## 31.9 Example: B12 deficiency

A study in New Zealand asked:

Among a certain group of women, are the odds of being vitamin B12 deficient different for women on a vegetarian diet compared to women on a non-vegetarian diet?

The population was ‘predominantly overweight/obese women of South Asian origin living in Auckland.’ The RQ could be worded in terms of odds ratios or proportions, too.

To test the claim, the hypotheses are:

• $$H_0$$: $$\text{population odds for vegetarians} = \text{population odds for non-vegetarians}$$: The odds of B12 deficiency are the same in both groups.
• $$H_1$$: $$\text{population odds for vegetarians} \ne \text{population odds for non-vegetarians}$$: The odds of B12 deficiency are not the same in both groups.

The parameter is the population OR, comparing the odds of being B12 deficient, for vegetarians to non-vegetarians.

Again (see Sect. 31.8), the RQ and the hypotheses could be written in terms of comparing the percentages, in terms of the odds ratio being equal to one, or in terms of relationships between the variables.

Here, the odds refer to the odds of a woman being B12 deficient. As with the RQ, the hypotheses could be worded in terms of odds ratios, proportions (or percentages), or relationships.

The data are shown in Table 31.7, and the numerical summary in Table 31.8. Since the RQ is about odds, a side-by-side bar chart is produced (Fig. 31.6) as the graphical summary.

TABLE 31.7: The number of vegetarian and non-vegetarian women who are (and are not) B12 deficient
B12 deficient Not B12 deficient Total
Vegetarians 8 26 34
Non-vegetarians 8 82 90
Total 16 108 124
TABLE 31.8: The odds and percentage of subjects that are B12 deficient
Odds B12 deficient Percentage B12 deficient Sample size
Vegetarians: 0.3077 23.5% 34
Non-vegetarians: 0.0976 8.9% 90
Odds ratio: 3.15

The software output (jamovi: Fig. 31.7; SPSS: Fig. 31.8) shows that the OR (and 95% CI) is $$3.154$$ ($$1.077$$ to $$9.238$$). The chi-square value is $$4.707$$, approximately equivalent to $$z$$-score of

$\sqrt{\frac{4.707}{1}} = 2.17;$ a small $$P$$-value is expected using the 68–95–99.7 rule. The software output shows that the two-tailed $$P$$-value is $$0.030$$, which is indeed ‘small.’

We conclude:

The sample provides moderate evidence ($$\chi^2 = 4.707$$; $$P=0.030$$) that the odds in the population of being vitamin B12 deficient is different for vegetarian women (odds: 0.3077) compared to non-vegetarian women (odds: 0.0976; OR: $$3.2$$; 95% CI: $$1.1$$ to $$9.2$$).

The statistically valid shoud be checked. The jamovi output (Fig. 31.7) shows that the smallest expected count is 4.39. Likewise, the text under the first table of SPSS output in Fig. 31.8 says that

1 cells (25.0%) have expected count less than 5. The minimum expected count is 4.39.

The smallest expected count is smaller than five, so the results may be statistically invalid. Nonetheless, only one cell has an expected count less than five, and only just under 5, so we shouldn’t be too concerned (but it should be noted).

### References

Gammon CS, von Hurst PR, Coad J, Kruger R, Stonehouse W. Vegetarianism, vitamin B12, and insulin resistance in a group of predominately overweight/obese South Africian women. Nutrition. 2012;28:20–4.