KOTORY
Robust Three-Group Tests for Heteroscedasticity in Linear Regression
Tests for heteroscedasticity in the linear regression model that
sort the data by a regressor, split them into three equal parts and compare
the error scale of the parts. The ordinary least squares version 'kah3.test()'
refers the ratio of the largest to the smallest residual mean square to its
exact null distribution, Hartley's maximum F-ratio with three groups; the
robust version 'kah.robust.test()' replaces the mean squares by least trimmed
squares scales, so that outliers neither create nor hide heteroscedasticity,
and refers the ratio to a maximum F-ratio with simulated effective degrees of
freedom, to a Monte Carlo reference or to a residual bootstrap. The
distribution, density, quantile and random generation functions of the
maximum F-ratio are provided, together with 'run.all.het()', which runs the
proposed tests next to the Goldfeld-Quandt, Breusch-Pagan, White and robust
modified Goldfeld-Quandt tests in one call. For more details see Hartley
(1950)
KOTORY
Robust three-group tests for heteroscedasticity in linear regression.
The tests sort the data by a regressor, split them into three equal parts, fit the regression in each part and compare the error scale of the parts.
| Function | What it does |
|---|---|
kah3.test() |
Least squares in each part. The ratio of the largest to the smallest residual mean square follows Hartley's maximum F-ratio with three groups exactly, so the p-value is exact at every sample size. |
kah.robust.test() |
Least trimmed squares (LTS) in each part, so outliers neither create nor hide heteroscedasticity as easily. Three references: an effective-degrees-of-freedom F-max approximation (default), a Monte Carlo reference, or a residual bootstrap for non-normal errors. |
run.all.het() |
Runs the KaH tests next to Goldfeld-Quandt, Breusch-Pagan (Koenker), White and the robust modified Goldfeld-Quandt test of Rana, Midi and Imon (2008) and, when skedastic is installed, three more recent tests. |
pfmax(), dfmax(), qfmax(), rfmax() |
Hartley's maximum F-ratio distribution for any number of groups and any positive degrees of freedom. |
Installation
# development version
# install.packages("remotes")
remotes::install_github("ebrahimkhaled/KOTORY")
Example
library(KOTORY)
# stopping distance of cars: the spread grows with speed
kah3.test(dist ~ speed, data = cars)
kah.robust.test(dist ~ speed, data = cars)
# all tests side by side
run.all.het(dist ~ speed, data = cars)
# an outlier fools least squares but not the robust test
set.seed(2)
x <- runif(60)
y0 <- 1 + x + rnorm(60)
y <- y0; y[which.min(x)] <- 12
c(clean = kah3.test(y0 ~ x)$p.value, outlier = kah3.test(y ~ x)$p.value)
c(clean = kah.robust.test(y0 ~ x)$p.value, outlier = kah.robust.test(y ~ x)$p.value)
Origin
The three-group tests were proposed in the doctoral thesis of Ahmed El-Kotory (Alexandria University), where their critical values were tabulated by simulation. This package replaces those tables: the least squares version follows Hartley's (1950) maximum F-ratio exactly, and the robust version is referred to an effective-degrees-of-freedom approximation, a Monte Carlo reference or a bootstrap.
References
- Hartley HO (1950). The maximum F-ratio as a short-cut test for heterogeneity of variance. Biometrika 37(3/4), 308-312. https://doi.org/10.1093/biomet/37.3-4.308
- Goldfeld SM, Quandt RE (1965). Some tests for homoscedasticity. JASA 60(310), 539-547. https://doi.org/10.1080/01621459.1965.10480811
- Rousseeuw PJ (1984). Least median of squares regression. JASA 79(388), 871-880. https://doi.org/10.1080/01621459.1984.10477105
- Rana MS, Midi H, Imon AHMR (2008). A robust modification of the Goldfeld-Quandt test for the detection of heteroscedasticity in the presence of outliers. Journal of Mathematics and Statistics 4(4), 277-283. https://doi.org/10.3844/jmssp.2008.277.283
Authors
Ahmed El-Kotory and Ebrahim Khaled Ebrahim (maintainer), Department of Statistics, Faculty of Business, Alexandria University.