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PM 515 Behavioral Epidemiology Measurement Modeling and Structural Equation Modeling

PM 515 Behavioral Epidemiology Measurement Modeling and Structural Equation Modeling. Ping Sun, Ph.D. Jennifer Unger, Ph.D. Topics. Review Measurement Modeling Correction for bias induced from measurement errors in predictors Structural equation modeling Growth curve modeling

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PM 515 Behavioral Epidemiology Measurement Modeling and Structural Equation Modeling

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  1. PM 515Behavioral EpidemiologyMeasurement Modeling and Structural Equation Modeling Ping Sun, Ph.D. Jennifer Unger, Ph.D.

  2. Topics • Review • Measurement Modeling • Correction for bias induced from measurement errors in predictors • Structural equation modeling • Growth curve modeling • Twin Data Analysis

  3. Review Let’s go back to the path analysis we conducted previously

  4. Stress Depression Hostility Economic and Psychological Determinants of Health Status in Chinese Adults Self Perceived Income Disparity Relative to Others -0.11 0.12 Self Reported General Health Status Past 0.08 -0.04 0.07 0.35 0.26 0.03 -0.03 Household Income -0.03 -0.03 -0.07 0.02 Note: adjusted for age, gender, parents' highest education, spouse's general health status, child's general health status, child's academic performance, rank of school, rank of district, and city yellow links designate U shape relationships, while white links designate linear relationships.

  5. Review The path directly links what we have actually measured Or say: All constructs in the pathway are manifested and measurable variables

  6. What if Some constructs are latent and are assessed by multiple indicators? Or Some constructs are latent and are correlated with other constructs but not subject to direct assessment?

  7. We will need to embed in the pathway the ‘measurement model’ and the ‘structural equations model’.

  8. Measurement Modelwhen there are two indicators

  9. Measurement Modelwhen there are three indicators

  10. Measurement Modelexample 1 1 Hostility1 (x11) e1 1 Hostility (ξ1) Hostility2 (x12) 1 e2 1

  11. X11 = ξ1 + e1 X12 = ξ1 + e2 Measurement Modelexample 1 1 Hostility1 (x11) e1 1 Hostility (ξ1) Hostility2 (x12) 1 e2 1 Assumptions: Var(ξ1) = Phi1 Var(e1)=E1, Var(e2)=E2, Cov (e1, e2) = Cov(ξ1,e1) = Cov(ξ1,e2)=0

  12. X11 = ξ1 + e1 X12 = ξ1 + e2 Measurement Modelexample 1 Cov (x11, x12) = Var(ξ1) + Cov(ξ1, e2) + Cov(ξ1,e1) + Cov(e1,e2) = Var(ξ1) = Phi1 Var(x11) = Var(ξ1) + Var(e1) + 2*Cov(ξ1,e1) = Phi1+E1 Var(x12) = Var(ξ1) + Var(e2) + 2*Cov(ξ1,e2) = Phi1+E2

  13. SAS Statement and Output proc calis data=youth12 cov pall all; lineqs hostility = f1 + d1 , bhostility = f1 + d2; std f1 d1 d2 = p1 p2 p3; run; SAS output for example 1

  14. What can Measurement Modeling Accomplish? To make use of the multiple (=>2) assessments of the predictors or outcomes to estimate the ‘true’ variance of the constructs.

  15. Why do we need to estimate the ‘true’ variance of the latent constructs? • To refine the dependent variables to increase power for hypothesis testing • To refine the independent variables to correct biases induced to estimated parameters due to the measurement errors in independent variables

  16. How Power can be enhanced if the DV can be measured more precisely? With more error With less error

  17. Effects of Random Error in Dependent VariablesA Schematic View Random error compounded in DV reduces power: make it more difficult to detect a relationship and draw a conclusion

  18. Effects of Random Error in Independent VariableA Schematic View Random error compounded in IV cause bias in estimation: It undershoots bivariate correlation, and under- or over estimate relationships in multivariate analysis

  19. Measurement Error in IV and the Induced Bias in Regressionexample 2 βbiased Hostility1 (x11) Cigarette smoking βbiased = Cov(X11, smoking) / Var(X11) d1 1 Hostility1 (x11) e1 1 Hostility (ξ1) β Cigarette smoking Hostility2 (x12) 1 e2 1 βtrue = Cov(ξ1, smoking) / Var(ξ1)

  20. Measurement Error in IVexample 2 Hostility1 (x11) d1 e1 1 Hostility (ξ1) 1 Cigarette smoking 1 Hostility2 (x12) e2 1 * Proc calis statement: ; proc calis data=youth12 cov pall all; lineqs hostility = f1 + d1 , bhostility = f1 + d2 , bymonthcig1 = b1 f1 + d3 ; std f1 d1 d2 d3 = p1 p2 p3 p4; run;

  21. Measurement Error in IVexample 2 Hostility1 (x11) d1 e1 1 Hostility (ξ1) 1 Cigarette smoking 1 Hostility2 (x12) e2 1 Proc calis output and interpretations: Example2.lst

  22. Measurement Error in IVexample 2 Hostility d1 Cigarette smoking βbiased = Cov(hostility, smoking) / Var(hostility) = 0.02 Results without adjustment for measurement error βtrue = Cov(ξ1, smoking) / Var(ξ1) = 0.06 βbiased = Cov(hostility, smoking) / Var(hostility) = 0.02

  23. Revealing The True RelationshipAfter adjusting for measurement error in the independent variables • In bivariate relationship, after adjusting for measurement error in the indepepdent variable, the true relationship should always be larger • In multiple regression where more than one independent variables are measures with random errors, the measurement error adjusted beta could be larger, and could also be smaller

  24. Measurement Error in Multiple IVsexample 3 1 1 Hostility1 (x11) e11 Hostility (ξ1) β1 1 Hostility2 (x12) e12 1 1 1 stress1 (x21) e21 Stress (ξ2) β2 Cigarette smoking 1 stress2 (x22) e22 1 β3 1 1 depression1 (x31) d1 e31 Depression (ξ3) 1 depression2 (x32) e32 1 Other Covariates

  25. Measurement Error in Multiple IVsexample 3 Proc Calis Statement proc calis data=youth12 cov pall all; lineqs hostility = f1 + e11 , bhostility = f1 + e12 , stress = f2 + e21 , bstress = f2 + e22 , depression = f3 + e31 , bdepression= f3 + e32 , bymonthcig1 = b1 f1 + b2 f2 + b3 f3 + d1 ; std f1 f2 f3 d1 = phi11 phi22 phi33 psi11, e11 e12 e21 e22 e31 e32 = td11 td12 td21 td22 td31 td32 ; cov f1 f2 = phi12, f1 f3 = phi13, f2 f3 = phi23, e11 e21 = td11_21, e11 e31 = td11_31, e21 e31 = td21_31, e12 e22 = td12_22, e12 e32 = td12_32, e22 e32 = td22_32 ; partial school_rank1 school_rank2 school_rank3 district1 district2 city1 city2 city3 city4 city5 city6; run;

  26. 1 1 Hostility1 (x11) e11 Hostility (ξ1) 1 Hostility2 (x12) 1 e12 1 1 depression1 (x21) e21 Depression (ξ2) Cigarette smoking 1 depression2 (x22) 1 e22 1 1 stress1 (x31) d1 e31 Stress (ξ3) 1 stress2 (x32) 1 e32 Other Covariates Measurement Error in Multiple IVsexample 3 Proc Calis output: example 3.lst β1 = 0.027 β2 = -0.059 β3 = 0.129

  27. Measurement Error in IVexample 3 Hostility d1 Depression Cigarette smoking Stress Results without adjustment for measurement error β1 = 0.01 β2 = -0.002 β3 = 0.022

  28. Measurement Error in IVexample 3 Findings by comparing the results with and without adjusting for measurement errors in the independent variables: When more than one IVs are compounded with measurement errors, the results could be biased toward lower or higher estimates !

  29. Measurement Error in IV Third Party software is available for such modeling. Amos, EQS, HLM, LISREL, Mplus, Mx, etc. A Dwyer-Sun SAS Macro can also be used for a special case where one or more of the independent variables were assessed twice

  30. To Use MDREG to correct for Bias Induced by Measurement Error 1 1 Hostility1 (x11) e11 Hostility (ξ1) β1 1 Hostility2 (x12) e12 1 1 1 stress1 (x21) e21 Stress (ξ2) β2 Cigarette smoking 1 stress2 (x22) e22 1 β3 1 1 depression1 (x31) d1 e31 Depression (ξ3) 1 depression2 (x32) e32 1 Other Covariates

  31. To Use MDREG to correct for Bias Induced by Measurement Error SAS Statement Example 4 %include 'C:\MDREG6.SAS'; %mdreg (maindata=youth12, varp=aymonthcig1, mainreg=Logistic, zin=hostility stress depression school_rank1 school_rank2 school_rank3 district1 district2 &city, xin=bhostility bstress bdepression, calidata=youth12, model=Model1A, lprint=1, dout=0);

  32. To Use MDREG to correct for Bias Induced by Measurement Error SAS Output: example 4 ***** PARAMETER ESTIMATION ****** PARAMETERs Intercept HOSTILITY STRESS DEPRESSION Original -2.3647 0.1814 -0.1690 0.3473 Original_StErr 0.2349 0.0795 0.0544 0.0635 Original_t -10.0668 2.2826 -3.1072 5.4658 Corrected -2.7777 0.3565 -0.8023 1.3700 Corrected_StErr 0.3881 0.2006 0.1933 0.2804 Corrected_t -7.1570 1.7777 -4.1515 4.8865 Click here to view Full output

  33. To use a particular software, one needs to get familiar with the syntax for each software

  34. Syntax for LISREL

  35. Structural Equation Modelingexample 4 Hostility1 (x11) 1 1 d11 Hostility (ξ1) y11 e11 Cigarette Smoking η1 Hostility2 (x12) 1 d12 1 y12 e12 depression1 (x21) 1 1 d21 Depression (ξ2) ζ1 ζ2 depression2 (x22) 1 d22 y21 e21 1 Alcohol Drinking η2 stress1 (x31) 1 1 d31 Stress (ξ3) y22 e22 stress2 (x32) 1 d32 1

  36. LISREL Formulas Structural Equation Modeling:

  37. LISREL FormulasMeasurement Modeling on X: assuming measurement errors in measures in the same survey / timing are correlated

  38. LISREL Statements DA NG=1 NI=6 NO=12000 MA=CM CM SY fi=CSCS01.COV LA Y11 y12 y21 y22 x11 x12 x21 x22 x31 x32 SE Y11 y12 y21 y22 x11 x12 x21 x22 x31 x32 MO NY=4 NE=2 NK=3 NX=5 LY=FU,FI LX=FU,FI BE=FU GA=FU,FR PH=SY,FR PS=SY,FR C TD=SY TE=SY LE Cigarette Alcohol LK Hostility Depression Stress PA BE 0 1 1 0 MA LX 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 PA TD 2 0 2 5 0 3 0 5 0 3 6 0 7 0 4 0 6 0 7 0 4 MA LE 1 0 1 0 0 1 0 1 PA LE 8 0 8 10 0 9 0 10 0 9 OU EF ND=3 SE TV SC IT=850 AD=OFF

  39. Syntax for Mplus

  40. Latent Growth Curve Modeling Condition Constant F1 Linear F2 Quadratic F3 1 3 1 1 9 1 2 1 4 1 V1 V2 V3 V4 e1 e2 e3 e3

  41. MPLUS Statement title: A three level growth modeling; DATA: file=C:\Work\TND\Programs\TND1\mplus\hard3.dat; VARIABLE: NAMES ARE idnumaSCLCODE cond_2 usehard0 usehard1 usehd2 usehd3; CATEGORICAL ARE ; MISSING IS .; USEVAR = cond_2 aSCLCODE usehard0 usehard1 usehd2 usehd3; CLUSTER=aSCLCODE; ANALYSIS: TYPE = TWOLEVEL; ITERATIONS = 1200; ESTIMATOR = MUMLM; MODEL: %BETWEEN% levelb BY usehard0 usehard1 usehd2 usehd3@1; trendb BY usehard0@0 usehard1@1 usehd2@2 usehd3@3; trendb2 BY usehard0@0 usehard1@1 usehd2@4 usehd3@9; [usehard0@0 usehard1@0 usehd2@0 usehd3@0]; [levelbtrendb trendb2]; levelb ON cond_2; trendb ON cond_2; trendb2 ON cond_2; %WITHIN% levelw BY usehard0@1 usehard1@1 usehd2@1 usehd3@1; trendw BY usehard0@0 usehard1@1 usehd2@2 usehd3@3; trend2w BY usehard0@0 usehard1@1 usehd2@4 usehd3@9; levelw ON cond_2; trendw ON cond_2; trend2w ON cond_2; OUTPUT: SAMPSTAT STANDARDIZED;

  42. Syntax for MxSpecifically made for Analysis of Twin Dataset

  43. ACE Modeling for Twin Data 1/.5 C E A E C A x z y x y z P1 P2

  44. Implied Covariance Matrices For ACE Model • Difference between MZ and DZ covariance ~ Genetic Variance / 2

  45. Simple MX code for twin data analysis • Separate the subject into multiple groups: (e.g. by male dz, male mz, female dz, female mz, and male-female dz) • Fit the covariance matrix in each model, while define some parameters to be the same across the multiple groups. • See the notes area for an example of the MX code.

  46. Another Example of Twin Data AnalysisRelationship between ACE and other variables • The relationship between ACE components in in BMI and family cohesion: QingDao Twin Data Analysis • See notes area for code

  47. Age Adjusted ACE Modeling with Moderation for Ordinal Phenotype r=1.0 rg Em Cm Am Af Cf Ef ecm+βcm*M ecf +βcf*M eem+βem *M eam+βam*M eaf+βaf*M eef+βef *M μm+β1*age+βm*M µf +β2*age+βf*M M Pm Pf M

  48. ACE Composition for BMIModerated by Binary (high /low) Family Cohesion Level Error bars designate 95% CI. The difference in A between high and Low FC groups was 67.8% (95%CI: 15.2%-88.6%)

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