LGM: Path Diagram I
The latent growth model (LGM) is used in longitudinal studies and extends CFA to the analysis of items across time. This dataset measures student gender and GPA across five semesters. The goal of this LGM is to assess the linear trajectory of GPA, with the question being, “Does the average student GPA in a particular school go up or down over time?”
Interpretation I
Parameter interpretation: intercept
The predicted GPA (i) in the first semester is 2.602.
Interpretation of the mean slope (s) parameter
For every semester (i.e., one-unit increase) in time, i.e. for every sequential semester, average GPA is expected to go up by 0.104 points.
lavaan 0.6-12 ended normally after 62 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 10
Number of observations 200
Model Test User Model:
Test statistic 12.202
Degrees of freedom 10
P-value (Chi-square) 0.272
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
i =~
gpa0 1.000
gpa1 1.000
gpa2 1.000
gpa3 1.000
gpa4 1.000
s =~
gpa0 0.000
gpa1 1.000
gpa2 2.000
gpa3 3.000
gpa4 4.000
Covariances:
Estimate Std.Err z-value P(>|z|)
i ~~
s -0.000 0.002 -0.073 0.942
Intercepts:
Estimate Std.Err z-value P(>|z|)
.gpa0 0.000
.gpa1 0.000
.gpa2 0.000
.gpa3 0.000
.gpa4 0.000
i 2.602 0.018 140.832 0.000
s 0.104 0.006 16.444 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.gpa0 0.069 0.009 7.263 0.000
.gpa1 0.068 0.008 8.628 0.000
.gpa2 0.056 0.006 9.195 0.000
.gpa3 0.036 0.005 7.997 0.000
.gpa4 -0.001 0.005 -0.121 0.903
i 0.034 0.007 4.577 0.000
s 0.006 0.001 5.880 0.000LGM vs. HLM I
Equivalence of the LGM to the hierarchical linear model (HLM)
The latent growth model in its basic form (i.e., intercept and linear slope) is equivalent to the hierarchical linear model (HLM) with random intercept and slope on the condition that the residual variances or terms in the latent growth model are constrained to be equivalent across time. If you do not have to use the intercept and slope to predict another latent variable, then you should prefer HLM to LGM.
Linear mixed model fit by REML ['lmerMod']
Formula: gpa ~ time + (time | student)
Data: longdat
REML criterion at convergence: 327.1
Scaled residuals:
Min 1Q Median 3Q Max
-3.0392 -0.5145 -0.0096 0.5421 2.9373
Random effects:
Groups Name Variance Std.Dev. Corr
student (Intercept) 0.043903 0.2095
time 0.006083 0.0780 -0.15
Residual 0.047025 0.2169
Number of obs: 1000, groups: student, 200
Fixed effects:
Estimate Std. Error t value
(Intercept) 2.600500 0.018989 136.95
time 0.105350 0.007344 14.35
Correlation of Fixed Effects:
(Intr)
time -0.423Path Diagram II
Adding a predictor (student gender) to the latent growth model (LGM)
Interpretation II
Parameter interpretation: intercept (i)
Gender does not significantly predict the intercept, but it predicts the linear slope. This implies that female and male students do not differ in their starting GPA’s in the first semester of school (females have an estimated 0.068 higher GPA compared to males but the standard error does not warrant statistical significance).
Interpretation of the mean slope (s) parameter
However, gender predicts the slope so that female students have a linear growth trajectory that is 0.035 GPA units higher than male students across these five semesters.
lavaan 0.6-12 ended normally after 55 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 12
Number of equality constraints 4
Number of observations 200
Model Test User Model:
Test statistic 107.504
Degrees of freedom 17
P-value (Chi-square) 0.000
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
i =~
gpa0 1.000
gpa1 1.000
gpa2 1.000
gpa3 1.000
gpa4 1.000
s =~
gpa0 0.000
gpa1 1.000
gpa2 2.000
gpa3 3.000
gpa4 4.000
Regressions:
Estimate Std.Err z-value P(>|z|)
i ~
sex 0.068 0.038 1.805 0.071
s ~
sex 0.035 0.014 2.396 0.017
Covariances:
Estimate Std.Err z-value P(>|z|)
.i ~~
.s -0.003 0.002 -1.343 0.179
Intercepts:
Estimate Std.Err z-value P(>|z|)
.gpa0 0.000
.gpa1 0.000
.gpa2 0.000
.gpa3 0.000
.gpa4 0.000
.i 2.565 0.027 94.080 0.000
.s 0.087 0.010 8.317 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.gpa0 (a) 0.047 0.003 17.321 0.000
.gpa1 (a) 0.047 0.003 17.321 0.000
.gpa2 (a) 0.047 0.003 17.321 0.000
.gpa3 (a) 0.047 0.003 17.321 0.000
.gpa4 (a) 0.047 0.003 17.321 0.000
.i 0.042 0.007 5.850 0.000
.s 0.006 0.001 5.315 0.000LGM vs. HLM II
Equivalence of the LGM to the hierarchical linear model (HLM) with a time predictor
Linear mixed model fit by REML ['lmerMod']
Formula: gpa ~ time + sex + time:sex + (time | student)
Data: longdat
REML criterion at convergence: 323.1
Scaled residuals:
Min 1Q Median 3Q Max
-3.02747 -0.52063 -0.00502 0.54474 2.97804
Random effects:
Groups Name Variance Std.Dev. Corr
student (Intercept) 0.043105 0.20762
time 0.005835 0.07639 -0.19
Residual 0.047025 0.21685
Number of obs: 1000, groups: student, 200
Fixed effects:
Estimate Std. Error t value
(Intercept) 2.56484 0.02740 93.609
time 0.08716 0.01053 8.276
sex 0.06792 0.03782 1.796
time:sex 0.03465 0.01454 2.384
Correlation of Fixed Effects:
(Intr) time sex
time -0.454
sex -0.725 0.329
time:sex 0.329 -0.725 -0.454