What is the difference between "conditional analysis" and "conditional logistic regression"?
It is also really hard to find out an easy example of that "conditional" means in this case...
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In addition to the comments by Antoine above, a conditional logistic regression is often used when the cases and controls are matched. A conditional logistic regression can account for the fact that there is a dependence between how the patients were selected. Another example of when conditional logistic regression would be helpful is in the context of a dataset that has levels of strata or centers. In this context estimating the association between exposure (within a center or center) and an outcome is the goal.
A conditional analysis can have many meanings. I would need additional information to address this point. You may referring to doing analysis under a condition such as doing an analysis on people above the age of 60, or female patients in a data set. People often refer to these as subgroup analyses, if your study was not originally powered for investigating these associations.