Exercises: Contingency tables - Risk, Odds and χ2 tests

Question 6.1

The following data are taken from the paper Caries prevalence in northern Scotland before and 5 years after, water defluoridation (Stephen et al., 1987, BDJ 163: 324-326). They show the social composition of children recruited to two arms of the study one before and one after water defluoridation. What is the probability that we don't know the social class of a child in the fluoridated arm? What is the probability that a child in the defluoridated arm is from social class III?

Social class Fluoridated Defluoridated
I & II 16 32
III 45 53
IV & V 32 22
Not known 13 19
Total 106 126


Question 6.2

A recent MDentSci project was looking at a number of risk factors thought to be associated with the health of oral implants in a population of elderly patients. One factor considered was smoking. The table below shows the number of healthy and non-healthy implants for smokers an non-smokers. Calculate and interpret the risk ratio and the odds ratio. The patients were selected for the study on the basis of the health of their implants (a case-control study). Which of the two ratios you have calculated would you use to report your results? Why?

A χ2 test was performed on these data, the results were: χ2 = 2·023, d.o.f. = 1, P = 0·16. Interpret these results.

Healthy implant
Yes No Total
Smoker 32 48 80
Non-smoker 10 7 17
Total 42 55 97


Question 6.3

A study (Erosion of dental enamel among competitive swimmers at a gas-chlorinated swimming pool, Centerwall et al., 1986, Am. J. Epid. 123: 641-647) was carried out to see if swimming in chlorinated water was linked to erosion of dental enamel. 49 swimmers with erosion of dental enamel (cases) were recruited along with 245 swimmers without erosion (controls). The data are summarised below.

Erosion of dental enamel
Hours of swimming per week Yes No Total
6 or more 32 118 150
less than 6 17 117 134
Total 49 235 284

Calculate the appropriate ratio to show the effect of excessive swimming on erosion of dental enamel. A χ2 test was performed on these data, the results were: χ2 = 4·802, d.o.f. = 1, P = 0·03. Interpet these results.


Question 6.4

The following data from a prospective study are taken from the paper Dental caries in pre-school children: associations with social class, toothbrushing habit and consumption of sugars and sugar-containing foods (Gibson & Williams 1999, Caries Research 33: 101-113). They show the number of children with caries according to three different risk factors: social class; toothbrushing frequency; and frequency of consumption of sugary foods. Which of these three factors has most impact on the likelihood of a child developing caries?

Social class Yes No Total
Manual 162 574 736
Non-manual 64 574 638
Total 226 1148 1374
Brushing frequency Yes No Total
0 or 1 per day 114 477 591
>1 per day 112 671 783
Total 226 1148 1374
Sugary foods Yes No Total
<3 times a day 61 347 408
3 or more times a day 165 801 966
Total 226 1148 1374