Contingency TablesExample – EMT Assessment of KidsPearson’s Chi-Square TestSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Example - SPSS OutputExample - Cyclones Near AntarcticaSlide 13Slide 14Slide 15Slide 16Slide 17SPSS Output - Cyclone ExampleData SourcesContingency 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 countsExample – EMT Assessment of Kids•Explanatory Variable – Child Age (Infant, Toddler, Pre-school, School-age, Adolescent)•Response Variable – EMT Assessment (Accurate, Inaccurate)AssessmentAge Acc Inac TotInf 168 73 241Tod 230 73 303Pre 254 53 307Sch 379 58 437Ado 652 124 776Tot 1683 381 2064Source: Foltin, et al (2002)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 variablePearson’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 stepPearson’s Chi-Square Test•Notation to obtain test statistic–Rows represent explanatory variable (r levels)–Cols represent response variable (c levels)1 2 … c Total1 n11n12 … n1c n1.2 n21n22 … n2c n2. … … … … … …r nr1 nr2 … nrcnr.Total n.1n.2… n.cn..Pearson’s Chi-Square Test•Marginal distribution of response and expected cell counts under hypothesis of no association:....^....^..1.1^)(nnnpnnEnnpnnpjijiijcc Pearson’s Chi-Square Test•H0: No association between variables•HA: Variables are associated)(value:..)())((:..222)1)(1(,222XPPXRRnEnEnXSTcri jijijij---Example – EMT Assessment of KidsAssessmentAge Acc Inac TotInf 168 73 241Tod 230 73 303Pre 254 53 307Sch 379 58 437Ado 652 124 776Tot 1683 381 2064AssessmentAge Acc Inac TotInf 197 44 241Tod 247 56 303Pre 250 57 307Sch 356 81 437Ado 633 143 776Tot 1683 381 2064 Observed ExpectedExample – 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:1972064)1683(241)(..1..111nnnnE• The contribution to the test statistic for this cell is27.4197)197168(2Example – EMT Assessment of Kids•H0: No association between variables•HA: Variables are associated488.9:..1.40143)143124(197)197168(:..24,05.2)12)(15(,05.2222--XRRXST Reject H0, conclude that the accuracy of assessments differs among age groupsExample - SPSS OutputAGE * ASSESS Crosstabulation168 73 241196.5 44.5 241.0230 73 303247.1 55.9 303.0254 53 307250.3 56.7 307.0379 58 437356.3 80.7 437.0652 124 776632.8 143.2 776.01683 381 20641683.0 381.0 2064.0CountExpected CountCountExpected CountCountExpected CountCountExpected CountCountExpected CountCountExpected CountInfantToddlerPre-schoolSchool ageAdolescentAGETotalAccurate InaccurateASSESSTotalChi-Square Tests40.073a4 .00037.655 4 .00029.586 1 .0002064Pearson Chi-SquareLikelihood RatioLinear-by-LinearAssociationN of Valid CasesValue dfAsymp. Sig.(2-sided)0 cells (.0%) have expected count less than 5. Theminimum expected count is 44.49.a.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 occurredSource: 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,pp519-537Example - Cyclones Near AntarcticaRegion\Season Autumn Winter Spring Summer Total40-49S370 452 273 422151750-59S526 624 513 1059272260-79S980 1200 995 17514926Total 1876 2276 1781 3232 9165For 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. Region\Season Autumn Winter Spring Summer Total% (n)40-49S24.4 29.8 18.0 27.8100.0 (1517)50-59S19.3 22.9 18.9 38.9100.0 (2722)60-79S19.9 24.4 20.2 35.5100.0 (4926)Example - Cyclones Near Antarctica40-49S50-59S60-79SregionBars show MeansAutumn Winter Spring Summerseason10.0020.0030.0040.00regpctGraphical Conditional Distributions for RegionsExample - Cyclones Near AntarcticaRegion\Season Autumn Winter Spring Summer Total40-49S370 452 273 422151750-59S526 624 513 1059272260-79S980 1200 995 17514926Total 1876 2276 1781 3232 9165Note 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: Region\Season Autumn Winter Spring Summer Total40-49S310.5 376.7 294.8 535.0151750-59S557.2 676.0 529.0 959.9272260-79S1008.3 1223.3 957.3 1737.14926Total 1876 2276 1781 3232 9165Observed Cell Counts (fo):Example - Cyclones Near AntarcticaRegion Season fo fe (fo-fe)^2 ((fo-fe)^2)/fe40-49S Autumn 370 310.5 3540.25 11.401771340-49S Winter 452 376.7 5670.09 15.052004240-49S Spring 273 294.8 475.24 1.6120759840-49S Summer 422 535.0 12769 23.867289750-59S Autumn 526 557.2 973.44 1.7470208250-59S Winter 624 676.0 2704 450-59S Spring 513 529.0 256 0.4839319550-59S Summer 1059 959.9 9820.81 10.231076260-79S Autumn 980 1008.3 800.89 0.7942973360-79S Winter 1200 1223.3 542.89 0.4437913860-79S Spring 995 957.3 1421.29 1.484686160-79S Summer 1751 1737.1 193.21 0.1112256171.2291706Computation of 2obsExample - Cyclones Near Antarctica•H0: Seasonal