Statistical Methods in Medical Research 4ed

230 Comparison of several groups
Table 8.2 Presence or absence of S. mansoni eggs in the stool.
Age (years) 0± 10± 20± 30± 40± Total
Positive 14 16 14 7 6 57
Negative 87 33 66 34 11 231
Total 101 49 80 41 17 288
The expected number of positives for the age group 0± is
…57†…101†=288 ˆ 19 99:
The set of expected numbers for the 10 cells in the table is
Total
19 99 9 70 15 838 113 6 57
81 01 393 0 64 17 32 89 13 64 231
Total 101 49 80 41 17 288
The fact that the expected numbers add to the same marginal totals as those observed is a
useful check.
The contribution to X 2 from the first cell is
and the set of contributions for the 10 cells is
giving a total of
…14 19 99† 2 =19 99 ˆ 1 79,
1 79 4 09 0 21 0 15 2 07
0 44 1 01 0 05 0 04 0 51
X 2 ˆ 10 36:
The degrees of freedom are k 1 ˆ 4, for which the 5% point is 9 49. The departures from
the null hypothesis are thus significant at the 5% level …P ˆ 0 035†.
In this example the column classification is based on a continuous variable, age, and it
would be natural to ask whether the proportions of positives exhibit any smooth trend with
p
age. The estimated proportions, with their standard errors calculated as …piqi=ni† , are
0 14 0 33 0 18 0 17 0 35,
0 03 0 07 0 04 0 06 0 12
the last being based on particularly small numbers. No clear trend emerges (a method for
testing for a trend is given in §15.2). About half the contribution to X 2 comes from the
second age group (10±19 years) and there is some suggestion that the proportion of
positives in this group is higher than in the neighbouring age groups.
To illustrate the use of (8.30), call the numbers of positives ri. Then
X 2 ˆ…14 2 =101 ‡ ...‡ 6 2 =17 57 2 =288†=…0 1979†…0 8021†,