VGAM (version 1.1-9)

R2latvar: R-squared for Latent Variable Models

Description

R-squared goodness of fit for latent variable models, such as cumulative link models. Some software such as Stata call the quantity the McKelvey--Zavoina R-squared, which was proposed in their 1975 paper for cumulative probit models.

Usage

R2latvar(object)

Value

The \(R^2\) value. Approximately, that amount is the variability in the latent variable of the model explained by all the explanatory variables. Then taking the positive square-root gives an approximate multiple correlation \(R\).

Arguments

object

A cumulative or binomialff fit using vglm. Only a few selected link functions are currently permitted: logitlink, probitlink, clogloglink. For models with more than one linear predictor, a parallelism assumption is needed also, i.e., the constraint matrices must be a 1-column matrix of 1s (except for the intercept). The model is assumed to have an intercept term.

Author

Thomas W. Yee

Details

Models such as the proportional odds model have a latent variable interpretation (see, e.g., Section 6.2.6 of Agresti (2018), Section 14.4.1.1 of Yee (2015), Section 5.2.2 of McCullagh and Nelder (1989)). It is possible to summarize the predictive power of the model by computing \(R^2\) on the transformed scale, e.g., on a standard normal distribution for a probitlink link. For more details see Section 6.3.7 of Agresti (2018).

References

Agresti, A. (2018). An Introduction to Categorical Data Analysis, 3rd ed., New York: John Wiley & Sons.

McKelvey, R. D. and W. Zavoina (1975). A statistical model for the analysis of ordinal level dependent variables. The Journal of Mathematical Sociology, 4, 103--120.

See Also

vglm, cumulative, propodds, logitlink, probitlink, clogloglink, summary.lm.

Examples

pneumo <- transform(pneumo, let = log(exposure.time))
(fit <- vglm(cbind(normal, mild, severe) ~ let, propodds, data = pneumo))
R2latvar(fit)