Measurement Equivalence Using Structural Equation Modeling T echniques Joseph J. Sudano, PhD Assistant Professor, Medicine and Population and Quantitative Health Sciences, Case Western Reserve University Center for Health Care Research and Policy at MetroHealth Medical Center Academy Health Annual Research Meeting New Orleans, LA June 26 th , 2017 Special Acknowledgments: Adam Perzynski, PhD;
Measurement Equivalence Using Structural Equation Modeling T echniques No financial conflicts of interest … yet … although I will mention that I am the current CEO of software startup Global Health Metrics, LLC.
Conceptual Model of Traditional OLS Regression Observed or manifest variables e1 Happy D1 Education e2 Feel Blue Depression Occupation e3 Tired e4 Income Enjoy Life Latent construct or factor 3
Causal Modeling and Confirmatory Factor Analysis (A special instance of Structural Equation Modeling) Education e1 Happy e2 Feel Blue Income Depression e3 Tired D1 D3 e4 Enjoy Life Occupation D2 4
Acknowledgement: This study was funded by Grant number R01-AG022459 from the NIH National Institute on Aging. ( Med Care 2011;49: 480 – 488). Measuring Disparities: Bias in Self-reported Health Among Spanish-speaking Patients J.J. Sudano 1,2 , A.T. Perzynski 1,2 , T.E. Love 2 , S.A. Lewis 1 ,B. Ruo 3 , D.W. Baker 3 1 The MetroHealth System, Cleveland, OH; 2 Case Western Reserve University School of Medicine, Cleveland, OH; 3 Northwestern University Feinberg School of Medicine
Measurement Models using Multiple Indicators Single items are unreliable Single cases prevent generalizability Use multiple indicators and large samples to estimate the values of the latent, unobserved variables or factors The SF-36v2 uses multiple indicators describing multiple factors in order to measure health more reliably.
Measurement Model of the SF-36v2 Modified
Objective & Significance Do observed differences in self-reported health reflect true differences in health? ◦ Cultural and language differences may create measurement bias ◦ If outcomes aren’t measuring the same thing in different groups, we have a problem
Measurement Equivalence & Factorial Invariance It is only possible to properly interpret group differences after measurement equivalence has been established (Horn & McArdle, 1992; Steenkamp & Baumgartner, 1998). “It may be the case that the groups differ … but it also may be the case that extraneous influences are giving rise to the observed difference.” Meredith & T eresi (2006 p. S69) The external validity of any conclusion regarding group differences rests securely on whether the measurement equivalence of the scale has been established (Borsboom, 2006).
Cross-sectional Study N= 1281 Medical patients categorized into four groups: White, Black, English-speaking Hispanic and Spanish-speaking Hispanic. Multigroup Confirmatory Factor Analysis (MGCFA)
T wo Types of Invariance Metric (Weak) Invariance ◦ Are the item factor loadings equivalent across groups? ◦ Is a one unit change in the item equal to a one unit change in the factor score for all groups? Scalar (Strong) Invariance ◦ Are the item intercepts equivalent across groups? ◦ Unequal intercepts results in unequal scaling of factor scores
T wo Types of Invariance Metric (Weak) Invariance ◦ Are the item factor loadings equivalent across groups? ◦ Is a one unit change in the item equal to a one unit change in the factor score for all groups? Scalar (Strong) Invariance ◦ Are the item intercepts equivalent across groups? ◦ Unequal intercepts results in unequal scaling of factor scores
T wo Types of Invariance Metric (Weak) Invariance ◦ Are the item factor loadings equivalent across groups? ◦ Is a one unit change in the item equal to a one unit change in the factor score for all groups? Scalar (Strong) Invariance ◦ Are the item intercepts equivalent across groups? ◦ Unequal intercepts results in unequal scaling of factor scores
Weak invariance health Self-rated health e4 What happens to the model fit when we constrain all of these paths (loadings) to be equal across groups?
Table 1: Goodness of Fit for SF-36 Multigroup Factorial Invariance Testing (N = 1281) B-S χ 2 * ∆RMSEA ∆CFI B- S ∆χ 2 ∆df Model Description RMSEA (95% CI) CFI df Ref 1 Unconstrained Model 0.028(.017 - .030) 0.936 3001 2172 2 Metric Invariance (Factor Weights) 0.029(.028 - .030) 0.931 3110 2253 1 0.001 -0.005 109 81 3 Scalar Invariance (Intercepts) 0.033(.032 - .034) 0.907 3215 2358 2 0.004 -0.024 105 105 4 Partial Scalar Invariance (B=W=HS not HE) 0.033(.032 - .034) 0.909 3179 2323 2 0.004 -0.022 69 70 5 Partial Scalar Invariance (B=W=HE not HS) 0.030(.029 - .032) 0.921 3180 2323 2 0.001 -0.010 70 70 6 2nd Order Structural Invariance** 0.030(.029 - .032) 0.921 3187 2333 2 0.001 -0.010 77 80 7 2nd & 3rd Order Structural Invariance** 0.030(.029 - .032) 0.921 3196 2339 2 0.001 -0.010 86 86 * The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non -normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The Unconstrained Model Fits the Data Well Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281) B-S χ 2 * ∆RMSEA ∆CFI B- S ∆χ 2 ∆df Model Description RMSEA (95% CI) CFI df Ref 1 Unconstrained Model 0.028(.017 - .030) 0.936 3001 2172 2 Metric Invariance (Factor Weights) 0.029(.028 - .030) 0.931 3110 2253 1 0.001 -0.005 109 81 3 Scalar Invariance (Intercepts) 0.033(.032 - .034) 0.907 3215 2358 2 0.004 -0.024 105 105 4 Partial Scalar Invariance (B=W=HS not HE) 0.033(.032 - .034) 0.909 3179 2323 2 0.004 -0.022 69 70 5 Partial Scalar Invariance (B=W=HE not HS) 0.030(.029 - .032) 0.921 3180 2323 2 0.001 -0.010 70 70 6 2nd Order Structural Invariance** 0.030(.029 - .032) 0.921 3187 2333 2 0.001 -0.010 77 80 7 2nd & 3rd Order Structural Invariance** 0.030(.029 - .032) 0.921 3196 2339 2 0.001 -0.010 86 86 * The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non -normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The model with factor The Unconstrained Model fits loadings constrained still fits the data well the data well. Table 1: Goodness of Fit for SF-36v2 Multigroup Factorial Invariance Testing (N = 1281) B-S χ 2 * ∆RMSEA ∆CFI B- S ∆χ 2 ∆df Model Description RMSEA (95% CI) CFI df Ref 1 Unconstrained Model 0.028(.017 - .030) 0.936 3001 2172 2 Metric Invariance (Factor Weights) 0.029(.028 - .030) 0.931 3110 2253 1 0.001 -0.005 109 81 3 Scalar Invariance (Intercepts) 0.033(.032 - .034) 0.907 3215 2358 2 0.004 -0.024 105 105 4 Partial Scalar Invariance (B=W=HS not HE) 0.033(.032 - .034) 0.909 3179 2323 2 0.004 -0.022 69 70 5 Partial Scalar Invariance (B=W=HE not HS) 0.030(.029 - .032) 0.921 3180 2323 2 0.001 -0.010 70 70 6 2nd Order Structural Invariance** 0.030(.029 - .032) 0.921 3187 2333 2 0.001 -0.010 77 80 7 2nd & 3rd Order Structural Invariance** 0.030(.029 - .032) 0.921 3196 2339 2 0.001 -0.010 86 86 * The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non -normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
Metric (Weak) Invariance was Confirmed
I forget what an intercept is Scalar (Strong) Invariance ◦ Are the item intercepts equivalent across groups? Intercept: the intercept in a multiple regression model is the mean for the response when all of the explanatory variables take on the value 0. Could be called the “starting point”
The model with factor The Unconstrained Model fits loadings constrained still fits the data well the data well. Table 1: Goodness of Fit for SF-36v2 Multigroup Factorial Invariance Testing (N = 1281) B-S χ 2 * ∆RMSEA ∆CFI B- S ∆χ 2 ∆df Model Description RMSEA (95% CI) CFI df Ref 1 Unconstrained Model 0.028(.017 - .030) 0.936 3001 2172 2 Metric Invariance (Factor Weights) 0.029(.028 - .030) 0.931 3110 2253 1 0.001 -0.005 109 81 3 Scalar Invariance (Intercepts) 0.033(.032 - .034) 0.907 3215 2358 2 0.004 -0.024 105 105 4 Partial Scalar Invariance (B=W=HS not HE) 0.033(.032 - .034) 0.909 3179 2323 2 0.004 -0.022 69 70 5 Partial Scalar Invariance (B=W=HE not HS) 0.030(.029 - .032) 0.921 3180 2323 2 0.001 -0.010 70 70 6 2nd Order Structural Invariance** 0.030(.029 - .032) 0.921 3187 2333 2 0.001 -0.010 77 80 7 2nd & 3rd Order Structural Invariance** 0.030(.029 - .032) 0.921 3196 2339 2 0.001 -0.010 86 86 * The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non -normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained). Constraining the intercepts results in a worsening of model fit
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