HLTH 5013 Module 3 Measures of Association Analysis Example

A Risk Ratio of 0.62 That Shrinks to 0.69: Computing and Interpreting Measures of Association for a County Diabetes Prevention Program

Student Name

American College of Education

HLTH5013: Epidemiology and Statistics

Module 3 Assignment

Instructor Name

October 19, 2026

The Data

The previous module recommended a retrospective cohort analysis of the composite county's lifestyle program for adults with prediabetes. Using the regional health information exchange, the county identified 2,400 adults whose A1c was in the prediabetes range in 2022 and who had no diabetes diagnosis. Of these, 620 attended at least four sessions of the lifestyle program in its first year, the group treated here as enrolled, and 1,780 did not. The outcome is a new diagnosis of type 2 diabetes, or an A1c of 6.5% or higher, within three years.

Among the 620 enrolled adults, 62 developed diabetes. Among the 1,780 not enrolled, 285 did. Arranged as a two-by-two table, enrolled adults form the exposed row, with 62 cases and 558 non-cases, and non-enrolled adults form the unexposed row, with 285 cases and 1,495 non-cases.

Risk in Each Group

Because this is a cohort, the three-year risk, or cumulative incidence, can be calculated directly for each group. Among enrolled adults, the risk was 62 divided by 620, or 10.0%. Among non-enrolled adults, it was 285 divided by 1,780, or 16.0%. One in ten enrolled adults developed diabetes within three years, compared with about one in six of those who did not enroll.

Relative and Absolute Measures

The risk ratio divides the risk in the exposed group by the risk in the unexposed group: 10.0% divided by 16.0% equals 0.62. Enrolled adults had about 62% of the risk of non-enrolled adults, or about 38% lower risk. For a protective exposure, 1 minus the risk ratio, here 0.38, is the preventive fraction among the exposed: the share of the risk that enrolled adults would have had, which apparently did not occur.

The risk difference subtracts one risk from the other: 10.0% minus 16.0% equals minus 6.0 percentage points. Turning that difference upside down, 1 divided by 0.060, yields roughly 17, the number needed to treat. In practical terms, if the association were entirely causal, the county would need to enroll about 17 adults with prediabetes to prevent one case of diabetes over three years. The ratio tells the board the program looks powerful; the difference tells it how many people must enroll to see that power once. Both are needed, because a large relative reduction in a rare outcome can mean very few cases prevented.

The Odds Ratio

The odds ratio compares the odds of the outcome rather than the risk. The odds of diabetes among enrolled adults were 62 to 558, or 0.111; among non-enrolled adults, 285 to 1,495, or 0.191. The odds ratio is 0.111 divided by 0.191, or 0.58. It is further from 1 than the risk ratio of 0.62 because the outcome is not rare. Zhang and Yu (1998) showed that the odds ratio exaggerates the relative risk whenever the outcome is frequent among the unexposed, and they proposed a correction using that group's risk. In a cohort, where risks can be calculated directly, the risk ratio is the more natural measure; the odds ratio matters mainly for case-control studies and logistic regression.

Precision and Chance

A 95% confidence interval for the risk ratio can be calculated on the log scale. The natural log of 0.62 is about minus 0.47, and its standard error comes from the four cell counts: take 1/62 minus 1/620 plus 1/285 minus 1/1,780 and find the square root, which is about 0.132. Adding and subtracting 1.96 times the standard error and converting back gives an interval of about 0.48 to 0.81. The interval excludes 1.0, so the association is statistically significant at the 5% level, and it suggests the true risk ratio, if unconfounded, lies somewhere between a 19% and a 52% reduction.

A chi-square test of independence on the same table gives a value of about 13.4 with one degree of freedom, corresponding to a p-value well below .001. Random variation by itself would rarely produce a gap this large. Neither the interval nor the test, however, addresses whether the difference is caused by the program.

Confounding and Adjustment

Grimes and Schulz (2002) caution that an association in observational research may reflect chance, bias or confounding before it reflects cause, and they describe confounding as a third factor associated with both the exposure and the outcome that distorts the apparent relationship. Baseline A1c is an obvious candidate. People nearer the lower end of the prediabetes range are less likely to progress, and they may also be more likely to enroll.

Stratifying by baseline A1c tests this. Among adults with A1c from 5.7% to 6.0%, the risk was 7.0% for the 400 enrolled and 11.0% for the 900 not enrolled, a risk ratio of 0.64. Among adults with A1c from 6.1% to 6.4%, it was 15.5% for the 220 enrolled and 21.1% for the 880 not enrolled, a risk ratio of 0.73. Enrollees were indeed concentrated in the lower stratum, 65% of them compared with 51% of non-enrollees. Combining the strata with the method of Mantel and Haenszel (1959), which weights each stratum's comparison by its size, gives an adjusted risk ratio of about 0.69. Part of the crude association, then, reflected the healthier starting point of those who enrolled; after adjustment, the program is still associated with about 31% lower risk.

Relative Versus Absolute Benefit by Stratum

The stratified results also show why relative and absolute measures can point to different priorities. The risk ratios in the two A1c strata were similar, 0.64 and 0.73, so the program's relative association with lower risk does not appear to differ much by starting A1c; a formal test of whether the stratum ratios differ would likely find no clear difference with these numbers. The absolute picture is different. In the lower stratum, the risk difference was 4.0 percentage points, a number needed to treat of 25. In the higher stratum, it was 5.6 points, a number needed to treat of about 18. Because people with higher starting A1c are at greater baseline risk, a similar relative reduction prevents more cases per person enrolled. For a county with a limited number of program slots, that argues for recruiting actively among adults at the upper end of the prediabetes range, the group that currently enrolls least often. It is an example of how an analysis intended to check for confounding can also inform where a program should focus.

What the Board Should Hear

The board should hear three things. First, adults who enrolled developed diabetes less often: 10% compared with 16% over three years. Second, part of that gap came from who enrolled, and after accounting for baseline A1c, the association remains meaningful at about 31% lower risk, similar in direction to the original trial, though smaller. Third, other differences between enrollees and non-enrollees, such as motivation, could not be measured, so the true effect may be smaller still. The next module will analyze the full data set with additional variables, and the randomized clinic rollout recommended earlier will provide stronger evidence over time.

References

Grimes, D. A., & Schulz, K. F. (2002). Bias and causal associations in observational research. The Lancet, 359(9302), 248-252. https://doi.org/10.1016/S0140-6736(02)07451-2

Mantel, N., & Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22(4), 719-748. https://doi.org/10.1093/jnci/22.4.719

Zhang, J., & Yu, K. F. (1998). What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA, 280(19), 1690-1691. https://doi.org/10.1001/jama.280.19.1690