1. Chapter 13: Chi-Square Test

2. Motivating Example
Research Question: Among all adults in the U.S. who were in a car accident, is there a relationship between cell phone use and injury severity?
Sample: 200 randomly-selected U.S. adults who were in car accidents
Results: See Table 11
1This example is entirely fictitious

3. Table 1: Bivariate Table
Injuries
Used
Cell?
Total
Sustained No
Yes
None
82 (82%)
66 (66%)
148
Minor
12 (12%)
18 (18%) 30
Severe
6 (6%)
16 (16%) 22
100
200
Relationship: There is a relationship in the sample; cell phone users are less likely than non-users to sustain no injuries (66% vs. 82%)

4. Table 2: “No Association” Table
Injuries
Used
Cell?
Total
Sustained No
Yes
None
74 (74%)
148 (74%)
Minor
15 (15%)
30 (15%)
Severe
11 (11%)
22 (11%)
100
200
No Relationship: Cell phone users are just as likely as non-users to sustain no injuries (74%)

5. Relationship in Sample Vs. Population
Sample: We found a relationship in the sample of 200 accident victims
Population: We want to know whether there is a relationship in the population
ALL adults in the U.S. who were in car accidents
We can use hypothesis testing procedures
The chi-square test is used to test hypotheses involving bivariate tables

6. Chi-Square (χ2) Test Procedure
State the null and research hypotheses
Compute a χ2 statistic
Determine the degrees of freedom
Find the p-value for the χ2 statistic
Decide whether there is evidence to reject the null hypothesis
Interpret the results

7. χ2 Test Assumptions
Assumption 1: The sample is selected at random from a population
Assumption 2: The variables are nominal or ordinal
Note: In this class, you won’t have to determine whether the assumptions have been met

8. χ2 Test: Hyptheses
Null Hypothesis (H0): The two variables are not related in the population
Research Hypothesis (H1): The two variables are related in the population
Alpha (α): This will be given to you in every problem (when it’s not given, assume α = 0.05)

9. χ2 Test: Hyptheses Cell Phone – Injuries Example
Null Hypothesis (H0): Cell phone use and injury severity are not related among all adults in the U.S. who were in a car accident
Research Hypothesis (H1): Cell phone use and injury severity are related among all adults in the U.S. who were in a car accident
Alpha (α): Use α = 0.05)

10. χ2 Test: Calculating the χ2 Statistic
Formula:
Two Components
Observed Frequencies (fo)
Expected Frequencies (fe)

11. Calculating the χ2 Statistic: fo and fe
Observed Frequencies (fo)
Definition: The actual frequencies in the sample
Example: In the cell phone – injuries example, these are given in Table 1
Expected Frequencies (fe)
Definition: The frequencies we would expect assuming the two variables were independent
In other words, assuming the null hypothesis was true
Example: In the cell phone – injuries example, these are given in Table 2

12. Calculating the χ2 Statistic: Logic Behind the Formula
We are comparing the observed and expected frequencies
We are comparing the results in our sample with what we would expect if the two variables were independent (i.e., assuming H0 is true)
We are doing this because we are “testing the null hypothesis (H0)”, which assumes that the two variables are independent in the population

13. Calculating the χ2 Statistic: Size of Difference
Small Difference
If the differences between the observed and expected frequencies are small, the χ2 statistic will be small
As a result, we will likely fail to reject H0
Large Difference
If the differences between the observed and expected frequencies are large, the χ2 statistic will be large
As a result, we will likely reject H0
What is Small or Large?
We will use Appendix D to decide what is small or large

14. Calculating the χ2 Statistic: Computing fe
Procedure: For each cell, multiply the corresponding column marginal and row marginal, then divide by the sample size:
Huh?!?!? Let’s do this for the cell phone – injuries example (next several slides)

15. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200
Begin with a table containing only the row and column totals

16. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200
For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)

17. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200

18. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200
For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)

19. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200

20. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200

21. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None
148
Minor 30
Severe 22
100
200

22. Calculating the χ2 Statistic: Computing fe
Injuries
Used
Cell?
Total
Sustained No
Yes
None 74
148
Minor 15 30
Severe 11 22
100
200
This is the complete table of expected frequencies (fe)

23. Calculating the χ2 Statistic
Injuries
Used
Cell?
Total
Sustained No
Yes
None 82 66
148
Minor 12 18 30
Severe 6 16 22
100
200
Observed
Frequencies (fo)
Injuries
Used
Cell?
Total
Sustained No
Yes
None 74
148
Minor 15 30
Severe 11 22
100
200
Expected
Frequencies (fe)

24. χ2 Test: Degrees of Freedom (df)
Formula:
r = number of rows
c = number of columns
Interpretation: The number of cells in the table that need to have numbers before we can fill in the remaining cells
Cell Phone – Injury Example

25. χ2 Test: Determining the P-Value
χ2 Distribution
The p-value will be based on the χ2 distribution
The χ2 distribution is positively skewed
This means that our hypothesis tests will always be one-tailed
Values of the χ2 statistic are always positive
Minimum = 0 (variables are completely independent)
Maximum = ∞
The shape of the χ2 distribution is dictated by its df
See figure on next slide

26. χ2 Test: Determining the P-Value

27. χ2 Test: Determining the P-Value
Steps
Find df in the first column of Appendix D
Read across the row until you find the χ2 value you computed
Read up to the first row to find the p-value
Cell Phone – Injury Example
χ2 = 7.46, df = 2
Reading across the row where df = 2, a value of 7.46 is between and 7.824
Reading up to the top row, the p-value is between 0.05 and 0.02

28. χ2 Test: Determining the P-Value
Additional practice finding p-values
χ2 = 0.446, df = 2
P-value = 0.80
χ2 = 4.09, df = 1
P-value is between 0.02 and 0.05
χ2 = 0.01, df = 2
P-value is greater than 0.99
χ2 = 15.00, df = 4
P-value is between and 0.01

29. χ2 Test: Evidence to Reject H0?
Decision Rule
If the p-value is less than α, we have evidence to reject H0 in favor of H1
If the p-value is greater than α, we do not have evidence to reject H0 in favor of H1
Cell Phone – Injury Example
The p-value (which is between 0.02 and 0.05) is less than α = 0.05
We have evidence to reject H0 in favor of H1

30. χ2 Test: Interpretation
If We Reject H0: We have evidence to suggest that the two variables are related in the population
If We Do Not Reject H0: We do not have evidence to suggest that the two variables are related in the population
Cell Phone – Injury Example: We have evidence that cell phone use and injury severity are related among all adults in the U.S. who were in a car accident