Performs Yuen's robust t-test for trimmed means. Compared with the classical t-test, the procedure is substantially less sensitive to outliers, heavy tails, and moderate departures from normality.
The test is based on:
trimmed means,
winsorized variances,
Welch-type degrees of freedom.
For paired tests, trimming is performed on the paired differences, i.e.
the one-sample trimmed t-test (Tukey & McLaughlin, 1963) is applied to
\(x - y\). This tests the trimmed mean of the differences, which in
general is not the difference of the trimmed means; the latter is what
e.g. WRS2::yuend() compares.
Usage
yuenTTest(x, ...)
# S3 method for class 'formula'
yuenTTest(formula, data, subset, na.action = na.pass, paired = FALSE, ...)
# Default S3 method
yuenTTest(
x,
y = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0,
paired = FALSE,
conf.level = 0.95,
trim = 0.2,
...
)Arguments
- x
numeric vector of observations. Non-finite values (
NA,NaN,Inf,-Inf) are removed; in the paired case the pair is removed.- ...
further arguments passed to methods.
- formula
a formula of the form
lhs ~ rhs.- data
optional data frame for the formula interface.
- subset
an optional expression specifying a subset of observations, evaluated in
data(subset = ID != 1), as int.test().- na.action
a function indicating what should happen when the data contain
NAs. Defaults tona.pass(): non-finite values are then removed by the default method.- paired
logical indicating whether a paired test is performed. Only available in the default method: the formula interface describes independent groups and does not identify pairs.
- y
optional second numeric vector.
- alternative
character string specifying the alternative hypothesis. One of
"two.sided","less", or"greater".- mu
hypothesized trimmed mean (or trimmed mean difference).
- conf.level
confidence level for the confidence interval.
- trim
fraction of observations trimmed from each tail. Must satisfy
0 <= trim < 0.5.
Details
Robust one-, two-, and paired-sample t-tests based on trimmed means and winsorized variances.
Winsorizing. With \(g = \lfloor \mathrm{trim} \cdot n \rfloor\),
the \(g\) smallest observations are set to the \((g+1)\)-th order
statistic and the \(g\) largest to the \((n-g)\)-th, as in Yuen
(1974) and Wilcox (2005). The winsorized variance must be taken this way,
at order statistics rather than at interpolated quantiles: the standard
error \(\sqrt{(n-1) s_w^2 / (h(h-1))}\) with \(h = n - 2g\) is
derived for exactly \(g\) replaced values in each tail, the same
\(g\) observations that mean() with trim removes. The results agree
with WRS2::yuen() and PairedData::yuen.t.test().
Standard error. In all three designs the squared standard error of a
trimmed mean is \((n-1) s_w^2 / (h(h-1))\), with \(s_w^2\) the
winsorized variance and \(h\) the number of observations left after
trimming; the degrees of freedom are \(h - 1\) (combined by Welch's
formula in the two-sample case). With trim = 0, or whenever
\(g = 0\), the three tests reduce exactly to the corresponding
t.test(): one-sample, paired, and Welch. The one-sample version in
WRS2::trimse() uses the asymptotically equivalent
\(s_w / ((1 - 2\,\mathrm{trim}) \sqrt{n})\); it differs in small
samples, where it inflates the standard error even if no observation is
trimmed (e.g. \(n = 4\), trim = 0.2).
The confidence interval is for the estimated parameter itself (the
trimmed mean, or the difference of trimmed means), independent of mu.
References
Wilcox, R. R. (2005). Introduction to Robust Estimation and Hypothesis Testing. Academic Press.
Tukey, J. W., & McLaughlin, D. H. (1963). Less vulnerable confidence and significance procedures for location based on a single sample: trimming/winsorization 1. Sankhya A, 25, 331–352.
Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61, 165–170.
See also
Other test.location:
brunnerMunzelTest(),
hotellingT2Test(),
moodMedianTest(),
signTest(),
tTestSummary(),
vanDerWaerdenTest(),
zTest()
Examples
x <- rnorm(25, 100, 5)
yuenTTest(x, mu = 99)
#>
#> Yuen One-Sample Trimmed Mean t-test
#>
#> data: x
#> t = 0.76501, df = 14.0, trim = 0.2, p-value = 0.457
#> alternative hypothesis: true trimmed mean is not equal to 99
#> 95 percent confidence interval:
#> 96.95275 103.31743
#> sample estimates:
#> trimmed mean of x
#> 100.1351
#>
with(sleep,
yuenTTest(extra[group == 1],
extra[group == 2]))
#>
#> Yuen Two-Sample Trimmed Mean t-test
#>
#> data: extra[group == 1] and extra[group == 2]
#> t = -1.6168, df = 8.2647, trim = 0.2000, p-value = 0.1434
#> alternative hypothesis: true trimmed mean difference is not equal to 0
#> 95 percent confidence interval:
#> -4.0306400 0.6973066
#> sample estimates:
#> trimmed mean of x trimmed mean of y
#> 0.5333333 2.2000000
#>
yuenTTest(extra ~ group, data = sleep)
#>
#> Yuen Two-Sample Trimmed Mean t-test
#>
#> data: extra ~ group
#> t = -1.6168, df = 8.2647, trim = 0.2000, p-value = 0.1434
#> alternative hypothesis: true trimmed mean difference is not equal to 0
#> 95 percent confidence interval:
#> -4.0306400 0.6973066
#> sample estimates:
#> trimmed mean of x trimmed mean of y
#> 0.5333333 2.2000000
#>