## 31.8 Example: Pet birds

A study examined people with lung cancer, and a matched set of similar controls who did not have lung cancer, and compared the proportion in each group that had pet birds .

These data were studied in Sect. 25.6; the data are shown again in Table 31.4, and the numerical summary in Table 31.5 (the computations are shown in Sect. 25.6).

TABLE 31.4: The pet bird data
Kept pet birds 98 101 199
Did not keep pet birds 141 328 469
Total 239 429 668

One RQ in the study was:

Are the odds of having a pet bird the same for people with lung cancer (cases) and for people without lung cancer (controls)?

The parameter is the population OR, comparing the odds of keeping a pet bird, for adults with lung cancer to adults who do not have lung cancer.

The RQ could also be written as:

• Is the percentage of people having a pet bird the same for people with lung cancer (cases) and for people without lung cancer (controls)?
• Is the odds ratio of people having a pet bird, comparing people with lung cancer (cases) and for people without lung cancer (controls), equal to one?
• Is there a relationship between having a pet bird and having lung cancer?
Of these, the first is probably the easiest to understand.

From this RQ (which is written in terms of odds), the hypotheses could be written as:

• $$H_0$$: The odds of having a pet bird is the same for people with lung cancer (cases) and for people without lung cancer (controls).
• $$H_1$$: The odds of having a pet bird is not the same for people with lung cancer (cases) and for people without lung cancer (controls).

The null hypothesis could also be written as:

• The percentage of people having a pet bird is the same for people with lung cancer (cases) and for people without lung cancer (controls).
• The odds ratio of people having a pet bird, comparing people with lung cancer (cases) and for people without lung cancer (controls), is equal to one.
• There is no relationship between having a pet bird and having lung cancer.
Of these, the first is probably the easiest to understand.

Begin by assuming the null hypothesis is true: no difference exists between the odds in the population. Based on this assumption, the expected counts can be found.

From the data (Table 31.4), overall $$199\div 668 = 29.79$$% of people own a pet bird. If there really was no difference in the odds (or the percentages) of owning a pet bird between those with and without lung cancer, about $$29.79$$% of the people in both lung cancer groups are expected to own a pet bird.

TABLE 31.5: The odds and percentage of subjects keeping pet birds
Odds of keeping pet bird Percentage keeping pet bird Sample size
With lung cancer: 0.6950 41.0% 239
Without lung cancer: 0.3079 25.5% 429
Odds ratio: 2.26

About 29.79% of the 239 lung-cancer cases (or 71.20) would be expected to have a pet bird, and about 29.79% of the 429 non-lung-cancer cases (or 127.80) would be expected to have a pet bird. A table of these expected counts (Table 31.6). shows that all expected counts are greater than five. In practice, you do not need to compute the expecte counts; software does this automatically.

TABLE 31.6: The expected counts for the pet bird data, if the proportion owning pet birds was the same for lung cancer cases, and non-lung-cancer cases
Kept pet birds 71.2 127.8 199
Did not keep pet birds 167.8 301.2 469
Total 239.0 429.0 668

The numbers in Table 31.6 are what is expected, if the percentage of people owning a pet bird is the same for lung cancer and non-lung cancer cases. How close are the expected and observed counts (in Table 31.4)?

To compare the sample statistic (what we observed) with the hypothesised population parameter, software is used to compute the value of $$\chi^2$$ (jamovi: Fig. 31.4; SPSS: Fig. 31.5): $$\chi^2=22.374$$, approximately equivalent to a $$z$$-score of

$\sqrt{22.374/1} = 4.730,$ which is very large. Hence, a small $$P$$-value is expected.

The software shows that the $$P$$-value is very small ($$P<0.001$$). As usual, a small $$P$$-value means that there is very strong evidence supporting $$H_1$$, if $$H_0$$ is assumed true. That is, the evidence suggests there is a difference in the odds in the population. We write:

The sample provides very strong evidence ($$\chi^2=22.374$$; two-tailed $$P<0.001$$) that the odds in the population of having a pet bird is not the same for people with lung cancer (odds: 0.695) and for people without lung cancer (odds: 0.308; OR: $$2.26$$; 95% CI from $$1.6$$ to $$3.2$$).

Think 31.2 (Interpretation) This doesn’t imply that owning a pet bird causes lung cancer. Why not?
Because the study observational. Confounders may explain the relationship (can you think of one?). In addition, may be having lung cancer means that people seek companionship in the form of a pet.

### References

Kohlmeier L, Arminger G, Bartolomeycik S, Bellach B, Rehm J, Thamm M. Pet birds as an independent risk factor for lung cancer: Case-control study. British Medical Journal. 1992;305(6860):986–9.