Beta coefficient>1 and confusion in interpreting a bidirectional relationship

[Blain Waan]

I have several binary variables ( at two-time points year 1 and 2 (say, Y1_yr1, Y1_yr2, Y2_yr1, Y2_yr2, Y3_yr1, Y3_yr2, Y4_yr1 and Y4_yr2) and a few covariates (e.g. X1, X2, X3).

I have made the Y variables ordered (to get DWLS) and ran the following codes in lavaan.

mod <- '
# Year 1
    Y1_yr1 ~ Y3_yr1 + X1 + X2 + X3
    Y2_yr1 ~ Y3_yr1 + X1 + X2 + X3
    Y3_yr1 ~ Y1_yr1 + Y2_yr1 + X1 + X2 + X3
    Y4_yr1 ~ Y1_yr1 + Y2_yr1 + Y3_yr1 + X1 + X2 + X3
# Year 2
    Y1_yr2 ~ Y3_yr2 + Y1_yr1 + X1 + X2 + X3
    Y2_yr2 ~ Y3_yr2 + Y2_yr1 + X1 + X2 + X3
    Y3_yr2 ~ Y1_yr2 + Y2_yr2 + Y3_yr1 + X1 + X2 + X3
    Y4_yr2 ~ Y1_yr2 + Y2_yr2 + Y3_yr2 + Y4_yr1 + X1 + X2 + X3
'
fit <- sem(mod, data=dat)
summary(fit, standardized=TRUE)

The model converges normally but I find some confusing results:

1) The path Y2_yr1 -> Y3_yr1 is positive and significant. But the path Y3_yr1 -> Y2_yr1 is negative and significant. In individual logistic regressions, both associations were positive and significant. I expected both paths to be positive and significant in SEM too. Why does it happen and am I making any mistake? Is the testing of the bidirectional relationship meaningful here given the information I have? Our intuition was that some people having a presence of Y3 (depression) will be more likely to show the presence of Y2 (addiction) and again some people having a presence of Y2 (addiction) will be more likely show a presence of Y3 (depression). 

2) The beta coefficient is >1 for the path Y2_yr1 -> Y2_yr2. Can it be greater than 1 as both variables are binary? How should I interpret this? 

[Nabil Awan]

Can a probit regression coefficient be greater than 1?

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[Terrence Jorgensen]

Can a probit regression coefficient be greater than 1?

Using the default parameterization="delta" means your the total latent-response variance is fixed at 1, but the only condition under which a slope should always be < 1 is if the predictor's variance is also <= 1 and there is only 1 predictor.  Standardized slopes in multiple regression can exceed 1. 

Not sure about the other issues, especially without access to the data.

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen