Comparing two groups: one-way ANOVA vs. regression analysis with dummy I am a bit puzzled with the following issue: the outcome of one-way ANOVA test shows that the mean difference of variable y between two country samples is statistically significant. However, after pooling the two samples together and running an OLS regression with y as dependent variable, some other IV and by including a country dummy, the effect of the country dummy appears to be statistically insignificant, implying that there is no country-related effect on dependent variable y. Any explanation for this outcome?
 A: If I'm right in guessing that the other IV was not included in the ANOVA, then the most likely reason is that the two countries differed on the other IV, and that the countries only look 'significant' if the other IV is omitted.  I wonder if the other IV is 'significant'?  With sufficient confounding, it may not be.  One last (but I think unlikely) possibility is that when you added the IV you lost 1 degree of freedom, and so if the IV were totally unrelated to the response, you would have lost a trivial amount of statistical power.  
A: In addition to the other answers, you wrote

the effect of the country dummy appears to be statistically
  insignificant, implying that there is no country-related effect on
  dependent variable y.

That is not what a non-significant result implies. In fact, unless the means on Y for the two countries were identical (extremely unlikely), you know that there was a difference in your sample. And the chance that there was a 0 difference in the population is infinitesimal. 
A non-significant result does not imply anything. It simply says that, if the difference in the population were 0, you could get a test statistic this high more than 5% of the time. 
A: This may be the same as gung's answer but it looks at it slightly differently and I think this could help understanding and interpretation.  The IV is highly correlated with country.  Therefore if the country indicator helps predict the response the IV could too.  So by themselves each would be significant but included together only one will.  It would therefore probably be best for prediction especially if the sample size is small to include just one of the two.  Which one to choose would only matter if you care about causation.  The problem this causes is often called confounding which is part of what gung was discussing.
