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Chapter 16: Chi Square

Chapter 16: Chi Square. PSY295-001—Spring 2003 Summerfelt. Overview. z, t, ANOVA, regression, & correlation have Used at least one continuous variable Relied on underlying population parameters Been based on particular distributions Chi square ( χ 2 ) is Based on categorical variables

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Chapter 16: Chi Square

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  1. Chapter 16: Chi Square PSY295-001—Spring 2003 Summerfelt

  2. Overview • z, t, ANOVA, regression, & correlation have • Used at least one continuous variable • Relied on underlying population parameters • Been based on particular distributions • Chi square (χ2) is • Based on categorical variables • Non-parametric • Distribution-free Chapter 16 Chi-Square

  3. Categorical Variables • Generally the count of objects falling in each of several categories. • Examples: • number of fraternity, sorority, and nonaffiliated members of a class • number of students choosing answers: 1, 2, 3, 4, or 5 • Emphasis on frequency in each category Chapter 16 Chi-Square

  4. Contingency Tables • Two independent variables • Can be various levels similar to two-way ANOVA • Gender identity, level of happiness Chapter 16 Chi-Square

  5. Intimacy and Depression • Everitt & Smith (1979) • Asked depressed and non-depressed women about intimacy with boyfriend/husband • Data on next slide Chapter 16 Chi-Square

  6. Data Chapter 16 Chi-Square

  7. What Do the Data Say? • It looks as if depressed women are more likely to report lack of intimacy. • What alternative explanations? • Is the relationship reliably different from chance? • Chi-square test Chapter 16 Chi-Square

  8. Chi-Square on Contingency Table • The formula • Expected frequencies • E = RT X CT GT • RT = Row total, CT = Column total, GT = Grand total Chapter 16 Chi-Square

  9. Expected Frequencies • E11 = (37*138)/419 = 12.19 • E12 = (37*281)/419 = 24.81 • E21 = (382*138)/419 = 125.81 • E22 = (382*281)/419 = 256.19 • Enter on following table Chapter 16 Chi-Square

  10. Observed and Expected Freq. Chapter 16 Chi-Square

  11. Degrees of Freedom • For contingency table, df = (R - 1)(C - 1) • For our example this is (2 - 1)(2 - 1) = 1 • Note that knowing any one cell and the marginal totals, you could reconstruct all other cells. Chapter 16 Chi-Square

  12. Chi-Square Calculation Chapter 16 Chi-Square

  13. Conclusions • Since 25.61 > 3.84, reject H0 • Conclude that depression and intimacy are not independent. • How one responds to “satisfaction with intimacy” depends on whether they are depressed. • Could be depression-->dissatisfaction, lack of intimacy --> depression, depressed people see world as not meeting needs, etc. Chapter 16 Chi-Square

  14. Larger Contingency Tables • Is addiction linked to childhood experimentation? • Do adults who are, and are not, addicted to substances (alcohol or drug) differ in childhood categories of drug experimentation? • One variable = adult addiction • yes or no • Other variable = number of experimentation categories (out of 4) as children • Tobacco, alcohol, marijuana/hashish, or acid/cocaine/other Chapter 16 Chi-Square

  15. Chapter 16 Chi-Square

  16. Chi-Square Calculation Chapter 16 Chi-Square

  17. Conclusions • 29.62 > 7.82 • Reject H0 • Conclude that adult addiction is related to childhood experimentation • Increasing levels of childhood experimentation are associated with greater levels of adult addiction. • e.g. Approximately 10% of children not experimenting later become addicted as adults. Chapter 16 Chi-Square Cont.

  18. Conclusions--cont. • Approximately 40% of highly experimenting children are later addicted as adults. • These data suggest that childhood experimentation may lead to adult addiction. Chapter 16 Chi-Square

  19. Tests on Proportions • Proportions can be converted to frequencies, and tested using c2. • Use a z test directly on the proportions if you have two proportions • From last example • 10% of nonabused children abused as adults • 40% of abused children abused as adults Chapter 16 Chi-Square Cont.

  20. Proportions--cont. • There were 566 nonabused children and 30 heavily abused children. Chapter 16 Chi-Square Cont.

  21. Proportions--cont. • z = 5.17 • This is a standard z score. • Therefore .05 (2-tailed) cutoff = +1.96 • Reject null hypothesis that the population proportions of abuse in both groups are equal. • This is just the square root of the c 2 you would have with c 2 on those 4 cells. Chapter 16 Chi-Square

  22. Independent Observations • We require that observations be independent. • Only one score from each respondent • Sum of frequencies must equal number of respondents • If we don’t have independence of observations, test is not valid. Chapter 16 Chi-Square

  23. Small Expected Frequencies • Assume O would be normally distributed around E over many replications of experiment. • This could not happen if E is small. • Rule of thumb: E> 5 in each cell • Not firm rule • Violated in earlier example, but probably not a problem Chapter 16 Chi-Square Cont.

  24. Expected Frequencies--cont. • More of a problem in tables with few cells. • Never have expected frequency of 0. • Collapse adjacent cells if necessary. Chapter 16 Chi-Square

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