Simple intercepts, simple slopes, and regions of significance in LCA 2-way interactions
Kristopher J. Preacher (Vanderbilt University)
Patrick J. Curran (University of North Carolina at Chapel Hill)
Daniel J. Bauer (University of North Carolina at Chapel Hill)
This page provides a fully worked example using the interactive table for probing significant interaction effects in latent curve analysis.
The data used in this example were drawn from the National Longitudinal Survey of Youth (NLSY). The primary variables of interest are four repeated measures of child antisocial behavior (y) taken at 2-year intervals. Linear growth in antisocial behavior was modeled by a conditional LCA model, with individual differences in slopes predicted by a variable measuring the cognitive support of the child in the home (x). Residual variances were constrained to equality over the four repeated measures. Sample size was N = 221. A path diagram representing this model is depicted to the right.
According to this model, 
1 represents the effect of cognitive support (x) on the intercept, while 2 represents the effect of primary interest - the ability of x to predict individual differences in change in antisocial behavior (y) over time. 
represents the mean intercept and 
represents the mean slope.
LISREL 8.54 was used to estimate model parameters. Model output can be viewed here. According to the output, the model fits well. In addition, homecog significantly predicts individual differences in y slope, denoting a significant 2-way interaction between cognitive support and time. It is the purpose of the LCA 2-way interaction table to aid the researcher in probing this interaction.
As part of the LISREL run, the asymptotic covariance matrix of parameter estimates was requested by inserting the option "EC" on the output (OU) line. The lower triangular elements of the ACOV matrix were output to a separate text file (called fort.1) in free format in scientific notation. The contents of this file are as follows:
0.15232D-02 -0.24661D-03 0.29531D-03 0.12224D-17 0.11493D-18 0.42328D-01
-0.44129D-18 -0.50758D-19 -0.82541D-02 0.50448D-02 0.21425D-18 -0.16909D-19
0.24729D-02 -0.18071D-02 0.18224D-02 0.94912D-18 0.27755D-18 0.17168D-17
0.24117D-18 0.27740D-18 0.33016D+00 -0.64594D-18 0.21656D-20 -0.75056D-02
0.32167D-02 -0.21445D-02 -0.54581D-18 0.10722D-01 -0.13860D-01 0.22440D-02
-0.11123D-16 0.40155D-17 -0.19496D-17 -0.86365D-17 0.58778D-17 0.13530D+00
0.22440D-02 -0.26872D-02 -0.10458D-17 0.46187D-18 0.15386D-18 -0.25256D-17
-0.19706D-19 -0.21906D-01 0.26232D-01 -0.36138D-33 0.13717D-33 -0.19797D-48
0.74386D-49 -0.48761D-49 -0.98400D-49 0.13847D-48 0.49444D-18 0.15324D-18
0.27393D-01
Rearranged in the form of a lower triangle, the ACOV matrix is as follows:
0.15232D-02
-0.24661D-03 0.29531D-03
0.12224D-17 0.11493D-18 0.42328D-01
-0.44129D-18 -0.50758D-19 -0.82541D-02 0.50448D-02
0.21425D-18 -0.16909D-19 0.24729D-02 -0.18071D-02 0.18224D-02
0.94912D-18 0.27755D-18 0.17168D-17 0.24117D-18 0.27740D-18 0.33016D+00
-0.64594D-18 0.21656D-20 -0.75056D-02 0.32167D-02 -0.21445D-02 -0.54581D-18 0.10722D-01
-0.13860D-01 0.22440D-02 -0.11123D-16 0.40155D-17 -0.19496D-17 -0.86365D-17 0.58778D-17 0.13530D+00
0.22440D-02 -0.26872D-02 -0.10458D-17 0.46187D-18 0.15386D-18 -0.25256D-17 -0.19706D-19 -0.21906D-01 0.26232D-01
-0.36138D-33 0.13717D-33 -0.19797D-48 0.74386D-49 -0.48761D-49 -0.98400D-49 0.13847D-48 0.49444D-18 0.15324D-18 0.27393D-01
In LISREL's ACOV matrix, rows and columns represent free model parameters in the order in which they are numbered in the output. Thus, in this example, the rows and columns of the ACOV matrix are ordered in following fashion:
LISREL Label Label in Figure
Row 1: BETA(1,3) gamma 1 ...the effect of x on the intercept
Row 2: BETA(2,3) gamma 2 ...the effect of x on the slope
Row 3: PSI(1,1)
Row 4: PSI(2,1)
Row 5: PSI(2,2)
Row 6: PSI(3,3)
Row 7: TE(1,1-4)
Row 8: AL(1,1) mu alpha ...the intercept mean
Row 9: AL(1,2) mu beta ...the slope mean
Row 10: AL(1,3)
Say that we are interested in determining the regions of significance associated with yt on 
t regressions at particular (conditional) values of x of 6.6, 9.1, and 11.6 (which represent values of x 1SD below the mean, at the mean, and 1SD above the mean respectively). We should use the first of the two tables on the interactive page. Drawing upon information in the LISREL output file and ACOV matrix above, the cells of the table should be filled in as follows:
Clicking on "Calculate" yields the following output:
LCA TWO-WAY INTERACTION SIMPLE SLOPES OUTPUT
Your Input
========================================================
lambda(1) = 0
lambda(2) = 4
x1(1) = 6.6
x1(2) = 9.1
x1(3) = 11.6
Intercept = 2.1301
lambda Slope = 0.5888
x1 Slope = -0.0633
lambda*x1 Slope = -0.0453
alpha = 0.05
Asymptotic (Co)variances
========================================================
var(g00) = 0.1353
var(g10) = 0.0262
var(g20) = 0.0015
var(g30) = 0.0003
cov(g00,g20) = -0.0139
cov(g10,g30) = -0.0027
Region of Significance
========================================================
x1 at lower bound of region = 10.7843
x1 at upper bound of region = 24.8526
(simple slopes are significant *outside* this region.)
Simple Intercepts and Slopes at Conditional Values
========================================================
At x1(1)...
simple intercept = 1.7123(0.1367), z=12.5222, p=0
simple slope = 0.2898(0.0602), z=4.8142, p=0
At x1(2)...
simple intercept = 1.5541(0.0958), z=16.2162, p=0
simple slope = 0.1766(0.0422), z=4.1866, p=0
At x1(3)...
simple intercept = 1.3958(0.1368), z=10.2046, p=0
simple slope = 0.0633(0.0602), z=1.0518, p=0.2929
Simple Intercepts and Slopes at Region Boundaries
========================================================
Lower Bound...
simple intercept = 1.4475(0.1162), z=12.4535, p=0
simple slope = 0.1003(0.0512), z=1.9602, p=0.05
Upper Bound...
simple intercept = 0.5569(0.6222), z=0.895, p=0.3708
simple slope = -0.537(0.274), z=-1.9602, p=0.05
Points to Plot
=======================================================
Line for x1(1): From {lambda=0, Y=1.7123} to {lambda=4, Y=2.8716}
Line for x1(2): From {lambda=0, Y=1.5541} to {lambda=4, Y=2.2604}
Line for x1(3): From {lambda=0, Y=1.3958} to {lambda=4, Y=1.6491}
The first two sections of the output repeat the information entered by the researcher. The third section, "Region of Significance," reports the values of cognitive support outside which (in this case) the slope of antisocial behavior on time is significant at 
= .05. In our sample, the highest score on the cognitive support scale is only 14, so values of cognitive support below 10 correspond to significant slopes.
The fourth section reports simple intercepts and simple slopes for the regression of yt on t at particular (conditional) values of x. Consistent with the region of significance, only the first two conditional values of x -- 6.6 and 9.1 -- correspond to significant simple slopes. The simple intercepts are not of interest in this example, but may be important in other contexts. The fifth section reports simple intercepts and simple slopes associated with conditional values of x at the boundaries of the region of significance. p = here because simple slopes at the upper and lower bounds of this region are just-significant. The sixth and final section reports line coordinates to help the user create plots of the interaction effect. A separate line can be drawn to represent the simple regression of yt on t at particular values of x.
The page will provide not only coordinates for lines, but R code for plotting the interaction online (R is a statistical computing language). Below the main output window is an R syntax window with syntax generated using user-input information. Clicking on "Submit above to Rweb" will generate a plot of the interaction effect:
Below the first Rweb syntax window is a second syntax window. Submitting this code results in a plot of the confidence bands. With minor adjustment of the default range of x values, the region of rejection can be seen to lie between roughly 10.8 and 24.9, indicated by vertical dotted lines:
Original version posted June, 2004.