡܣܟ̈ܝܐ: AMOS ܟܚܫܒ χ² ܥܡ ܡܢܝܢܐ ܬܚܬܝܐ N − 1 (Byrne, p. 93)، ܘܡܘܠܕܢܐ ܕܡܢܝ̈ܢܐ ܕܝܠܗ ܐܚܪܢܐ ܝܠܗ – ܬܚܘ̈ܡܐ ܒܕ ܗܘܝ ܩܪܝܒ̈ܐ ܐܠܐ ܠܐ ܫܘ̈ܝܐ.

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

ܓܘ lavaan ܟܦܪܫܚ ܪܘ̈ܫܡܐ (labels) ܩܐ ܫܒ̈ܝܠܐ (a*trust)، ܡܥܒܕ̈ܢܘܝܬܐ ܒ-:=، ܘܟܒܥܚ se = "bootstrap" (Rosseel, 2012, Section 7.4)؛ iseed ܟܦܪܫ ܙܪܥܐ. ܬܚܘ̈ܡܐ ܕܬܘܟܠܢܐ – ܓܘ parameterEstimates ܥܡ boot.ci.type = "perc" ܝܢ "bca.simple". button "ܡܚܬܬܐ ܕܣܩܪܝܦܬ R" ܓܘ ܡܦܩܢܐ ܟܟܬܒ ܡܘܕܝܠ ܕܦܪܫܠܘܟܘܢ ܥܡ ܗܘ ܡܢܝܢܐ ܕܕܘܡ̈ܝܐ ܘܗܘ ܙܪܥܐ. ܣܩܪܝܦܬ ܕܠܬܚܬ ܪܗܝܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ (lavaan 0.6-17)، ܘܦܐܬܐ، ܥܡ 1,000 ܕܘܡ̈ܝܐ ܘܙܪܥܐ 2026، ܟܝܗܒܐ ܗܢܝ̈ ܡܢܝ̈ܢܐ.

# ܡܨܥܝܘܬܐ ܒܝܢܝ ܥܠ̈ܠܬܐ ܟܣ̈ܝܬܐ ܥܡ bootstrap – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)   # ܩܪܝܢܐ ܕܐܩܣܠ
library(lavaan)   # ܦܘܩܕ̈ܢܐ sem, parameterEstimates

# ܛܥܢܐ: ܛܘܦܣܐ ܬܩܝܢܐ – ܣܕܪܐ = ܡܫܬܘܬܦܢܐ، ܥܡܘܕܐ = ܦܣܘܩܐ؛ ܓܠܝܘܢܐ "נתונים"
d <- read_excel("C:/Rdata/DATA_cfa_example_heb.xlsx", sheet = "נתונים")
items <- c(paste0("שייכות_", 1:4), paste0("אמון_", 1:4), paste0("השתתפות_", 1:4))
d <- as.data.frame(d[, items])
names(d) <- c(paste0("b", 1:4), paste0("t", 1:4), paste0("p", 1:4))   # ܫܡܗ̈ܐ ܩܛܝ̈ܢܐ ܒܐܢܓܠܝܙܝܐ ܩܐ ܡܘܕܝܠ

# ܡܘܕܝܠ: ܬܘܟܠܢܐ ← ܕܝܠܢܝܘܬܐ ← ܫܘܬܦܘܬܐ، ܘܐܦ ܬܘܟܠܢܐ ← ܫܘܬܦܘܬܐ؛ ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ، ܟܠܢܝܬܐ ܘܦܘܚܡܐ ܕܡܨܥܝܘܬܐ
model <- "
  belong =~ b1 + b2 + b3 + b4
  trust  =~ t1 + t2 + t3 + t4
  partic =~ p1 + p2 + p3 + p4
  belong ~ a*trust
  partic ~ b*belong + c*trust
  ind := a*b
  tot := c + a*b
  pm  := (a*b) / (c + a*b)
"
# 1000 ܕܘܡ̈ܝܐ ܕ-bootstrap؛ iseed – ܙܪܥܐ (ܐܝܟ ܓܘ ܦܐܬܐ: ܗܘ ܙܪܥܐ، ܗܢܝ̈ ܕܘܡ̈ܝܐ)
fit <- sem(model, data = d, se = "bootstrap", bootstrap = 1000, iseed = 2026)
pe <- parameterEstimates(fit, boot.ci.type = "perc")
pe[pe$op %in% c("~", ":="), c("lhs", "op", "rhs", "label", "est", "se", "ci.lower", "ci.upper")]
# ܛܝ̈ܡܐ: a 0.531 [0.402, 0.681]; b 0.529 [0.363, 0.712]; c -0.014 [-0.187, 0.158]
# ܛܝ̈ܡܐ: ind 0.281 (SE 0.061) [0.174, 0.412]; tot 0.267 [0.121, 0.438]; pm 1.051 (SE 1.104) [0.574, 2.252]
pb <- parameterEstimates(fit, boot.ci.type = "bca.simple")       # ܡܬܘܪܨܐ ܡܢ ܡܣܒ ܒܐܦ̈ܐ
pb[pb$op == ":=", c("lhs", "est", "ci.lower", "ci.upper")]          # ܛܝ̈ܡܐ: ind [0.175, 0.413]

# ܩܐ ܦܘܚܡܐ – ܐܘܪܚܐ ܕ-delta (ܕܠܐ bootstrap)
fit_ml <- sem(model, data = d)
parameterEstimates(fit_ml)[parameterEstimates(fit_ml)$op == ":=", c("lhs", "est", "se", "pvalue")]
# ܛܝ̈ܡܐ: ind SE 0.060; pm SE 0.332 – ܩܘܒܠ 1.104 ܓܘ bootstrap: delta ܠܐ ܟܚܙܝܐ ܠܐ ܩܝܡܘܬܐ ܕܦܘܚܡܐ

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܟܬܢܚ ܡܘܕܝܠ (ܡܢܒܝܢܐ، ܡܨܥ̈ܝܐ، ܦܐܪܐ، ܫܒܝܠܐ ܬܪܝܨܐ ܝܢ ܠܐ)، ܦܐܝܘܬܐ ܕܡܘܕܝܠ (model fit)، ܫܒ̈ܝܠܐ (b ܥܡ ܛܥܝܘܬܐ ܬܩܝܢܬܐ، ܘ-β)، ܘܩܐ ܟܠ ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ – ܫܘܥܪܐ (estimate)، ܬܚܘܡܐ ܕܬܘܟܠܢܐ ܕ-bootstrap ܘܙܢܐ ܕܝܠܗ (ܕܪ̈ܓܐ ܡܢ ܡܐܐ ܝܢ ܡܬܘܪܨܐ ܡܢ ܡܣܒ ܒܐܦ̈ܐ)، ܡܢܝܢܐ ܕܕܘܡ̈ܝܐ ܘܙܪܥܐ؛ ܥܡ ܟܡܐ ܡܨܥ̈ܝܐ – ܐܦ ܟܠ ܐܘܪܚܐ ܒܦܪܝܫܘܬܐ. Kline (2023، ܦܐܬܐ 128) ܟܡܥܗܕ ܕܠܐ ܝܠܗ ܣܢܝܩܐ ܠܡܩܪܝܐ ܬܚܘܡܐ ܐܝܟ ܒܘܚܪܢܐ: "ܟܠ ܛܝ̈ܡܐ ܕܓܘ ܬܚܘܡܐ ܕܬܘܟܠܢܐ ܒܥܩܪܐ ܫܘ̈ܝܐ ܝܢܐ"، ܘܨܦܪܐ ܕܓܘܗ "ܠܐ ܝܠܗ ܝܬܝܪ ܝܚܝܕܝܐ ܡܢ ܟܠ ܛܝܡܐ ܐܚܪܢܐ" ܕܓܘܗ. Kline (ܦܐܬܐ 422): "ܛܥܝܘܬܐ ܝܠܗ ܠܡܬܢܝܐ ܒܣ ܦܘܫܪܐ ܡܬܩܢܐ". APA (2020، ܦܣܩܐ 6.36) – ܕܠܐ ܨܦܪܐ ܩܕܡ ܢܘܩܙܐ ܩܐ β ܘܩܐ p، ܘܥܡ ܨܦܪܐ ܩܐ b؛ ܠܘܚܐ ܥܡ ܣܕܪܐ ܩܐ ܟܠ ܡܥܒܕܢܘܬܐ – ܐܝܟ ܠܘܚܐ 3 ܓܘ ܡܦܩܢܐ. ܘܓܘ ܨܚܚܐ ܕܡܬܘܪܬܐ – ܬܘܩܢܐ (ܚܕ ܙܒܢܐ ܝܢ ܐܘܪܟܐ ܕܙܒܢܐ) ܘܡܘ̈ܕܝܠܐ ܫܘ̈ܝܐ، ܕܕܠܐ ܐܢܝ̈ "ܡܨܥܝܘܬܐ" ܛܥܢܬܐ ܝܠܗ ܚܝܠܬܢܝܬܐ ܡܢ ܝܬܝܪ ܡܢ ܝܗܝ̈ܒܬܐ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

"ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܪܫܝܡܬܐ ܝܠܗ – ܡܨܥܝܘܬܐ ܚܘܝܬܐ ܝܠܗ". ܓܘ ܬܘܩܢܐ ܕܚܕ ܙܒܢܐ ܐܗܐ ܗܘܦܘܬܣܝܣ ܝܠܗ ܕܝܗܝ̈ܒܬܐ ܟܦܐܝܝ ܥܡܗ، ܠܐ ܚܘܝܐ (Kline, 2023, pp. 349-350).

ܠܡܒܥܝܐ ܩܕܡ ܐܣܘܪܐ ܪܫܝܡܐ ܒܝܢܝ ܡܢܒܝܢܐ ܘܦܐܪܐ. ܓܘ ܡܨܥܝܘܬܐ ܠܐ ܫܠܡܬܐ ܟܠܢܝܬܐ ܡܨܝܐ ܕܗܘܝܐ ܨܦܪܐ ܐܦ ܐܡܬܝ ܕܠܐ ܬܪܝܨܬܐ ܪܒܬܐ ܝܠܗ (Kline, p. 127).

ܠܡܬܟܠܐ ܥܠ Sobel. ܦܘܠܓܐ ܕ-product ܠܐ ܫܘܝܐ ܝܠܗ ܠܬܪܝܢ ܓܢ̈ܒܐ؛ ܟܬܢܚ ܬܚܘܡܐ ܕ-bootstrap.

ܠܡܫܚܠܦܐ ܙܪܥܐ ܗܠ ܕܬܚܘܡܐ ܠܐ ܟܡܚܘܐ ܨܦܪܐ. ܙܪܥܐ ܟܦܪܫܚ ܡܢ ܩܕܡ ܘܟܬܢܚ؛ ܐܢ ܦܐܪܐ ܬܠܝܐ ܝܠܗ ܒܙܪܥܐ – ܥܠ ܬܚܘܡܐ ܝܠܗ.

ܠܡܩܪܝܐ P_M ܐܝܟ "ܡܢܬܐ ܒܡܐܐ ܕܡܨܥܝܘܬܐ". ܗܘ ܡܨܐ ܕܣܠܩ ܠܥܠ ܡܢ 1، ܐܝܟ ܓܘ ܡܬܠܐ؛ ܟܬܢܚ ܓܢܒܗ ܠܐ ܬܪܝܨܬܐ ܡܬܩܢܬܐ.

ܠܡܗܦܟܐ ܓܐܪ̈ܐ ܘܠܓܒܝܐ ܡܘܕܝܠ ܥܡ ܡܥܒܕܢܘܬܐ ܠܐ ܬܪܝܨܬܐ ܪܒܬܐ. ܡܘ̈ܕܝܠܐ ܫܘ̈ܝܐ ܠܐ ܡܨܝ ܕܦܣܩܝ ܣܛܐܛܝܣܛܝܩܐܝܬ (Thoemmes, 2015, as cited in Kline, pp. 352-353).

ܡܒܘ̈ܥܐ

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

Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). Guilford Press.

Byrne, B. M. (2016). Structural equation modeling with AMOS: Basic concepts, applications, and programming (3rd ed.). Routledge.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6(1), 1-55. https://doi.org/10.1080/10705519909540118

Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford Press.

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

Rhemtulla, M., Brosseau-Liard, P. É., & Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17(3), 354-373. https://doi.org/10.1037/a0029315

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1-36. https://doi.org/10.18637/jss.v048.i02

Efron (1987)، Hayes (2022)، Iacobucci ܘܐܚܪ̈ܢܐ (2007)، Jackson (2003)، Lachowicz ܘܐܚܪ̈ܢܐ (2018)، Pearl (2009)، Preacher ܘ-Kelley (2011)، Sobel (1982) ܘ-Thoemmes (2015) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Kline (2023)؛ Arbuckle (2007)، Hancock ܘ-Nevitt (1999)، Ichikawa ܘ-Konishi (1995) ܘ-Yung ܘ-Bentler (1996) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Byrne (2016)؛ Sobel (1982) – ܐܦ ܓܘ Rosseel (2012). ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܒܢܘܣܚܐ ܫܪܫܝܐ. ܥܝ̈ܕܐ ܕܣܐܝܛ: ܓܘܒܝܐ ܡܩܕܡܐ ܕ-1,000 ܕܘܡ̈ܝܐ ܘܙܪܥܐ 2026، ܡܚܘܝܐ ܕܬܪܝܢ ܙܢ̈ܐ ܕܬܚܘܡܐ ܚܕ ܓܢܒ ܚܕ، ܘܒܣ marker. ܚܘܫܒܢܐ ܓܘ ܦܐܬܐ ܫܪܝܪܐ ܝܠܗ ܩܘܒܠ lavaan 0.6-17 ܓܘ R – ܗܢܝ̈ ܕܘܡ̈ܝܐ، ܗܢܝ̈ ܬܚܘ̈ܡܐ.