metaSEM and control variables

Dear all,

I use R (2.15.3) with Mike Cheung’s metaSEM package (0.8-4) and thus, OpenMx (1.3.2-2301). I apply Mike Cheung’s tssem1 and tssem2 commands to perform random effects MASEM. My data set is very similar to Becker and Schram (1994), so I will use the Becker and Schram (1994) data set as it is displayed on Mike Cheung’s website (data set) and make one minor adjustment to this data set: I assume that there was an additional variable, namely class size (SIZE). I also assume that the correlation coefficients between class size and the three other variables were available in the original 10 data sets which Becker and Schram (1994) used. This data set would look like this, for example (all class size data are made up):


Becker94test <- structure(list(data = structure(list(
`Becker (1978) Females` = structure(c(1, 0.47, -0.21, 0.12, 0.47, 1, -0.15, -0.05, -0.21, -0.15, 1, 0.47, 0.12, -0.05, 0.47, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Becker (1978) Males` = structure(c(1, 0.28, 0.19, 0.23, 0.28, 1, 0.18, 0.3, 0.19, 0.18, 1, 0.18, 0.23, 0.3, 0.18, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Berry (1957) Females` = structure(c(1, 0.48, 0.41, -0.1, 0.48, 1, 0.26, 0.17, 0.41, 0.26, 1, 0.34, -0.1, 0.17, 0.34, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Berry (1957) Males` = structure(c(1, 0.37, 0.4, 0.05, 0.37, 1, 0.27, 0.45, 0.4, 0.27, 1, -0.23, 0.05, 0.45, -0.23, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Rosenberg (1981) Females` = structure(c(1, 0.42, 0.48, 0.34, 0.42, 1, 0.23, 0.23, 0.48, 0.23, 1, 0.31, 0.34, 0.23, 0.31, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Rosenberg (1981) Males` = structure(c(1, 0.41, 0.74, 0.27, 0.41, 1, 0.44, 0.14, 0.74, 0.44, 1, 0.56, 0.27, 0.14, 0.56, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Weiner A (1984) Females` = structure(c(1, 0.26, 0.72, -0.03, 0.26, 1, 0.36, 0.35, 0.72, 0.36, 1, 0.24, -0.03, 0.35, 0.24, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Weiner A (1984) Males` = structure(c(1, 0.32, 0.52, 0.56, 0.32, 1, 0.1, 0.22, 0.52, 0.1, 1, 0.08, 0.56, 0.22, 0.08, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Weiner B (1984) Females` = structure(c(1, 0.58, 0.64, 0.08, 0.58, 1, 0.4, 0.63, 0.64, 0.4, 1, 0.6, 0.08, 0.63, 0.6, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))),
`Weiner B (1984) Males` = structure(c(1, 0.34, 0.28, 0.42, 0.34, 1, -0.03, 0.46, 0.28, -0.03, 1, 0.21, 0.42, 0.46, 0.21, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE")))),
.Names = c("Becker (1978) Females", "Becker (1978) Males", "Berry (1957) Females", "Berry (1957) Males", "Rosenberg (1981) Females", "Rosenberg (1981) Males", "Weiner A (1984) Females", "Weiner A (1984) Males", "Weiner B (1984) Females", "Weiner B (1984) Males")),
n = c(74, 153, 48, 55, 51, 18, 27, 43, 35, 34),
gender = c("Females", "Males", "Females", "Males", "Females", "Males", "Females", "Males", "Females", "Males")), .Names = c("data", "n", "gender"))


The model (based on Cheung, 2013) would look like this (see attachment if not displayed here):

I want to include SIZE as a control variable and treat the variable as follows. Apart from SIZE, everything is similar to (code).


## Stage 1 analysis: A diagonal matrix for random effects
random1 <- tssem1(Becker94test$data, Becker94test$n, method="REM", RE.type="Diag")
summary(random1)
coef(random1)

## Stage 2 analysis (A1, S1)
random2 <- tssem2(random1, Amatrix=A1, Smatrix=S1, diag.constraints=TRUE, intervals="LB")
summary(random2)


Results (please do not concentrate on the values as I made up the SIZE data for explanatory reasons):


> summary(random2)

Goodness-of-fit indices:
Value
Sample size 538.0000
Chi-square of target model 35.1684
DF of target model 2.0000
p value of target model 0.0000
Number of constraints imposed on "Smatrix" 1.0000
DF manually adjusted 0.0000
Chi-square of independence model 152.0658
DF of independence model 6.0000
RMSEA 0.1757
SRMR 0.1796
TLI 0.3188
CFI 0.7729
AIC 31.1684
BIC 22.5927
OpenMx status1: 0 ("0" and "1": considered fine; other values indicate problems)


Because I assume that SIZE was used as a control variable in the original studies, I interpret the results of Amatrix[1,4] as regression coefficient from SIZE to Math (similarly to Cheung (2013), Table 2). Am I right to treat SIZE in a similar way as I treat the independent variables Spatial and Verbal for this purpose, although SIZE is a control variable?

Thanks for your help in advance!

Regards, Eberhard

References
Becker BJ, Schram CM. 1994. Examining explanatory models through research synthesis. In The handbook of research synthesis. Cooper H, Hedges L (eds.), Sage: New York, NY.
Cheung MW-L. 2013. Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods: 1-12.