Chi-Square Test

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Chi-Square Test

• Mon, Apr 19th, 2004

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Chi-Square (?2)

• Are 2 categorical variables related (correlated)
  or independent of each other?
• Compares in categories that would be expected
  by chance (E) to in categories actually
  observed (O)
• Null hyp (Ho) no relationship between 2
  variables (theyre independent of ea.other)
• Alternate (Ha) the 2 variables are related (not
  independent)

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?2 Formula

• ?2 ? (fo fe)2
• fe
• So well compare observed and expected
  frequencies for each cell in the table

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Example

• Is age (lt30 v. gt30) related to preference for
  analog/digital watches?

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Step 1 Compute Marginals
Marginals are the row and column totals
140
60
100
20
80
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Step 2 Compute Expected Frequencies (fE)

• fE (Column marginal Row marginal ) / N
• For people under 30
• fe (digital) 100 140 / 200 70
• fe (analog) 80140 / 200 56
• fe (undec) 20140 / 200 14
• Over 30
• fe (digital) 10060 / 200 30
• fe (analog) 8060 / 200 24
• fe (undec) 2060 / 200 6

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Step 3 Compute X2

• Find difference (residual) betw observed
  expected for each cell (fo fe)
• Square those differences
• Divide squared differences by fe
• Sum the results

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Summary
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(cont.)

• Last step Add up (fo-fe)2 / fe
• ?2 5.71 4.57 1.14 13.33 10.67 2.67
  38.09
• Step 4 Compare to ?2 critical with df (
  columns 1) ( rows 1)
• Here df (2-1)(3-1) 2 df, ? .05, critical
  5.99

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Hypothesis Test

• If ?2 observed gt ?2 critical, reject Ho
• Reject Ho ? conclude there is a relationship
  between the 2 variables
• Here, 38.09 gt 5.99, reject Ho, there is a
  relationship between age watch preference

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In SPSS

• Analyze ? Descriptive Stats ? Crosstabs
• Choose whichever variable youd like for row
  variable and the other for column variable
• Click Statistics button, and check chi-squared
  option
• Click Cells Button, choose expected count

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SPSS (cont.)

• Output look for Pearson chi-sq and Asymp
  Sig column gives significance value for chi-sq
  test
• If Asymp Sig value is lt .05 (alpha), reject Ho
• Note there is an option for clustered graphing,
  read this example in the lab