Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 248can still be used as a test statistic after we have estimated each p ij in order to compute the“expected” counts. Again we use the data and the assumptions to estimate the p ij .Basically, the data is used to estimate the marginal probabilities, and the assumption ofindependence allows us to estimate the p ij from there.Table 9.6 Seat-belt usage in California withmarginal distributionsChildParent buckled unbuckled marginalbuckled 56 8 64unbuckled 2 16 18marginal 58 24 82The marginal probabilities are estimated by the marginal distributions of the data. For ourexample these are given in Table 9.6. The estimate forisand for it is Similarly, for we have andAs usual, we’ve used a “hat” for estimated values. With these estimates, wecan use the relationship to find the estimate For our seat-belt datawe have the estimates in Table 9.7. In order to show where the values comes from, thevalues have not been simplified.Table 9.7 Seat-belt usage in California withestimates for the corresponding p ijParentbuckledunbuckledChildbuckled unbuckled marginalmarginal 1With this table we can compute the expected amounts in the ijth cell with This isoften written R i C j /n, where R i is the row sum and C i the column sum, as this simplifiescomputations by hand.With the expected amounts now known, we form the χ 2 statistic as:(92)Under the hypothesis of multinomial data and the independence of the variables, thesampling distribution of χ 2 will be the chi-squared distribution with (n r −1)·(n c −1) degreesof freedom. Why this many? For the row variable we have n r −1 unspecified values thatthe marginal probabilities can take (not n r , as they sum to 1) and similarly for the columnvariable. Thus there are (n r −1)·(n c −1) unspecified values.