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")
#> Fitting the model...done. Took: 0.2 s
#> Computing standard errors...done. Took: 0.1 s
m1 <- glm(dv ~ z11, data = example01, family = poisson())
summary(m0)
#> lhs op rhs dest type group par_free par SE
#> 1 % w groupw <NA> 1 1 6.7691671 0.033891754
#> 2 dv ~ 1 regcoef <NA> 1 2 2.7590148 0.014636007
#> 3 dv ~ z11 regcoef <NA> 1 3 -0.1376593 0.008095389
#> 4 z11 ~ 1 z mean 1 4 1.5792611 0.041792618
#> 5 z11 ~~ z11 z var 1 5 1.5213083 0.072912680
#> 6 dv ~~ dv sigmaw size 1 0 0.0000000 NA
summary(m1)
#>
#> Call:
#> glm(formula = dv ~ z11, family = poisson(), data = example01)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -4.7673 -1.0555 -0.1332 0.9342 4.3367
#>
#> 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: 0.3 s
#> Fitting the model...done. Took: 4 s
#> Computing standard errors...done. Took: 2.5 s
summary(m2)
#> lhs op rhs dest type group par_free par SE
#> 1 % w groupw <NA> 1 1 6.76964197 0.03388371
#> 2 dv ~ 1 regcoef <NA> 1 2 2.68663313 0.02379601
#> 3 dv ~ eta1 regcoef <NA> 1 3 -0.08359125 0.01188202
#> 4 z41 ~ 1 mm nu 1 0 0.00000000 NA
#> 5 eta1 =~ z41 mm lambda 1 0 1.00000000 NA
#> 6 z42 ~ 1 mm nu 1 4 -0.11917309 0.11548397
#> 7 eta1 =~ z42 mm lambda 1 5 1.29327899 0.05719989
#> 8 z43 ~ 1 mm nu 1 6 -0.44339473 0.12211127
#> 9 eta1 =~ z43 mm lambda 1 7 1.34991108 0.06193403
#> 10 eta1 ~ 1 lv_grid mean 1 8 1.62188325 0.06037340
#> 11 eta1 ~~ eta1 lv_grid var 1 9 1.93859883 0.16490032
#> 12 dv ~~ dv sigmaw size 1 10 9.77343464 0.87386838
#> 13 z41 ~~ z41 sigmaw veps 1 11 1.45156140 0.09244556
#> 14 z42 ~~ z42 sigmaw veps 1 12 1.45590733 0.12307052
#> 15 z43 ~~ z43 sigmaw veps 1 13 1.27641140 0.13764269In 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: 1.3 s
#> Fitting the model...done. Took: 8.5 s
#> Computing standard errors...done. Took: 11.3 s
summary(m3)
#> lhs op rhs dest type group par_free par SE
#> 1 treat % w groupw <NA> 1 1 6.05912528 0.04833676
#> 2 dv ~ 1 regcoef <NA> 1 2 2.78210242 0.02747382
#> 3 dv ~ z11 regcoef <NA> 1 3 -0.12670638 0.01261108
#> 4 dv ~ eta1 regcoef <NA> 1 4 -0.10132733 0.01543693
#> 5 z41 ~ 1 mm nu 1 0 0.00000000 NA
#> 6 eta1 =~ z41 mm lambda 1 0 1.00000000 NA
#> 7 z42 ~ 1 mm nu 1 5 -0.05967178 0.11116074
#> 8 eta1 =~ z42 mm lambda 1 6 1.26310127 0.05478004
#> 9 z43 ~ 1 mm nu 1 7 -0.38713853 0.11690551
#> 10 eta1 =~ z43 mm lambda 1 8 1.32299844 0.05897220
#> 11 eta1 ~ 1 lv_grid mean 1 9 1.58142694 0.07981689
#> 12 eta1 ~~ eta1 lv_grid var 1 10 1.91143485 0.19823917
#> 13 z11 ~ 1 z mean 1 11 1.59253246 0.06253208
#> 14 z11 ~~ z11 z var 1 12 1.67760248 0.11425722
#> 15 eta1 ~~ z11 z cov_z_lv 1 13 0.47538478 0.09925245
#> 16 dv ~~ dv sigmaw size 1 0 0.00000000 NA
#> 17 z41 ~~ z41 sigmaw veps 1 14 1.51170895 0.13360365
#> 18 z42 ~~ z42 sigmaw veps 1 15 1.46559039 0.15896507
#> 19 z43 ~~ z43 sigmaw veps 1 16 1.48943922 0.16644346
#> 20 treat % w groupw <NA> 2 17 6.09358579 0.04751105
#> 21 dv ~ 1 regcoef <NA> 2 18 2.87281686 0.02317606
#> 22 dv ~ z11 regcoef <NA> 2 19 -0.10529937 0.01236109
#> 23 dv ~ eta1 regcoef <NA> 2 20 -0.04073700 0.01174385
#> 24 z41 ~ 1 mm nu 2 0 0.00000000 NA
#> 25 eta1 =~ z41 mm lambda 2 0 1.00000000 NA
#> 26 z42 ~ 1 mm nu 2 21 -0.05967178 0.11116074
#> 27 eta1 =~ z42 mm lambda 2 22 1.26310127 0.05478004
#> 28 z43 ~ 1 mm nu 2 23 -0.38713853 0.11690551
#> 29 eta1 =~ z43 mm lambda 2 24 1.32299844 0.05897220
#> 30 eta1 ~ 1 lv_grid mean 2 25 1.64428240 0.07442472
#> 31 eta1 ~~ eta1 lv_grid var 2 26 2.17708130 0.23636010
#> 32 z11 ~ 1 z mean 2 27 1.55289387 0.05473276
#> 33 z11 ~~ z11 z var 2 28 1.37709205 0.09381845
#> 34 eta1 ~~ z11 z cov_z_lv 2 29 0.64066362 0.10118807
#> 35 dv ~~ dv sigmaw size 2 0 0.00000000 NA
#> 36 z41 ~~ z41 sigmaw veps 2 30 1.34201922 0.11910312
#> 37 z42 ~~ z42 sigmaw veps 2 31 1.52475385 0.15060507
#> 38 z43 ~~ z43 sigmaw veps 2 32 1.12820338 0.17208252