Generalised Link Transformation

Transform a numeric variable using a GLM link function, or return an estimate of same.

Usage

glt(x, family = NULL, force.est = FALSE)

Arguments

x: a positive numeric vector, corresponding to a variable to be transformed. Can also be a list or nested list of such vectors.
family: Optional, the error distribution family containing the link function which will be used to transform x (see family() for specification details). If not supplied, it is determined from x (see Details).
force.est: Logical, whether to force the return of the estimated rather than direct transformation, where the latter is available (i.e. does not contain undefined values).

Value

A numeric vector of the transformed values, or an array, list of vectors/arrays, or nested list.

Details

glt() can be used to provide a 'generalised' transformation of a numeric variable, using the link function from a generalised linear model (GLM) fit to the variable. The transformation is generalised in the sense that it can extend even to cases where a standard link transformation would produce undefined values. It achieves this by using an estimate based on the 'working' response variable of the GLM (see below). If the error distribution family is not specified (default), then it is determined (roughly) from x, with binomial(link = "logit") used when all x <= 1 and poisson(link = "log") otherwise. Although the function is generally intended for variables with a binomial or Poisson distribution, any variable which can be fit using glm() can be supplied. One of the key purposes of glt() is to allow the calculation of fully standardised effects (coefficients) for GLMs (in which case x = the response variable), while it can also facilitate the proper calculation of SEM indirect effects (see below).

Estimating the direct link transformation

A key challenge in generating fully standardised effects for a GLM with a non-gaussian link function is the difficulty in calculating appropriate standardised ranges (typically the standard deviation) for the response variable in the link scale. This is because a direct transformation of the response will often produce undefined values. Although methods for circumventing this issue by indirectly estimating the variance of the response on the link scale have been proposed – including a latent-theoretic approach for binomial models (McKelvey & Zavoina, 1975) and a more general variance-based method using pseudo R-squared (Menard, 2011) – here an alternative approach is used. Where transformed values are undefined, the function will instead return the synthetic 'working' response from the iteratively reweighted least squares (IRLS) algorithm of the GLM (McCullagh & Nelder, 1989). This is reconstructed as the sum of the linear predictor and the working residuals – with the latter comprising the error of the model on the link scale. The advantage of this approach is that a relatively straightforward 'transformation' of any non-gaussian response is readily attainable in all cases. The standard deviation (or other relevant range) can then be calculated using values of the transformed response and used to scale the effects. An additional benefit for piecewise SEMs is that the transformed rather than original response can be specified as a predictor in other models, ensuring that standardised indirect and total effects are calculated correctly (i.e. using the same units).

To ensure a high level of 'accuracy' in the working response – in the sense that the inverse-transformation is practically indistinguishable from the original response variable – the function uses the following iterative fitting procedure to calculate a 'final' working response:

A new GLM of the same error family is fit with the original response variable as both predictor and response, and using a single IRLS iteration.
The working response is reconstructed from this model.
The inverse transformation of the working response is calculated.
If the inverse transformation is 'effectively equal' to the original response (tested using all.equal() with the default tolerance of 1.5e-8), the working response is returned; otherwise, the GLM is refit with the working response now as the predictor, and steps 2-4 are repeated – each time with an additional IRLS iteration.

This approach will generate a very reasonable transformation of the response variable, which will also be practically indistinguishable from the direct transformation, where this can be compared (see Examples). It also ensures that the transformed values, and hence the standard deviation, are the same for any GLM fitting the same response (provided it uses the same link function) – facilitating model comparisons, selection, and averaging.

Note

As we often cannot directly observe the GLM response variable on the link scale, any method estimating its values or statistics will be wrong to some degree. The heuristic approach described here aims to reduce this error as far as (reasonably) possible, while also generating standardised effects whose interpretation most closely resembles those of the ordinary linear model. The solution of using the working response from the GLM to scale effects is a practical, but reasonable one, and one that takes advantage of modern computing power to minimise error through iterative model fitting. An added bonus is that the estimated variance is constant across models fit to the same response variable, which cannot be said of previous methods (Menard, 2011). The overall approach would be classed as 'observed-empirical' by Grace et al. (2018), as it utilises model error variance (the estimated working residuals) rather than theoretical distribution-specific variance.

References

Grace, J. B., Johnson, D. J., Lefcheck, J. S., & Byrnes, J. E. K. (2018). Quantifying relative importance: computing standardized effects in models with binary outcomes. Ecosphere, 9, e02283. doi:10/gdm5bj

McCullagh P., & Nelder, J. A. (1989). Generalized Linear Models (2nd Edition). Chapman and Hall.

McKelvey, R. D., & Zavoina, W. (1975). A statistical model for the analysis of ordinal level dependent variables. The Journal of Mathematical Sociology, 4(1), 103-120. doi:10/dqfhpp

Menard, S. (2011). Standards for Standardized Logistic Regression Coefficients. Social Forces, 89, 1409-1428. doi:10/bvxb6s

Examples

# Compare estimate with a direct link transformation
# (test with a poisson variable, log link)
set.seed(13)
y <- rpois(30, lambda = 10)
yl <- unname(glt(y, force.est = TRUE))

# Effectively equal?
all.equal(log(y), yl)
#> [1] TRUE
# TRUE

# Actual difference...
all.equal(log(y), yl, tolerance = .Machine$double.eps)
#> [1] "Mean relative difference: 2.489317e-10"
# "Mean relative difference: 2.489317e-10"