Contingency Tables (cross tabs)

Generally used when variables are nominal
and/or ordinal


Even here, should have a limited number of variable
attributes (categories)
Inside the cells of the table are frequencies
(number of cases that fit criteria)
 To examine relationships within the sample,
most use percentages to standardize the cells
Example

A survey of 2883 U.S. residents
 Is one’s political ideology (liberal, moderate,
conservative) related to their satisfaction with
their financial situation?
 Null = ideology is not related to satisfaction
with financial status (the are independent)
 Convention for bivariate tables


IV (Ideology) is on the top of the table (dictates
columns)
The DV ($ status)is on the side (dictates rows).
Are these variable related within
the sample?
Satisfaction
With
Current $
Situation
Total
Political Ideology
Liberal
Moderate
Conservative
Satisfied
242
300
334
876
More or Less
329
499
439
1267
Unsatisfied
213
294
233
740
Total
784
1093
1006
2883
The Test Statistic for Contingency
Tables

Chi Square, or χ2

Calculation
• Observed frequencies (your sample data)
• Expected frequencies (UNDER NULL)


Intuitive: how different are the observed cell
frequencies from the expected cell frequencies
Degrees of Freedom:
• 1-way = K-1
• 2-way = (# of Rows -1) (# of Columns -1)
Calculating
 χ2



= ∑ [(fo - fe)2 /fe]
Where Fe= Row Marginal X Column Marginal
N
So, for each cell, calculate the difference between the
actual frequencies (“observed”) and what frequencies
would be expected if the null was true (“expected”).
Square, and divide by the expected frequency.
Add the results from each cell.
Satisfaction
With
Current $
Situation
Satisfied
Total
Political Ideology
Liberal
Moderate
Conservative
242 (238)
300 (332)
334 (305)
876
More or Less
329
499
439
1267
Unsatisfied
213
294
233
740
Total
784
1093
1006
2883
FIND EXPECTED FREQUENCIES UNDER NULL
Example: 876(784)/2883 = 238
Satisfaction
With
Current $
Situation
Political Ideology
Liberal
Moderate
Total
Conservative
Satisfied
242 (238)
300 (332)
334 (305)
876
More or Less
329 (344)
499 (480)
439 (442)
1267
Unsatisfied
213 (201)
294 (280)
233 (258)
740
784
1093
1006
2883
Total
FIND EXPECTED FREQUENCIES UNDER NULL
Example: 876(784)/2883 = 238
Calculating χ2
 χ2





= ∑ [(fo - fe)2 /fe]
[(242-238) 2 / 238 ] = .067
[(300-332) 2 / 332 ] = 3.08
Do the same for the other seven cells…
Calculate obtained χ2
Figure out appropriate df and then Critical χ2
(alpha = .05)
• Would decision change if alpha was .01?
Interpreting Chi-Square
 Chi-square
has no intuitive meaning, it can
range from zero to very large

As with other test statistics, the real interest is
the “p value” associated with the calculated
chi-square value
• Conventional testing = find χ2 (critical) for stated
“alpha” (.05, .01, etc.)

Reject if χ2 (observed) is greater than χ2 (critical)
• SPSS: find the exact probability of obtaining the χ2
under the null (reject if less than alpha)
SPSS Procedure

Analyze Descriptive Statistics  Crosstabs

Rows = DV
 Columns = IV
 Cells

Column Percentages
 Statistics

Chi square
Agenda
 Today

Group based assignment (review for exam,
review chi-square)
 Monday

More “hands on” review for exam + final
project time
 Wednesday

Go over HW#4, review conceptual stuff, more
practice