tt {tcftt}

The transformation based test

Description

This test is suitable for testing the equality of two-sample means for the populations having unequal variances. When the populations are not normally distributed, the sampling distribution of the Welch's t-statistic may be skewed. This test conducts transformations of the Welch's t-statistic to make the sampling distribution more symmetric. For more details, please refer to Zhang and Wang (2020).

Usage

tt(x1, x2, alternative = "greater", effectSize = 0, alpha = 0.05, type = 1)

Arguments

x1

the first sample.

x2

the second sample.

alternative

the alternative hypothesis: "greater" for upper-tailed, "less" for lower-tailed, and "two.sided" for two-sided alternative.

effectSize

the effect size of the test. The default value is 0.

alpha

the significance level. The default value is 0.05.

type

the type of transformation to be used. Possible choices are 1 to 4. They correspond to the TT1 to TT4 in Zhang and Wang (2020). Which type provides the best test depends on the relative skewness parameter A in Theorem 2.2 of Zhang and Wang (2020). In general, if A is greater than 3, type =3 is recommended. Otherwise, type=1 or 4 is recommended. The type=2 transformation may be more conservative in some cases and more liberal in some other cases than the type=1 and 4 transformations. For more details, please refer to Zhang and Wang (2020).

Value

test statistic, critical value, p-value, reject decision at the given significance level.

References

Zhang, H. and Wang, H. (2020). Transformation tests and their asymptotic power in two-sample comparisons Manuscript in review.

Examples

x1 <- rnorm(20, 1, 3)
x2 <- rnorm(21, 2, 3)
tt(x1, x2, alternative = 'two.sided', type = 1)

#Negative lognormal versus normal data
 n1=50;  n2=33
 x1 = -rlnorm(n1, meanlog = 0, sdlog = sqrt(1)) -0.3*sqrt((exp(1)-1)*exp(1))
 x2 = rnorm(n2, -exp(1/2), 0.5)
 tt(x1, x2, alternative = 'less', type = 1)
 tt(x1, x2, alternative = 'less', type = 2)
 tt(x1, x2, alternative = 'less', type = 3)
 tt(x1, x2, alternative = 'less', type = 4)

#Lognormal versus normal data
 n1=50;  n2=33
 x1 = rlnorm(n1, meanlog = 0, sdlog = sqrt(1)) + 0.3*sqrt((exp(1)-1)*exp(1))
 x2 = rnorm(n2, exp(1/2), 0.5)
 tt(x1, x2, alternative = 'greater', type = 1)
 tt(x1, x2, alternative = 'greater', type = 2)
 tt(x1, x2, alternative = 'greater', type = 3)
 tt(x1, x2, alternative = 'greater', type = 4)