Welch's t test

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In statistics, Welch's t-test (or unequal variances t-test) is a two-sample location test, and is used to test the hypothesis that two populations have equal means. Welch's t-test is an adaptation of Student's t-test,[1] and is more reliable when the two samples have unequal variances and unequal sample sizes.[2] These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping. Given that Welch's t-test has been less popular than Student's t-test[2] and may be less familiar to readers, a more informative name is "Welch's unequal variances t-test" or "unequal variances t-test" for brevity.

Assumptions[edit]

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.[1] Welch's t-test is an approximate solution to the Behrens-Fisher problem.

Calculations[edit]

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


t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,

where \overline{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 \nu  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:


\nu \quad  \approx \quad 
 {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over
 { \quad {s_1^4 \over N_1^2 \nu_1} \; + \; {s_2^4 \over N_2^2 \nu_2 } \quad }}

Here \nu_1 = N_1-1, the degrees of freedom associated with the 1st variance estimate.

Welch's t-test can also be calculated for ranked data and might then be named Welch's U-test.[3]

Statistical test[edit]

Once t and \nu 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[edit]

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.[2]

It is not recommended to pre-test for equal variances and then choose between Student's t-test or Welch's t-test.[4] 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.[5] Reliability decreases for skewed distributions and smaller samples, where one could possibly perform Welch’s t-test on ranked data.[3] Note that performing a variety of tests and choosing one that gives the desired P-value is cheating!

Examples[edit]

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 \mu_{1} = 20 and \mu_{2} = 22.

The first example is for equal variances (\sigma_{1}^2 = \sigma_{2}^2 = 4) and equal sample sizes (N_{1} = N_{2} = 15). Let A1 and A2 denote two random samples:

A1 = {27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4}

A2 = {27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4}

The second example is for unequal variances (\sigma_{1}^2 = 16, \sigma_{2}^2 = 1) and unequal sample sizes (N_{1} = 10, N_{2} = 20). The smaller sample has the larger variance:

A1 = {17.2, 20.9, 22.6, 18.1, 21.7, 21.4, 23.5, 24.2, 14.7, 21.8}

A2 = {21.5, 22.8, 21.0, 23.0, 21.6, 23.6, 22.5, 20.7, 23.4, 21.8, 20.7, 21.7, 21.5, 22.5, 23.6, 21.5, 22.5, 23.5, 21.5, 21.8}

The third example is for unequal variances (\sigma_{1}^2 = 1, \sigma_{2}^2 = 16) and unequal sample sizes (N_{1} = 10, N_{2} = 20). The larger sample has the larger variance:

A1 = {19.8, 20.4, 19.6, 17.8, 18.5, 18.9, 18.3, 18.9, 19.5, 22.0}

A2 = {228.2, 26.6, 20.1, 23.3, 25.2, 22.1, 17.7, 27.6, 20.6, 13.7, 23.2, 17.5, 20.6, 18.0, 23.9, 21.6, 24.3, 20.4, 24.0, 13.2}

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

Sample A1 Sample A2 Student's t-test Welch's t-test
Example N_{1} \overline{X}_{1} s_{1}^{2} N_{2} \overline{X}_{2} s_{2}^{2} t \nu P P_{sim} t \nu P P_{sim}
1 15 20.8 7.9 15 23.0 3.8 -2.46 28 0.021 0.021 -2.46 25.0 0.021 0.017
2 10 20.6 9.0 20 22.1 0.9 -2.10 28 0.045 0.150 -1.57 9.9 0.149 0.144
3 10 19.4 1.4 20 21.6 17.1 -1.64 28 0.110 0.036 -2.22 24.5 0.036 0.042

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.

Software implementations[edit]

Language/Program Function Notes
LibreOffice TTEST(Data1; Data2; Mode; Type) See [1]
MATLAB ttest2(data1, data2) See [2]
Microsoft Excel pre 2010 TTEST(array1, array2, tails, type) See [3]
Microsoft Excel 2010 and later T.TEST(array1, array2, tails, type) See [4]
Python scipy.stats.ttest_ind(a, b, axis=0, equal_var=False) See [5]
R t.test(data1, data2, alternative="two.sided", var.equal=FALSE) See [6]

See also[edit]

References[edit]

  1. ^ a b Welch, B. L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika 34 (1–2): 28–35. doi:10.1093/biomet/34.1-2.28. MR 19277. 
  2. ^ a b c Ruxton, G. D. (2006). "The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test". Behavioral Ecology 17: 688–690. doi:10.1093/beheco/ark016. 
  3. ^ a b Fagerland, M. W.; Sandvik, L. (2009). "Performance of five two-sample location tests for skewed distributions with unequal variances". Contemporary Clinical Trials 30: 490–496. doi:10.1016/j.cct.2009.06.007. 
  4. ^ Zimmerman, D. W. (2004). "A note on preliminary tests of equality of variances". British Journal of Mathematical and Statistical Psychology 57: 173–181. doi:10.1348/000711004849222. 
  5. ^ Fagerland, M. W. (2012). "t-tests, non-parametric tests, and large studies—a paradox of statistical practice?". BioMed Central Medical Research Methodology 12: 78. doi:10.1186/1471-2288-12-78.