R: Hauck-Donner Effects: A Detection Test for Wald Tests

hdeff {VGAM} | R Documentation

## Hauck-Donner Effects: A Detection Test for Wald Tests

### Description

A detection test for the Hauck-Donner effect on each regression coefficient of a VGLM regression.

### Usage

    hdeff(object, ...)
    hdeff.vglm(object, derivative = NULL, se.arg = FALSE,
               subset = NULL, hstep = 0.005, fd.only = FALSE, ...)

### Arguments

`object` | A `vglm` object. Although only a limited number of family functions have an analytical solution to the HDE detection test (`binomialff`, `borel.tanner`, `cumulative`, `erlang`, `felix`, `lindley`, `poissonff`, `topple`, `uninormal`, `zipoissonff`, and `zipoisson`; hopefully some more will be implemented in the short future!) the finite-differences (FDs) method can be applied to almost all VGAM family functions to get a numerical solution.
`derivative` | Numeric. Either 1 or 2. Currently only a few models having one linear predictor are handled analytically for `derivative = 2`, e.g., `binomialff`, `poissonff`. However, the numerical method can return the first two derivatives for almost all models.
`se.arg` | Logical. If `TRUE` then the derivatives of the standard errors are returned as well, because usually the derivatives of the Wald statistics are of central interest. Requires `derivative` to be assigned the value 1 or 2 for this argument to operate.
`subset` | Logical or vector of indices, to select the regression coefficients of interest. The default is to select all coefficients. Recycled if necessary if logical. If numeric then they should comprise elements from `1:length(coef(object))`. This argument can be useful for computing the derivatives of a Cox regression (`coxph`) fitted using artificially created Poisson data; then there are many coefficients that are effectively nuisance parameters.
`hstep` | Positive numeric and recycled to length 2; it is the so-called step size when using finite-differences and is often called h in the calculus literature, e.g., f'(x) is approximately (f(x+h) - f(x)) / h. For the 2nd-order partial derivatives, there are two step sizes and hence this argument is recycled to length 2. The default is to have the same values. The 1st-order derivatives use the first value only. It is recommended that a few values of this argument be tried because values of the first and second derivatives can vary accordingly. If any values are too large then the derivatives may be inaccurate; and if too small then the derivatives may be unstable and subject to too much round-off/cancellation error (in fact it may create an error or a `NA`).
`fd.only` | Logical; if `TRUE` then finite-differences are used to estimate the derivatives even if an analytical solution has been coded, By default, finite-differences will be used when an analytical solution has not been implemented.
`...` | currently unused but may be used in the future for further arguments passed into the other methods functions.

### Details

Almost all of statistical inference based on the likelihood assumes that the parameter estimates are located in the interior of the parameter space. The nonregular case of being located on the boundary is not considered very much and leads to very different results from the regular case. Practically, an important question is: how close is close to the boundary? One might answer this as: the parameter estimates are too close to the boundary when the Hauck-Donner effect (HDE) is present, whereby the Wald statistic becomes aberrant.

Hauck and Donner (1977) first observed an aberration of the Wald test statistic not monotonically increasing as a function of increasing distance between the parameter estimate and the null value. This "disturbing" and "undesirable" underappreciated effect has since been observed in other regression models by various authors. This function computes the first, and possibly second, derivative of the Wald statistic for each regression coefficient. A negative value of the first derivative is indicative of the HDE being present. More information can be obtained from `hdeffsev` regarding HDE severity: there may be none, faint, weak, moderate, strong and extreme amounts of HDE present.

In general, most models have derivatives that are computed numerically using finite-difference approximations. The reason is that it takes a lot of work to program in the analytical solution (this includes a few very common models, such as `poissonff` and `binomialff`, where the first two derivatives have been implemented).

### Value

By default this function returns a labelled logical vector; a `TRUE` means the HDE is affirmative for that coefficient (negative slope). Hence ideally all values are `FALSE`. Any `TRUE` values suggests that the MLE is too near the boundary of the parameter space, and that the p-value for that regression coefficient is biased upwards. When present a highly significant variable might be deemed nonsignificant, and thus the HDE can create havoc for variable selection. If the HDE is present then more accurate p-values can generally be obtained by conducting a likelihood ratio test (see `lrt.stat.vlm`) or Rao's score test (see `score.stat.vlm`); indeed the default of `wald.stat.vlm` does not suffer from the HDE.

Setting `deriv = 1` returns a numerical vector of first derivatives of the Wald statistics. Setting `deriv = 2` returns a 2-column matrix of first and second derivatives of the Wald statistics. Then setting `se.arg = TRUE` returns an additional 1 or 2 columns.

Some 2nd derivatives are `NA` if only a partial analytic solution has been programmed in.

For those VGAM family functions whose HDE test has not yet been implemented explicitly (the vast majority of them), finite-difference approximations to the derivatives will be used—see the arguments `hstep` and `fd.only` for getting some control on them.

### Note

The function `summaryvglm` conducts the HDE detection test if possible and prints out a line at the bottom if the HDE is detected for some regression coefficients. By “if possible”, only a few family functions are exempt and they have an `infos` slot with component `hadof = FALSE`; such as `normal.vcm`, `rec.normal` because it uses the BFGS-IRLS method for computing the working weights. For these few a `NULL` is returned by `hdeff`.

If the second derivatives are of interest then it is recommended that `crit = "c"` be added to the fitting so that a slightly more accurate model results (usually one more IRLS iteration). This is because the FD approximation is very sensitive to values of the working weights, so they need to be computed accurately. Occasionally, if the coefficient is close to 0, then its Wald statistic's second derivative may be unusually large in magnitude (this could be due to something such as roundoff error).

This function is currently under development and may change a little in the short future. For HDE severity measures see `hdeffsev`.

Yee (2018) gives details about HDE detection for the entire VGLM class, and proves a tipping point theorem with tipping points 1/4 and 3/5. The HDE severity measures allow partitioning of the parameter space into 6 regions from the interior and going outwards towards the boundary edges. It is also shown that with 1-parameter binary regression the HDE cannot occur unless the log odds ratio is at least 2.40, which corresponds to an odds ratio of 11.0 or more.

### Author(s)

Thomas W. Yee.

### References

Hauck, J. W. W. and A. Donner (1977). Wald's test as applied to hypotheses in logit analysis. Journal of the American Statistical Association, 72, 851–853. Corrigenda: JASA, 75, 482.

Yee, T. W. (2018) On the Hauck-Donner effect in Wald tests: Detection, and parameter space characterization (submitted for publication).

### See Also

`summaryvglm`, `hdeffsev`, `vglm`, `lrt.stat`, `score.stat`, `wald.stat`, `confintvglm`, `profilevglm`.

### Examples

    pneumo <- transform(pneumo, let = log(exposure.time))
    fit <- vglm(cbind(normal, mild, severe) ~ let, data = pneumo,
                trace = TRUE, crit = "c",  # Get some more accuracy
                cumulative(reverse = TRUE,  parallel = TRUE))
    cumulative()@infos()$hadof  # Analytical solution implemented
    hdeff(fit)
    hdeff(fit, deriv = 1)  # Analytical solution
    hdeff(fit, deriv = 2)  # It is a partial analytical solution
    hdeff(fit, deriv = 2, se.arg = TRUE,
          fd.only = TRUE)  # All derivatives solved numerically by FDs

[Package VGAM version 1.1-3 Index]