Contingency Tables Tables representing all combinations of levels

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 • Explanatory Variable – Child Age (Infant, Toddler, Pre-school, School-age, Adolescent) • Response Variable – EMT Assessment (Accurate, Inaccurate) Source: Foltin, et al (2002) Assessment Age Acc Inac Tot Inf 168 73 241 Tod 230 73 303 Pre 254 53 307 Sch 379 58 437 Ado 652 124 776 Tot 1683 381 2064

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 (H 0: No association between the variables) • r = # of levels of explanatory variable • c = # of levels of response variable

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

Pearson’s Chi-Square Test • Notation to obtain test statistic – Rows represent explanatory variable (r levels) – Cols represent response variable (c levels) 1 2 … c Total 1 n 12 … n 1 c n 1. 2 n 21 n 22 … n 2 c n 2. … … … r nr 1 nr 2 … nrc nr. Total n. 1 n. 2 … n. c n. .

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

Pearson’s Chi-Square Test • H 0: No association between variables • HA: Variables are associated

Example – EMT Assessment of Kids Observed Expected Assessment Age Acc Inac Tot Inf 168 73 241 Inf 197 44 241 Tod 230 73 303 Tod 247 56 303 Pre 254 53 307 Pre 250 57 307 Sch 379 58 437 Sch 356 81 437 Ado 652 124 776 Ado 633 143 776 Tot 1683 381 2064

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

Example – EMT Assessment of Kids • H 0: No association between variables • HA: Variables are associated Reject H 0, conclude that the accuracy of assessments differs among age groups

Example - SPSS Output

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, pp 519 -537

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.

Example - Cyclones Near Antarctica Graphical Conditional Distributions for Regions

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 -49 S band, we would expect (0. 205)(1517)=310. 5 to have occurred in the first cell of the table. The full table of fe:

Example - Cyclones Near Antarctica Computation of

Example - Cyclones Near Antarctica • H 0: 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(c 2 22. 46)=. 001 P<. 001

SPSS Output - Cyclone Example P-value

Data Sources • Foltin, G. , D. Markinson, M. Tunik, et al (2002). “Assessment of Pediatric Patients by Emergency Medical Technicians: Basic, ” Pediatric Emergency Care, 18: 8185. • 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