Chi Square Test Vocabulary

The Chi-Square test is typically used to analyze the relationship between two variables under the following conditions:

1. Both variables should be qualitative in nature, meaning they consist of categories rather than numerical measurements.

2. The two variables must have been measured on the same individuals, ensuring consistent conditions for analysis.

3. The observations pertaining to each variable are derived from different subjects, also known as between-subjects design.

Chi-Square test

• Is nonparametric in nature

• Is designed to analyze the relationship between two variables using frequency information.

Two-Way Contingency Tables

• The basis of analysis for the chi-square test

• Called this because it examines two variables

Cell

In a contingency table, each unique combination of values from the two variables is referred to as a

Observed Frequencies

The entries within the cells represent the number of individuals who are characterized by the corresponding values of the variables and are referred to as this

Marginal Frequencies

• The numbers in the last column and the bottom row indicate how many individuals have each separate characteristic

• Are the sums of the frequencies in the corresponding rows or columns.

Expected frequencies

• The logic underlying the chi-square test focuses on this concept.

• These expected frequencies can be compared with the frequencies that are observed when you actually flip the coin

Inference of a relationship using the Chi-Square Test:

1. State a null hypothesis indicating no relationship between the variables.

2. Obtain a set of observed frequencies from the data.

3. Derive the expected frequencies based on the assumption that the null hypothesis holds.

4. Compare the expected frequencies against the observed frequencies to identify any significant differences.

5. Reject the null hypothesis if a substantial difference exists between observed and expected frequencies, according to a predetermined alpha level.

Null and Alternative Hypotheses

one asserts that there is no relationship between the two variables in the population, while the other states that a relationship does exist.

Chi-square statistic

Reflects the overall differences between the observed and expected frequencies

Expected frequencies GOF

• Application of the chi-square test requires the computation of an expected frequency for each cell of a contingency table under assumption that there is no relationship between the two variables in the population.

• Sample size multiplied by each claimed proportion

Sampling distribution of the chi-square statistic

• If two variables are unrelated in the population, the population value of the chi square statistic will equal 0.

• However, because of sampling error, a chi-square statistic score that is computed from sample data might be greater than 0 even when the null hypothesis is true.

Chi-square distribution

• Takes on different shapes depending on how many degrees of freedom are associated with it.

• Becomes less skewed as n increases

Assumptions of the Chi-Square test

The chi-squared test is typically used to analyze the relationship between two qualitative variables, however, it an also be applied when one or both variables are quantitative.
Assumptions:
1) The observations are independently and randomly sampled from the population of all possible observations.
2) The expected frequency for each cell is >5.

Computational formula for 2 X 2 tables

If both variables have only two levels, the chi-square statistic can be derived using a different computational formula.

Chi-Squared goodness-of-fit Test

1 sample, 1 variable

Chi-Squared Independence Test

Used to test the independence of two variables
1 sample, 2 variables

Chi-squared homogeneity test

Used to test the independence of populations are the same
multiple samples, 1 variable

Expected frequencies Independence and Homogeneity

row x column / total