It depends: 1. If using hand calculation, one must be careful about the Sum of Squares formulae.

2. The SS formulae are well taken care of, if one uses statistical software. 

3. If the difference between the sample sizes is too large, this can affect the homogeneity of variance assumption. As such, there isn’t a good rule of thumb for the point at which unequal sample sizes make heterogeneity of variance a problem. However, ANOVA is robust if the difference is smaller!!! 

4. It can be a real issue, if one deals with factorial ANOVA, and so on !!!

It depends: 1. If using hand calculation, one must be careful about the Sum of Squares formulae.

2. The SS formulae are well taken care of, if one uses statistical software. 

3. If the difference between the sample sizes is too large, this can affect the homogeneity of variance assumption. As such, there isn’t a good rule of thumb for the point at which unequal sample sizes make heterogeneity of variance a problem. However, ANOVA is robust if the difference is smaller!!! 

4. It can be a real issue, if one deals with factorial ANOVA, and so on !!!

If the variance homogeneity is affected, it is possible to use a non-parametric test such as the Kruskal-Waliis test.   

You can run a regression of the dependent variable on dummies for groups. Use a dummy for each group, and no intercept.

You perform tests with the usual F-test in regressions.

(If you happen to have an equal number of sample sizes, this amounts to the usual ANOVA with equal sample sizes.)

If you worry about heteroskedasticity, then employ (White's) "robust errors".

Cheers -- Harald

There is nothing different about unequal sample sizes when it comes to variance heterogeneity. The standard anova or regression approach includes an assumption of equal variance across groups, regardless of the sample sizes. The main implication of unequal sample sizes is in the interpretation of results: for example, there is no longer a common standard error for each group mean, or a common standard error of difference for each pair of groups. Any standard software should cope with this without difficulty.

Annadurai,

As Peter says, the assumption does not change. The rule of thumb I was taught back in the 80s relates to Charles's and Mario's points: ANOVA is/was deemed relatively "robust" to heterogeneity of variances in the case of equal cell sizes, but violation of the assumption can have greater consequences with disparate cell sizes.

I honestly have no idea how this has held up to more recent scrutiny and simulations, but that's my guess as to the pattern of answers you're getting.

Pat

Thank you all

@ Charles : « If the difference between the sample sizes is too large, this can affect the homogeneity of variance assumption » is false, variance are equal or not whatever the difference between the sample is (since we're interested in _individual_ variances, and not _mean_ variances). However, as said by others also, ANOVA is indeed more sensitive to departure to homoscedasticity when sample sizes are very different.

@ Ignacio : beware that, since the Kruskal-Wallis test compares distributions at whole [null hypothesis: all samples come from the same distribution function], if variances are really different it can be expected to give significant p-values, even if means are the same, so results must be interpretated with care... (even if, in practice, power will be small to detect such cases)

@Emmanuel: The methods used for quantitative data have relied on relative complex assumptions about distributional forms. These assumptions are often likely to be clearly untrue. To overcome this problem we can argue that methods are robust, that is, not very sensitive to nonnormality. If sample sizes are unequal, the homogeneity of variance assumption should be validated. In distribution-free methods, such as the Kruskal-Wallis test, there is little emphasis on population parameters, such as the sample estimate of variance. The whole object is to avoid a particular functional form for a population distribution. Can be used when there may be obvious nonnormality or may be possible nonnormality, perhaps to a very marked extent, but the sample sizes may be too small to establish whether or not this is so. Furthermore, the K-W test has a relative efficiency of 95.5% against the F test.

In my opinion, the K-W test is a technique to fall back on when standard assumptions have particularly doubtful validity. It is useful to be able to confirm the results of a normal-theory significance test by performing an appropriate non-parametric test.

@Ignacio: I agree with the first part of your comment. However, if distributions do have a variance, and if this variance chance between the samples, distributions do differ so the KW test should be significant, precisely because it only rely on the global shape of the unknown distribution and not on hypotheses on some parameters.

Is it a problem or not is another debate: if variance is different, something differ between groups anyway, so it is interesting. My point was just to warn that, whereas ANOVA focuses on changes in expectations (position), KW does not so interpretating a significant result of a KW test is not as obvious as « means are different ».

For the last point: everything is fine if both tests lead to the same conclusion, but if not, what do you decide? Having several tests to answer to the same question may be confusing...

@Emmanuel: I agree with you, the result is not obvious. From my understanding, if both tests show different outcomes, the result is inconclusive or at least questionable. I would try making a new study increasing the sample size.

If you want to compare two groups A and B with unequal sample size, why not do an unpaired t-test instead? The mathematical rigor for two samples is better with t-tests rather than ANOVA, You may want to do an exploratory analysis to check the inequality of variances in the two groups and then choose a t-test accordingly.

@Reginald: If there are two groups only you are right.

I think Reginald is mistaken. ANOVA has the same rigour as the unpaired t-test for two samples, and the two will give the same result based on the same assumptions.

Dear Basharat Ali, If you've unequal replicates, you can use Type II ANOVA.  You can find rules and methods to do Type II ANOVA in many references.

You can use Welch's ANOVA instead. This test is easily accessible in SPSS and other stats software. See https://psyarxiv.com/wnezg/