Chi-Square Versus Fisher's Test Versus Welch T-Test I am having a hard time to understanding when to use which of these tests. Fisher's test is useful for small sample sizes to compare categorical variables. Ok. Take a simple example, you have data such that in R code:
    sampledata = data.frame("male" = sample(0:1, r = T, size = 1000), 
            "obese" = sample(0:1, r = T, size = 1000), 
            "heightShort" = sample(0:1, r = T, size = 1000), 
            heightTall = sample(0:1, r = T, size = 1000))

And if you wished to do comparisons to see if the proportion of obese, heightShort and heightTall is different for male and female what is the appropriate test? Since this is binary and I want to compare proportion I think also Welch's T test statistic may be appropriate also but I do not know if there is a right or wrong or best choice to do. Again I wish to compare proportions.
 A: The difference between Fisher's Exact test and the Chi-Square test is that Fisher's Exact test requires calculating all possible permutations and is thus exact. As the number of dimensions grows, the calculations become cumbersome and eventually intractable, so the approximation inherent to the Chi-Square test makes it appropriate.
However, both of these tests are intended to test if there is a relationship between different classes, such as expected values and observed results. A Welch t-test is testing something else. It tests specifically if two populations with unknown true means and variances (but both assumed to be normal) have the same mean. More precisely, the null hypothesis is that the two sets of observations come from a population with the same mean, even when the sample sizes are different and the true mean and variance are unknown.
Though all the tests are addressing a null hypothesis that "two things are the same" versus the alternative hypothesis that "these two things are different" they are geared to different problems.
Fisher's test works very well with small sample sizes and is exactly correct for those. It was also developed within the context of categorical data.
The Chi-Square test works well for large sample sizes using some basic assumptions. It is itself an approximation to the G-test, and with modern statistical software, both the G-test and Fisher's exact are usable in many more scenarios than the early 20th century.
Both these tests are geared to analyzing similarity or differences in classes based on the frequency of members in that class. For example, to uses these tests on numeric data, once has to bin the values into classes. The Welch's test, on the other hand, is meant to be applied to sets of observations. Its null hypothesis is that the population MEANS of the two sets of observations are the same. Nothing about the variance. With the assumption of normality, the mean and variance are sufficient statistics, so two populations with the same mean AND variance are the same; but to the best of my understanding, Welch is only testing for the mean.
