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)
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.
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 \(R\).
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.
# NOT RUN {pneumo <- transform(pneumo, let = log(exposure.time))
(fit <- vglm(cbind(normal, mild, severe) ~ let, propodds, data = pneumo))
R2latvar(fit)
# }