Appendix 1 : Initial SEM Equations

Technical Appendix

Appendix [1]: Initial SEM Equations

<txt>Path diagram 2a is used to represent a simultaneous equations where the individual differences (i = 1 to N) are organized as

(1) Near[1]i = n1 + ntTrainingi + n[1]i and

Far[1]i = f1 + ftTrainingi + fnNear[1]i + f[1]i

where the Near[1] and Far[1] variables indicated measured scores post-training, the Training variable is coded as 0 = Control or 1 = Trained. In this model both variables have intercepts (n1 and f1) and residual terms (n[1] and f[1]). The Near and Far transfer scores are directly impacted by the Training (with slopes nt and nt). In addition, the variation in the Far scores are impacted by the Near scores (with slope fn). In path regression terms, this parameter adds the hypothesis that the training will be carried over, or transferred, from the Near to Far scores.

In Path diagram 2b we eliminate the grouping variable (Training) but add a group coding (g) as a superscript on both the variables and the model parameters. Here we will write a model for two separate groups (g=T or C) where the equations for any group are written as

(2) Near[1]i(g) = n1(g) + n[1]i(g) and

Far[1]i(g) = f1(g) + fn(g)Near[1]i(g) + f[1]i(g)

where the superscript g indicates that each of these scores or coefficients can differ over grous. By writing this model as a multiple group SEM we can test the hypothesis of invariance of any parameter over groups using the standard chi-square test for nested models. In this case, the parameters of the impact of training are implicit (no  included), but we can tests for mean differences due to training by constraining the model intercepts to be the same over groups (i.e., n1(c)=n1(t) and/or f1(c)=f1(t)). Similarly, to test for an interaction of the Near Far effect, we constrain the two regression slopes to be equal (i.e., fn(c)=fn(t)).

Pre-Post SEM Alternatives

<txt>To utilize all pre-post data we write a linear change model for two separate groups (g=T or C) where the superscript g indicates that each of these scores or coefficients can differ over groups. A path diagram of this SEM is presented in Figure 3 for one group, but a two-group model is assumed. This kind of model can be understood in three parts. First, the pre-test means and deviations for both Near and Far indicators are decomposed as means and deviations as

(3) Near[0]i = n + n[0]i and

Far[0]i = f + f[0]i ,

So the parameters do not vary over groups, and the pretest variables are allowed to covary (nf). Next, each variable at the post-test is thought to be composed of the pre-test score plus an unobserved change score. This is accomplished by incorporating a set of fixed unit weights (=1) and writing

(4) Near[1]i(g) = Near[0]i + Near[1]i(g) and

Far[1]i(g) = Far[0]i + Far[1]i(g)

so we add two latent changes (Near[1] or Far[1]). Now that these latent change scores as part of the model they can be the dependent variables in a simultaneous equation written as

(5) Near[1]i(g) = n1(g) + n(g)Near[0]i + fn(g)Far[0]i+ n[1]i(g) and

Far[1]i(g) = f1(g) + f(g)Far[0]i + fn(g)Near[0]i + fn(g)Near[1]i(g) + f[1]i(g)

In this form, the latent changes (Near[1] or Far[1]) can have intercepts (n1(g), f1(g)), lagged auto-regressions from the pre-test scores (n(g), f(g)), crossed and lagged regressions from the pre-test scores (fn(g), nf(g)), and the NearFar is included for the post-test (fn(g)) scores. Of course, one benefit of the multiple group SEM approach is that any of these parameters may be examined for invariance over groups without creating product terms.

Multivariate Pre-Post SEM Models

<txt>In path diagram 4 a common factor is defined for three measures of near transfer by writing

(6) Letter Series[t]i = 1Near([t)i + n1([t)i

Letter Sets[t]i = 2Near[t]i + n2[t]i

Word Series[t]i = 3Near[t]i + u3[t]i

where the Near[t] is a common factor with three factor loadings (1,2,3) at each occasion (t = 0,1). This model allows a separation of the variance of each measure into one part due to the common factor (n) and one part due to the unique factors (j2). In order to identify the scale of measurement of the unobserved common factor we will fix the first loading (1=1). We can also add specific covariances for each measure (jj) to eliminate potentially spurious covariation (<cr46Meredith & Horn, 2001).

A similar model is subsequently posed for the measures of Far transfer, and we write

(7) EPT[t]i = 4 Far[t]i + u4[t]i

OTDL[t]i = 5 Far[t]i + u5[t]i

IATDL[t]i = 6 Far[t]i + u6[t]i

using the same constraints to yield the same separation of common and unique variance. One potential complication here is that the OTDL[1] variable is not measured, so the OTDL[1] is assumed to be a latent variable and the common factor Far[1] is only formed using two indicators at the post-test.

It is important to point out that, if we do have metric invariance, we can then use the latent difference model of Equation (4) applied at the common factor level, as presented in Figure 4. One benefit of modeling change at this common factor level is that if the common factors (Near[0] and Near[1]) are considered free of measurement error, then the latent change score (Near) must also be considered free of measurement error.

<acpt>

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