Contingency Tables

1
Contingency Tables

• Tables representing all combinations of levels of
  explanatory and response variables
• Numbers in table represent Counts of the number
  of cases in each cell
• Row and column totals are called Marginal counts

2
Example EMT Assessment of Kids

• Explanatory Variable Child Age (Infant,
  Toddler, Pre-school, School-age, Adolescent)
• Response Variable EMT Assessment (Accurate,
  Inaccurate)

Source Foltin, et al (2002)
3
Pearsons Chi-Square Test

• Can be used for nominal or ordinal explanatory
  and response variables
• Variables can have any number of distinct levels
• Tests whether the distribution of the response
  variable is the same for each level of the
  explanatory variable (H0 No association between
  the variables)
• r of levels of explanatory variable
• c of levels of response variable

4
Pearsons Chi-Square Test

• Intuition behind test statistic
• Obtain marginal distribution of outcomes for the
  response variable
• Apply this common distribution to all levels of
  the explanatory variable, by multiplying each
  proportion by the corresponding sample size
• Measure the difference between actual cell counts
  and the expected cell counts in the previous step

5
Pearsons Chi-Square Test

• Notation to obtain test statistic
• Rows represent explanatory variable (r levels)
• Cols represent response variable (c levels)

6
Pearsons Chi-Square Test

• Marginal distribution of response and expected
  cell counts under hypothesis of no association

7
Pearsons Chi-Square Test

• H0 No association between variables
• HA Variables are associated

8
Example EMT Assessment of Kids
Observed
Expected
9
Example EMT Assessment of Kids

• Note that each expected count is the row total
  times the column total, divided by the overall
  total. For the first cell in the table

• The contribution to the test statistic for this
  cell is

10
Example EMT Assessment of Kids

• H0 No association between variables
• HA Variables are associated

Reject H0, conclude that the accuracy of
assessments differs among age groups
11
Example - SPSS Output
12
Example - Cyclones Near Antarctica

• Period of Study September,1973-May,1975
• Explanatory Variable Region (40-49,50-59,60-79)
  (Degrees South Latitude)
• Response Season (Aut(4),Wtr(5),Spr(4),Sum(8))
  (Number of months in parentheses)
• Units Cyclones in the study area
• Treating the observed cyclones as a random
  sample of all cyclones that could have occurred

Source Howarth(1983), An Analysis of the
Variability of Cyclones around Antarctica and
Their Relation to Sea-Ice Extent, Annals of the
Association of American Geographers,
Vol.73,pp519-537
13
Example - Cyclones Near Antarctica
For each region (row) we can compute the
percentage of storms occuring during each season,
the conditional distribution. Of the 1517
cyclones in the 40-49 band, 370 occurred in
Autumn, a proportion of 370/1517.244, or 24.4
as a percentage.
14
Example - Cyclones Near Antarctica
Graphical Conditional Distributions for Regions
15
Example - Cyclones Near Antarctica
Observed Cell Counts (fo)
Note that overall (1876/9165)10020.5 of all
cyclones occurred in Autumn. If we apply that
percentage to the 1517 that occurred in the
40-49S band, we would expect (0.205)(1517)310.5
to have occurred in the first cell of the table.
The full table of fe
16
Example - Cyclones Near Antarctica
Computation of
17
Example - Cyclones Near Antarctica

• H0 Seasonal distribution of cyclone occurences
  is independent of latitude band
• Ha Seasonal occurences of cyclone occurences
  differ among latitude bands
• Test Statistic
• P-value Area in chi-squared distribution with
  (3-1)(4-1)6 degrees of freedom above 71.2
• Frrom Table 8.5, P(c2??22.46).001 ? Plt .001

18
SPSS Output - Cyclone Example
P-value
19
Data Sources

• Foltin, G., D. Markinson,M. Tunik, et al
  (2002). Assessment of Pediatric Patients by
  Emergency Medical Technicians Basic, Pediatric
  Emergency Care, 1881-85.
• Howarth, D.A. (1983), An Analysis of the
  Variability of Cyclones around Antarctica and
  Their Relation to Sea-Ice Extent, Annals of the
  Association of American Geographers, 73519-537