Outline. Session A: Various Definitions. 1. Basics of Path Diagrams and Path Analysis

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1 Session A: Basics of Structural Equation Modeling and The Mplus Computer Program Kevin Grimm University of California, Davis June 9, 008 Outline Basics of Path Diagrams and Path Analysis Regression and Structural Regression Structural Expectations Covariance Expectations Mean Expectations The Common Factor Model 5 Steps of SEM The Mplus Computer Program Various Definitions. Basics of Path Diagrams and Path Analysis Formal statistical statement about the relations among chosen variables Hypothesized pattern of (linear) relationships among a set of variables Collection of statistical techniques that allow the examination of a set of relationships between one or more IV s (continuous or discrete) and one or more DV s (continuous or discrete)

2 Some Advantages of SEM Explicit representation of theory (no default model) Representation of complex multivariate theories involving latent entities are possible Direct and indirect effects can be teased apart Analysis of multiple groups Variables can be outcomes (DV) and predictors (IV) Missing Data Some Disadvantages of SEM Need for strong substantive theory regarding relationship of variables Sample Size Need for some basic understanding of statistics Yield to temptation, when the tail wags the dog Equivalent models Yields to statements of causality easily Special SEM Some SEM Software General linear model (t-test, regression, multiple regression, ANOVA, etc.) Path model Confirmatory factor analysis Latent variable path model Latent growth curve analysis LISREL Mplus AMOS SAS Proc Calis Mx COSAN Sepath LisComp Systat Ramona

3 Underlying Principles Because SEM concerns relationships among variables, the emphasis is mainly on moment matrices (i.e., the structure of the data) The hypothesized model attempts to explain the structure of the data, with parsimony and accuracy Some statistical statement is needed about the match between the observed data and the hypothesized model (e.g. Fit Statistics) Some SEM Terminology Manifest variables: Observed (i.e., measured) variables defining the structure we wish to model Latent variables: Unobserved (i.e., unmeasured) variables implied by the covariance among two or more manifest variables Specification: Exercise of formally stating a model Some SEM Terminology () SEM Path Diagrams A Key Association: Non- (or bi-) directional (reciprocal) Y Squares = Observed Variables relation between variables Direct effect: (uni-) Directional (non-reciprocal) U f Circles = Latent Variables relation between variables (IV and DV) Indirect effect: effect of an IV on a DV through one or more intervening or mediating variables Double-Headed Arrows = Variances/Covariances (Association) Single-Headed Arrows = Regressions (Direct Effect) Total effect: sum of direct and indirect effects of an IV on a DV Triangle = Assigned variable = Constant (=.0) for modeling Means

4 Regression & Structural Regression Y X e n 0 n n n 0 = intercept, predicted value of Y when predictors (X) are zero = slope coefficient, predicted amount of change in Y for a unit change in X e = residual, part of Y not predicted by X, uncorrelated with X Variance of Y is decomposed into variance explained by X and unexplained variance (e) X Structural Regression X Y e X 0 e Y X e n 0 n n n Structural Regression (Short-hand) Structural Expectations X X X 0 Y e Y X e n 0 n n n Every Structural Model has a set of Structural Expectations. Variance/Covariance Expectations Mean Expectations Used to estimate parameters Difference between Structural Expectations and Observed Statistics (Covariance/Mean) is the model misfit Expectations can be calculated based on a path diagram or computed algebraically.

5 Calculating Covariance Expectations X X X X Covariance Expectations Variance Expectations (Select) EX X [, ] X EX [, X] X X X,Y Covariance Expectations (Select) Y Y EX [, Y] X Y, EX X [, ] X Y Y EX [, Y] X, Y Calculating Mean Expectations Mean Expectations (Select) X X X EX [ ] X Y EX [ ] X Y Y

6 X X Y e e X 0 Expected Variances EXX [ ] X EYY [ ] b b X e Confirmatory Factor Analysis Expected Covariance EXY [ ] b X Expected Means EX [ ] X E[ Y] b X b 0 Confirmatory Factor Analysis (CFA) Used to study how well a hypothesized structure fits to a sample of measurements Hypothesis-driven Explicitly test a priori hypotheses (theory) about the structures that underlie the data Number of, characteristics of, and interrelations among underlying factors Specify a common measurement base for comparisons across groups/occasions (factorial invariance) Confirmatory Factor Analysis (CFA) Testing an a-priori hypothesis about the structures in the data Requires specific expectations regarding The number of factors Which variables reflect given factors How the factors are related to one another

7 One Factor Model Covariance Expectations for the Single Common Factor f = Y Y Y 3 Y + 3 Y + Y Y Y 3 Y U U U 3 3 Path Tracing Rules Note: The main diagonal variances include the unique variances, j, but the offdiagonal covariances do not. All terms include the variance of the common factor,. Structural Expectations: Identification Constraints Additional constraints are often needed to obtain a unique set of estimates, and we typically assume the latent variable has no meaningful scaling, so we assume it has variance E{f, f } = =.0 After this scaling each pair covariance is simply a product of the pairs of loadings: E{ } = One Factor Model f 3 Y Y Y 3 U U U 3 3

8 Covariance Expectations for the Single Common Factor with = Numerical Expectations for a Population Common Factor with = and = [.8,.7,.6] = Y Y Y 3 = Y Y Y 3 Y + Y + Y +.36 =.0 Y Y *.8 = =.0 Y 3 *.8 =.48 *.7 = =.0 Note: Each covariance, ij, expectation follows a simple pattern determined by the product of the respective loadings i j Structural Equation Modeling Covariance expectations from a single common factor model are a simple product of the loadings Introducing Means into the CFA Model Consider The Common Factor Model f The extension to multiple factors is straightforward However, as models become more complicated (read realistic) the structural expectations become more complex as well 3 Structural Equation Modeling (covariance analysis) is simply the method by which we test our expectations against our data Y u y Y u y Y 3 u y3

9 Introducing Means into the CFA Model Observed Variable Means Structural Expectations: Means Observed Variable Means f Y Y Y 3 3 = Y Y Y3 Y Y Y3 Y Y Y 3 u y u y u y3 Path Tracing Rules Introducing Means into the CFA Model Latent Variable Mean Structural Expectations: Means Latent Variable Mean f f Y Y Y 3 3 = f f f Y Y Y 3 u y u y u y3 Path Tracing Rules

10 5 Steps in SEM Analyses SEM is often viewed as an advanced and novel form of analysis but this approach is not new: 5 Steps in SEM Analysis. Theory-Data: form some basic ideas of merging theory and data. Specification: form explicit hypotheses using regression and factor analysis concepts to form structural restrictions 3. Estimation: use specialized computer software to estimate coefficients, standard errors and various statistical indicators 4. Evaluation: compare alternative structural restrictions in a series of statistical tests 5. Re-evaluation & Extension: exploring new ideas/models Step : Theory-Data The purpose of statistical procedures is to assist in establishing the plausibility of a theoretical model (Cooley, 978) SEM is a general statistical framework that allows researchers to be explicit about theory and how it might be reflected in one s data Statistical models are where theories and data collide However, in their assumptions, statistical models invoke a particular notion of reality that may or may not match one s theoretical ideas Step : Theory-Data SEM is a confirmatory framework for testing an a-priori hypotheses about the structures in the data Requires specific expectations regarding One s theory How one s theory may be reflected in one s data Selection of persons Selection of variables Selection of occasions

11 Step : Model Specification There are many ways to use SEM programs (e.g., Mplus, AMOS, LISREL, Mx) to produce the same result Specify a set of expectations that match the theory to be tested Path Diagram, Matrix, or Multiple Equation specifications are functionally equivalent Step 3: Parameter Estimation A series of computational steps is taken, each successive step to minimize the value of the fit function equation a weighted distance between the observations and expectations. At each step in the minimization process, a vector of model parameter estimates is updated so that the model reproduces the observed covariance matrix as closely as possible. When these parameter estimates no longer improve the fit over the previous estimates, the process is said to have converged. The estimates at convergence are the parameter estimates. Example of a Fit Function F(ml) = ln S -ln +tr(s - )-p, where S is the sample covariance matrix and is the estimated covariance matrix based on the assumed model. Step 4: Evaluating Fit of SEM Relative Fit: A variety of structural models are typically fitted to the same observations (data) Nested models Fit Indices: Lots of fit indices are available: simple residuals, standard errors, and likelihood ratio and chi-square tests Lawley & Maxwell, 97; Browne, 985; Browne & Cudeck, 993

12 Step 4: Evaluating Fit of SEM Fit Indices (cont): If we calculate the parameters based on the principles of maximum likelihood estimation (MLE) we obtain a likelihood ratio statistic (L ) of misfit Under standard assumptions (e.g., normality of residuals) L follows a distribution with df = N s -N p Step 4: Evaluating Fit of SEM Testing Fit: We use L type tests to ask Should we reject the hypothesis of this model? Often this gets rephrased... : Are the observed data consistent with the hypothetical model? Is the model plausible? Does the model fit? Step 4: Evaluating Fit of SEM Probability of Close Fit: Probability models based on normal distribution theory are available e.g., p(perfect fit) and p(close fit) These same statistical analyses can be used even when models are complex e.g., when latent variables, multiple groups, or incomplete data are the focus of study Major Bonus of SEM Step 4: Evaluating Model Fit How good is the model? How well does the model represent the data? How well does the model represent the theory? Fit to the data Measures of how well the estimated covariance matrix derived from the model matches the observed covariance matrix Fit to the theory Subjective interpretation

13 Confirmatory Hypothesis Tests When restrictions are stated in advance of the estimation, the hypothesis is clear and we can use statistical probability models These models have degrees-of-freedom and are rejectable Examine overall fit, standard errors and residuals We do not conclude the model fits the data, but we can conclude the model does not fit the data, or that one model fits better than another Relative fit: We need to examine most restrictions via the comparison of at least two alternative models Model Fit Statistics (or -LL) df = degrees of freedom Null hypothesis Estimated covariance matrix = Observed covariance matrix (sensitive to sample size) RMSEA Range: 0.00 to.00 lower values indicate better fit M. Browne s rule of thumb: RMSEA <.05 indicates good fit CFI (Comparative Fit Index) NFI (Normed Fit Index) TLI (Tucker-Lewis Index) Range: 0.00 to.00+ higher values indicate better fit Nested Hypothesis Tests Two alternative models may be nested Parameters are said to be nested when they are included in one model (M 0 ) and then can be removed to form the alternative model (M ) The hierarchy of restrictions makes it easy to use statistical probability tests Under typical assumptions the difference between two nested models can be evaluated using a chi-square test Relative Fit of Nested Models difference tests (for nested models) [(Model B ) - (Model A )]/ df B -df A Information criteria for non-nested model comparisons (using same data) AIC (Aikake Information Criteria) BIC (Bayes Information Criteria) Lower values are better Should be used in conjunction with judgments about the theoretical interpretation of the models

14 Evaluating Relative Fit Relative Fit of Nested Models =.0 = =.0 Evaluate Fit for Model A f f f Add restrictions to construct Model B Evaluate Fit for Model B Y Y Y 3 Y 4 Y 5 Y 6 Y Y Y 3 Y 4 Y 5 Y 6 Evaluate difference in fit = /df Is the restricted (parsimonious) model of significantly worse fit than the less restrictive (more complex) model or is this complexity needed? U U U 3 U 4 U 5 U = 55, df = 9, RMSEA =.4 U U U 3 U 4 U 5 U =, df = 8, RMSEA =.053 Model Comparison: / df = 44/ p <.05 Step 4: Evaluating Model Fit How good is the model? How well does the model represent the data? How well does the model represent the theory? Examine the relative fit of multiple models Reject those models that fit relatively worse Carry forward those models that fit relatively well Step 5: Re-evaluation & Extension Moving from Confirmation to Exploration A philosophical debate In most confirmatory analyses some results suggest alterations of the original concepts (specifying a different theory ) Often the model is modified because the original model does not fit An exploratory phase begins Note: If there is a lack of prior directional hypotheses, probability models based on normal distribution theory are no longer available

15 A General Latent Variable Framework & Analysis Software TITLE: Mplus Language I Factor Model; Muthén & Muthén, DATA: FILE = wisc3raw.dat; VARIABLE: NAMES = id verb verb verb4 verb6 perfo perfo perfo4 perfo6 info comp simi voca info6 comp6 simi6 voca6 momed grad constant; USEVARIABLES = info comp simi voca; ANALYSIS: TYPE = MEANSTRUCTURE; Y Squares = observed variables USEVAR = var var ; MPlus Language II f Circles = latent variables U ANALYSIS: TYPE = MEANSTRUCTURE;! TYPE = BASIC; gives sample statistics! TYPE = NOMEANSTRUCTURE; allows for model without means Double-Headed Arrows = Covariances var WITH var; Double-Headed Arrows = Variances var*; or factor@; Single-Headed Arrows = Regressions var ON var; or factor BY var var var3; Triangle = Assigned variable = Constant (=.0) for modeling Means [var]; MODEL:!Factor Loadings verb BY info@ comp simi voca;!factor Variance verb*;!mean of Factor [verb@0];!means of Observed [info comp simi voca];!variances of Observed info comp simi voca; OUTPUT: SAMPSTAT STANDARDIZED;

16 Session B: Alternative Structural Equation Models for Change over Two-Occasions Kevin Grimm University of California, Davis June 9, 008 Overview. Practical Preliminaries. Two-occasion longitudinal data 3. Type-A auto-regression models 4. Type-D difference score models 5. Combining alternative models 6. Summary & Discussion Practical Preliminaries Data set formatting Preliminary data examination Correlations (covariances) over time Means over time Longitudinal plots Examine for Shapes, Outliers, Possible time bases, etc. Longitudinal Data Formats Two common data formats Single-record per person (wide form data) all the data associated with one person appears in a single record id adhd adhd4 adhd5 adhd

17 Multiple-record per person (long form data, person-period data, relational data) the data associated with one person appear in multiple records indexed by id and time variables. id age read Preliminary Examination of Longitudinal Data Sample Statistics Correlations (covariances) over time Stability coefficients Means over time Plots of intraindividual change over time Sample Statistics in SAS/SPSS Sample Statistics in Mplus *Examining Correlations Across Time; PROC CORR DATA=wiscraw; VAR adhd adhd4 adhd5 adhd7; RUN; *Examining Means Across Time; PROC MEANS DATA=wiscraw; VAR adhd adhd4 adhd5 adhd7; RUN; *Correlations over time. CORRELATIONS /VARIABLES= adhd adhd4 adhd5 adhd7 /MISSING=PAIRWISE. *Means over time. DESCRIPTIVES VARIABLES= adhd adhd4 adhd5 adhd7 /STATISTICS=MEAN STDDEV MIN MAX. TITLE: Descriptive Sample Stats; DATA: FILE = adhd_uncg.dat; VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_tot adhd_tot4 adhd_tot5 adhd_tot7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =. USEVAR = adhd_tot - adhd_tot7; ANALYSIS: TYPE = BASIC; OUTPUT: SAMPSTAT; *Note: Data must be in single-record per person (wide) format *Note: Data must be in single-record per person (wide) format

18 Sample Statistics Longitudinal Correlations & Means Plot of Intraindividual Change in SAS adhd adhd4 adhd5 adhd7 Means N adhd adhd adhd adhd PROC GPLOT DATA = temp_long (where = (new_id < 50)); SYMBOL REPEAT=500 I=join V=dot H=.5 W= C=black; AXIS LABEL = (A=90 F=SWISSX H=.3 'ADHD Total Score') ORDER = (0 to 60 by 0) MINOR = none OFFSET = (); AXIS LABEL = (F=SWISSX H=.3 'Age') ORDER = ( to 7 by ) MINOR = none OFFSET = (); PLOT adhd_t * age = id /NOLEGEND VAXIS=AXIS HAXIS=AXIS; RUN; *Note: Data must be in multiple-record per person (long) format Plot of Intraindividual Change in SPSS Longitudinal Plot of ADHD Total Score (N = 50) igraph /x=var(age) type = scale /y=var(adhd_t) type=scale /style=var(id) /line(mode) key=off style=line interpolate=straight /scalerange=var(adhd_t) min=0 max=60. *Note: Data must be in multiple-record per person (long) format

19 . Two-Occasion Longitudinal Data Two-Occasion Data Are Valuable Two-occasions are the first case of longitudinal data collections There are several special properties of repeated measures data Most analyses deal with basic questions of representing change over time Different problems seem to suggest different models and methods of analysis Example of Two-Occasion Data Data from the RIGHT (Research Investigating Growth and Health Trajectories) Track Research Project Focus on the development and developmental trajectories of early disruptive behavior Study participants N=43 children Measured at age, 4, 5 & 7 years of age In this illustration the Attention Deficit Hyperactivity Total Scores from the age 4 & age 7 assessments are used

20 Summary Statistics from the RIGHT Track Study The CORR Procedure Variables: adhd_to4 adhd_to7 Univariate Histograms of ADHD Age 4 and Age 7 Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum adhd_to adhd_to Pearson Correlation Coefficients Prob > r under H0: Rho=0 Number of Observations adhd_to4 adhd_to7 adhd_to < adhd_to < Quotes from Bachman et al (00, p.3) There are two approaches to the prediction of change in such analyses, and we use both. The first approach involves computing a change score by subtracting the before measure from the after measure, and then using the change score as the dependent variable. The second approach uses the after measure as the dependent variable and include the before measure as one of the predictors (i.e., as a covariate). In either case one could say that the earlier score is being controlled, but the means of controlling differ and the results of the analysis also can differ --- sometimes in important ways. Bivariate Scatterplot X=Age 4 versus Y=Age 7

21 Plotting Individual Trajectories (n=50) SEM and Two-Occasion Data Analyses We can use use regular regression or ANOVA programs to get reasonable answers to some change questions. Alternatively, we can use any standard SEM computer program and any available options for input and output. SEMs are based on: (a) an algebraic model, (b) a corresponding path diagram, (c) input the model to a SEM program, and (d) examine expectations generated. Empirical models can be fit using any technique for means and covariances or raw data (e.g., in M+). SEM provide a framework for dealing with more than two alternative models, including those based on latent variable and requiring incomplete data analyses. Most Common Linear Regression Models A linear model is expressed for n= to N as 3. Type-A Auto-Regressive Models for Repeated Measures Y[] n = 0 + Y[] n + e n 0 is the intercept term -- the predicted score of Y[] when Y[]=0 is the coefficient term -- the change in the predicted score of Y[] for a one unit change in Y[] e is the residual score -- an unobserved and random score which is uncorrelated with Y[] but forms part of the variance of Y[]. The ratio of the variance of e to Y[] ( e / y = -R ) can be a useful index of forecast efficiency. In our notation, Greek letters used for estimated parameters.

22 Y Typical Autoregression Path Model for Two Repeated Measures Y[] Y 0 Y[] e e SAS/SPSS Autoregression Input Script SAS *Type A - Autoregressive Model for Two Occasion; PROC REG DATA = adhd; MODEL adhd_to7 = adhd_to4 / STB; RUN; SPSS *Autoregressive Model. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.0) /NOORIGIN /DEPENDENT adhd_to7 /METHOD=ENTER adhd_to4. Auto-Regression Results (SAS) Dependent Variable: adhd_to7 Number of Observations Read 43 Number of Observations Used 34 Number of Observations with Missing Values 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square Dependent Mean.0654 Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < TITLE: DATA: VARIABLE: ANALYSIS: MODEL: Autoregressive Model in Mplus ADHD Autoregressive Model; FILE = adhd_uncg_wide.dat; LISTWISE=ON; NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7; TYPE = MEANSTRUCTURE; adhd_to7 ON adhd_to4; adhd_to4 adhd_to7; [adhd_to4 adhd_to7]; OUTPUT: SAMPSTAT;

23 Mplus Output Mplus Output SUMMARY OF ANALYSIS Number of groups Number of observations 34 Number of dependent variables Number of independent variables Number of continuous latent variables 0 Observed dependent variables Continuous ADHD_TO7 Observed independent variables ADHD_TO4 Means Covariances SAMPLE STATISTICS ADHD_TO7 ADHD_TO ADHD_TO7 ADHD_TO4 ADHD_TO ADHD_TO Correlations ADHD_TO7 ADHD_TO4 ADHD_TO7.000 ADHD_TO Mplus Output TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 0 P-Value CFI/TLI CFI.000 TLI.000 Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 5 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= Mplus Output Two-Tailed Estimate S.E. Est./S.E. P-Value ADHD_TO7 ON ADHD_TO Means ADHD_TO Intercepts ADHD_TO Variances ADHD_TO Residual Variances ADHD_TO

24 Results from Autoregression Model Graphic Result of the Autoregression ADHD[4] = ADHD[7] e * Note: Fully saturated model so =0 with df=0; Asterisk indicates t= p/se(p) >. = 4. Type D Latent Difference Score Models for Repeated Measures Statistical Features of Difference Scores Using the same repeated measures scores, the difference scores have useful properties for the means, such as [] = [] + D so D = [] - [] n And also for the variances and covariances = + D + D so D = + - and D = - The statistics of the difference scores are a transformation of the statistics in the measured variables, and this is useful.

25 Classic Critique of Difference Scores There have been major critics of the use of difference scores (e.g., Cronbach & Furby, 970). A typically model for a set of observed repeated measures is Y[] n = y n + e[] n and Y[] n = y n + e[] n where y = an unobserved true score for both occasions, and e[t]=an unobserved random error that is independent over each occasion. In this model the true score remains the same and all changes are based on the random noise. Classic Critique of Difference Scores If the previous model holds then the simple difference score can be rewritten as D n = Y[] n - Y[] n = (y n + e[] n )- (y n + e[] n ) = (y n -y n ) + (e[] n - e[] n ) = e[] n - e[] n So the variance of the difference score is entirely based on the differences in the random error scores. This also implies that the reliability of the difference score is zero. For these and other reasons the use of the simple difference score has been frowned upon in much of developmental research. Classic Resolution of Difference Critique Other researchers (e.g., Nesselroade, 97, 974) defined a model for a set of observed repeated measures as Y[]n = y[] n + e[] n and Y[] n = y[] n + e[] n where y 0 = an unobserved true score for he first occasion, y = an unobserved true score for the second occasion, and e[t]=an unobserved random error. It is also possible to redefine this model as a gain and write Y[] n = y[] n + e[] n and Y[] n = y[] n + y n + e[] n where y[] n = an unobserved true score for both occasions, y = an unobserved true change at the second occasion, and e[t]=an unobserved random error. Classic Resolution of Difference Critique However, iff this model holds then the difference score D n = Y[] n - Y[] n = (y[] n + e[] n ) - (y[]+ e[] n ) = (y[] n -y[] n ) + (e[] n - e[] n ) = y n + (e[] n - e[] n ) = y n + e n So the variance of the difference score is party based on the differences in the random error scores but also partly on the gain in the true score. The relative size of the true-score gain determines variance and reliability of the difference. This implies the difference score may be a very good way to consider measuring change, and researchers should consider them carefully, especially latent scores without accumulated errors.

26 Models with Difference Scores Using the same repeated measures scores, we can write the alternative difference score for any person n as Y[] n = Y[] n + y n where the y is an implied or latent difference score. This can be verified simply by rewriting the model as y n =Y[] n -Y[] n We can also add features of the means and covariances as y n = d + y n * and E{y* y*}= and E{Y[]* y*}= Difference Score Model in SAS/SPSS *Type D - Difference Score Model for Two Occasion; DATA diff; SET adhd; delta = adhd_to7 - adhd_to4; constant = ; RUN; PROC REG DATA = diff; MODEL delta = constant/ NOINT; RUN; *Computing Difference Score & Difference Score Model. COMPUTE delta = adhd_to7 - adhd_to4. COMPUTE constant =. EXECUTE. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.0) /ORIGIN /DEPENDENT delta /METHOD=ENTER constant. SAS/SPSS Output Path Diagram of the Latent Difference Score Model Dependent Variable: delta Number of Observations Read 43 Number of Observations Used 34 Number of Observations with Missing Values 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Uncorrected Total Root MSE R-Square 0.05 Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t constant Y[] Y[] y Note: The y is an unobserved variable whose moments (,, ) are implied by the formula Y[]=Y[]+y.

27 Mplus Latent Difference Score TITLE: ADHD Difference Model; DATA: FILE = adhd_uncg_wide.dat; LISTWISE=ON; VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7; ANALYSIS: MODEL: TYPE = MEANSTRUCTURE; adhd_to7 ON adhd_to4@; adhd_to4 adhd_to7@; [adhd_to4 adhd_to7@0]; delta BY adhd_to7@; delta; [delta]; SUMMARY OF ANALYSIS Mplus Output Number of groups Number of observations 34 Number of dependent variables Number of independent variables Number of continuous latent variables Observed dependent variables Continuous ADHD_TO7 Observed independent variables ADHD_TO4 Continuous latent variables DELTA OUTPUT: SAMPSTAT; Mplus Output Chi-Square Test of Model Fit Value Degrees of Freedom 0 P-Value CFI/TLI CFI.000 TLI.000 Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 5 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= Mplus Output MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value DELTA BY ADHD_TO ADHD_TO7 ON ADHD_TO ADHD_TO4 WITH DELTA Means ADHD_TO DELTA Intercepts ADHD_TO Variances ADHD_TO DELTA Residual Variances ADHD_TO

28 Results of the Latent Difference Score Model Y[] Y[] y Note: Fully saturated model so =0 with df=0; Asterisk indicates t= p/se(p) >. Summary of Latent Difference Models The use of latent difference score (LDS) in SEM is based on the same statistical information in the twooccasion data or the calculated difference score model. This means that variations in the parameters of each change model can be evaluated in the same way -- by the difference in goodness-of-fit tests. However, by avoiding the direct calculation of the different score we can now consider models where we attempt a model-based separation of the errors of measurement from the systematic change. The previous point will be emphasized in the next set of models when we measure: (a) more variables, (b) more time points, or (c) more of both. 5. Combining Features of Alternative Repeated Measures Models Interpreting Change From Autoregression Assume scores over time for multiple variables have been fitted using this form of regression over time Y[] n = 0 + Y[] n + e n. Rewrite this expression as a residual change (Y[] n - Y[] n ) = 0 + e n. Or rewrite this expression as a direct change (Y[] n - Y[] n ) = 0 + Y[] n + e n - Y[] n = 0 +( ) Y[] n + e n = 0 + Y[] n + z n 3. Or rewrite this expression as a historical change Y[] n = 0 + Y[] n + e[] n BUT if Y[] n = 0 + Y[0] n + e[] n then (Y[] n - Y[] n ) = (Y[] n - Y[0] n ) + (e[] n - e[] n )

29 Models with Latent Difference Scores Assuming we can write the alternative difference score for any person n as Y[] n = Y[] n + y n where the y is an implied or latent difference score. So suppose we consider features of a dual change prediction system with y n = 0 + Y[] n + z n This model of y makes it easy to see the possible to plots the latent difference scores. To obtain the autoregression from the difference score coefficients we write 0 = 0 and = ( -). This is a non-trivial resolution of the fundamental question of the alternative models. A path diagram for the prediction of the Latent Difference Score Y[] Y[] y z Note: The y is an unobserved variable whose moments (,, ) are implied by the formula Y[]=Y[]+y AND y = 0 + Y[] + z. Mplus Latent Difference Score TITLE: ADHD Difference Model; DATA: FILE = adhd_uncg_wide.dat; LISTWISE=ON; VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL: adhd_to7 ON adhd_to4@; adhd_to4 adhd_to7@; [adhd_to4 adhd_to7@0]; delta BY adhd_to7@; delta; [delta]; delta ON adhd_to4;!prediction of Change OUTPUT: SAMPSTAT; Mplus Output Chi-Square Test of Model Fit Value Degrees of Freedom 0 P-Value CFI/TLI CFI.000 TLI.000 Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 5 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <=

30 Mplus Output MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value DELTA BY ADHD_TO DELTA ON ADHD_TO ADHD_TO7 ON ADHD_TO Means ADHD_TO Intercepts ADHD_TO DELTA Variances ADHD_TO Residual Variances ADHD_TO DELTA Autoregression vs. Change Estimates. We previous fit the auto-regression over time with MLE of Y[] n = 0 + Y[] n + e n = + Y[] n + e n. We now use the same data and rewrite this expression as a direct change with MLE of (Y[] n - Y[] n ) = 0 +( ) Y[] n + z n = 0 + Y[] n + z n = + *Y[] n + z n 3. The explained variance is the same residual variance ( e ) but it is compared to the variance at time ( ) in auto-reg model, but it is compared to the variance of the difference ( ) in latent-difference model. Testing Hypotheses with Difference Scores Hypothesis : No change has occurred Hypothesis : ADHD at age 4 is not predictive of change in ADHD from age 4 to age 7 No Change TITLE: ADHD Difference Model; DATA: FILE = adhd_uncg_wide.dat; LISTWISE=ON; VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL: adhd_to7 ON adhd_to4@; adhd_to4 adhd_to7@; [adhd_to4 adhd_to7@0]; delta BY adhd_to7@; delta; [delta@0];!no Average Change delta WITH adhd_to4; OUTPUT: SAMPSTAT;

31 Mplus Output TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 6.89 Degrees of Freedom P-Value CFI/TLI CFI TLI Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 4 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= No Prediction of Change TITLE: ADHD Difference Model; DATA: FILE = adhd_uncg_wide.dat; LISTWISE=ON; VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL: adhd_to7 ON adhd_to4@; adhd_to4 adhd_to7@; [adhd_to4 adhd_to7@0]; delta BY adhd_to7@; delta; [delta]; delta ON adhd_to4@0;!no Prediction of Change OUTPUT: SAMPSTAT; Mplus Output TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom P-Value CFI/TLI CFI TLI Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 4 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) 6. Summary & Discussion RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <=

32 Summary of Two Occasion Models Given any two-occasion repeated measures data, we can write many alternative structural change models. It is not easy to distinguish these models by goodnessof-fit tests, because some can be exactly identified, rotated, and fit as well as one another. Our interpretations of change from these models are fundamentally restricted by these initial choices -- so choose carefully and match the problem at hand. Measurement error can have direct impacts on the lowering the determination of individual differences in changes. Note: The two models discussed here, auto-regression and difference-scores can be distinguished when more than t > 3 repeated measures are available. Alternative Indices of Change Assuming A[t]=The index at a particular time, we can calculate:. The Difference score [-]= A[] - A[]. The Ratio score []= A[] / A[] 3. The Change Ratio score []= {A[]-A[]} / A[] 4. The Change Ratio score []= {A[]-A[]} / A[] 5. The Residual Change Ratio [] = A[] - { 0 + A[]} with 0 and to be determined from the data References on Change Issues Re-considering Lord s Paradox? Holland, P.W. & Rubin, D.B. (983). In H. Wainer & S. Messick (Eds.), Principals of modern psychological measurement (pp. 3 5). Hillsdale, NJ: Erlbaum. (also see Laird, N, 983, Am Stat., 37, ). Additional problems due to Regression to the Mean? (Nesselroade et al, 98) The problems of un-reliability of difference scores? (Burr and Nesselroade, 990) Alternative change measures in Pretest-Postest? (Bonate, 000, Analysis of Pretest-Posttest Designs) Issues in Significance Testing? Harlow, L.L. Mulaik, S.A. & Steiger, J.H. (997). What if there were no significance tests? Hillsdale, NJ: Erlbaum. References on SEM Baltes, P.B., Dittmann-Kohli, F. & Kliegl, R. (986). Reserved capacity of the elderly in aging-sensitive tests of fluid intelligence: replication and extension. Psychology and Aging, (), Joreskog, K.G. & Sorbom, D. (979). Advances in factor analysis and structural equation models. In J. Magdson (Ed.), Cambridge, MA: Abt Books. Loehlin, John C. (998). Latent Variable Models: An Introduction to Factor, Path, and Structural Analysis. 3rd ed. Mahwah, N.J.: Lawrence Erlbaum Associates. McArdle, J.J. (996). Current directions in structural factor analysis. Current Directions in Psychological Science, 5 (), -8. McDonald, R.P. (985). Factor Analysis and Related Methods, New Jersey: Lawrence Erlbaum, Publishers.

33 Session C: Group Differences over Two Occasions Kevin Grimm University of California, Davis June 9, 008 Overview. Group Differences in Longitudinal Data. Traditional ANCOVA in Two Occasions 3. ANOVA Difference Score Regression 4. Multiple Group Structural Equation Modeling 5. Latent Class Mixture Modeling 6. Summary & Discussion Group differences in change. Group Differences in Longitudinal Data The alternative models of change are often considered in the context of a group difference Differences between groups can be considered for any feature of the model (e.g., means, regressions, etc.) If assignment to groups is based on random selection, then the inferences about the source of the subsequent changes is clear the impact is due to the assignment The SEM approach can add power to the model If assignment to groups is pre-existing or based on non-random selection (e.g., self selection) then the inferences about the source of the changes is often ambiguous the changes may be due to other features related to the initial selection The SEM approach does not counteract the selection mechanism, but it can help clarify some of the ambiguity.

34 Traditional concept of group differences on the distribution of a variable Group information on ADHD Data Gender Coded 0 for male (n = 45), for female (n = 69) Group A Group B Questions related to group information Do males/females tend to display more symptoms of ADHD? Are there more/less between-person differences in ADHD for males/females? Do males/females change more in their symptoms of ADHD? Are males/females more variable in their patterns of change in ADHD? Descriptive Information in ADHD by Gender Statistic ADHD Age 4 Mean (SD) Overall (N=34) 3.35 (9.39) Females (n f =69).7 (9.58) Males (n m =45) 4.60 (9.03) A note on Lord s paradox Lord, F.M. (967) A paradox in the interpretation of group comparisons, Psychological Bulletin, 68, A large university is interested in investigating the effects on the students of the diet provided in the university dining halls.various types of data are gathered. In particular the weight of each student at the time of his arrival in September and his weight in the following June are recorded ADHD Age 7 Mean (SD).07 (0.06) 0.68 (9.97) 3.68 (9.95) Lord suggested that Analyst A used ANCOVA to remove the preexisting differences and, and Analyst B used Repeated Measures ANOVA to examine changes Age 4/Age 7 Correlation The paradox Analyst A obtained significant group differences but Analyst B did not, and then Lord simply ended the paper

35 Multiple Regression ANCOVA. Traditional ANCOVA in SEM A linear model is expressed (for n= to N) as Y[] n = b 0 + b. Y[] n + b. G n + e n where G is a binary variable Iff coded in dummy (0,) form we can write Y[] n [: G n =0] =b 0 + b. Y[] n + b. 0 n + e n Y[] n [: G n =] =b 0 + b. Y[] n + b. n + e n so b 0 is the intercept for the group coded 0, b is the slope for the group coded 0, b is the change in the intercept for the group coded An Auto-Regression model with Group Differences Y[] Y[] e 0 g g G e Autoregression Input Script (SAS) TITLE Structural Equation Models of Change'; TITLE Auto-Regression and Difference Score Models of Change with Group Information'; *Type A with Group - Traditional ANCOVA Model with Repeated Measures; DATA adhd; SET adhd; adhd_by_girl = adhd_to4 * girl; RUN; PROC REG DATA = adhd; ModA: MODEL adhd_to7 = adhd_to4 / STB; ModA: MODEL adhd_to7 = adhd_to4 girl / STB; ModA3: MODEL adhd_to7 = adhd_to4 girl adhd_by_girl / STB; RUN; g

36 Autoregression Input Script (SPSS) *Computing Interaction between gender and ADHD at age 4. COMPUTE adhd_by_girl = adhd_to4 * girl. EXECUTE. *Autoregressive Model with Group Information. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.0) /NOORIGIN /DEPENDENT adhd_to7 /METHOD=ENTER adhd_to4 /METHOD=ENTER girl adhd_to4 /METHOD=ENTER girl adhd_to4 adhd_by_girl. SAS Autoregression The REG Procedure Number of Observations Read 43 Number of Observations Used 34 Number of Observations with Missing Values 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square Dependent Mean.0654 Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < Autoregression with Intercept Difference Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square Dependent Mean.0654 Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < girl Mplus Autoregression & Group Input VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7 girl; ANALYSIS: MODEL: TYPE = MEANSTRUCTURE; adhd_to7 ON adhd_to4; adhd_to4 adhd_to7; [adhd_to4 adhd_to7]; adhd_to7 ON girl; [girl*.5] (M_girl); girl (V_girl); girl WITH adhd_to4; MODEL CONSTRAINT: v_girl = M_girl * ( - M_girl); OUTPUT: SAMPSTAT TECH;

37 Mplus output Means ADHD_TO7 ADHD_TO4 GIRL Covariances ADHD_TO7 ADHD_TO4 GIRL ADHD_TO ADHD_TO GIRL Correlations ADHD_TO7 ADHD_TO4 GIRL ADHD_TO7.000 ADHD_TO GIRL TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom P-Value.0000 CFI/TLI Mplus output CFI.000 TLI.04 Loglikelihood H0 Value H Value Information Criteria Number of Free Parameters 8 Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + ) / 4) RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= Mplus output MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value ADHD_TO7 ON ADHD_TO GIRL GIRL WITH ADHD_TO Means ADHD_TO GIRL Intercepts ADHD_TO Variances ADHD_TO GIRL Residual Variances ADHD_TO Autoregression Model with Group Difference Y[] Y[] e[] G

38 Predicted Regression Lines Autoregression Model Group and Pre-Existing Initial Differences Males Females e Y[] Y[] e[] e 0 g g 0 G g g Two Occasion ANCOVA with Interaction The model for multiple groups can be written as Y[] n = b 0 + b. Y[] n + b g. G n + b i. (Y[] n.g n ) + e n where G is a binary variable -- yes or no and the product variable is created as (Y[] n. G n ) Iff coded in dummy (0,) form we can write Y[] n [: G n =0]=b 0 + b Y[] n + b 0 n + b 3 0 n + e n Y[] n [: G n =]=b 0 + b Y[] n + b n + b 3. Y[] n + e n b 0 is the intercept for the group coded 0 b is the slope for the group coded 0 b is the change in the intercept for the group b 3 is the change in the slope for the group coded Autoregression Model with Group Differences and Interaction Y[] Y[] e 0 g g G 3 G*Y[] g g e

39 Autoregression with Int+Slope Differences Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square Dependent Mean.0654 Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < girl adhd_by_girl Mplus Autoregression & Interaction VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7 girl g_by_adhd; DEFINE: g_by_adhd = girl * adhd_to4; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL: adhd_to7 ON adhd_to4; adhd_to4 adhd_to7; [adhd_to4 adhd_to7]; adhd_to7 ON girl; [girl*.5] (M_girl); girl (V_girl); girl WITH adhd_to4;!new Additions adhd_to7 ON g_by_adhd; [g_by_adhd*.5]; g_by_adhd; g_by_adhd WITH girl adhd_to4; MODEL CONSTRAINT: v_girl = M_girl * ( - M_girl); OUTPUT: SAMPSTAT TECH; Mplus Output MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value ADHD_TO7 ON ADHD_TO GIRL G_BY_ADHD GIRL WITH ADHD_TO G_BY_ADH WITH GIRL ADHD_TO Means ADHD_TO GIRL G_BY_ADHD Intercepts ADHD_TO Variances ADHD_TO GIRL G_BY_ADHD Residual Variances ADHD_TO Predicted Regression Lines Males Females

40 3. Repeated Measures ANOVA and Difference Score Regression Difference Score ANOVA / ANCOVA A difference model is expressed (for n= to N) as Y[] n = Y[] n + Dy n and Dy n = m d + Dy n * now we add G as a binary variable ( yes or no ) Dy n = g 0 + g G n + d n g o is the intercept (average difference) for the group coded 0, and g is the change in the intercept for the group coded We can add an interaction here simply by writing Dy n = g 0 + g G n + g {Y[] n G n }+d n so g is difference in the slope for group coded A Latent Difference Score Model with Group Differences d Y[] Y[] y d d SAS/SPSS Input for Difference Score Model PROC REG DATA = diff; ModD0a: MODEL delta = / STB; ModDa: MODEL delta = adhd_to4 / STB; ModDb: MODEL delta = girl / STB; ModD: MODEL delta = adhd_to4 girl / STB; ModD3: MODEL delta = adhd_to4 girl adhd_by_girl / STB; RUN; g g 0 G *Difference Model with Group Information. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.0) /NOORIGIN /DEPENDENT delta /METHOD=ENTER adhd_to4 /METHOD=ENTER girl adhd_to4 /METHOD=ENTER girl adhd_to4 /METHOD=ENTER girl adhd_to4 adhd_by_girl. g

41 Side-by-Side Distributions of Differences 0 = Boys = Girls Group Difference Score - SAS Output Dependent Variable: delta Number of Observations Read 43 Number of Observations Used 34 Number of Observations with Missing Values 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq 0.48 Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < girl Mplus Latent Difference Score with Group Information VARIABLE: NAMES = id girl minority sesyr dadhp7 doddp7 adhd_to adhd_to4 adhd_to5 adhd_to7 adhd_in adhd_in4 adhd_in5 adhd_in7 adhd_hy adhd_hy4 adhd_hy5 adhd_hy7; MISSING =.; USEVAR = adhd_to4 adhd_to7 girl; ANALYSIS: MODEL: TYPE = MEANSTRUCTURE; adhd_to7 ON adhd_to4@; adhd_to4 adhd_to7@; [adhd_to4 adhd_to7@0]; delta BY adhd_to7@; delta; [delta]; delta ON adhd_to4 girl; OUTPUT: SAMPSTAT; Mplus Output MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value DELTA BY ADHD_TO DELTA ON ADHD_TO GIRL ADHD_TO7 ON ADHD_TO GIRL WITH ADHD_TO Means ADHD_TO Intercepts ADHD_TO DELTA Variances ADHD_TO Residual Variances ADHD_TO DELTA

42 Results of Latent Difference Score Model with Group Differences -.36 * Predicted Change in ADHD by Groups * Y[] Y[] y.96 * Males Females 3.35 * -.58 * 4.37 * G.5 Latent Difference Score Results By Group The latent difference model implies m[] -m[] = b 0 + b *m[] + b * G n since G is a binary variable then for two groups m[] -m[]{ G=0} = *m[] -.5*0 = -.43 m[] -m[]{ G=} = *m[] -.5*= -.94 The latent difference model implies m[] = m[] + b *m[] + b 0 + b * G n since G is a binary variable then for two groups m[] { G=0} = *m[] -.5*0 =.9 m[] { G=} = *m[] -.5* =.4 Difference Score, Group & Interaction SAS output Dependent Variable: delta Number of Observations Read 43 Number of Observations Used 34 Number of Observations with Missing Values 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE R-Square 0.66 Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Estimate Intercept < adhd_to < girl adhd_by_girl