1. Statistical Methods Lecture 26

2. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
The table below shows the importance of personal appearance for several age groups.
Are Age and Appearance independent, or is there a relationship? 2

3. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
A stacked barchart suggests a relationship:
Test for independence using a chi-square test of independence. 3

4. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
The test is mechanically equivalent to the test for homogeneity, but with some differences in how we think about the data and the results:
Homogeneity Test: one variable (Card Type) measured on two or more populations (Mailings).
Independence Test: Two variables (Appearance and Age) measured on a single population.
We ask the question “Are the variables independent?” rather than “Are the groups homogeneous?” This subtle distinction is important when drawing conclusions. 4

5. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
Assumptions and Conditions
Counted Data Condition – Data must be counts
Independence Assumption – Counts need to be independent from each other. Check for randomization
Randomization Condition – Random sample needed
Sample Size Assumption – There must be enough data so check the following condition.
Expected Cell Frequency Condition – Expect at least 5 individuals per cell. 5

6. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
Example : Automobile Manufacturers
Consumer Reports uses surveys to measure reliability in automobiles. Annually they release survey results about problems that consumers have had with vehicles in the past 12 months and the origin of manufacturer. Is consumer satisfaction related to country of origin?
State the hypotheses.
Find the test statistic.
Given the p-value, state your conclusion. 6

7. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
Example : Automobile Manufacturers 7

8. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
Example : Automobile Manufacturers
Consumer Reports uses surveys to measure reliability in automobiles. Annually they release survey results about problems that consumers have had with vehicles in the past 12 months and the origin of manufacturer. Is consumer satisfaction related to country of origin?
State the hypotheses.
Find the test statistic.
Given p-value = 0.231, state your conclusion.
There is not enough evidence to conclude there is an association between vehicle problems and origin of vehicle. 8

9. Chi-Square Test of Independence
QTM1310/ Sharpe
Chi-Square Test of Independence
For the Appearance and Age example, we reject the null hypothesis that the variables are independent.
So, it may be of interest to know how differently two age groups (teens and 30-something adults) select the “very important” category (Appearance response 6 or 7).
Construct a confidence interval for the true difference in proportions…
Then point out, after last point, that you’re back in the “difference of two proportions” land. 9

10. Don’t use chi-square methods unless you have counts.
QTM1310/ Sharpe
Don’t use chi-square methods unless you have counts.
Beware large samples! With a sufficiently large sample size, a chi-square test can always reject the null hypothesis.
Don’t say that one variable “depends” on the other just because they’re not independent. 10

11. Goodness-of-Fit Tests
QTM1310/ Sharpe
Goodness-of-Fit Tests
The Chi-Square Calculation
Find the expected values. These come from the null hypothesis value.
Compute the residuals,
Square the residuals,
Compute the components. Find for each cell.
Find the sum of the components,
Find the degrees of freedom
Test the hypothesis, finding the p-value or comparing the test statistic from 5 to the appropriate critical value. 11

12. Comparing Chi-Square Tests
Type of Data
Type of Question
Example
Good-ness of Fit
Counts in several categories
Do the data match my model?
M&Ms
Homogeneity
One variable, across multiple populations
Is the variable similar across all populations?
Types of credit cards (variable) across mailings (populations)
Independence
Two variables, one population
Are the two variables independent or not?
Problems and country of origin (variables) within cars (population)

13. Looking back
Recognize when a chi-square test of goodness of fit, homogeneity, or independence is appropriate.
Assumptions—same!
Mechanics—same!
Interpretation—(slightly) different!
Assumptions: quantitative data from incr. normal popn as sample gets smaller, r. sample
Particular form of the test statistic
Two-tailed test: effect on the p-value (have to double tail prob) or rejection region (have to take given alpha & divide it between two tails)