We have shown via a simulation study that the simple product-based estimator (ESR

_{S}) that has been calculated in previous studies only performs well in certain situations. Mainly, those situations are when the exposure probability is low (~5%) and the magnitude of the RR is small (~3.0). There are two reasons for this and they can easily be seen by deconstructing the overall risk of disease using the law of total probability.

Recall that for the product-based estimator of the ESR to be unbiased that what we really need is an estimate of the risk of disease in the unexposed and not the overall risk. When the exposure probability is low, less weight is put on the probability of disease among the exposed. Put this together with a small RR and most of the overall risk of disease is being influenced by those who are unexposed. However, increasing the exposure probability puts more weight on the risk of disease among the exposed, which will give you a much more biased estimate of the risk of disease among the unexposed. We also showed that ESR_{R} provides a substantial improvement over the ESR_{S} in terms of observed relative bias. We found that the observed relative bias of ESR_{R} was near 0% in almost all cases.

Coverage probabilities for the 95% CI for ESR_{S} were inversely related to the observed relative bias of ESR_{S}. As the observed relative bias increased, the coverage probability decreased. The overestimation of the ESR using existing methodology (ESR_{S}) led to 95% CIs that were less likely to cover the true ESR. Also, the expected lengths given coverage for these 95% CIs were usually longer than the lengths produced for ESR_{R} using either the log-based variance or the binomial variance rendering this method of point and interval estimation to be sub-optimal.

Coverage probabilities for the 95% CI for ESR_{R} using a log-based variance exhibited greater than 95% coverage in most cases. The exception was when the sample size for the overall risk was 1,000 and the sample size for the RR was 5,000. Paradoxically, this was the only situation in which the 95% CI of ESR_{R} using a binomial variance exhibited greater than 95% coverage. In terms of expected length given coverage, neither of these two methods of interval estimation of ESR_{R} performed better than the other in all situations. The coverage probability and expected length given coverage depended on the variance estimate that was employed. From equation 11, we can see that the log-based variance of ESR_{R} took into account variability from the overall risk and the RR. We also assumed that the two measures were independent and had a covariance of zero, which is a reasonable assumption because the two measures come from two independent samples. From equation 12, we can see that the binomial variance of ESR_{R} probability of exposure from sample 2 so that the variance would not be under-estimated. However, in most cases the variability still was under-estimated. When the sample sizes were equal, the under-estimation was very little since the coverage probabilities ranged from 87%-95% in most cases. However in Scenario 1, when the sample size combination was 1,000/1,000 and the RR was 1.0, 1.5, and 2.0 the coverage probabilities were 66%, 78%, and 83% respectively.

The four scenarios, which were defined by the combinations of two different exposure probabilities (.05 and.20) and two different probabilities of disease in the unexposed (.02 and.09), did not affect the observed relative bias of ESR_{R}. However, as we increased these two parameters, the observed relative bias of ESR_{S} increased. This phenomenon was also demonstrated when comparing coverage probabilities based on the log-based variance for ESR_{R} and ESR_{S}. When comparing coverage probabilities based on the binomial variance for ESR_{R}, the scenario does matter with larger values of the probability of exposure and/or probability of disease in the unexposed increased coverage probabilities. This is not surprising because the estimate of the binomial variance will increase with increasing exposure probabilities and increasing probability of disease among the unexposed.

Results from our case study most closely resemble scenario two where the magnitude of the RR is 2.0. In scenario two, we assumed an exposure probability of 0.20 and a probability of disease in the unexposed of.02. In our case study the RR was 1.91, the exposure probability (probability of being obese) was 0.371, and the overall risk of disease (symptomatic knee OA) ranged from 0.0087 to 0.0132. While the simulations suggest that the estimator would be biased, the overall risk of disease is small so the difference between the two estimates in absolute terms is not large with the largest over-estimation occurring in those ages 70-79 by 0.71%.

It is likely that the estimates produced by Horsburgh and Stewart et al. were accurate. In the article by Horsburgh et al on tuberculosis, he estimated the ESR of tuberculosis for those with advanced HIV infection; old, healed tuberculosis; and immunosuppressive therapy[1]. While the RR of obtaining a new case of tuberculosis is high for those with advanced HIV infection and old, healed tuberculosis, the probability of exposure is so low for these exposures that the impact of the large RR would be muted. For those with immunosuppressive therapy, the RR of a new case of tuberculosis is modest (2.0) and the probability of exposure is low so the overall probability of disease is a good estimate of the probability of disease among those who are not on immunosuppressive therapy [1]. In the Stewart article, the largest RR is 4.62, but this corresponds to an exposure probability of 0.001. When the exposure probabilities are large enough to possibly impact the estimate of the ESR, the RR is low enough (< 2.0) to offset the possible bias [2].

An article by Cupples et al. calculated risk curves for first-degree relatives of patients with Alzheimer's disease. Their method used the odds ratio instead of the relative risk and included converting probabilities to odds [6]. Our method will allow clinicians and other researchers to find the ESR in one step, provided the summary statistics needed for the calculation (P_{1}(D), RR_{2}, and P_{2}(E)) are available.

We acknowledge that there are limitations with this study. The first is that simulation studies can not be considered a proof. However, we did show mathematically that the proposed estimator of the ESR is unbiased and the results of our simulation confirm this finding. It would be important to show mathematically what the true coverage probabilities are for our 95% CIs across different RRs, exposure probabilities, and probabilities of disease among the unexposed. We also acknowledge that our simulations showed coverage probabilities that well exceed 95% when we are calculating 95% CIs for ESR_{R} using a log-based variance.

We also evaluated the properties of our point and interval estimators when the sample size was small. We observed that one should only consider carrying out these calculations in smaller samples if the prevalence of exposure and disease among the unexposed is sufficiently large. If one of these values is small than the validity of the estimate of the RR may be questionable. Thus, we recommend that investigators using this methodology only use estimates that are of the highest quality.

The implications of our study are substantial. Clinicians can use these estimates to better explain risk of disease to patients. Many times clinicians and patients can misinterpret the meaning of having a certain RR of disease. Interpreting the probability of disease given a certain exposure (the ESR) is much more transparent. Future studies that examine the calculation of ESRs may look at the impact of having the odds ratio (OR) rather than the RR. Also, the consideration of under which study designs and magnitudes of the exposure/disease would an approximation using the OR be valid is an important question to answer. It is likely that the OR would be valid when the prevalence of the outcome is less than 10% but examining this rigorously would be of great importance [7]. Lastly, re-sampling and bootstrapping techniques may be a useful method of obtaining CIs with appropriate coverage.