Chapter 13: Chi-Square Test 1. Motivating Example Research Question: Among all adults in the U.S. who were in a car accident, is there a relationship.
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Chapter 13: Chi-Square Test
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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
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Table 1: Bivariate TableInjuries Used Cell? Total
Sustained
No Yes
None 82 (82%) 66 (66%) 148Minor 12 (12%) 18 (18%) 30
Severe 6 (6%) 16 (16%) 22Total 100 100 200
Relationship: There is a relationship in the sample; cell phone users are less likely than non-users to sustain no injuries (66% vs. 82%)
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Table 2: “No Association” Table
Injuries Used Cell? TotalSustaine
dNo Yes
None 74 (74%) 74 (74%) 148 (74%)
Minor 15 (15%) 15 (15%) 30 (15%)Severe 11 (11%) 11 (11%) 22 (11%)Total 100 100 200
No Relationship: Cell phone users are just as likely as non-users to sustain no injuries (74%)
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Relationship in Sample Vs. PopulationSample: 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
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Chi-Square (χ2) Test ProcedureState 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
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χ2 Test AssumptionsAssumption 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
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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)
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χ2 Test: HypthesesCell 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)
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χ2 Test: Calculating the χ2 Statistic
Formula:
Two Components◦Observed Frequencies (fo)
◦Expected Frequencies (fe)
CellsAll e
eo
f
ff 22
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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
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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
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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
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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)
N
)marginalrow)(marginalcolumn(ef
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
Begin with a table containing only the row and column totals
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)
74200
)100)(148(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
74200
)100)(148(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
For each cell, multiply the corresponding row and column total, then divide by the total sample size (200 here)
15200
)100)(30(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
15200
)100)(30(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
11200
)100)(22(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 148
Minor 30
Severe 22
Total 100 100 200
11200
)100)(22(
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Calculating the χ2 Statistic: Computing feInjuries Used Cell? Total
Sustained No Yes
None 74 74 148
Minor 15 15 30
Severe 11 11 22
Total 100 100 200
This is the complete table of expected frequencies (fe)
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Calculating the χ2 StatisticInjuries Used Cell? Total
Sustained No Yes
None 82 66 148
Minor 12 18 30
Severe 6 16 22
Total 100 100 200
46.7
27.227.260.060.086.086.011
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11
25
15
9
15
9
74
64
74
64
11
5
11
)5(
15
3
15
)3(
74
)8(
74
8
11
1116
11
116
15
1518
15
1512
74
7466
74
7482
222222
22222222
CellsAll e
eo
f
ff
Injuries Used Cell? Total
Sustained No Yes
None 74 74 148
Minor 15 15 30
Severe 11 11 22
Total 100 100 200
ObservedFrequencies
(fo)
ExpectedFrequencies
(fe)
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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
)1()1( crdf
2)1)(2()12()13()1()1( crdf
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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
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χ2 Test: Determining the P-Value
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χ2 Test: Determining the P-ValueSteps
◦ 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 5.991 and 7.824
◦ Reading up to the top row, the p-value is between 0.05 and 0.02
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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 0.001 and 0.01
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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
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χ2 Test: InterpretationIf 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
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