Bounds for selection bias using outcome probabilities

Objectives: Determining the causal relationship between exposure and outcome is the goal of many observational studies. However, the selection of subjects into the study population, either voluntary or involuntary, may result in estimates that suffer from selection bias. To assess the robustness of the estimates as well as the magnitude of the bias, bounds for the bias can be calculated. Previous bounds for selection bias often require the speciﬁcation of unknown relative risks, which might be difficult to provide. Here, alternative bounds based on observed data and unknown outcome probabilities are proposed. These unknown probabilities may be easier to specify than unknown relative risks. Methods: I derive alternative bounds from the deﬁnitions of the causal estimands using the potential outcomes framework, under speciﬁc assumptions. The bounds are expressed using observed data and unobserved out-comeprobabilities.Theboundsarecomparedtopreviouslyreportedboundsinasimulationstudy.Furthermore, a study of perinatal risk factors for type 1 diabetes is provided as a motivating example. Results: I show that the proposed bounds are often informative when the exposure and outcome are sufficiently common, especially for the risk difference in the total population. It is also noted that the proposed bounds can be uninformative when the exposure and outcome are rare. Furthermore, it is noted that previously proposed assumption-free bounds are special cases of the new bounds when the sensitivity parameters are set to their most conservative values. Conclusions: Depending on the data generating process and causal estimand of interest, the proposed bounds can be tighter or wider than the reference bounds. Importantly, in cases with sufficiently common outcome and exposure, the proposed bounds are often informative, especially for the risk difference in the total population. It is also noted that, in some cases, the new bounds can be wider than the reference bounds. However, the proposed bounds based on unobserved probabilities may in some cases be easier to specify than the reference bounds based on unknown relative risks.


Introduction
Selections, either voluntary or involuntary, of subjects in a study can bias estimates of causal effects of an exposure on an outcome.Bias due to these selections is commonly referred to as selection bias and can impact the estimates for both the subgroup under examination and the broader population.In empirical studies, it is often of interest to perform a sensitivity analysis to assess the robustness of the estimates and the magnitude of different types of biases.A frequently used method is to calculate bounds for the bias, and several suggestions have been made for selection bias (for example [1][2][3][4][5]).
Bounds for the selection bias can be derived based on the observed data [5].These bounds do not require any additional assumptions nor the specification of any unobserved sensitivity parameters.However, they can be non-informative meaning that they can sometimes be too large to give any useful information about the size of the selection bias.Alternatively, less conservative bounds can be derived under specific assumptions and require the user to provide plausible values on unobserved sensitivity parameters [1][2][3][4].Sensitivity parameters are often some type of relative risk based on unobserved variables and can be difficult to specify.
Here, I suggest alternative bounds for selection bias, following the same principle as [6] does for bounds for unmeasured confounding by combining both data and sensitivity parameters.These bounds are derived under the same assumptions as [4] (here referred to as SV) and are based both on the data and sensitivity parameters that the user must specify.The sensitivity parameters are probabilities instead of relative risks, and can therefore be easier to specify for the user since probabilities can only take values between zero and one as opposed to relative risks which can be any positive value.Additionally, the assumption-free (AF) bounds in [5] are obtained as a special case of the bounds when the most conservative sensitivity parameters are used.It is worth noting that the new bounds for the total population reduce to an improved version of the AF bounds when the most conservative sensitivity parameters are used.
The remainder of this article is outlined as follows.First, the model and notation are described.Secondly and thirdly, the bounds for the causal estimands in the selected subpopulation and total population are derived.Next, a simulation study is performed, and an empirical example is presented.Lastly, the paper is summarized.

Notation and estimands
The bounds are derived in the framework of the Neyman-Rubin causal model [7,8].Two potential outcomes, Y 1 and Y 0 , are defined for each subject.Additionally, each subject is either exposed, E = 1, or unexposed, E = 0, and has a set of confounders, X. Inclusion into the study population is indicated with a binary selection variable, S, which is equal to 1 if the subject is included in the study population and 0 otherwise.The selection variable can be comprised of several inclusion criteria.The observed data is (E, X, Y|S = 1), where the observed outcome, Y, is assumed to be equal to the potential outcome under the exposure, Y = EY 1 − (1 − E)Y 0 .Furthermore, it is assumed that there is no unmeasured confounding in the total population (Y e ⫫ E | X, e = 0, 1), overlap ( < P(E = 1|X) < 1 − , for some  > 0), and that conditional independence between Y e and E does not hold in the subpopulation (Y e ⫫ ∕E | X, S = 1, e = 0, 1).For simplicity, it is assumed that the analysis is conditioned on X throughout, and thus X is suppressed in all the formulas.
The causal estimands considered are the relative risk and the risk difference in both the subpopulation and total population, However, what is observed is the estimate of the observational relative risk or risk difference, In the presence of selection bias, the causal estimands and the observational estimands are not the same.In the following sections, I derive bounds for the causal estimands which can be compared to the observational estimand.

Derivation of the bounds
Bounds for the causal estimands are found by first deriving bounds for the probabilities of the potential outcomes.The potential outcomes can be deconstructed [9] as , where e = 0, 1, and all probabilities except for the counterfactual probabilities P(Y e = 1|E = 1 − e, S = 1) are observed from the data.Following a similar procedure as [6]; these unobserved probabilities can be bounded if, for an unmeasured variable(s), U, the independence property Y e ⊥ ⊥ E|U, S = 1 ( 7 ) holds.Since the interest lies in the subpopulation, i.e. conditional on S = 1, there is no need for a conditional independence assumption that excludes the selection variable.There are several causal structures for which this independence property holds, e.g., the two structures in Figure 1A and B The full derivation of these inequalities is presented in Appendix A. As a result of the above inequality, the potential outcome probabilities can be bounded from below as (8) and from above as where and are sensitivity parameters.With these bounds on the potential outcome probabilities, the relative risk, RR S , can be bounded as with the lower bound defined as and the upper bound Similarly, the risk difference, RD S , can be bounded as with and These bounds are referred to as the generalized assumption-free (GAF) bounds for the subpopulation.This name is chosen since the bounds reduce to the AF bounds in [5] when the sensitivity parameters are set to their most conservative values, i.e., m S = 0 and M S = 1.However, it is noted that the GAF bounds are not assumptionfree when m S > 0 or M S < 1.

Properties of the bounds
Below follows a discussion of the properties of the derived bounds.First, I define the feasible regions for the sensitivity parameters and bounds.Secondly, I discuss the non-sharpness of the bounds and lastly, I present a different bound for cases when an unmeasured variable U is not assumed.

Feasible regions
For the GAF bounds to be valid, the sensitivity parameters must be chosen so that they are in their feasible regions.Firstly, since m S and M S are probabilities, they must be between 0 and 1, and m S < M S , from their definitions.Additionally, the sensitivity parameters are further restricted by the observed data since (15) Furthermore, the intervals in ( 12) and (13) always cover the null effect, i.e., LB S < 1 < UB S and LB ′ S < 0 < UB ′ S , see Appendix A for details.

Discussion on sharpness of the bounds
A bound is sharp if it can be equal to the causal estimand.Therefore, for sharpness to hold, it must be possible for the sensitivity parameters m S =min e,u P(Y=1|E=e, U=u, S=1) and M S =max e,u P(Y=1|E=e, U=u, S=1) to be equal to P(Y e = 1|E = 1 − e, S = 1), e = 0, 1.From the derivation of the bounds, the counterfactual probabilities can be written as Let u min be the value of U such that P(Y=1|E=e, U=u min , S=1)=min u P(Y=1|E=e, U=u, S=1).The right-hand side of the above equation can then be written as ∑

Counterfactual bounds
The GAF bounds in (12) and ( 13) rely on the assumption of an unmeasured variable, U, fulfilling Y e ⫫ E|U, S = 1.
However, if no such variable exists, e.g., if there is a causal relation between Y and S (as exemplified in Figure 1C), the GAF bounds are not valid.As an alternative, the counterfactual probabilities P(Y e = 1|E = 1 − e, S = 1) can instead be used in the sensitivity analysis.As opposed of providing guesses on the causal estimands right away, guessing on the counterfactual probabilities exploits some observed data in the calculation of the bounds, which can reduce the uncertainty in the result.The sensitivity parameters for the bounds based on the counterfactual probabilities are m *

Bounds for the causal estimands in the total population Derivation of the bounds
Bounds for the causal estimands in the total population are found in a similar way as the bounds for the causal estimands in the subpopulation.Under the assumption of no unmeasured confounding in the total population, the probabilities P(Y e = 1) = P(Y = 1|E = e) and they can be decomposed as where e = 0, 1.The probabilities P(S = 1|E = e) cannot be calculated unless data is available on the nonselected part of the population as well.However, P(S = 1|E = e) = P(E = e|S = 1)P(S = 1)∕P(E = e) ≥ P(E = e|S = 1)P(S = 1) and the bounds are constructed such that P(S = 1|E = e) can be substituted with P(E = e|S = 1)P(S = 1).To bound the probabilities of the potential outcomes, the proportion of selected subjects, P(S = 1), must be known.It is noted that P(S = 1) is not the probability of being included in the sample, but the probability of having the characteristics such that the subject is selected.In some situations, this probability is known, e.g., when the number of dropouts in a study is known or when the selection variable is a disease with a known prevalence, but in other situations, the user might have to provide an estimate.In the expression above, holds is assumed, e.g. Figure 1A and B, it implies that The full derivation of these inequalities is presented in Appendix A. Thus, the potential outcome probabilities are bounded from below as and from above as and are sensitivity parameters.The unobserved sensitivity parameters M T and m T are inserted in the bounds instead of unobserved probabilities.However, it can aid the reasoning about the values of the sensitivity parameters when the variable U is taken into account.
With these bounds on the potential outcome probabilities, the relative risk, RR T , can be bounded as with the lower bound defined as and the upper bound Similarly, the risk difference, RD T , can be bounded as These bounds are referred to as the GAF bounds for the total population.

Properties of the bounds
Below follows a discussion of the properties of the derived bounds.First, I define the feasible regions for the sensitivity parameters and bounds.Secondly, I discuss the non-sharpness of the bounds and lastly, I present a different bound for cases when an unmeasured variable U is not assumed.

Feasible regions
The feasible regions for the GAF bounds in the total population are found in a similar way as for the GAF bounds in the subpopulation.First, since M T and m T are probabilities, they must be between 0 and 1, and m T < M T from their definitions.Additionally, the sensitivity parameters are further restricted by the observed data in the same way as the sensitivity parameters for the subpopulation, meaning that the feasible regions for the sensitivity parameters are 0 ≤ m T < min (20) It is noteworthy that the GAF bounds in (17) and (18) reduces to improved versions of the AF bounds in [5] when the sensitivity parameters are set to their most conservative values, i.e., m T = 0 and M T = 1.See Appendix B for more details on the improved AF bounds.Furthermore, the intervals in (17) and (18) always cover the null effect, i.e.LB T < 1 < UB T and LB ′ T < 0 < LB ′ T , see Appendix A for details.

Discussion on sharpness of the bounds
In the GAF bounds in the total population both P(E = 1) = 1 and P(E = 0) = 1, which is logically impossible.This is done to ensure that the bounds can be used even when data is not available on the subjects such that S = 0.
However, setting both the exposure probabilities to 1 results in bounds that are not sharp, i.e. the causal estimand cannot be equal to the GAF bounds.

Counterfactual bounds
The GAF bounds in (17) and (18) rely on the assumption of an unmeasured variable, U, fulfilling Y ⫫ S|E, U.However, if no such variable exists, e.g., if there is a causal relation between Y and S (as exemplified in Figure 1C), the T and m T with m * T in the formulas for LB T , UB T , LB ′ T and UB ′ T , and they are referred to as the CAF bounds to make the notation match the bounds in the subpopulation.These bounds do not necessarily cover the null effect for RR T and RD T , since the data do not restrict the sensitivity parameters.The CAF bounds are not sharp since, similarly to the GAF bounds, P(E = 1) = 1 and P(E = 0) = 1, which is logically impossible.It is worth noting that the CAF bounds are also valid under the assumption of an unmeasured variable U for which relation (16) holds.Furthermore, the improved AF bounds in Appendix B are based only on observed data and can be used.

Simulation study
In this section, the GAF and the (improved) AF bounds are compared to SV's bounds for the risk difference and the relative risk in both the subpopulation and total populations.Note again that the (improved) AF bounds are obtained from the GAF bounds when using the most conservative sensitivity parameters.The causal structure is seen in Figure 1B, and the distributions are generated from the model where expit(x) = 1∕(1 + e −x ) is the inverse logit function.The coefficients for E and U (, ,  and ) are independently drawn from N(0,  2 ), for  = 1 and  = 3.The constants ( 1 ,  2 ,  and ) are then set to obtain the specified marginal probabilities, P(U = 1) = 0.20, 0.50 P(E = 1) = 0.05, 0.20 P(S = 1) = 0.50, 0.80 Thus, there are 32 combinations of probabilities and standard deviations.For each combination, 1,000 distributions are generated.However, only distributions where the observational estimand is larger than the causal estimand are considered since this is a requirement for SV's bounds, and thus around 500 distributions are considered for every combination.It is worth noting that if the observational estimand is smaller than the causal estimand, the exposure can be recoded, but this is not done here.For each valid distribution, the causal estimand, the observational estimand, and the GAF, (improved) AF, and SV lower bounds are calculated.The bounds are evaluated by comparing the proportions, p, when SV's bounds are tighter than the other two bounds, respectively.Since the AF bound is the most conservative case of the GAF bound, the GAF bound is always tighter and corresponding comparisons are not necessary.Furthermore, the absolute mean distance between the causal estimand and the bounds, Δ, is calculated.The selection bias is evaluated on the same scale as the causal estimands.This means that Δ is calculated on a difference scale for RD T and RD S (Δ = |RD − bound|).On the other hand, for RR T and RR S , the bias is instead calculated as the absolute mean difference between the logarithms of the causal estimand and the bounds (Δ = | log RR − log bound|).The bounds are compared both when the true sensitivity parameters and when 15 % more conservative sensitivity parameters are used.The bounds based on 15 % more conservative sensitivity parameters are included since, in practice, the sensitivity parameters are not known, and the user might want to have a safety margin.The simulation is performed using R [10] and the code is available in the Online Supplementary Material.
The comparison of the bounds for the risk difference and relative risk in the subpopulation with true sensitivity parameters and  = 1 are presented in Tables 1 and 2. In this case, SV's bounds are always tighter than the AF bounds (p AF = 1 for both causal estimands) and almost always tighter than the GAF bounds (p GAF ≥ 0.97 for RD S and p GAF ≥ 0.98 for RR S ).This corresponds to the results in [5].When comparing the mean distances between the bounds and the causal estimands, one notices that SV's bounds are tighter for all combinations of probabilities, and even equal to zero for almost all combinations (when rounded to two decimals) for the risk  difference.The results are very similar when the conservative sensitivity parameters are used, as presented in Tables 5 and 6 in Appendix C. The mean distances between the bounds and the causal estimand increase, but SV's bounds are still tighter.Furthermore, the results for  = 3 follow the same pattern but vary more, see the Online Supplementary Material for more details.Table 3 reports the results for the risk difference in the total population with true sensitivity parameters and  = 1.Here, SV's bounds are almost never tighter than the GAF bound (p GAF ≤ 0.02) and often less tight than the AF bound (p AF ≤ 0.10).This corresponds to the results in [5].When comparing the mean distances between the causal risk difference and the bounds, the GAF bound is the tightest.From the definitions of the sensitivity parameters, the GAF lower bound is always smaller than the null effect.Since it is the tightest over all the distributions, and almost always tighter than SV's bound, the other two bounds are also smaller than the null effect.The results are very similar when the conservative sensitivity parameters are used, see Table 7 in Appendix C. Furthermore, the results for  = 3 vary more but follow the same pattern as for  = 1, see the Online Supplementary Material for details.Table 4 reports the results for the relative risk in the total population with true sensitivity parameters and  = 1.In this case, the results are reversed compared to the risk difference in the total population.Specifically, SV's bound is always tighter than the AF bound (p AF = 1), and often tighter than the GAF bound (p GAF ≥ 0.83).
Furthermore, looking at the mean difference between the causal estimand and the bounds, it is seen that SV's bound is much tighter compared to the other two bounds.Again, this is in line with the results in [5].The GAF lower bound is always smaller than the null effect, so the AF bound is also smaller than the null effect.However, since SV's bound is tighter than the GAF bound, it is not necessarily smaller than the null effect.The results are similar when the conservative sensitivity parameters are used, see Table 8 in Appendix C, and the results for  = 3 follow the same pattern, although with a bit more variation, see the Online Supplementary Material for details.

Empirical example
In this section, the GAF bounds for the causal relative risk in the subpopulation are applied to an empirical example, where the effect of preterm birth on type 1 diabetes is investigated.The purpose of this example is to demonstrate how the GAF bounds can be used in practice.The calculations are performed using R and the code is available in the Online Supplementary Material.
A study by [11] investigated the causal effect of preterm birth (E) on type 1 diabetes (Y).It is a case-control study, with data from several Swedish registers.The study population was restricted by three inclusion criteria, Nordic mothers, singleton births, and non-diabetic mothers, which comprise the selection variable, S. It can arguably exist an unmeasured variable, U, which influences both S as well as the outcome (such as a genetic factor), and another unmeasured variable, V, which influences both S and the exposure (such as socioeconomic status), thus making S a collider and introducing selection bias, see Figure 2 [12].Furthermore, it is plausible that conditioning on the genetic factor, there is no causal pathway between S to Y, and the assumption of an unmeasured variable U such that Y e ⫫ E|U, S = 1 is plausible.
Since this is a case-control study, the probabilities P(Y = 1|E = e, S = 1) are not known, and the causal odds ratio is instead estimated.However, the relative risk can be approximated by the odds ratio due to the low prevalence of the outcome.Thus, RR obs = 0.53, and, for the sake of illustration, the outcome probabilities are assumed to be P(Y = 1|E = 1, S = 1) = 0.00013 and P(Y = 1|E = 0, S = 1) = 0.00025 [13].The probabilities of the exposure among the selected are known and equal to P(E = 1|S = 1) = 0.005 and P(E = 0|S = 1) = 0.995.Thus, the GAF bounds for the relative risk in the subpopulation are  where the sensitivity parameters are 0.00025 < M S ≤ 1 and 0 ≤ m S < 0.00013.
Since the maximum value of m S is very small, UB S will be dominated by the value of M S .Here, m S is fixed to 0.000065, which is half the minimum value of the observed outcome probability.The larger sensitivity parameter, M S , is varied because its range is wider, and in Figure 3, I report the GAF bounds for M S ∈ {0.00026, 0.005}.
I restrict the values of M S to be smaller than 0.005 because the upper bound becomes uninformative as M S increases.The lower bound varies between 0.24 and 0.26 for all values of M S and the minimum value of the upper bound is 1.04.Thus, when bounding the causal relative risk from below, it can at most be twice the size compared to the estimated relative risk, assuming the values on the sensitivity parameters.The upper bound is always larger than 1, and thus the causal effect might be the reverse of the estimated effect.
The usefulness of the bounds depends on the purpose of the sensitivity analysis, and if the interest lies in bounding the causal estimand from below, i.e. to put a limit on the strongest protective effect preterm birth can have, the GAF bound is informative and can be used.However, if subject-matter experts estimate M S to be large, and the interest lies in bounding the causal estimand from above, some other type of sensitivity analysis may be better suited.

Conclusions
Here, new bounds for selection bias are proposed.The bounds are constructed using both the observed data distribution and sensitivity parameters based on unobserved probabilities.It is highlighted that sensitivity parameters based on probabilities can be easier to specify than the more commonly used sensitivity parameters based on unobserved relative risks.Two versions of the bound are proposed (the GAF and CAF bounds) for the causal estimands in both the subpopulation and total population, each valid under different assumptions.The GAF bounds require the presence of an unmeasured variable fulfilling specified conditional independence assumptions.The CAF bounds require the user to assume values on counterfactual probabilities or probabilities in the non-selected part of the population, depending on the population of interest.It is worth noting that the GAF bounds in the total and subpopulations always cover the null effect.
I compare the proposed GAF bounds for the relative risk and risk difference in the subpopulation and total population to the previously reported SV bound.I also include a comparison with the (improved) AF bounds, which is obtained as a special case of the GAF and CAF bounds when the most conservative sensitivity parameters are used.For the subpopulation, the SV bound is almost always tighter than the GAF bound, for both the risk difference and the relative risk.For the risk difference in the total population, the GAF bound is almost always tighter than the SV bound.For the relative risk in the total population, the SV bound is often tighter than the GAF bound.Lastly, I apply the GAF bound for the relative risk in the subpopulation on the result of a study on the effects of preterm birth and the development of type-1 diabetes.In this example, the lower GAF bound is informative but the upper bound is non-informative since the exposure (preterm birth) and outcome (type-1 diabetes) are rare.
The bounds proposed here are an addition to the previous literature on sensitivity analysis for selection bias.The GAF bounds are useful in some settings, e.g., for the risk difference in the total population, and in other settings, some other sensitivity analysis might be more appropriate, e.g., when the exposure and/or outcome probabilities are small.Which method should be used in practice depends on the application at hand and which assumptions that can be confidently motivated.

B Improved AF bounds
The AF bounds for the causal estimands in the total population in [5] are not sharp and can be improved by taking the assumption of no unmeasured confounding into account.The maximum value of the probability P(Y 0 = 1), under the assumption of no unmeasured confounding, is This is an improvement compared to P(Y 0 = 1) max = P(E = 1, S = 1) + P(Y = 1, E = 0, S = 1) + 2P(S = 0) in [5]; and should be used instead in the AF bounds.When the assumption of no unmeasured confounding is taken into account for the minimum value of P(Y 1 = 1), the result is = P(Y 1 = 1) min which is the same as in the original article.The probabilities P(Y 1 = 1) max and P(Y 0 = 1) min are found in similar ways.Note that the AF bounds for the total population are not sharp since both P(E = 0) = 1 and P(E = 1) = 1, which is logically impossible.However, by setting these probabilities to 1, the user is not required to provide values on probabilities in the non-observed part of the population.
e = 1|E = 1 − e, S = 1) and M * S = max e P(Y e = 1|E = 1 − e, S = 1) and their feasible regions are 0 ≤ m * S ≤ M * S ≤ 1.These lower and upper bounds for RR S and RD S are found by replacing M S with M * S and m S with m * S in the formulas for LB S , UB S , LB ′ S and UB ′ S , and they are referred to as the counterfactual assumption-free (CAF) bounds to match the notation of the GAF bounds.Note that the strict inequalities change to non-strict.These bounds do not necessarily cover the null effect for RR S and RD S , since the sensitivity parameters are not restricted by the data.The CAF bounds are sharp since the lower bound equals the causal estimand if the correct counterfactual values M * S = P(Y 0 = 1|E = 1, S = 1) and m * S = P(Y 1 = 1|E = 0, S = 1) are used.Similarly, the upper bound is equal to the causal estimand if the correct counterfactual probabilities M * S = P(Y 1 = 1|E = 0, S = 1) and m * S = P(Y 0 = 1|E = 1, S = 1) are used.It is worth noting that the CAF bounds are also valid under the assumption of an unmeasured variable U for which relation (7) holds.
bounds are not valid.As an alternative, the unobserved probabilities P(Y = 1|E = e, S = 0) can instead be used in the sensitivity analysis.The sensitivity parameters for the bounds based on the unobserved probabilities are m * T = min e P(Y = 1|E = e, S = 0) and M * T = max e P(Y = 1|E = e, S = 0) and their feasible regions are 0 ≤ m * T ≤ M * T ≤ 1.These lower and upper bounds for RR T and RD T are found by replacing M T with M *

Figure 2 :
Figure 2: DAG representing the possible causal structure in the preterm birth/type 1 diabetes example.

Figure 3 :
Figure 3: Lower (dashed blue line) and upper (solid green line) GAF bounds and null effect (dotted black line) for the preterm birth/type 1 diabetes example.

Table 1 :
Simulation results for RD S with  = 1 and true sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between RD S and the GAF, the AF, and SV's

Table 2 :
Simulation results for RR S with  = 1 and true sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between the log of RR S and the log of GAF, the AF, and SV's bound.

Table 3 :
Simulation results for RD T with  = 1 and true sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between RD T and the GAF, the AF, and SV's

Table 4 :
Simulation results for RR T with  = 1 and true sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between the log of RR T and the log of GAF, the AF, and SV's bound.

Table 5 :
Simulation results for RD S with  = 1 and 15 % more conservative sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between RD S and the GAF, the AF, and SV's bound.

Table 6 :
Simulation results for RR S with  = 1 and 15 % more conservative sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between the log of RR S and the log of GAF, the AF, and SV's bound.

Table 7 :
Simulation results for RD T with  = 1 and 15 % more conservative sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between RD T and the GAF, the AF, and SV's bound.

Table 8 :
Simulation results for RR T with  = 1 and 15 % more conservative sensitivity parameters.p GAF and p AF are the proportions that SV's bound are tighter than the GAF bound and the AF bound.Δ GAF , Δ AF , and Δ SV are the mean distance between the log of RR T and the log of GAF, the AF, and SV's bound.