Case–control design and the 2×2 table
This outbreak investigation is a case–control study because the team started with people who already had the outcome — the
46 cases of acute bloody diarrhea — and compared them with
92 well controls who did not develop illness. There was no complete list of everyone exposed, so a cohort study was not possible. In this design, the measure of association is the
odds ratio (OR), not the relative risk.
The data are arranged as follows:
| Exposure (flavored ice) | Cases (diarrhea) | Controls (well) | Total |
|---|
| Ate the ice | 36 (a) | 23 (b) | 59 |
| Did not eat the ice | 10 (c) | 69 (d) | 79 |
| Total | 46 | 92 | 138 |
Calculating the odds ratio
The odds ratio compares the odds of exposure among cases with the odds of exposure among controls. The formula is:
OR = (a × d) ÷ (b × c)
Substituting the values:
- a × d =
36 ×
69 =
2,484
- b × c =
23 ×
10 =
230
- OR = 2,484 ÷ 230 =
10.8
The same result is obtained by comparing the odds directly. Among cases, the odds of having eaten the ice were
36 ÷
10 =
3.6. Among controls, the odds were
23 ÷
69 =
0.333. Dividing the two odds gives 3.6 ÷ 0.333 =
10.8.
Interpreting the odds ratio
The odds of having eaten the flavored ice were about 11 times higher among cases than among controls. This strongly suggests that the street-vended flavored ice was associated with the outbreak of bloody diarrhea.
An OR of
1.0 would mean no association. An OR greater than
1.0 indicates that exposure is more common among cases, supporting a possible causal link. An OR less than
1.0 would suggest a protective effect.
Key point! In a case–control study, the odds ratio is the correct measure of association because the investigator fixes the number of cases and controls. The relative risk cannot be validly calculated from this design because the true incidence of disease in the exposed and unexposed populations is unknown.
Why the odds ratio is used here
Simon (2001) explains that some study designs permit only the calculation of the odds ratio, not the relative risk. A case–control study is the classic example. The relative risk requires knowing the incidence of disease in exposed and unexposed groups, which requires following a defined cohort forward in time. In this outbreak, there was no complete roster of everyone who bought food from the many different sources, so a cohort could not be assembled. The investigators therefore worked backward from identified cases and used the odds ratio to estimate the strength of the association.
The odds ratio also has the advantage of being
invariant to the labeling of the outcome. Whether the analysis treats “diarrhea” or “no diarrhea” as the outcome of interest, the magnitude of the odds ratio remains the same, which reduces ambiguity in interpretation
[1].
Common calculation errors to avoid
A frequent mistake is misplacing the cells in the 2×2 table. The formula OR = (a × d) ÷ (b × c) depends on correct labeling: a and b must both represent the exposed group, while c and d represent the unexposed group. Reversing cases and controls would invert the result.
Another error is confusing the odds ratio with the relative risk.
Watch out! The relative risk is easier to interpret intuitively, but it cannot be computed from a case–control study. Using the odds ratio formula with case–control data is appropriate; using a relative risk formula would produce a meaningless number
[1].
Rounding to one decimal place, the odds ratio is
10.8, which corresponds to option 4.
References (research sources)