## Parallel designs: Don’t use the (conventional) t-test! [Power / Sample Size]

Hi Sereng,

Since in this post (based on the

According to the FDA’s guidance (Section IV.B.1.d.):

Though you had equally sized groups, variances were

This calls for the Welch-test with Satterthwaite’s approximation

&\approx\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{1}{n_1-1}\left(\frac{s_1^2}{n_1}\right)^2 + \frac{1}{n_2-1}\left(\frac{s_2^2}{n_2}\right)^2}}

$$ For good reasons in ,

@Divyen: If the confidence interval based on my derivation does not match the reported one, it is evident that the Welch-test was used. In such a case calculating the \(\small{MSE}\) is not that trivial. Maybe I will try it later.

❝ […] the reference drug Cmax had almost twice the CV of the Test drug.

❝ Parallel Group Design

❝ Two Groups (n=70/group)

❝ Ratio (90% CI): 109.00 (87.00-135.00)

Since in this post (based on the

*t*-test assuming equal variances) I could reproduce your results:According to the FDA’s guidance (Section IV.B.1.d.):

For parallel designs, the confidence interval for the difference of means in the log scale can be computed using the total between-subject variance.^{1} […] equal variances should not be assumed.

(my emphasis)

Though you had equally sized groups, variances were

*not*equal.This calls for the Welch-test with Satterthwaite’s approximation

^{2}of the degrees of freedom:^{3,4}$$\eqalign{\nu&\approx\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{s_1^4}{n_1^2\,(n_1-1)} + \frac{s_2^4}{n_2^2\,(n_2-1)}}\\&\approx\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{1}{n_1-1}\left(\frac{s_1^2}{n_1}\right)^2 + \frac{1}{n_2-1}\left(\frac{s_2^2}{n_2}\right)^2}}

$$ For good reasons in ,

**SAS**

, and other software packages it is the default.- Using a pre-test (
*F*-test, Levene’s test, Bartlett’s test, Brown–Forsythe test) – as recommended in the past – is bad practice because it will inflate the Type I Error.^{5}

- If \({s_{1}}^{2}={s_{2}}^{2}\;\wedge\;n_1=n_2\), the formula given above reduces to the simple \(\nu=n_1+n_2-2\) anyhow.

- In all other cases the Welch-test is conservative, which is a desirable property.

**SPSS**

*both*the conventional*t*-test*and*the Welch-test are performed. Always use the second row of the table of results.@Divyen: If the confidence interval based on my derivation does not match the reported one, it is evident that the Welch-test was used. In such a case calculating the \(\small{MSE}\) is not that trivial. Maybe I will try it later.

- Misleading terminology. There is no ‘total between-subject variance’. In a parallel design only the
*total*variance – which is*pooled*from the between- and within-subject variances – is accessible.

- Satterthwaite FE.
*An Approximate Distribution of Estimates of Variance Components.*Biom Bull. 1946; 2(6): 110–4. doi:10.2307/3002019.

- Both formulas are given in the literature. They are equivalent.

- Allwood M.
*The Satterthwaite Formula for Degrees of Freedom in the Two-Sample t-Test.*College Board. 2008. Open access.

- Zimmermann DW.
*A note on preliminary tests of equality of variances*. Br J Math Stat Psychol. 2004; 57(1): 173–81. doi:10.1348/000711004849222.

—

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

*Dif-tor heh smusma*🖖🏼 Довге життя Україна!_{}Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

### Complete thread:

- Back Calculating Sample Size Sereng 2022-05-12 17:40 [Power / Sample Size]
- Back Calculating Sample Size ElMaestro 2022-05-12 22:29
- Back Calculating Sample Size Helmut 2022-05-12 23:10
- Back Calculating Sample Size ElMaestro 2022-05-13 01:38
- Back Calculating Sample Size Sereng 2022-05-18 05:32
- PowerTOST: Total sample size Helmut 2022-05-18 08:55

- Back Calculating Sample Size dshah 2022-05-13 11:54

- Back Calculating Sample Size Helmut 2022-05-12 23:10
- Parallel designs: Don’t use the (conventional) t-test!Helmut 2022-05-17 14:17

- Back Calculating Sample Size ElMaestro 2022-05-12 22:29