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Chapter 16: Chi Square

This chapter delves into the Chi-Square (χ2) test, a fundamental tool for analyzing categorical variables in psychological research. It outlines the distinction between parametric tests and non-parametric methods, emphasizing the importance of frequency counts within categories. The chapter also covers Chi-Square calculations, expected frequencies, and the implications of observed results on various psychological phenomena, such as the relationships between intimacy and depression, and childhood experimentation leading to adult addiction. Key concepts like contingency tables and independence of observations are discussed.

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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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