# Extremely unequal samples sizes for Welch t-test

#### Igoor

##### New Member
Hi folks,

I am currently comparing some gene expression results from tumor and normal cases and would like to determine for a small number of genes (3 in total), whether their population means are different between tumor and normal case.

I am not assuming equal variances, so I was opting for Welch's t-test.

However, the issue I am facing now is that I have extremely unbalanced samples sizes (group1 = 79 samples and group2 = 9 samples).

Apparently, Welch-t test performs worse in terms of type I error for increasingly unbalances samples sizes, while the regular t-test is more robust here. However, I am not able to assume equal variances, which leaves me unsure whether or not it is at all feasible to run a t-test here. Is it even meaningful at all to want to run statistics when dealing with such unequal sample size? Is there in that case any particular test that should be employed rather than Welch's t-test?

Cheers!

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#### Karabiner

##### TS Contributor
However, the issue I am facing now is that I have extremely unbalanced samples sizes (group1 = 79 samples and group2 = 9 samples).
That's the very reason to use the Welch test: inhomogenous variances
together with different sample sizes. With equal sample sizes you could
just use the t-test, regardless of the unequal variances.

With kind regards

K.

#### Igoor

##### New Member
That's the very reason to use the Welch test: inhomogenous variances
together with different sample sizes. With equal sample sizes you could
just use the t-test, regardless of the unequal variances.

With kind regards

K.
Thank you so much for the prompt reply.

According to what I have read sofar about the different options when using t-tests regarding equal/unequal variances and equal/unequal variances, i totally agree with you. However, dealing with these extremely unbalanced group sizes, I am concerned about type I and II errors. Will the Welch's t-test be too conservative and have low power or instead have increased type I error?

But your advice would be to carry on with the analysis using the Welch's t-test?

Thanks!
/I.

#### Karabiner

##### TS Contributor
If the larger variance is in the cell with the larger n, then the t-test becomes more conservative
(less power, more type II errors). If the larger variance is in the cell with the smaller n, then it
is too liberal (more type I errors than indicated by the p value). The Welch test corrects this.

With kind regards

K.

#### ricardo.moura.rpm

##### New Member
Hi folks,

I am currently comparing some gene expression results from tumor and normal cases and would like to determine for a small number of genes (3 in total), whether their population means are different between tumor and normal case.

I am not assuming equal variances, so I was opting for Welch's t-test.

However, the issue I am facing now is that I have extremely unbalanced samples sizes (group1 = 79 samples and group2 = 9 samples).

Apparently, Welch-t test performs worse in terms of type I error for increasingly unbalances samples sizes, while the regular t-test is more robust here. However, I am not able to assume equal variances, which leaves me unsure whether or not it is at all feasible to run a t-test here. Is it even meaningful at all to want to run statistics when dealing with such unequal sample size? Is there in that case any particular test that should be employed rather than Welch's t-test?

Cheers!
Good afternoon,

I was researching around the forum and saw your question about the aplication of the Welch's test to small unequal samples.

I'm currently elaborating a paper where we found the exact distribution to the case the samples are odd, and also a better aproximation than Welch's to the other cases.

If you're interested we can apply you case study as an aplication to our research.

Kind Regards,

Ricardo