CHI-SQUARE TESTS FOR LARGER TABLES 159Table 11.8 Association Between Health Status and Affording Medical Care
Afford Medical Carea
Health Status Almost Never Not Often Often Always Total
Excellent 4(8.40) 20(22.32) 2 l(24.72) 99(88.56) 144
Good 12( 18.02) 43(47.90) 59(53.04) 195(190.04) 309
Fair ll(6.13) 21( 16.27) 15( 18.02) 58(64.57) 105
Poor 8(2.45) 9(6.5 1) 8(7.21) 17(25.83) 42
Total 35 93 103 369 600
a Expected frequencies are shown in parentheses.or
(4 - 8.40)’ (20 - 22.32)’ (17 - 25.83)2
= 30.7078
= 8.40 4- 22.32 + ’ ’ ’ + 25.83
The d.f.’s are always given by (T - l)(c - 1) or in this example by (4 - 1)(4 - 1) = 9.
Here T is the number of rows and c is the number of columns. The formula for the
d.f.’s can be obtained by noting that we are free to fill in T - 1 times c - 1, or in this
case 9 of the cells, and still get the same row and column totals. For example, if we
know that we had 4, 20, and 21 observations in the first three columns of the first
row and that the total number in the first row was 144, we would know that the last
observation in the first row must be 99.
To determine if the computed value of chi-square is significant, we look in Table
A.4 using the row with 9 d.f. Note that the tabled value of chi-square that is needed
for a small P value gets larger as the d.f. increases. For 9 d.f. we need a computed
chi-square 2 16.92 to reject the null hypothesis of no association at the a = .05 level;
we need a tabled value of only 3.84 for 1 d.f. Since our computed value of 30.7078
is larger than the tabled value of 23.59 for the column headed .995, we know that our
P value is < .005.
Note that in this example the data are ordinal data. The chi-square test does not
take advantage of the data being ordinal, interval, or ratio rather than nominal. In
medical studies, authors often group data that are interval or ratio. For example,
body mass index (BMI) and age are often grouped into three or more groups and
then tested using the chi-square test. This may make the explanation simpler but it
is not using all the information available in the data set. However, two-way tables
are a common method of presenting the results. Use of the correlation coefficient to
measure association between interval and ratio variables is explained in Chapter 12.
In Chapter 13, Spearman’s rho, which can be used for ordinal, interval, or ratio data
to measure association, is described briefly.
11.4.2 Interpreting a Significant TestWe now know that there is a statistically significant association between health status
and being able to afford medical care. But we are left with the problem of interpreting