# Using t-test/ANOVA on bounded data

Can a t-test/ANOVA approach to hypothesis testing be carried out when data has well-defined upper and lower limits and is not continuous?

Let's assume a situation where data are generated by some phenomenon resembling some Gaussian/normal distribution (i.e. $X \sim N(\mu,\sigma)$), but such that the data is integer-valued (i.e. $X \in Z$ ) and that the data lies within a finite range (i.e. $[a, b], \infty < a < b < \infty$).

Obviously, there are a few differences between the distribution of the values generated by the phenomenon and a Gaussian/normal distribution:

• $X$ is not continuous.
• $X$ does not have a support that extends into infinity in either direction.

However, the data generated by this distribution will still look kind of Gaussian/normal in many ways.

(a) Now, assume we have a similar phenomenon in which data are generated in a seemingly similar but unknown process.

We want to compare the distributions of both phenomena to see if there is a statistically significant difference between the phenomena distributions.

Can we use a t-test to perform this analysis?

(b) Now, assume we have several similar phenomena which generate data in a seemingly similar but unknown process.

We want to compare the distributions of all the phenomena to see if there is a statistically significant difference between at least one of the phenomena distributions and the other distributions.

Can we use ANOVA to perform this analysis?

(EDIT: I have changed the framing of this question in order to better capture the issue)

You should use a count regression model (such as Poisson or negative binomial), or possibly logistic regression, with "school" or "class" as the independent variable.

First, I find it strange that every classroom has exactly 25 kids. That argues for either using logistic regression on the proportion of left handers (rather that the count) or else using a count regression model and accounting for total number of kids in the class.

Second, since left handedness is relatively rare (about 10% of people) you are going to get a lot of classrooms with small counts. This argues against using t-test/ANOVA.

Third (as an aside) what about ambidextrous kids? Handedness is not really a dichotomy. There are people who use different hands for different things in all sorts of combinations. This argues for using a better measure of handedness.

• Thanks for your reply. This was an entirely fictitious situation. It was just a---poorly---contrived situation where data was integer-valued and bounded from above and below. I just wanted to know, in general, whether or not a t-test can be used where data is bounded from above and/or below and is integer-valued. I have changed the question to represent a more general scenario. Perhaps this will better encapsulate the question. – wilf badge Apr 15 '16 at 12:35

It is NOT necessary to check the data for symmetry or normalcy when conducting a t test or ANOVA. These tests are inferences on the means, not the actual data. The Central Limit Theorem says that the means (given a large enough sample size) will follow a normal distribution even if the data they summarize are highly skewed. The 'large enough' sample size is determined by the distribution of the data, but it is silly to attempt to model the distribution of the data to estimate the needed sample size. This is a misunderstood aspect of statistics, and there are many great papers on this including this one: "The Importance of the Normality Assumption in Large Public Health Data Sets" by Lumley, Diehr, Emerson and Chen.

This is definitely not applicable to t-test and ANOVA, because your dependent variable is not continuous and not anything close to normal. In this example, you don't even have to check the QQ plot because it can't be right. If your data can't go negative, it's not a symmetric distribution thus you can't use t-test or ANOVA.

Apart from what @Peter_Flom has already mentioned, you can also use a chi-square test to test the association for your 2x2 contingency table (school vs handlers). Possibilities include chi-square independence, test for odd-ratios, and Fischer's exact test.

• @wilfbadge Please consider to give myself and Peter a upvote. This is how this site works. – SmallChess Apr 15 '16 at 12:04
• I have given an upvote to both of you, but my reputation is too low for it to show publicly. – wilf badge Apr 15 '16 at 12:29
• I have changed the question to represent a more general scenario. Perhaps this will better encapsulate the question. – wilf badge Apr 15 '16 at 12:31