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Contingency Tables

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. Example – EMT Assessment of Kids.

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Contingency Tables

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  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. Pearson’s 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. Pearson’s 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. Pearson’s Chi-Square Test • Notation to obtain test statistic • Rows represent explanatory variable (r levels) • Cols represent response variable (c levels)

  6. Pearson’s Chi-Square Test • Marginal distribution of response and expected cell counts under hypothesis of no association:

  7. H0: No association between variables HA: Variables are associated Pearson’s Chi-Square Test

  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)100%=20.5% 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  P< .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, 18:81-85. • 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, 73:519-537

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