The package can be used to estimate latent variable count regression models in one or multiple groups. In its simplest form, it can estimate ordinary Poisson or negative binomial regression models with manifest covariates in one group (similar to the glm()-function from the stats package or the glm.nb()-function from the MASS package). In its most complex form, it can regress a count variable on multiple manifest and latent covariates within multiple groups. Let’s see, how it works!
As said before, the simplest case that can be estimated with lavacrag is an ordinary Poisson regression model, regressing a count outcome Y on a manifest covariate Z with $$ \begin{align*} E(Y|Z) &= \mu_Y = \exp(\beta_0 + \beta_1 \cdot Z)\\ Y &\sim \mathcal{P}(\lambda = \mu_Y) \end{align*} $$ In our example dataset, we can fit this model and compare it to the output of the glm()-function from the stats package:
# Usage of main function: countreg(y ~ z, data = d, family = "poisson")
m0 <- countreg(dv ~ z11, data = example01, family = "poisson")
#> Computing starting values...done. Took: 0 s
#> Fitting the model...done. Took: 0.1 s
#> Computing standard errors...done. Took: 0 s
m1 <- glm(dv ~ z11, data = example01, family = poisson())
summary(m0)
#>
#>
#> --------------------- Group 1 -----------------------
#>
#> Regression:
#> Estimate SE Est./SE p-value
#> 1 2.759 0.0146 189 0
#> z11 -0.138 0.0081 -17 0
#>
#> Means:
#> Estimate SE Est./SE p-value
#> z11 1.58 0.0418 37.8 0
#>
#> Variances:
#> Estimate SE Est./SE p-value
#> z11 1.52 0.0729 20.9 0
summary(m1)
#>
#> Call:
#> glm(formula = dv ~ z11, family = poisson(), data = example01)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 2.759062 0.014636 188.51 <2e-16 ***
#> z11 -0.137692 0.008095 -17.01 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for poisson family taken to be 1)
#>
#> Null deviance: 2144.8 on 870 degrees of freedom
#> Residual deviance: 1844.0 on 869 degrees of freedom
#> AIC: 5588.4
#>
#> Number of Fisher Scoring iterations: 4In the next step, we add a latent covariate to the model. That is, we use the option lv to specify a list of latent variables giving the names of the latent variables and a character vector of indicators measuring the latent variable. We can use the name of the latent variable within the forml option. In addition, we change family to be “nbinom” in oder to estimate a negative binomial regression, that is, adding a dispersion parameter to the model:
m2 <- countreg(dv ~ eta1,
lv = list(eta1 = c("z41", "z42", "z43")),
data = example01,
family = "nbinom"
)
#> Computing starting values...done. Took: 0.1 s
#> Fitting the model...done. Took: 0.9 s
#> Computing standard errors...done. Took: 0.4 s
summary(m2)
#>
#>
#> --------------------- Group 1 -----------------------
#>
#> Regression:
#> Estimate SE Est./SE p-value
#> 1 2.6862 0.0238 113.06 0.00e+00
#> eta1 -0.0834 0.0119 -7.03 2.05e-12
#>
#> Estimate SE Est./SE p-value
#> Dispersion 9.75 0.87 11.2 0
#>
#> Means:
#> Estimate SE Est./SE p-value
#> eta1 1.62 0.0605 26.8 0
#>
#> Variances:
#> Estimate SE Est./SE p-value
#> eta1 1.95 0.166 11.8 0
#>
#> Measurement Model:
#> Intercepts:
#> Estimate SE Est./SE p-value
#> z41 ~ 1 0.000 NA NA NA
#> z42 ~ 1 -0.114 0.115 -0.987 0.323458
#> z43 ~ 1 -0.435 0.122 -3.581 0.000343
#>
#> Loadings:
#> Estimate SE Est./SE p-value
#> eta1 =~ z41 1.00 NA NA NA
#> eta1 =~ z42 1.29 0.0571 22.6 0
#> eta1 =~ z43 1.35 0.0617 21.8 0
#>
#> Residual Variances:
#> Estimate SE Est./SE p-value
#> z41 ~~ z41 1.46 0.0929 15.69 0
#> z42 ~~ z42 1.45 0.1231 11.80 0
#> z43 ~~ z43 1.27 0.1376 9.26 0In this final model, we use a combination of manifest and latent covariates in the forml option, that is, one of the covariates is defined in the lv and the other is observed in the dataset. In addition, we specify a multi-group structural equation model using the group option.
m3 <- countreg(dv ~ eta1 + z11,
lv = list(eta1 = c("z41", "z42", "z43")),
group = "treat",
data = example01,
family = "poisson"
)
#> Computing starting values...done. Took: 0.9 s
#> Fitting the model...done. Took: 4.1 s
#> Computing standard errors...done. Took: 2.5 s
summary(m3)
#>
#>
#> --------------------- Group 1 -----------------------
#>
#> Regression:
#> Estimate SE Est./SE p-value
#> 1 2.783 0.0276 100.74 0.00e+00
#> z11 -0.127 0.0126 -10.07 0.00e+00
#> eta1 -0.102 0.0155 -6.55 5.91e-11
#>
#> Means:
#> Estimate SE Est./SE p-value
#> eta1 1.58 0.0794 20.0 0
#> z11 1.59 0.0627 25.4 0
#>
#> Variances:
#> Estimate SE Est./SE p-value
#> eta1 1.88 0.196 9.59 0
#> z11 1.69 0.115 14.61 0
#>
#> Covariances:
#> Estimate SE Est./SE p-value
#> eta1 ~~ z11 0.468 0.0988 4.74 2.1e-06
#>
#> Measurement Model:
#> Intercepts:
#> Estimate SE Est./SE p-value
#> z41 ~ 1 0.000 NA NA NA
#> z42 ~ 1 -0.083 0.113 -0.736 0.461901
#> z43 ~ 1 -0.417 0.119 -3.497 0.000471
#>
#> Loadings:
#> Estimate SE Est./SE p-value
#> eta1 =~ z41 1.00 NA NA NA
#> eta1 =~ z42 1.27 0.0556 22.9 0
#> eta1 =~ z43 1.34 0.0602 22.2 0
#>
#> Residual Variances:
#> Estimate SE Est./SE p-value
#> z41 ~~ z41 1.52 0.133 11.38 0
#> z42 ~~ z42 1.47 0.159 9.20 0
#> z43 ~~ z43 1.48 0.167 8.87 0
#>
#>
#> --------------------- Group 2 -----------------------
#>
#> Regression:
#> Estimate SE Est./SE p-value
#> 1 2.873 0.0233 123.47 0.000000
#> z11 -0.105 0.0124 -8.51 0.000000
#> eta1 -0.041 0.0118 -3.47 0.000528
#>
#> Means:
#> Estimate SE Est./SE p-value
#> eta1 1.65 0.0737 22.3 0
#> z11 1.55 0.0546 28.4 0
#>
#> Variances:
#> Estimate SE Est./SE p-value
#> eta1 2.12 0.2279 9.31 0
#> z11 1.37 0.0931 14.74 0
#>
#> Covariances:
#> Estimate SE Est./SE p-value
#> eta1 ~~ z11 0.631 0.0988 6.39 1.71e-10
#>
#> Measurement Model:
#> Intercepts:
#> Estimate SE Est./SE p-value
#> z41 ~ 1 0.000 NA NA NA
#> z42 ~ 1 -0.083 0.113 -0.736 0.461901
#> z43 ~ 1 -0.417 0.119 -3.497 0.000471
#>
#> Loadings:
#> Estimate SE Est./SE p-value
#> eta1 =~ z41 1.00 NA NA NA
#> eta1 =~ z42 1.27 0.0556 22.9 0
#> eta1 =~ z43 1.34 0.0602 22.2 0
#>
#> Residual Variances:
#> Estimate SE Est./SE p-value
#> z41 ~~ z41 1.34 0.118 11.37 0.00e+00
#> z42 ~~ z42 1.53 0.151 10.10 0.00e+00
#> z43 ~~ z43 1.12 0.174 6.43 1.24e-10