Chapter 16: Chi Square

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Chapter 16 Chi Square

• PSY295-001Spring 2003
• Summerfelt

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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 (?2) is
• Based on categorical variables
• Non-parametric
• Distribution-free

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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

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Contingency Tables

• Two independent variables
• Can be various levels similar to two-way ANOVA
• Gender identity, level of happiness

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Intimacy and Depression

• Everitt Smith (1979)
• Asked depressed and non-depressed women about
  intimacy with boyfriend/husband
• Data on next slide

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Data
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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

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Chi-Square on Contingency Table

• The formula
• Expected frequencies
• E RT X CT GT
• RT Row total, CT Column total, GT Grand
  total

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Expected Frequencies

• E11 (37138)/419 12.19
• E12 (37281)/419 24.81
• E21 (382138)/419 125.81
• E22 (382281)/419 256.19
• Enter on following table

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Observed and Expected Freq.
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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.

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Chi-Square Calculation
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Conclusions

• Since 25.61 gt 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--gtdissatisfaction, lack of
  intimacy --gt depression, depressed people see
  world as not meeting needs, etc.

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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

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Chi-Square Calculation
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Conclusions

• 29.62 gt 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.

Cont.
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Conclusions--cont.

• Approximately 40 of highly experimenting
  children are later addicted as adults.
• These data suggest that childhood experimentation
  may lead to adult addiction.

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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

Cont.
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Proportions--cont.

• There were 566 nonabused children and 30 heavily
  abused children.

Cont.
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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.

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Independent Observations

• We require that observations be independent.
• Only one score from each respondent
• Sum of frequencies must equal number of
  respondents
• If we dont have independence of observations,
  test is not valid.

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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 gt 5 in each cell
• Not firm rule
• Violated in earlier example, but probably not a
  problem

Cont.
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Expected Frequencies--cont.

• More of a problem in tables with few cells.
• Never have expected frequency of 0.
• Collapse adjacent cells if necessary.