Welch's t-Statistic

A Welch's t-Statistic is a test statistic used in a Welch's t-test.

• AKA: Unpooled Variance t-Test Statistic.
• Context:
  • Input:
    • [math]\displaystyle{ \hat{\theta}_1 }[/math] : a point estimate based on the first sample.
    • [math]\displaystyle{ \hat{\theta}_2 }[/math] : a point estimate based on the second sample.
    • [math]\displaystyle{ s_w }[/math] : samples unpooled standard deviation.
    • [math]\displaystyle{ D = \theta_1 - \theta_2 }[/math] : a constant based on the difference between each population parameters (estimands theoretical or hypothesized numerical values).
  • Output:
    • [math]\displaystyle{ t }[/math]: a numerical value produced by the Welch's t-Statistic Calculating System.
  • Mathematical Expression: [math]\displaystyle{ t=\frac{(\hat{\theta}_1-\hat{\theta}_2)-D}{s_w} }[/math]
  • It can be interpreted as a modification of Independent Two-Sample t-Test Statistic when Population Variances are not equal.
• Example(s):
  • A Welch's t-statistic for testing the null hypothesis "populations means are equal across two groups, i.e [math]\displaystyle{ \mu_1=\mu_2 }[/math]" is

[math]\displaystyle{ t= \frac{(\overline{x}_1-\overline{x}_2)}{s_w} }[/math]

where [math]\displaystyle{ \overline{x}_1,\overline{x}_2 }[/math] are the individual sample mean values and [math]\displaystyle{ D=\mu_1-\mu_2 = 0 }[/math].

• • …
• Counter-Example(s):
  • An Independent Two-Sample t-Statistic.
  • An One-Sample t-Statistic.
• See: Student's t-Test, t-Statistic, Pooled Standard Deviation, Welch–Satterthwaite Equation.

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References

2017

• (Wikipedia, 2017A) ⇒ https://en.wikipedia.org/wiki/Welch's_t-test# Retrieved:2016-9-14.
  • Welch's t-test defines the statistic t by the following formula:

[math]\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 }}\, }[/math]

where [math]\displaystyle{ \overline{X}_1 }[/math], [math]\displaystyle{ s_1^2 }[/math] and [math]\displaystyle{ N_1 }[/math] 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 [math]\displaystyle{ \nu }[/math]  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:

[math]\displaystyle{ \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 }} }[/math]

Here [math]\displaystyle{ \nu_1 = N_1-1 }[/math], the degrees of freedom associated with the first variance estimate. [math]\displaystyle{ \nu_2 = N_2-1 }[/math], 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.

2017b

• (SPSS,2017) ⇒ http://libguides.library.kent.edu/SPSS/IndependentTTest
  • When the two independent samples are assumed to be drawn from populations with unequal variances (i.e., [math]\displaystyle{ \sigma_{12}\neq\;\sigma_{22} }[/math]), the test statistic t is computed as:

[math]\displaystyle{ t= \frac{\overline{x}_1-\overline{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} }[/math]

where

[math]\displaystyle{ \overline{x}_1 }[/math]= Mean of first sample

[math]\displaystyle{ \overline{x}_1 }[/math] = Mean of second sample

[math]\displaystyle{ n_1 }[/math] = Sample size (i.e., number of observations) of first sample

[math]\displaystyle{ n_1 }[/math] = Sample size (i.e., number of observations) of second sample

[math]\displaystyle{ s_1 }[/math] = Standard deviation of first sample

[math]\displaystyle{ s_1 }[/math] = Standard deviation of second sample

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