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Analyzing Contingency Tables using Pearson’s Chi-Square Test

Learn how to use Pearson’s Chi-Square Test to analyze contingency tables and determine statistical associations between variables. Understand the test statistic, expected cell counts, hypothesis testing, and interpretation of results with practical examples.

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Analyzing Contingency Tables using Pearson’s Chi-Square Test

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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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