hdeffsev {VGAM}

Hauck-Donner Effects: Severity Measures

Description

Computes the severity of the Hauck-Donner effect for each regression coefficient of a VGLM regression.

Usage

hdeffsev(x, y, dy, ddy, allofit = FALSE, eta0 = 0, COPS0 = eta0,
         severity.table = c("None", "Faint", "Weak",
             "Moderate", "Strong", "Extreme", "Undetermined"))

Arguments

x, y

Numeric vectors; x are the estimates (sorted), and y are the signed Wald statistics.

dy, ddy

Numeric vectors; the first and second derivatives of the Wald statistics. They can be computed by hdeff.

allofit

Logical. If TRUE then other quantities are returned in a list. The default is a vector with elements selected from the argument severity.table.

severity.table

Character vector with 6 values. The last value is used for initialization. Usually users should not assign anything to this argument.

eta0

Numeric. The hypothesized value. The default is appropriate for most symmetric binomial links,and also for Poisson regression with the natural parameter.

COPS0

Numeric. See Yee (2021).

Details

This function is rough-and-ready. It is possible to use the first two derivatives obtained from hdeff to categorize the severity of the the Hauck-Donner effect (HDE). It is effectively assumed that, starting at the origin and going right, the curve is made up of a convex segment followed by a concave segment and then the convex segment. Midway in the concave segment the derivative is 0, and beyond that the HDE is really manifest because the derivative is negative.

For "none" the estimate lies on the convex part of the curve near the origin, hence there is very little HDE at all.

For "weak" the estimate lies on the concave part of the curve but the Wald statistic is still increasing as estimate gets away from 0, hence it is only a mild form of the HDE.

Previously "faint" was used but now it has been omitted.

For "moderate", "strong" and "extreme" the Wald statistic is decreasing as the estimate gets away from eta0, hence it really does exhibit the HDE. It is recommended that lrt.stat be used to compute LRT p-values, as they do not suffer from the HDE.

Value

By default this function returns a labelled vector with elements selected from severity.table. If allofit = TRUE then Yee (2022) gives details about some of the other list components, e.g., a quantity called zeta is the normal line projected onto the x-axis, and its first derivative gives additional information about the position of the estimate along the curve.

Note

This function is likely to change in the short future because it is experimental and far from complete. Improvements are intended.

Currently, in order for "Strong" to be assigned correctly, at least one such value is needed on the LHS and/or RHS each. From those, two other boundary points are obtained so that it creates two intervals.

Author(s)

Thomas W. Yee.

References

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, 117, 1763–1774. doi:10.1080/01621459.2021.1886936.

Yee, T. W. (2022). Some new results concerning the Wald tests and the parameter space. In review.

See Also

seglines, hdeff.

Examples

deg <- 4  # myfun is a function that approximates the HDE
myfun <- function(x, deriv = 0) switch(as.character(deriv),
  '0' = x^deg * exp(-x),
  '1' = (deg * x^(deg-1) - x^deg) * exp(-x),
  '2' = (deg*(deg-1)*x^(deg-2) - 2*deg*x^(deg-1) + x^deg)*exp(-x))

xgrid <- seq(0, 10, length = 101)
ansm <- hdeffsev(xgrid, myfun(xgrid), myfun(xgrid, deriv = 1),
                 myfun(xgrid, deriv = 2), allofit = TRUE)
digg <- 4
cbind(severity = ansm$sev, 
      fun      = round(myfun(xgrid), digg),
      deriv1   = round(myfun(xgrid, deriv = 1), digg),
      deriv2   = round(myfun(xgrid, deriv = 2), digg),
      zderiv1  = round(1 + (myfun(xgrid, deriv = 1))^2 +
                       myfun(xgrid, deriv = 2) * myfun(xgrid), digg))