Course Notes - Department of Mathematics and Statistics

• We can continue to calculate the the expected values for all ofthe values. Since the row and column totals must be met we cancalculate some of the values by subtraction (in square [] brackets).Pain ScoreIRS Improve No Change Worse TotalPlacebo 6 9 [5] r 1 = 20Single Dose 12 18 [10] r 2 = 40Double Dose [12] [18] [10] r 3 = 40c 1 = 30 c 2 = 45 c 3 = 25 n = 100Calculating the χ 2 statistic• We now need to compare the Observed and Expected counts.• Under the null hypothesis there should not be a big differencebetween these counts. But how closely must they agree?• We will use the χ 2 statistic:no.ofrows∑i=1no.ofcolumns∑j=1χ 2 = (O ij − E ij ) 2E ijχ 2 =(10 − 6)2 (5 − 9)2 (5 − 5)2+ +6 9 5(15 − 12)2 (20 − 18)2 (5 − 10)2+ + +12 18 10(5 − 12)2 (20 − 18)2 (15 − 10)2+ + +12 18 10= 14.72• Note χ 2 will always be positive.• In repeated sampling these χ 2 values are distributed as chi-squaredistribution which has ν degrees of freedom, whereν = (number of rows - 1) × (number of columns -1)155