When selection into a study is based on convenience or poor sampling, or the resulting bias is due to loss to follow-up, we are usually interested in making some inference about a larger population from which we would have liked to be able to sample.

However, sometimes selection bias results when we are trying to study a subgroup within a larger population (e.g., in the obesity paradox example, where we are trying to infer something about people with heart disease). In the latter case, we make slightly different assumptions and get a different bound based on different parameters. This result corresponds to Result 5 in Smith & VanderWeele 2019).

With no additional assumptions, the computed bound for a risk ratio or risk difference will be for general selection bias (Result 1 in Smith & VanderWeele 2019). The options correspond to the additional results.

Unmeasured factor a defining characteristic of selection

• \(S = U\). This occurs when the factor responsible for selection bias is assumed to be common to the entire selected population (Result 2).

Selection associated with increased risk of outcome

• \(P(Y=1\mid A=1,S=1)\big/P(Y=1\mid A=1,S=0)\) and \(P(Y=1\mid A=0,S=1)\big/P(Y=1\mid A=0,S=0)\) are both greater than 1 (Result 3).

Selection associated with decreased risk of outcome

• \(P(Y=1\mid A=1,S=1)\big/P(Y=1\mid A=1,S=0)\) and \(P(Y=1\mid A=0,S=1)\big/P(Y=1\mid A=0,S=0)\) are both less than 1.

The first and second or first and third assumptions may also be combined (Result 4).

Mathematical definitions of all of the possible parameters follow. Only certain parameters will be applicable in any given situation.

For general selection bias:

\[ \text{RR}_{UY |(A = 1)} = \frac{\max_u P(Y = 1 | A = 1, u)}{\min_u P(Y = 1 | A = 1, u)} \] \[ \text{RR}_{UY |(A = 0)} = \frac{\max_u P(Y = 1 | A = 0, u)}{\min_u P(Y = 1 | A = 0, u)} \] \[ \text{RR}_{SU |(A = 1)} = \max_u \frac{P(U = u | A = 1, S = 1)}{P(U = u | A = 1, S = 0)} \] \[ \text{RR}_{SU |(A = 0)} = \max_u \frac{P(U = u | A = 0, S = 0)}{P(U = u | A = 0, S = 1)} \;. \]

These values can be interpreted as the maximum relative risks comparing any two values of \(U\) on \(Y\) within strata of \(A = 1\) and \(A = 0\), respectively; and the maximum factors by which selection increases the prevalence of some value of \(U\) within the stratum \(A = 1\) and by which non-selection increases the relative prevalence of some value of \(U\) within stratum \(A = 0\).

When the selected population is the target population:

\[ \text{RR}_{UY |(S = 1)} = \max_a \frac{\max_u P(Y = 1 | A = a, S = 1, U = u)}{\min_u P(Y = 1 | A = a, S = 1, U = u)} \] \[ \text{RR}_{AU |(S = 1)} = \max_u \frac{P(U = u | A = 1, S = 1)}{P(U = u | A = 0, S = 1)} \;. \]

The first value can be interpreted as the maximum factor by which the unmeasured variable increases risk of the outcome within some exposure group, among the selected population. The second is less intuitive, describing the relationship between the exposure and the unmeasured factor that's induced when conditioning on selection. However, it can generally be interpreted as referring to the extent to which \(U\) increases the probability of selection.

Control selection bias:

\[ \text{RR}_{UA_1} = \frac{\max_u P(A = 1 | Y = 0, u, c)}{\min_u P(A = 1 | Y = 0, u, c)} \] \[ \text{RR}_{UA_0} = \frac{\max_u P(A = 0 | Y = 0, u, c)}{\min_u P(A = 0 | Y = 0, u, c)} \] \[ \text{RR}_{S_1U} =\max_u \frac{P(U = u | Y = 0, S = 1, c)}{P(U = u | Y = 0, S = 0, c)} \] \[ \text{RR}_{S_0U} = \max_u \frac{P(U = u | Y = 0, S = 0, c)}{P(U = u | Y = 0, S = 1, c)} \;. \]

The first two values describe how much more likely controls are to be exposed (or unexposed) given two different values of \(U\). The second two values describe the maximum extent to which some value of \(U\) is more likely among the selected controls compared to the general population of possible controls, and vice versa.

With no additional assumptions, the computed bound for a risk ratio or risk difference will be for general selection bias (Result 1 in Smith & VanderWeele 2019). The options correspond to the additional results.

Unmeasured factor a defining characteristic of selection

• \(S = U\). This occurs when the factor responsible for selection bias is assumed to be common to the entire selected population (Result 2).

Selection associated with increased risk of outcome

• \(P(Y=1\mid A=1,S=1)\big/P(Y=1\mid A=1,S=0)\) and \(P(Y=1\mid A=0,S=1)\big/P(Y=1\mid A=0,S=0)\) are both greater than 1 (Result 3).

Selection associated with decreased risk of outcome

• \(P(Y=1\mid A=1,S=1)\big/P(Y=1\mid A=1,S=0)\) and \(P(Y=1\mid A=0,S=1)\big/P(Y=1\mid A=0,S=0)\) are both less than 1.

The first and second or first and third assumptions may also be combined (Result 4).

Mathematical definitions of all of the possible parameters follow. Only certain parameters will be applicable in any given situation.

For general selection bias:

\[ \text{RR}_{UY |(A = 1)} = \frac{\max_u P(Y = 1 | A = 1, u)}{\min_u P(Y = 1 | A = 1, u)} \] \[ \text{RR}_{UY |(A = 0)} = \frac{\max_u P(Y = 1 | A = 0, u)}{\min_u P(Y = 1 | A = 0, u)} \] \[ \text{RR}_{SU |(A = 1)} = \max_u \frac{P(U = u | A = 1, S = 1)}{P(U = u | A = 1, S = 0)} \] \[ \text{RR}_{SU |(A = 0)} = \max_u \frac{P(U = u | A = 0, S = 0)}{P(U = u | A = 0, S = 1)} \;. \]

These values can be interpreted as the maximum relative risks comparing any two values of \(U\) on \(Y\) within strata of \(A = 1\) and \(A = 0\), respectively; and the maximum factors by which selection increases the prevalence of some value of \(U\) within the stratum \(A = 1\) and by which non-selection increases the relative prevalence of some value of \(U\) within stratum \(A = 0\).

When the selected population is the target population:

\[ \text{RR}_{UY |(S = 1)} = \max_a \frac{\max_u P(Y = 1 | A = a, S = 1, U = u)}{\min_u P(Y = 1 | A = a, S = 1, U = u)} \] \[ \text{RR}_{AU |(S = 1)} = \max_u \frac{P(U = u | A = 1, S = 1)}{P(U = u | A = 0, S = 1)} \;. \]

The first value can be interpreted as the maximum factor by which the unmeasured variable increases risk of the outcome within some exposure group, among the selected population. The second is less intuitive, describing the relationship between the exposure and the unmeasured factor that's induced when conditioning on selection. However, it can generally be interpreted as referring to the extent to which \(U\) increases the probability of selection.

Control selection bias:

\[ \text{RR}_{UA_1} = \frac{\max_u P(A = 1 | Y = 0, u, c)}{\min_u P(A = 1 | Y = 0, u, c)} \] \[ \text{RR}_{UA_0} = \frac{\max_u P(A = 0 | Y = 0, u, c)}{\min_u P(A = 0 | Y = 0, u, c)} \] \[ \text{RR}_{S_1U} =\max_u \frac{P(U = u | Y = 0, S = 1, c)}{P(U = u | Y = 0, S = 0, c)} \] \[ \text{RR}_{S_0U} = \max_u \frac{P(U = u | Y = 0, S = 0, c)}{P(U = u | Y = 0, S = 1, c)} \;. \]

The first two values describe how much more likely controls are to be exposed (or unexposed) given two different values of \(U\). The second two values describe the maximum extent to which some value of \(U\) is more likely among the selected controls compared to the general population of possible controls, and vice versa.

When selection into a study is based on convenience or poor sampling, or the resulting bias is due to loss to follow-up, we are usually interested in making some inference about a larger population from which we would have liked to be able to sample.

However, sometimes selection bias results when we are trying to study a subgroup within a larger population (e.g., in the obesity paradox example, where we are trying to infer something about people with heart disease). In the latter case, we make slightly different assumptions and get a different bound based on different parameters. This result corresponds to Result 5 in Smith & VanderWeele 2019).