A detection test for the Hauck-Donner effect on each regression coefficient of a VGLM regression or 2 x 2 table.
hdeff(object, ...)
hdeff.vglm(object, derivative = NULL, se.arg = FALSE,
subset = NULL, theta0 = 0, hstep = 0.005,
fd.only = FALSE, ...)
hdeff.numeric(object, byrow = FALSE, ...)
hdeff.matrix(object, ...)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.
Usually 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.
Alternatively object may represent a 2 x 2 table of
positive counts.
If so, then the first row corresponds
to \(x2=0\) (baseline group)
and the second row \(x2=1\). The first column
corresponds to \(y=0\) (failure) and
the second column \(y=1\) (success).
Another alternative is that object is
a numerical vector
of length 4, representing a 2 x 2 table
of positive counts.
If so then it is fed into hdeff.matrix
using
the argument byrow, which matches
matrix.
See the examples below.
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.
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.
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.
Numeric. Vector recycled to the necessary length which is
the number of regression coefficients.
The null hypotheses for the regression coefficients are that
they equal those respective values, and the alternative
hypotheses are all two-sided.
It is not recommended that argument subset be used
if a vector of values is assigned here because
theta0[subset] is implied and might not work.
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).
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.
It is possible that NAs are returned.
If so, and if fd.only = FALSE, then a warning
is issued and a recursive
call is made with fd.only = TRUE---this is more
likely to return an answer without any NAs.
Logical;
fed into matrix if object is
a vector of length 4 so that there are two choices in the
order of the elements.
currently unused but may be used in the future for further arguments passed into the other methods functions.
Thomas W. Yee.
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).
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. (2022) On the Hauck-Donner effect in Wald tests: Detection, tipping points and parameter space characterization, Journal of the American Statistical Association, in press.
Yee, T. W. (2021). Some new results concerning the Hauck-Donner effect. Manuscript in preparation.
summaryvglm,
hdeffsev,
vglm,
lrt.stat,
score.stat,
wald.stat,
confintvglm,
profilevglm.
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
# 2 x 2 table of counts
R0 <- 25; N0 <- 100 # Hauck Donner (1977) data set
mymat <- c(N0-R0, R0, 8, 92) # HDE present
(mymat <- matrix(mymat, 2, 2, byrow = TRUE))
hdeff(mymat)
hdeff(c(mymat)) # Input is a vector
hdeff(c(t(mymat)), byrow = TRUE) # Reordering of the data
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