HYPOTHESIS TESTING - Chi-Square Tests

Why are Chi-Square tests used?

Compares frequencies across categories in the data

Distribution

Shape depends on the degrees of freedom

• computed using number of groups within the data

As df increases, gets closer to a normal distribution

As number of comparison groups increases, distribution curve flattens

• larger x2 values more probable

• wider range of x2 values likely

Begins at 0

Data Requirements + Assumptions

Categorical Data

Assumptions

• Expected count > 5

• Observations independent (only in a single cell)

What are the two kinds of Chi-Squared tests?

Goodness of Fit

Test of Independence

What is a Goodness of Fit test?

Tests whether relative frequencies are consistent with expected ones

Distribution across a single category

Formula for a Goodness of Fit test

If results are significant

• look at Pearsons / standardised residuals

• to find out which levels within the category had the biggest difference

Standardised Residual results meanings

Positive

• Indicate observed frequency of corresponding level is higher than expected frequency

Negative

• Indicate observed frequency of corresponding level is lower than expected frequency

Magnitude Values ≤ -2 = observed frequency much lower than expected

Magnitude Values ≥ 2 = observed frequency much higher than expected

What is a Test of Independence

Checks whether 2 categorical variables from a single population are independent of each other

Specifically whether there is any dependence

H0 = A + B are independent

H1 = There is an association between A + B

i + j = specific cells

t-statistic for a Test of Independence

Column total - row total divided by total number of observations

Find t-statistic on the distribution for a Test of Independence

Phi Coefficient

Magnitude notations

• small effect = 0.1

• medium effect = 0.3

• large effect = 0.5

Cramer’s V

Cramer’s V Interpretation