Welch's t-test

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In statistics, Welch t -test , or unequal variant t -test , is a two-sample location test which is used to test the hypothesis that two populations have the same means. Welch's t -test is an adaptation of Student's t -test, which is more reliable when two samples have unequal variances and unequal sample sizes. These tests are often referred to as "unpaired" or "independent samples" t -tests, since they are usually applied when the underlying statistical unit of the two samples being compared does not overlap. Given that Welch's t -test has been less popular than Student's t -test and may be less familiar to readers, a more informative name is "unequal variants of Welch < i> -test "or" unequal variances t -test "for brevity.

Video Welch's t-test

Assumption

Students t -test assume that two populations have a normal distribution and with the same variance. Welch's t -test is designed for unequal variances, but the assumption of normality is maintained. Welch's t -test is the approximate solution to the Behrens-Fisher problem.

Maps Welch's t-test

Calculation

Welch's t -test mendefinisikan statistik t dengan rumus berikut:

${\ displaystyle t \ quad = \ quad {\; {\ overline {X}} _ {1} - {\ overline {X}} _ {2} \; \ over {\ sqrt {\; {s_ {1} ^ {2} \ over N_ {1}} \; \; {s_ {2} ^ {2} \ over N_ {2}} \ quad}}} \,}  ÂÂ$

di mana ${\ displaystyle {\ overline {X}} _ {1}}  ÂÂ$ , ${\ displaystyle s_ {1} ^ {2}}  ÂÂ$ dan ${\ displaystyle N_ {1}}  ÂÂ$ adalah sampel pertama, varians sampel dan ukuran sampel, masing-masing. Tidak seperti di Student t -test, penyebutnya tidak berdasarkan pada estimasi varians gabungan.

Di sini ${\ displaystyle \ nu _ {1} = N_ {1} -1}  ÂÂ$ , derajat kebebasan yang terkait dengan estimasi varian pertama. ${\ displaystyle \ nu _ {2} = N_ {2} -1}  ÂÂ$ , derajat kebebasan yang terkait dengan estimasi varian ke-2.

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

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Test stats

After t and ${\ displaystyle \ nu}$ t -distribution to test the null hypothesis that the two populations mean the same (the using a two-sided test), or an alternate hypothesis that one of the populations means is greater than or equal to the other (using a one-tailed test). The approximate degree of freedom is rounded down to the nearest integer.

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Advantages and limitations

Welch's t -test is stronger than Student t -test and maintains the type I error type close to nominal for unequal variances and for unequal sample sizes under normality. In addition, Welch's t -test strength is close to Student's t -test, even when the population variance is equal and the sample size is balanced. Welch's t -test can be generalized to more than 2-samples, which is stronger than one-way variance analysis (ANOVA).

This is not recommended to pre-test for the same variance and then choose between t -test or Welch's t -test. In contrast, Welch's t -test can be applied directly and without substantial loss to Student's t -test as mentioned above. Welch's t -test remains strong for tilted distribution and large sample sizes. Decreased reliability for smaller skewed distributions and samples, where one can perform Welch's t -test in rank data.

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Example

The following three examples compare Welch's t -test and Student's t -test. The sample comes from a random normal distribution using the R programming language.

Untuk ketiga contoh, mean populasi adalah ${\ displaystyle \ mu _ {1} = 20}  ÂÂ$ dan ${\ displaystyle \ mu_ {2} = 22}  ÂÂ$ .

Contoh pertama adalah untuk varians yang sama ( ${\ displaystyle \ sigma _ {1} ^ {2} = \ sigma _ {2} ^ {2} = 4}  ÂÂ$ ) dan ukuran sampel yang sama ( ${\ displaystyle N_ {1} = N_ {2} = 15}  ÂÂ$ ). Misalkan A1 dan A2 menunjukkan dua sampel acak:

${\ displaystyle A_ {1} = \ {27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21,7, 21.4 \}}  ÂÂ$

${\ displaystyle A_ {2} = \ {27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5, 24.4 \}}  ÂÂ$

Referensi p-nilai diperoleh dengan mensimulasikan distribusi statistik t untuk hipotesis nol dari mean populasi yang sama ( ${\ displaystyle \ mu_ {1} - \ mu_ {2} = 0}  ÂÂ$ ). Hasilnya dirangkum dalam tabel di bawah ini, dengan nilai p dua arah:

Welch's t -test and Student t -test gives identical results when two samples have identical variants and sample sizes (Example 1). But note that if you take a sample of data from a population with identical variance, the sample variance will be different, as well as the results of two t-tests. So with actual data, two tests almost always give different results.

For unequal variance, Student t -test provides a low p-value when smaller samples have larger variants (Example 2) and a high-p value when larger samples have more variants large (Example 3). For unequal variances, Welch's t -test gives a p-value close to the p-simulation value.

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Implementation of the software

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See also

• Students t -test
• Z -test
• Factorial experiments
• One way variance analysis
• Two T-squared Hotelling stats, Multivariate extensions from Welch's t -test

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References

Source of the article : Wikipedia