The code generated by this utility can be pasted directly into an R console window. R (a free, open-source statistical computing environment) may be obtained here: http://cran.r-project.org/.
This web page calculates simple intercepts, simple slopes, and the region of significance to facilitate the testing and probing of two-way interactions estimated in latent curve analysis (LCA) models. In LCA, repeated measures of a variable y are modeled as functions of latent factors representing aspects of change or latent curves, typically an intercept factor and one or more slope factors. We use the standard structural equation modeling (SEM) notation to define equations, and we assume that the user is knowledgeable both in the general SEM and in the testing, probing, and interpretation of interactions in multiple linear regression (e.g., Aiken & West, 1991). The following material is intended to facilitate the calculation of the methods presented in Curran, Bauer, and Willoughby (2004), and we recommend consulting this paper for further details.
Let yit represent repeated measures of variable y for i = 1, 2, ..., N individuals at t = 1, 2, ..., T occasions (all of these techniques generalize to times varying over i, but for simplicity we assume that all individuals are measured at the same occasions; see Curran et al., 2004, p. 222 for details). In matrix notation, the general form of an LCA measurement model is
where y is a T x 1 vector of repeated measures for individual i, is a T x k matrix of factor loadings (where k is the number of latent curve factors, here 2 to define a linear trajectory), is a k x 1 vector of latent curve factors, and is a T x 1 vector of time-specific residuals.
In most applications of LCA, the elements of are constrained to reflect linear growth, e.g.:
The first column of contains loadings on the intercept factor. In LCA models, time is not explicitly included as a variable, but rather is incorporated in the model as elements of the second column of . The variance of the slope factor represents individual differences in the slope of the latent trajectory.
An expression for the latent curve factors is:
where is a k x 1 vector of latent curve factor means and is a k x 1 vector of residuals. Scalar expressions for elements in with no exogenous predictors are:
A typical element of y is:
One of the primary advantages of the LCA framework is that the factors representing intercept and slope can serve as endogenous (dependent) variables in other model equations. The figure above represents just such a conditional LCA model, in which the intercept and slope representing the latent trajectory of the repeated measures of y are modeled as dependent variables regressed on x. In such cases, the latent curve factors may be expressed as functions of the exogenous predictor x:
where is a k x p matrix of regression parameters linking the k latent curve factors to the p exogenous predictors and x is a p x 1 vector of exogenous predictors. Substituting Equation 6 into Equation 1 yields a reduced form equation for y:
The first parenthetical term in Equation 8 is referred to as the fixed component and the second parenthetical term as the random component.
The prediction of with time-invariant predictors x represents an interaction with time. To see why this is so, consider the scalar expressions for elements in when there is only one exogenous predictor x:
a typical element of y is then:
The fixed component of Equation 10 can be seen to contain an intercept term (i.e., ), conditional main effects for time (i.e., ) and the exogenous predictor x (i.e., 1), and the interaction of time and x (i.e., 2). Thus, the effect of time on y depends in part on the level of x. Given this, we can draw upon classical techniques for testing and plotting conditional effects in multiple regression. See our supporting material for probing interactions in standard regression here.
yt on t regressions at x1. The regression of y on time for specific values of x we term yt on t regressions at x1. Taking the expectation of Equation 10 and rearranging clarifies the role of x when x moderates the magnitude of the regression of y on time:
Note that Equation 11 has the form of a simple regression of y on t where the first parenthetical term is the intercept of the simple regression and the second parenthetical term is the slope of the simple regression. We will refer to the first parenthetical term as the simple intercept and the second term as the simple slope. It can be seen that the simple intercept and simple slope are compound coefficients that result from the linear combination of other parameters. To further explicate this, we can re-express Equation 11 in terms of sample estimates such that
These general expressions for the simple intercept (0) and simple slope (1) define the conditional regression of y on t as a function of x. Because these are sample estimates, we must compute standard errors to conduct inferential tests of these effects. The computation of these standard errors is one of the key purposes of our calculators.
yt on x1 regressions at t. Conversely, the effect of x on y can be seen to depend on time. This regression of y on x for specific values of time we term yt on x1 regressions at t. Rearranging Equation 11 clarifies the role of time when time moderates the magnitude of the regression of y on x:
Note that Equation 14 has the form of a simple regression of y on x where the first parenthetical term is a simple intercept and the second parenthetical term is a simple slope. As with yt on t regressions at x1, yt on x1 regressions at t may be expressed in terms of compound coefficients:
The sample estimates of the simple intercept (0) and simple slope (1) define the conditional regression of y on x as a function of t. Again, 0 and 1 are general expressions for simple intercepts and simple slopes for the regression of y on x conditional on t and, despite similarity in notation, are not to be confused with the simple intercept and simple slope of the regression of y on t conditional on x.
We are primarily interested in two cases: (1) the estimation of the simple intercept (0) and the simple slope (1) of the conditional regression of outcome y on time as a function of the moderator x or (2) the estimation of the simple intercept and the simple slope of the conditional regression of outcome y on x as a function of time. When comparing the calculation of the simple intercepts and slopes across these two cases, it is clear that they share a common computational form, and this is why we have used the same notation to define the simple intercept and slope for each case. However, to simplify the use of our tables in practice, we have developed calculators separately for yt on t regressions at x1 and yt on x1 regressions at t, although the underlying analytics are all identical (see Curran, Bauer, & Willoughby, 2004 for details). We now turn to a brief description of the values that can be calculated using our tables below.
For yt on t regressions at x1, the first available output is the region of significance of the simple slope describing the relation between the outcome y and time as a function of a moderator x. We do not provide the region of significance for the simple intercept given that this is rarely of interest in practice. The region of significance defines the specific values of x at which the regression of y on time transitions from non-significance to significance. There are lower and upper bounds to the region. In many cases, the regression of y on time is significant at values of x that are less than the lower bound and greater than the upper bound, and the regression is non-significant at values of the moderator falling within the region. However, there are some cases in which the opposite holds (i.e., the significant slopes fall within the region). Consequently, the output will explicitly note how the region should be interpreted in terms of the significance and non-significance of the simple slopes. There are also instances in which the region cannot be mathematically obtained, and an error is displayed if this occurs for a given application. By default, the region is calculated at = .05, but this may be changed by the user. Finally, the point estimates and standard errors of both the simple intercepts and the simple slopes are automatically calculated precisely at the lower and upper bounds of the region.
The region of significance is also available for yt on x1 regressions at t, in which case the region defines the specific values of time at which the slope of the regression of y on x transitions from non-significance to significance.
The second available output is the calculation of point estimates and standard errors for up to three simple intercepts and simple slopes of the regression of y on time at specific levels of x (or the regression of y on x at specific levels of time). In the tables we refer to these specific values as conditional values. There are a variety of potential conditional values of the moderator that may be chosen for the computation of the simple intercepts and slopes. If x is dichotomous (e.g., 0 or 1 to denote gender), we could select the first and second conditional values to be equal to 0 and 1 to compute the regression of y on time for males and for females (leaving the third conditional value blank). If the moderator is continuous, we might select values of x that are one standard deviation above the mean, equal to the mean, and one standard deviation below the mean. For yt on x1 regressions at t it probably makes the most sense to choose values of t actually used in the model, although this is not strictly required. Whatever the conditional values chosen, these specific values are entered in the section labeled "Conditional Values," and this will provide the corresponding simple intercepts and simple slopes of the regression of y on time at those specific values of x (or the regression of y on x at those specific values of time). The calculation of simple intercepts and slopes at specific values is optional; the user may leave any or all of the conditional value fields blank.
Given the calculation of one or more simple slopes, it is common to plot these relations graphically to improve interpretability of effects. The final available output is the calculation of a lower and upper value associated with each of the simple slopes to aid in the graphing of these using any standard software package (e.g., Excel, SPSS, etc.). These are provided to simply aid in the graphing of effects; no inferential tests apply here. For the regression of y on x at specific levels of time, the user enters any two values of x in order to plot the regression line between y and x at specific values of t. Although any pair of moderator values can be used, we recommend using the lower and upper specified values of t. However, many other specific values can be chosen that may be more appropriate for a particular research application. For the regression of y on time at specific levels of x, the user enters any two values of t in order to plot the regression line between y and t at specific values of x. Although any pair of values can be used, we recommend using either the lower and upper observed values of x, the lower and upper possible values of x, or one sd below and above the mean of x. However, again, many other specific values can be chosen.
Simple intercepts, simple slopes, and the region of significance can be obtained by following these seven steps. Use as many significant digits as possible for optimal precision.
Once all of the necessary information is entered into the table, simply click "Calculate." The status box will identify any errors that might have been encountered. If no errors are found, the results will be presented in the output window. The results in the output window can be pasted into any word processor for printing.
Below the output window are two additional windows. If conditional values of x and t are entered, clicking on "Calculate" will also generate R code for producing a plot of the interaction effect (R is a statistical computing language). This R code can be submitted to a remote Rweb server by clicking on "Submit above to Rweb." A new window will open containing a plot of the interaction effect. The user may make any desired changes to the generated code before submitting, but changes are not necessary to obtain a basic plot. Indeed, this window can be used as an all-purpose interface for R.
Assuming enough information is entered into the interactive table, the second output window below the table will include R syntax for generating confidence bands. The user is expected to supply lower and upper values for either x or t (-10 and +10 by default). As above, this R code can be submitted to a remote Rweb server by clicking on "Submit above to Rweb." A new window will open containing a plot of confidence bands.
Assuming enough information is entered into the interactive table, the second output window below the table will include R syntax for generating confidence bands, continuously plotted confidence intervals for simple slopes corresponding to all conditional values of the moderator. The x-axis of the resulting plot will represent conditional values of the moderator (x), and the y-axis represents values of the simple slope of y regressed on time.
If the moderator is dichotomous, only two values along the x-axis (corresponding to the codes used for grouping) would be interpretable. Therefore, in cases where the focal predictor is continuous and the moderator is dichotomous, we suggest using the lower table instead, treating time as the moderator for the confidence bands / regions of significance plot. Regardless of what variable is treated as the moderator, the user is expected to supply lower and upper values for the moderator (-10 and +10 by default). As above, this R code can be submitted to a remote Rweb server by clicking on "Submit above to Rweb." A new window will open containing a plot of confidence bands.
Click here to see a fully worked example.
Curran, P. J., Bauer, D. J, & Willoughby, M. T. (2004). Testing main effects and interactions in latent curve analysis. Psychological Methods, 9, 220-237.
Preacher, K. J., Curran, P. J., & Bauer, D. J. (2006). Computational tools for probing interaction effects in multiple linear regression, multilevel modeling, and latent curve analysis. Journal of Educational and Behavioral Statistics, 31, 437-448.