Welch's t test

Assumptions

Student's t-test assumes that the two populations have normal distributions and with equal variances. Welch's t-test is designed for unequal variances, but the assumption of normality is maintained. Welch's t-test is an approximate solution to the Behrens–Fisher problem.

Calculations

Welch's t-test defines the statistic t by the following formula:

where X ¯ 1 , s 1 2 and N 1 are the 1st sample mean, sample variance and sample size, respectively. Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.

The degrees of freedom ν   associated with this variance estimate is approximated using the Welch–Satterthwaite equation:

Here ν 1 = N 1 − 1 , the degrees of freedom associated with the first variance estimate. ν 2 = N 2 − 1 , the degrees of freedom associated with the 2nd variance estimate.

Welch's t-test can also be calculated for ranked data and might then be named Welch's U-test.

Statistical test

Once t and ν have been computed, these statistics can be used with the t-distribution to test the null hypothesis that the two population means are equal (using a two-tailed test), or the alternative hypothesis that one of the population means is greater than or equal to the other (using a one-tailed test). The approximate degrees of freedom is rounded down to the nearest integer.

Advantages and limitations

Welch's t-test is more robust than Student's t-test and maintains type I error rates close to nominal for unequal variances and for unequal sample sizes. Furthermore, the power of Welch's t-test comes close to that of Student's t-test, even when the population variances are equal and sample sizes are balanced. Welch's t-test can be generalized to more than 2-samples, which is more robust than one-way analysis of variance (ANOVA).

It is not recommended to pre-test for equal variances and then choose between Student's t-test or Welch's t-test. Rather, Welch's t-test can be applied directly and without any substantial disadvantages to Student's t-test as noted above. Welch's t-test remains robust for skewed distributions and large sample sizes. Reliability decreases for skewed distributions and smaller samples, where one could possibly perform Welch's t-test on ranked data.

Examples

The following three examples compare Welch's t-test and Student's t-test. Samples are from random normal distributions using the R programming language.

For all three examples, the population means were μ 1 = 20 and μ 2 = 22 .

The first example is for equal variances ( σ 1 2 = σ 2 2 = 4 ) and equal sample sizes ( N 1 = N 2 = 15 ). Let A1 and A2 denote two random samples:

The second example is for unequal variances ( σ 1 2 = 16 , σ 2 2 = 1 ) and unequal sample sizes ( N 1 = 10 , N 2 = 20 ). The smaller sample has the larger variance:

The third example is for unequal variances ( σ 1 2 = 1 , σ 2 2 = 16 ) and unequal sample sizes ( N 1 = 10 , N 2 = 20 ). The larger sample has the larger variance:

Reference p-values were obtained by simulating the distributions of the t statistics for the null hypothesis of equal population means ( μ 1 − μ 2 = 0 ). Results are summarised in the table below, with two-tailed p-values:

Welch's t-test and Student's t-test gave practically identical results for the two samples with equal variances and equal sample sizes (Example 1). For unequal variances, Student's t-test gave a low p-value when the smaller sample had a larger variance (Example 2) and a high p-value when the larger sample had a larger variance (Example 3). For unequal variances, Welch's t-test gave p-values close to simulated p-values.