Some non-statisticians may have difficulties in interpreting the RD, RR, PAR, or AFe, but may prefer measures, such as the EIN, CIN, or ECIN. Thus, impact numbers may help to communicate study results. Furthermore, the calculation of confidence intervals for impact numbers can add to the interpretation of study results by providing a measure of estimation uncertainty. This is important because estimated impact numbers may be used by policy makers in decision-making procedures in health care.
We considered the situation of prospective cohort studies and we used standard methods which are adequate for large sample sizes. For example, we chose an interval estimator using Wald's test statistic proposed by Walter to show the principle of calculation confidence intervals for PAR [3
]. There exist more methods to calculate confidence intervals for the PAR, for instance Walter [15
] proposed formulas for estimating the variance of the PAR for different study designs. These methods are used in a web page presented by Buchan for point and interval estimation of PAR and RR [17
]. Lui [18
] compared 5 methods to calculate confidence intervals for PAR and presented an overview of the adequacy of these methods in different situations, for instance varying sample sizes, varying exposure effects, or varying exposure probabilities. The use of one of these alternative methods for interval estimation of PAR should be considered in dependence on the actual study design.
We considered the situation of prospective cohort studies without confounders. The basic principle of inverting and exchanging the confidence limits of the standard effect measures is also applicable to studies investigating confounders or other designs such as case-control studies, so long as adequate methods for adjusted point and interval estimation of RD, PAR, and AFe are available.
We assumed a fixed follow-up time, no persons lost to follow-up, and no censoring. In the case of varying follow-up times, more complicated methods based upon survival time techniques have to be developed.
The following limitation of impact numbers should be considered. It may be difficult for users to understand positive and negative values of effect measures. In the case of the risk difference it is possible to switch between ARI and ARR. However, negative results for PAR and AFe
are not useful in practice. Thus, in the case of protective exposures, alternative effect measures such as the preventable fraction are applied in practice [7
]. This procedure leads to easily interpretable point estimators in practice but does not solve the problem of difficulties with confidence intervals. In the case of statistically non-significant results, the lower confidence limits for ARI, PAR, and AFe
would be negative. As the point of the zero effect of these three parameters is zero, the "point" of the zero effect for the corresponding impact numbers is infinity. Thus, the confidence intervals for statistically non-significant impact numbers consist of two regions, which is hard to understand for users. This issue created a lot of discussion with respect to the presentation of confidence intervals for NNTs. The most satisfactory solution seems to be the proposal of Altman who introduced the additional terminology "number needed to treat for one person to benefit" (NNTB) and "number needed to treat for one person to be harmed" (NNTH) [19
]. By using this terminology, confidence intervals for statistically non-significant NNTs can be presented as, e.g. "NNTB = 10 (NNTB 4 to ∞ to NNTH 20)", which clearly indicates that the estimation uncertainty is so large that both benefit and harm is compatible with the considered data. This approach was also used for NNEs in epidemiological studies [6
]. As EIN is equivalent to NNE, in principle, the same approach is applicable to EINs. The only difficulty is to find a terminology describing benefit and harm for EINs in an intuitive way.
Unfortunately, the approach of extending the name of the effect measure to distinguish between benefit and harm is not applicable to PAR and AFe. As the domain for both measures in the case of protective exposures is the interval ]-∞, 0 [and in the case of harmful exposures the interval ]0, 1[, the scales describing benefit and harm are different. We consider example 2 for illustration of the problem. If the total sample size of the study would be N = 1534 rather than N = 15337, the effect of smoking would be not significant at the 5% level in the resulting 2 × 2 table. With the same risks for stroke as in Table , 42 cases in 1052 smokers and 15 cases in 482 never-smokers are expected. In this table, for example, the result for PAR would be 0.16 with 95% confidence interval of [-0.20, 0.52]. By using formula (4.14) the result CIN = 6.1 with 95% confidence interval of [1.9, -5.0] would be obtained. It is important to know that not the values between -5 and 1.9 form the confidence interval for CIN, but the values between 1.9 and ∞ and the values between -∞ and -5. The confidence limits have the following meaning. It is compatible with the observed data that among 2 persons with stroke 1 case is attributable to smoking (harmful exposure) as well as that for each group of 5 persons with stroke 1 additional case will occur if smoking is eliminated from the population (protective exposure). Therefore, the results are interpretable, but the easiness of the impact number is lost. Mathematically, the impact numbers provide no other information than the corresponding classical epidemiological effect measures. The impact numbers are just the reciprocals of the epidemiological effect measures and describe the exposure effect in terms of whole numbers rather than percentages. In the case of statistically non-significant study results, the interpretation of the impact numbers is difficult and therefore the goal of presenting the study results in an intuitive way is not reached. Thus, we recommend to use the impact numbers for the presentation of study results in public health research only in the case of studies showing statistically significant exposure effects.
In the situation of statistically non-significant study results, just the absolute and relative frequencies should be presented complemented by point and interval estimates of a relation effect measure, which can be interpreted easily in all situations, e.g. the risk ratio. The impact numbers are only useful in the situation of significant exposure effects where it is helpful to describe the effect in different ways.