Szroeter test for ordered alternatives

Detects monotonic changes in variance when observations can be meaningfully ordered—typically by time or another covariate—using the cumulative weighting scheme proposed by Szroeter (1978).

Usage

performSzroeterTest(model, data, order_by,
  alternative = c("greater", "two.sided", "less"))

Arguments

model: A fitted stats::lm object representing the mean equation under study.
data: A base::data.frame (or compatible tibble) containing the variables used to fit model and the ordering variable supplied via order_by.
order_by: Character scalar naming the column that defines the ordering of observations prior to computing the statistic. The column must be numeric or coercible to an orderable vector.
alternative: Character scalar specifying the alternative hypothesis: "greater" (the default) tests for variance increasing with order_by, "less" for variance decreasing, and "two.sided" for a change in either direction. Szroeter's test targets monotone alternatives, so the one-sided form is the usual choice.

Value

An object of class htest reporting Szroeter's standardised statistic Q, the sample size, the underlying rank-weighted statistic h in estimate, and the p-value from the asymptotic normal reference distribution.

Details

After ordering the residuals $\hat{e}_{(i)}$ by order_by, the test forms the rank-weighted average of the squared residuals $$h = \frac{\sum_{i = 1}^n i \, \hat{e}_{(i)}^2}{\sum_{i = 1}^n \hat{e}_{(i)}^2},$$ which is Szroeter's (1978) class of statistics evaluated at the canonical weights $h_i = i$. Under homoskedasticity $h$ is centred on the mean rank $(n + 1) / 2$ with variance $(n^2 - 1) / (6n)$, so $$Q = \frac{h - (n + 1) / 2}{\sqrt{(n^2 - 1) / (6n)}}$$ is asymptotically standard normal. Variance that grows with order_by shifts weight onto the high ranks and drives $Q$ upwards.

The implementation relies on the package validation helpers to align the supplied data with the model residuals, check that the ordering variable is present, and ensure sufficient sample size and variability in the squared residuals.

Validation

The statistic and its null variance are checked against an independent reconstruction of Szroeter (1978) in tests/testthat/test-pass-a-reference.R. Releases before 0.7.0 divided the centred statistic by a further $\sqrt{n}$, shrinking $Q$ by a factor of roughly $2 / \sqrt{n}$ and leaving the test with essentially no power against any alternative; see NEWS.md.

References

Szroeter, J. (1978). A class of parametric tests for heteroscedasticity in linear econometric models. Econometrica, 46(6), 1311–1327. doi:10.2307/1913833

Godfrey, L. G. (1988). Misspecification Tests in Econometrics. Cambridge University Press. Section 5.4 outlines the Szroeter test.

Examples

data(mtcars)
mod <- lm(mpg ~ wt + qsec, data = mtcars)
performSzroeterTest(mod, mtcars, order_by = "wt")
#> [INFO] Running Szroeter test
#> 
#> 	Szroeter test for ordered heteroscedasticity
#> 
#> data:  mod
#> Q = -0.64937, n = 32, p-value = 0.742
#> alternative hypothesis: variance increases with 'wt'
#> sample estimates:
#>        h 
#> 15.00108 
#> 

# Detect ordered heteroscedasticity in simulated data with variance increasing
# in an index variable
set.seed(404)
n <- 150
x <- sort(runif(n))
y <- 2 + 0.5 * x + rnorm(n, sd = 0.4 + 0.8 * x)
df <- data.frame(y, x)
performSzroeterTest(lm(y ~ x, data = df), df, order_by = "x")
#> [INFO] Running Szroeter test
#> 
#> 	Szroeter test for ordered heteroscedasticity
#> 
#> data:  lm(y ~ x, data = df)
#> Q = 4.2696, n = 150, p-value = 9.793e-06
#> alternative hypothesis: variance increases with 'x'
#> sample estimates:
#>        h 
#> 96.84732 
#> 

# Two-sided version when the direction of the change is not known in advance
performSzroeterTest(mod, mtcars, order_by = "wt", alternative = "two.sided")
#> [INFO] Running Szroeter test
#> 
#> 	Szroeter test for ordered heteroscedasticity
#> 
#> data:  mod
#> Q = -0.64937, n = 32, p-value = 0.5161
#> alternative hypothesis: variance changes monotonically with 'wt'
#> sample estimates:
#>        h 
#> 15.00108 
#>