Non-parametric test (distribution-free)

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Chi-Square

• Non-parametric test (distribution-free)
• Nominal level dependent measure

 
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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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One-way Classification

• Observations sorted on only one dimension
• Example
• Observe children and count red, green, yellow, or
  orange Jello choices
• Are these colors chosen equally often, or is
  there a preference for one over the other?

Cont.
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One-way--cont.

• Want to compare observed frequencies with
  frequencies predicted by null hypothesis
• Chi-square test used to compare expected and
  observed
• Called goodness-of-fit chi-square (c2)

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Goodness-of-Fit Chi-square

• Fombonne (1989) Season of birth and childhood
  psychosis
• Are children born at particular times of year
  more likely to be diagnosed with childhood
  psychosis
• He knew the normal children born in each month
• e.g. .8.4 normal children born in January

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Fombonnes Data
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Chi-Square (c2)

• Compare Observed (O) with Expected (E)
• Take size of E into account
• With large E, a large (O-E) is not unusual.
• With small E, a large (O-E) is unusual.

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Calculation of c2
?2.05(11) 19.68
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Conclusions

• Obtained ?2 14.58
• df c - 1, where c categories
• Critical value of ?2 on 11 df 19.68
• Since 19.68 gt 14.58, do not reject H0
• Conclude that birth month distribution of
  children with psychoses doesnt differ from
  normal.

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Jello Choices

• Red Green Yellow Orange
• 35 20 25 20
• Is there a significant preference for one color
  of jello over other colors?

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• Red Green Yellow Orange
• O 35 20 25 20
• E 25 25 25 25
• X2 (35-25)2/25 (20-25)2/25 (25-25)2/25
  (20-25)2/25 6
• There was not one jello color chosen
  significantly more often than any other jello
  color, X2 (3, N 100) 6, p gt .05

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Contingency Tables

• Two independent variables
• Are men happier than women?
• Male vs. Female X Happy vs Not Happy
• Intimacy (Yes/No) X Depression/Nondepression

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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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Chi-Square on Contingency Table

• Same 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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Chi-Square Calculation
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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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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

• Jankowski Leitenberg (pers. comm.)
• Does abuse continue?
• Do adults who are, and are not, being abused
  differ in childhood history of abuse?
• One variable adult abuse (yes or no)
• Other variable number of abuse categories (out
  of 4) suffered as children
• Sexual, Physical, Alcohol, or Personal violence

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Chi-Square Calculation
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Conclusions

• 29.62 gt 7.82
• Reject H0
• Conclude that adult abuse is related to childhood
  abuse
• Increasing levels of childhood abuse are
  associated with greater levels of adult abuse.
• e.g. Approximately 10 of nonabused children are
  later abused as adults.

Cont.
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Nonindependent 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

• Rule of thumb E gt 5 in each cell
• Not firm rule
• Violated in earlier example, but probably not a
  problem
• More of a problem in tables with few cells.
• Never have expected frequency of 0.
• Collapse adjacent cells if necessary.

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.

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Effect Size

• Phi and Cramers Phi
• Define N and k
• Not limited to 2X2 tables

Cont.
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Effect Sizecont.

• Everitt cc data

Cont.
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Effect SizeOdss Ratio.

• Odds DepLack Intimacy
• 26/112 .232
• Odds Dep No Lack
• 11/270 .041
• Odds Ratio .232/.041 5.69
• Odds Depressed 5.69 times great if experiencing
  lack of intimacy.

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Effect SizeRisk Ratio.

• Risk Depression/Lack Intimacy
• 26/138 .188
• Risk Depression No Lack
• 11/281 .039
• Odds Ratio .188/.039 4.83
• Risk of Depressed 4.83 times greater if
  experiencing lack of intimacy.