3.2 Methodology
3.2.3 Eliminating confounders
The next step is eliminating insignificant confounders that have no interactions in the model. Note that the exposure should never be eliminated even if it has no interactions. Any confounder which was retained as being significantly interacting with other confounders should be retained in the model.
The process of eliminating confounders consists of two sub-processes. The first sub- process deals with OR of the gold-standard model and the reduced model (i.e., after re- moving the insignificant confounder(s)), which should not have discrepancies. The second sub-process involves the CI of the same two models, which also should not have meaningful differences between them. The smaller the differences between the upper and lower CIs, the better the model is. OR and CI are calculated from the exposure and its interactions only in
the two models. Considering our example (see Equation 3.3), X1 and X2 must be retained
in the model because they have significant interactions. X3 and X4 can be removed because
they do not have interactions and it is assumed that they are statistically insignificant. As illustrated in the list below, we have four possible scenarios for the final model.
• Scenario 1: ModelG with no change, the gold standard model.
• Scenario 2: ModelG withoutX3
• Scenario 3: ModelG withoutX4
• Scenario 4: ModelG withoutX3 and X4
The process of eliminating confounders is explained by algorithm 1. In this algorithm, we start by comparing the fourth scenario with the first scenario. OR of the two models are calculated as in Equation 3.4. The results of OR and CIs of all scenarios can be illustrated as in Table 3.2 and Table 3.3. The first test is to compare OR of the model of the fourth scenario (ORXi
3,4) with OR of ModelG (ORGi). If there are no major discrepancies between the first two columns and the last two columns of Table 3.2, then the test is successful and we can go to test the CI. In the CI assessment, we compare betweenCIGi and CIxi
3,4 as in Table 3.3. IfCIXi
3,4 values are either less than or equal toCIGi, then both confounders X3 andX4 can be eliminated; otherwise, we need to move on to the next test in Algorithm 1. The next step is to compare betweenORGi and ORXi
3 and betweenCIGi and CIxi3. If the test in this step fails, we move on to compare between the gold-standard model ModelG and the model after removing X4. In this step, we compare between ORGi and ORXi
4 and between CI i X4 and CIGi. In case all tests fail, we will keep all confounders, and the final model will be the
same as the gold-standard model. Assuming that we did not find any meaningful differences between the first and fourth scenarios and X3 and X4 were eliminated, then we would have
the final model as shown in equation 3.5.
Modelf inal
Y =β1E +β2X1+β3X2+β6E×X1+β7E×X2+β8(X1×X2)
(3.5)
if Gold-standard odds ratios of the Model with X3 and X4 (ORGi) almost the same
as odds ratios of the model after reducing both X3 and X4 (ORXi
3,4) then
if CIX34 ≤ CIG then Eliminate X3 and X4
end
else if Gold-standard odds ratios of the Model with X3 and X4 (ORGi) the
almost same as odds ratios of the model without X3 (ORXi
3) then
if CIX3 ≤ CIG then Eliminate X3
end
else if Gold-standard odds ratios of the Model with X3 and X4 (ORGi)
almost the same as odds ratios of the model without X4 (ORXi
4) then
if CIX4 ≤ CIG then Eliminate X4
end else
Do not eliminate any confounder and Exit end
end end end ;
Table 3.2: Odd ratios test for eliminating confounders X1= 0 X1= 1 X1= 0 X1= 1 X1= 0 X1= 1 X1= 0 X1= 1 X2= 1 ORG1 ORG4 ORX1 3 ORX34 ORX41 ORX44 ORX13,4 ORX34,4 X2= 2 ORG2 ORG5 ORX2 3 ORX35 ORX42 ORX45 ORX23,4 ORX35,4 X2= 3 ORG3 ORG6 ORX3 3 ORX36 ORX43 ORX46 ORX33,4 ORX36,4
• ORGi is the golden-standard odds ratio calculated before eliminating any of the confounders
• ORxi
3 is the odds ratio calculated after eliminating confounderX3
• ORxi
4 is the odds ratio calculated after eliminating confounderX4
• ORXi
3,4 is the odds ratio calculated after eliminating X3 and X4
Table 3.3: Confidence intervals test for eliminating confounders
X1= 0 X1= 1 X1= 0 X1= 1 X1= 0 X1= 1 X1= 0 X1= 1 X2= 1 CIG1 CIG4 CIx1 3 CIx43 CIx14 CIx44 CIx13,4 CIx43,4 X2= 2 CIG2 CIG5 CIx2 3 CIx53 CIx24 CIx45 CIx23,4 CIx53,4 X2= 3 CIG3 CIG6 CIx3 3 CIx63 CIx34 CIx46 CIx33,4 CIx63,4
• CIGi is the confidence intervals before eliminating any of the confounders
• CIxi
3 is the confidence intervals after eliminating confounderX3
• CIxi
4 is the confidence intervals after eliminating confounderX4
• CIxi
3,4 is the confidence intervals after eliminating confoundersX3 andX4