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A recent method for sensitivity analysis is the E-value (VanderWeele and Ping, 2017). Yet, I'm still struggling with the interpretation of such a value.

Coming from outside of epidemiology, where risk ratios are rarely used, I struggle with interpreting what an association measure on the risk ratio scale between unobserved confounders and both the treatment and outcome means, and I would like to translate this using odds-ratio.

Let's say we look at the effect of the number of siblings one has on one's high school graduation. We get an odds ratio of 0.76, meaning that an additional sibling reduces the odds of graduating by 24%. The corresponding E-value is 1.56.

In the hypothetical case where we have only one binary unobserved confounders, $Z = \{0,1\}$; is it correct to say that individuals $Z = 1$ must be at least 56% more likely (or unlikely) than individuals with $Z=0$ to graduate and to have an additional sibling?

If not, what would be an interpretation formulated in similar terms?

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    $\begingroup$ It would be helpful to know what are the things that are 'hardly understandable'. Do you know what is a risk ratio? Do you know what is an association measure? Do you understand the concept of unobserved confounders? $\endgroup$
    – Kuku
    Nov 22, 2021 at 16:54
  • $\begingroup$ Edited. Hope it makes more sense now! I think the biggest problem is the interpretation of a risk ratio. $\endgroup$
    – Maël
    Nov 27, 2021 at 11:47

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If you do not have "real" rates, i.e. a ratio of events per unit time, then you cannot form a rate ratio. A rate ratio would be one rate, i.e. 23 deaths/100,00 population/year divided by another rate, say 10 deaths/100,000 population/year. So the rate ratio would be 2.3 and its interpretation would be that the first population have 2.3 times the yearly risk of death compared to the second population. Poisson regression is an available option for such a data situation.

If you have no time of exposure or time of observation information to analyze, the usual fallback is to use the ratio of events to number of individuals exposed or observed in treatment groups of control groups. The odds of an event are easier to analyse statistically with logistic regression than using what might be seen as the more natural choice of analyzing the proportions. The odds are number of events divide by the number of non-events.

The first number could be formed from data like "161 deaths from a population of 700,000 in one year" or "322 death from the same population over 2 years". Similarly the second rate could be derived from data that accumulated "5 deaths from a population of 50,000 in one year". Obviously there are assumptions of uniformity in risk over time and within populations that might need examination.

Your hypothesized data is really only susceptible to analysis of differences in proportions or odds. Logistic regression lets you estimate log-odds and log-odds ratios from the original counts. Beta regression lets you analyze ratios, i.e., proportions.

The fact that the article is behind a paywall is a barrier to my further comment on it.

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  • $\begingroup$ This is question is about interpreting the E-value, but your answer seems to be about modeling rates and converting model results to rate ratios. You don't need to change your model at all or compute any risk ratios to be able to interpret the E-value. $\endgroup$
    – Noah
    Dec 8, 2023 at 16:34
  • $\begingroup$ The question appeared to be about translating to risk ratios from an undefined and novel terminology documented behind a paywall. The data situation described did not support calculating a rate ratio. It did not sound to me that the questioner had any analysis in hand. $\endgroup$
    – DWin
    Dec 8, 2023 at 23:54
  • $\begingroup$ @Noah As I read the online documentation from TJ Vanderweele's publications, it appeared that pretty much any of the commonly used risk measures (rate ratios, odds ratios, hazard ratios) could be used, but I don't understand why there would be no need to calculate a risk ratio. It also appeared that this would only apply to situations where some sort of regression analysis had been performed and you had some measures of the contribution of confounders to the estimate $\endgroup$
    – DWin
    Dec 9, 2023 at 0:10

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