Lecture 13: Chi-Square

Chi-square test

Statistical test used to analyze categorical data (counts) by comparing what you observe to what you would expect under some assumption

• GOF test: does one variable follow a specific distribution?

• Independence test: are two categorical variables related?

What is the difference between a chi-square test and a t-test?

Chi-square test

• Uses categorical data (counts/frequencies)

• Looks at patterns in proportions

T-test

• Uses continuous data (means)

• Compares average values between groups

What is the chi-square distribution and how does it differ from the normal distribution?

• Shape is right-skewed 

• Values are always ≥0 

• Distribution changes shape depending on df

How does the chi-square distribution change as a function of degrees of freedom / sample size?

As df increases:

• Distribution becomes less skewed

• Starts to look more symmetrical

• The peak shifts to the right

Can a chi-square test be one-sided and/or two-sided?

Always one-sided (right-skewed)

What kind of data must we have in order to conduct a chi-square test?

• Categorical variables

• Data in the form of frequencies/counts 

• Independent observations 

• Setup is usually a contingency table (ex: 2x3)

What is a chi-square goodness-of-fit test? What type of question does it test?

examines whether the observed distribution of a single categorical variable matches a theoretical or expected distribution

What do the null and alternative hypotheses look like for GOF?

Ho: The observed frequencies match the expected frequencies

Ha: The observed frequencies do NOT match the expected frequencies

How do we compute a chi-square test statistic?

Numerator: squared difference between Observed (O) and Expected (E) → (O - E)²

Denominator: expected frequency (E), which scales the difference

A sum of standardized squared deviations between observed and expected counts

What happens to chi-square when the difference between the observed frequencies and the expected frequencies (as specified under the null) increases (assuming all other things stay equal)?

If (O - E) gets larger → the numerator increases → chi-square increases → more evidence against Ho

What happens to the chi-square value when the sample size increases (assuming all other things stay equal)?

• Expected counts (E) increases, even small proportional differences can produce larger chi-square values

• Larger samples make it easier to detect significant differences

What do we compare the chi-square statistic to?

A critical value from the chi-square distribution OR use a p-value

What are the assumptions of a GOF test?

• Categorical data 

• Independent observations

• Expected frequencies are sufficiently large

How do we compute degrees of freedom for GOF?

df = k - 1

• k: number of categories

R output for a GOF test

• Refers to one variable only

• No mention of rows/columns

• Hypothesis is about distribution matching expected proportions

chi-square test of independence

Examine whether two categorical variables are related or if they’re independent of each other

• Tests questions like: “is gender related to political preference?”

Null and alternative hypotheses (chi-square test of independence)

Ho: The two variables are independent (no relationship exists)

Ha: The two variables are not independent (there’s an association)

How do we compute a chi-square test statistic?

Numerator: squared difference between observed and expected counts → (O - E)²

Denominator: expected counts (which standardizes the difference) → E

What happens to chi when the difference between the observed frequencies and the expected frequencies (as specified under the null) increases (assuming all other things stay equal)?

If (O - E) increases:

• The numerator increases

• So chi-square increases

What happens to the chi-square value when the sample size increases (assuming all other things stay equal)?

• Expected frequencies increase

• Even small proportional differences can produce larger chi-square values

What do we compare the chi-square statistic to?

A critical value from the chi-square distribution OR a p-value

What are the assumptions of a chi-sq test of independence?

• Categorical variables 

• Independent observations (no repeated measures in the same cell)

• Expected cell frequencies are sufficiently large

  • Rule of thumb: each expected count ≥ 5

• Data are in a contingency table 

How do we compute degrees of freedom for this test?

df = (r - 1)(c - 1)

What is Yate’s correction? When is it used?

An adjustment applied to a chi-square test to make it more accurate when working with small samples and discrete data

• Chi-square test uses a continuous distribution to approximate results from discrete count data

• Yates’ correction compensates for this mismatch by slightly shrinking the difference between observed and expected values before squaring it (by 0.5)

• It reduces the chi-square value, increases p-value, and makes the test more conservative (harder to reject Ho)

• Primarily used for a test of independence (2x2 tables)

R output for chi-square test of independence with Yates’ continuity correction

• Label becomes: "Pearson's Chi-squared test with Yates’ continuity correction”

• Chi-squared value is smaller

• P-value is larger (more conservative)

• Mainly used for 2x2 tables

• Yates’ correction: adjusts for the fact that chi is a continuous approximation but data are discrete counts

R output for chi-square test of independence w/o Yates’ continuity correction

• Based on a contingency table (2+ variables)

• You can extract expected counts ($expected) and residuals ($residuals)

• df = (r-1)(c-1)

• Tests association