SEM Day 3 Lab Exercises SPIDA 2007 Dave Flora

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1 SEM Day 3 Lab Exercises SPIDA 2007 Dave Flora 1 Today we will see how to estimate SEM conditional latent trajectory models and interpret output using both SAS and LISREL. Exercise 1 Using SAS PROC CALIS, reproduce the conditional linear latent trajectory model presented in today s lecture, where the math achievement latent growth factors were regressed on gender. This SAS code will do it. Note that math.dat is already in wide format. proc calis data=home.math ucov aug pshort stderr; lineqs math2 = 1f1 + 0f2 + e1, math3 = 1f1 + 1f2 + e2, math4 = 1f1 + 2f2 + e3, math5 = 1f1 + 3f2 + e4, f1 = al1 intercept + ga11 female + d1, f2 = al2 intercept + ga21 female + d2; std d1 d2 = ps11 ps22, e1 e2 e3 e4 = te11 te22 te33 te44; cov d1 d2 = ps21; run; Note that this CALIS code is nearly identical to the code used yesterday for the unconditional model; except now, we have added the female dummy code, multiplied by the gamma regression coefficients, to the LINEQS for the between-persons model. Examine the output. What is the value of the regression coefficient (gamma-11) relating gender to the intercept factor? How would you interpret this value conceptually? Is it significant? What is the value of the regression coefficient (gamma-21) relating gender to the slope factor? How would you interpret this value conceptually? Is it significant? How would you interpret the values for the intercept terms in the between-persons equations? Is the simple slope for males significant? With gender included in the model, is there significant residual heterogeneity in the intercept and slope factors? Exercise 2 Just as you did yesterday for the unconditional model, re-parameterize the conditional model so that the intercept factor represents math achievement at Grade 3 rather than Grade 2. Which parameter estimates have changed? Does this make sense?

2 Exercise 3 2 Following from the lecture notes, probe the interaction between gender and time by reverse-coding the female dummy code to be a male dummy code. You will need to do so inside a simple SAS data step. For example, here is one of several possible ways to do it: data math2; set home.math; if female = 0 then male = 1; else male = 0; run; Re-estimate the model from either Exercise 1 or Exercise 2, but using the male dummy code instead of the female dummy code. Remember to change the data option in your PROC CALIS statement: proc calis data=math2... Which parameter estimates have changed? Does this make sense? Is the simple slope for females significant? NOW for LISREL

3 Exercise 4 3 Using LISREL, reproduce the conditional linear trajectory model you estimated in Exercise 1 using PROC CALIS. You can easily do so by expanding yesterday s LISREL code for the unconditional model: Linear Conditional LTM for Math Achievement DA NI=8 NO=300 MA=CM LA math2 math3 math4 math5 female black hisp retain RA FI=' h:\courses\spida\sem \math.dat' SE / MO NY=4 NE=2 NX=1 NK=1 LY=FU,FI LX=FU,FI PS=SY,FR TE=SY,FI TD=FI TY=FU,FI AL=FR GA=FI KA=FR LE int slp LK gender VA 1.0 LY(1,1) LY(2,1) LY(3,1) LY(4,1) VA 0.0 LY(1,2) VA 1.0 LY(2,2) VA 2.0 LY(3,2) VA 3.0 LY(4,2) VA 1.0 LX(1,1) FR TE(1,1) TE(2,2) TE(3,3) TE(4,4) FR GA(1,1) GA(2,1) PD OU Notice that the SE line now selects the first 5 variables in the data file, where the 5 th is female Following from today s lecture, we have to create a phantom latent independent variable, which is perfectly measured by the observed variable female : NK=1 tells LISREL that there will be one ksi, ξ, or latent independent variable. LX=FU,FI tells LISREL to set up a lambda-x matrix. TD=FI tells LISREL that the theta-delta, Θ δ, matrix will be fixed to have values of zero. KA=FR tells LISREL that the latent independent variable has a non-zero mean (where KA stands for kappa, κ, the vector of means for ξ). LK gender indicates that the label of this ksi latent variable is gender. VA 1.0 LX(1,1) assigns a factor loading of value 1.0 to represent the relationship between the observed female variable and the latent gender variable. Thus, by giving female a factor loading (i.e., lambda) value of one and allowing zero residual measurement variance (i.e., by leaving theta-delta as fixed to zero), we establish the gender latent variable as perfectly measured by female. To include the relationships between gender and the latent growth factors, we tell LISREL to set up a gamma matrix with GA=FI. The elements of this matrix that relate gender to the growth factors are then freed with FR GA(1,1) GA(2,1). These correspond to the ga11 and ga21 parameters in the PROC CALIS code given above. Examine the output. Are the relationships between gender and the growth factors the same as you found with SAS? Are the residual variances of the growth factors the same as you found with SAS?

4 Exercise 5: Multiple Group Models 4 Use LISREL to reproduce the multiple-group analysis described in today s lecture. (Unfortunately, multiple-group modeling is not available in PROC CALIS.) Unlike the previous analyses, LISREL can t do this type of analysis by directly reading the text file of raw data. Instead, the input data needs to be separate sets of summary statistics (i.e., covariance matrix and vector of means) for the two groups (i.e., male and female). These statistics are easily produced in SAS, using PROC CORR. Because we need separate means and covariances by gender, we include the by statement in PROC CORR. However, it is important to sort the data by female first: proc sort data=home.math; by female; proc corr cov; by female; var math2-math5; run; The cov option tells SAS to produce the covariance matrix. (Or, alternatively, you could generate them by reading the raw data file into PRELIS; but doing it in SAS and pasting into LISREL is easy enough, if tedious.) The LISREL code to produce the first multiple-group model is on the next page. Note that you can copy-paste the means and covariance matrices directly from the file twogroup.txt.

5 Group 1: Male DA NG=2 NI=4 NO=136 MA=CM CM ME LA math2 math3 math4 math5 MO NY=4 NE=3 LY=FU,FI PS=SY,FR TE=SY,FI AL=FR TY=FI LE int slp quad VA 1.0 LY(1,1) LY(2,1) LY(3,1) LY(4,1) VA 0.0 LY(1,2) LY(1,3) VA 1.0 LY(2,2) LY(2,3) VA 2.0 LY(3,2) VA 3.0 LY(4,2) VA 4.0 LY(3,3) VA 9.0 LY(4,3) FR TE(2,2) TE(3,3) TE(4,4) OU 5 Group 2: Female DA NO=164 CM ME LA math2 math3 math4 math5 MO LY=IN PS=IN TE=SP AL=IN TY=FI LE int slp quad VA 0.0 TE(1,1) OU First, note that there is one set of code for each group, but both are typed into a single syntax file. After the first title ( Group 1: Male ), on the DA line, there is a new specification, NG=2, which tells LISREL that the number of groups is two (instead of the default one). Also on the DA line, note that we have given the male sample size, NO=136. Next, note that the input data for the male group is a covariance matrix (CM), followed by a list of means (ME). What follows is the same code for the quadratic model that we saw yesterday. Next comes the model specification for Group 2: Female. On the DA line, we only need to give the female sample size, NO=164. Then comes the covariance matrix and mean vector for the female group, and the labels of the observed variables. Now, we only need to tell LISREL which parts of the model are the same for the female group and which are different. On the MO line, LY=IN tells LISREL that the female lambda-y matrix is invariant relative to the male lambda-y matrix; thus, the same quadratic functional form of growth is specified for both groups. Similarly, PS=IN means that the psi matrix (variances and covariances among growth factors) is invariant, i.e., the same across groups.

6 6 However, by setting TE=SP, we are allowing the female theta-epsilon matrix (within-person residual variances) to have the same pattern as the male theta-epsilon matrix (i.e., symmetric), but to take on different estimated values. Finally, by setting AL=IN, we are also constraining alpha (the vector of growth factor means) to be equal, or invariant, across groups. (This model specification is actually slightly different from the first two-group model given in the lecture notes. Here, the complete Ψ matrix is constrained to be equal across groups, whereas in the lecture notes, the growth factor variances are equal across groups, but the covariances are not. Also, in the lecture notes, the residual variance for female Grade 5 math is constrained to zero, but not here.) Run the code and inspect the output. Find the mean and variance of the growth factors (intercept, linear component, quadratic component) for the male group and for the female group. Note how there is one chi-square value for the male group and a second chi-square value for the female group, and the sum of these two chi-squares equals the Global chi-square value given toward the end of the output. Record this global chi-square value (and corresponding degrees of freedom). You will need it for the next exercise. Exercise 6 Next, change the code above so that the growth factor means are allowed to vary across groups by changing AL=IN to AL=SP in the female group. Run the code and inspect the output. Be aware that this model specification creates an improper solution (Ψ is nonpositive definite). For the purposes of this exercise, ignore the problem. (But otherwise, it is something that should be fixed.) Again, find the mean and variance of the growth factors (intercept, linear component, quadratic component) for the male group and for the female group. How have these parameter estimates changed? Conduct the chi-square difference test to see whether freeing the factor means significantly improved model fit. Exercise 7 Now, change the code so that the growth factor variances (and covariances) are free to vary across groups. Conduct the chi-square difference test to see whether this modification significantly improved model fit. Exercise 8 Challenge: Following the lecture notes, modify the model specification so that the functional form of growth for females remains quadratic, but becomes linear for males. You will have to introduce several changes to both the alpha and psi matrices! (hint: In the male group, make alpha and psi fixed, and then free individual elements. In the female group, make psi and alpha completely free.)

7 Exercise 9 7 Using either SAS PROC CALIS or LISREL or (even better) both, estimate a conditional linear latent trajectory model for the ALCUSE variable in the alcohol1 data set you analyzed previously using multilevel modeling. Remember that you will need to reformat the data into wide format as described yesterday (unless you were wise enough to save the wide-format data set that you created). Regress the intercept and slope factors on the coa, male, and peer variables simultaneously. Which significantly predicts the intercept and slope factors? If any predictor is related to the slope factor, probe the relationship using methods described previously.