Let's say we have 10 possibly good predictor variables and one Y variable. We want to make a model for each possible combination of predictors (for example, there would be ten models of just each predictor variable alone, with an intercept). This way, we can see what the best combination of variables is. How many models would we have to create to create every possible model?

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BlackTom AI · Accepted Answer

To determine how many models there are when considering all possible combinations of 10 predictor variables, we use the powerset principle. Each predictor can be either included in a model or not included, giving 2 choices per predictor.

Option 1: 1024. This equals 2^10, which corresponds to all possible subsets of the 10 predictors, including the subset with no predictors (the intercept-only model). Since the statement asks for every possible combination, this count aligns with including the model that has no predictors as well as all non-empty subsets.

Option 2: 512. This is 2^9 and would imply fixing one predictor to always be included or excluded, or otherwise restricting the combinations. It does not represent all possible subsets of 10 predictors.

Option 3: 100. This number does not align with the combinatorial count of subsets of 10 items; there is no standard subset count that yields exactly 100 in this context.

Option 4: 20. This is far too small and does not reflect the exponential growth in the number of predictor subsets as the number of predictors increases, even for modest counts like 10.

In summary, when forming models for all possible combinations of 10 predictor variables, each predictor has two states (in or out), leading to 2^10 total subsets, i.e., 1024 possible models if we include the intercept-only model.

MISM2510.12719.202610 Exam 1

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