Unequal sample size one way ANOVA I want to compare the means of X between BMI groups, but have unequal sample sizes between my groups. 
group A (n=20, mean=16.2);
group B (n=90, mean=12.8);
group C (n=30, mean=10.8);
group D (n=8,  mean=11.2)



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*Can I use a one way ANOVA test when the sample sizes are extremely unequal?

*I performed a one way ANOVA test. The results showed significant differences for A-B and A-C, but not A-D. However when I did paired t-test for A-D, the result was significant. Which part am I doing wrong, the one way ANOVA or the t-test? Is an ANOVA test valid in this case?

 A: You have a total N = 148, distributed into 4 groups.  If you had 37 in each group instead, you would have greater statistical power.  Otherwise, a one-way ANOVA is just as valid here as anywhere else (given that the normal assumptions are met).  (To understand this better, it may help to read my answer here: How should one interpret the comparison of means from different sample sizes?)  So to answer 1. explicitly, yes, you can use a one-way ANOVA when the sample sizes are extremely unequal.  
However, your description in 2. seems odd to me, so let me add a few notes:  


*

*If the groups (A through D) were formed by categorizing BMI (a continuous variable), you would be better off using regression with BMI as your predictor; categorizing continuous variables is not a good thing to do.  

*It isn't clear what you mean when you say that A-B and A-C were significant, but A-D wasn't.  An ANOVA doesn't tell you that.  An ANOVA only tells you if there is a difference somewhere amongst your groups.  Did you run some post-hoc test to get those results?  

*I don't see how you could have run a paired t-test to compare A and D when they do not have the same ns.  Did you mean an unpaired t-test?  Under the assumption that you used some proper test for post-hoc comparisons with the ANOVA, that was probably the appropriate option as a t-test would not take into account that you have multiple comparisons, for example.  

