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Chapter 13: Chi-Square Test

Chapter 13: Chi-Square Test. 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 1 1.

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Chapter 13: Chi-Square Test

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  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 • Relationship: There is a relationshipin the sample; cell phone users are less likely than non-users to sustain no injuries (66% vs. 82%)

  4. Table 2: “No Association” Table • 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 χ2statistic • 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. χ2Test: 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: 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)

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

  11. Calculating the χ2 Statistic: foand 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 Begin with a table containing only the row and column totals

  16. Calculating the χ2 Statistic: Computing fe 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

  18. Calculating the χ2 Statistic: Computing fe 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

  20. Calculating the χ2 Statistic: Computing fe

  21. Calculating the χ2 Statistic: Computing fe

  22. Calculating the χ2 Statistic: Computing fe This is the complete table of expected frequencies (fe)

  23. Calculating the χ2 Statistic Observed Frequencies (fo) 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 • χ2Distribution • 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 5.991 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 0.001 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 haveevidence 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

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