Fall Homework Chapter 4

Transcription

1 Fall 18 1 Homework Chapter 4 1) Starting values do not need to be theoretically driven (unless you do not have data) 2) The final results should not depend on starting values 3) Starting values can be obtained from the estimates of simple (single factor) model 4) Adding correlated errors (identified using /LMtest set=pee;) tend to be more effective in improving model fit than adding cross loadings 5) Non-nested models can be compared using BIC, AIC (relative sense), CFI (each model is compared against the independence model), RMSEA (each model is compared against the saturated model)

2 Fall 18 2 Oct 24 Modeling Means (chapter 8 of EQS manual) Regression with means (Fig 8.1, p205) X = μx + Ex 1 (=V999) Y = α + βx + Ey α (intercept) (V1 = V999 + E1) (V2 = V999 + E2) μx Y (=V2) ε(=e2) (mean of X) β X (=V1) Xd (=E1) μy = α + βμx (is NOT a parameter of the model) X is a dependent variable V999 = 1 is an independent variable that has no variance or covariance Use Mardia et al s data as an example for running EQS

3 Fall 18 3 Figure 8.2 (EQS manual, p212) E1 E2 E3 E4 V1 V2 V3 V4 1 1 F1 F2 V999 D1 D2 Mean Structure model: means of V1, V2, V3, V4 V5 explained by the means of F1 & F2 E1 E2 E3 E4 E4 mechanics vectors algebra analysis statistics 1 1 F2 F1 D1 D2 V999 Covariance structure model 4 means of V1, V2, V3, V4 are parameters (for 4 sample means)

4 Fall 18 4 E1 E2 E3 E4 V1 V2 V3 V4 1 1 F1 F2 V999 D2 Mean Structure parameters = regression coefficients, variances and covariances of IV, intercepts of DV the coefficient of regression on V999 is an intercept the intercept will be the mean if there is no indirect effect from V999 V999 has no variance or covariances (in X = V999 + E1, the covariances of X and any other variables are through E) model should be identified without mean structure (at least should be saturated) (an unidentified model won t become identified if you simply add the mean) /specifications Analysis = moments; /print Effect = yes; (the total effect of V999 on a DV is the mean of that DV) /print covariance = yes; (this gives the predicted matrix) V F (Σ represents a variance / covariance) V ΣVV ΣVF F ΣFV ΣFF (F1, V999) = # the mean of F (i.e. if you want to make a constraint)

5 Default likelihood ratio statistic 2lnR =n[ tr(sσ -1 (θˆ)) log SΣ -1 (θˆ) - p+ (x- μ(θˆ)) Σ -1 (θˆ) (x- μ(θˆ))] Fall 18 5

6 Fall 18 6 Muthen & Khoo (1998) Younger cohort (male, N = 1393) 1989 data on education V1 = math7 V2 = math8 V3 = math9 V4 = math10 V5 = mother education V6 = home resources Sample means: V1 V2 V3 V4 V5 V Sample correlations: V1 V2 V3 V4 V5 V6 V1 -- V V V V V Sample SD: V1 V2 V3 V4 V5 V Linear Growth Model for 4 time points

7 Fall 18 7 E1 Vt1 1 Dα 1 α E2 Vt2 1 μα 1 V999 E3 Vt3 1 μβ 2 3 β E4 Vt4 Dβ All factor loadings are known simplified version of the 2 factor model Different people have different intercepts (α) Dα = how much kids differ at age 7 Dβ = difference in growth rate / slope Growth curve model is a special case also (more restricted) of the factor model What is mean change explained by this model? Mean of V1 = μα1 + E1 Mean of V2 = μα1 + μβ1 + E2 Mean of V3 = μα1 + μβ2 + E3 Mean of V4 = μα1 + μβ3 + E4 μα = average intercept μβ = average slope (average change over one year) Muthen1.EQS (Linear growth, younger male, without background variable) Muthen2.EQS (Linear growth, younger female, without background variable) Muthen3.EQS (Nonlinear growth, younger male, without background variable) Muthen4.EQS (Nonlinear growth, younger female, without background variable)

8 Fall 18 8 Growth Curve Model Let yit be the math score for student i at grade t t = 1, 2,, p; i = 1, 2,, N we might use a linear model to describe the growth pattern within individual: yit = αi + βi(t 1) + ζit or: yi1 = αi + ζi1 yi2 = αi + βi + ζi2 yi3 = αi + 2βi + ζi3 yi4 = αi + 3βi + ζi4 Between individuals: αi = μα + δαi βi = μβ + δβi With Time-varying Covariate vit and time-invariant covariates wi, xi Within individual (level 1): yit = αi + βi(t 1) + πtvit + ζit Between individuals (level 2): αi = μα + γαwi + καxi + δαi βi = μβ + γβwi + κβxi + δβi Non-linear growth: yit = αi + g(t)βi + πtvit + ζit a) for linear growth: g(t) = t 1 b) for quadratic growth: g(t) = (t 1) 2

9 Fall 18 9 Variance Components within individual o Yit = αi + βi(t-1) + ζit same model as V1=F1+E1; V2=F1+F2+E2 V3=F1+2F2+E3 Between individuals o αi = μα + δαi Put it together: o Yit = μα + μβ(t-1) + [δαi + (t-1)δβi + ζit] Mean and Variance o E(Yit) = μα + μβ(t-1) o Variance(Yit) = σ 2 α + 2(t-1)σαβ + (t-1) 2 σ 2 β + σ 2 ζ o σαβ positive for the 2 nd graph Plots of individual trajectory exhibit a fan shape Muthen5.EQS (Nonlinear growth, younger male, with background variables) Muthen6.EQS (Linear growth, younger female, with background variables)

10 Fall Can also leave factor loadings for the slope unknown (): Non-linear growth model E1 Vt1 1 Dα 1 α E2 Vt2 1 μα 1 V999 E3 Vt3 1 μβ β E4 Vt4 Dβ Non-linear growth model for 4 time points with 2 time-invariant covariates: E1 Vt1 1 1 α Dα HR E2 Vt2 1 mean of HR mean int. 1 V999 mean slope E3 Vt3 1 mean of MED β MED E4 Vt4 Dβ If these = 1, 2, 3, then it is a linear growth model Residuals (Dα and Dβ) should be smaller with these added predictors (MED and HR)

11 Fall Sequential Cohort V1 1 HR 1 α V2 1 1 V999 1 V3 β V4 MED V4 1 HR V5 V6 1 α 1 1 β V999 MED 2 Cohorts: combine so dotted are constrained to be equal everything on right side equal in the two models non-linear growth model Muthen7.EQS (Equality of means and variance at the connection, male, with background variables)

12 Muthen8.EQS (Modeling continuity using data from two cohorts, male, with background variables) Fall 18 12

13 Fall V1 = α + E1 V2 = α + β + E2 V3 = α + 2β + E3 V4 = α + 3β + E4 α = a + c11hr + c12med + Dα β = b + c21hr + c22med + Dβ a = between α and V999 b = between β and V999 c11 = between α and HR c12 = between α and MED c21 = between β and HR c22 = between β and MED HR = μhr + E5 (HR = E5) (observed error variance) MED = μmed + E6 (MED = E6) μhr = between V999 and HR μmed = between V999 and MED Muthen1.EQS (Linear growth, younger male, without background variable) For Growth Curve Models SEM allows consideration of measurement error in the predictor while HLM does not HLM can handle non-rectangular data, SEM cannot For the example considered, the two approaches are identical because HR and MED are not considered with errors, and the data are rectangular.

14 Fall Multiple Groups with Mean Structure Head Start Program example (EQS manual, p.228) V1 = Mother education, V2=Father education, V3 = Father Occupation, V4 = Income, V5 = Metropolitan Readiness Test, V6 = Illinois Test of Psycholinguistic Ability Control group N = 155 Head Start group N = 148 Sample means: V1 V2 V3 V4 V5 V6 Control HS F1 = demographics (covariates) F2 = achievement factors

15 Fall free parameter E1 = free and equal between groups,0 1 group free, 1 group fixed to 0 V1 1.0 E2 = D1 = V2 F1 E3 = =,0 V3 = = E4 V999 (=1) = = V4 E5 = D2,0 V5 = 1.0 F2 E6 = V6

16 Fall E1 Solution: χ 2 23 = 27.45, CFI =.987 V1 1.0 E2 3.9 D1.85 V2 F1 E V E4 V999 (=1) 6.4 V4 E D2.18 V5 1.0 F2 E6.85 V6

17 Fall Latent mean comparison with experimental data Example came from Bentler & Woodward (1978) p. 493 Education Quarterly (also in the EQS manual, p. 235) Olsson s Experiment: Verbal Ability Training V1, V3 = Synonyms V2, V4 = Antonyms Pre Post V1 V2 V3 V4 Control Experimental Covariance Matrices: Control Experimental V1 V2 V3 V4 V1 V2 V3 V

18 Fall E1 1.0 E2 D1 = = V2 = F1 (0,) (=) D2 = 0, V999 E3 = F2 1.0 V3 = = E4 Tests if your groups were equal to start with Tests if your groups were different after training V1 V4 Multisample with Mean and Covariance Structures not identified model with 1 group may be identified for multigroup situation set the intercepts for the measured variables = across groups set the factor intercepts of 1 group (reference group) to 0 the intercepts for the reference group are the expected means of this group set the initial value (starting values) of the intercepts for the measured variables at the sample means of the reference group Reference: Sorbom, D. (1974). A general method for studying differences in factor means and factor structures between groups. British Journal of Mathematical and Statistical Psychology, 27,