# Chi-square and two-sample t-test

This post explains a basic question I encountered, and the statistical concepts behind it.

The real-life problem

Someone asks me to construct data to prove that a treatment is useful for 1) kindergarten and 2) elementary school kids in preventing winter cold.

Chi-square and student’s t test

First, decide how to formulate the problem using statistical tests. This includes deciding the quantity and statistic to compare.

Basically, I need to compare two groups. Two tests come to mind: Pearson’s chi-square test, and two-sample t-test. This article summarizes main difference between the two tests, in terms of Null Hypothesis, Types of Data, Variations and Conclusions. The following section is largely based on that article.

Null Hypothesis

• Pearson’s chi-square test: test the relationship between two variables, or whether something has effects on the other thing (?). e.g. men and women are equally likely to vote for Republican, Democrat, Others, or Not at all. Here the two variables are “gender” and “voting choice”. The null is “gender does not affect voting choice”.
• Two-sample t-test : whether two sample have the same mean. Mathematically, this means $\mu_1 = \mu_2$ or $\mu_1 - \mu_2 = 0$. e.g. boys and girls have the same height.

Types of Data

• Pearson’s chi-square test: usually requires two variables. Each is categorical and can have many number of levels. e.g. one variable is “gender”, the other is “voting choice”.
• two sample t-test: requires two variables. One variable has exactly two levels (two-sample), the other is quantitively calculatable. e.g. in the example above, one variable is gender, the other is height.

Variations

• Pearson’s chi-square test: variations can be when the two variables are ordinal instead of categorical.
• two-sample t-test: variations can be that the two samples are paired instead of independent.

Transform the real-life problem into a statistical problem

Using chi-square test

Variable 1 is “using treatment”. Two levels: use or not.

Variable 2 is “getting winter cold”. Two levels: get cold or not.

For kindergarten kids and for pre-school kids, I thus have two 2 * 2 tables.

(question: can I do a chi-square test on three variables? The third one being “age”.)

Using two-sample t-test

Variable 1 is “using treatment”. Two levels: use or not

Variable 2 is supposed to be a numerical variable —- here use disease rate. But then there is no enough number of samples.

Thus, conclude that Chi-square test should be used here.

# Brief explanation of statistical learning framework

This post explains what is a statistical learning framework, and common results under this framework.

Problem

We have a random variable X, another random variable Y. Now we want to determine the relationship between X and Y.

We define the relationship by a prediction function f(x). For each x, this function produces an “action” a in the action space.

Now how do we get the predictive function f? We use a loss function l(a, y), that for each a and y, we produce a “loss”. Note since X is a random variable, f(x) is a transformation, so a is a random variable, too.

Also, l(a, y) is a transformation of (a, y), so l(a, y) is a random variable too. It’s distribution is based on both X and Y.

We then calculate f by minimizing the expectation of the loss, which is called “risk”. Since the distribution of l(a, y) is based both on the distribution of X and Y, to get this expectation, we need to do integration both on X and on Y. In the case of discrete variables, we do summation based on the pmf of (x, y).

The above are about theoretical properties of Y, X, loss function and prediction function. But we usually do not know the distribution of (X, Y). Thus, we choose to minimize empirical risk instead. We calculate empirical risk by summing all the empirical loss together, divided by m. (q: does this resemble Monte Carlo method? is this about computational statistics? Need a review.)

Results

In the case of square-loss, we have the result, a = E(y|x).

In the case of 0-1 loss, we have the result, a = arg max P(y|x)

Example:

We want to predict a student’s mid-term grade (Y). We want to know the relationship between predicted value, and whether she is hard-working (X).

We use square-loss for this continuous variable Y. Since we know that to minize square loss, for any random variable we should predict the mean value of the variable (c.f. regression analysis, in OLS scenerio we calculate the MSE — but need further connection to this framework).

Now we just observed unfortunately the student is not hard-working.

We know for a not-hardworking student the expectation of mid-term grade is 40.

We then predict the grade to be 40, as a way to minimize square-loss.