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 Logistic models with no intercept

When performing logistic regression, it’s quite uncommon to choose a model that lacks an intercept (β0) term, so uncommon that Prism displays a warning to alert you to make sure you made that decision for good reasons.

If you exclude the intercept term from your model, you’re making the assumption that the probability of observing a success (P(Y=1)) is equal to 0.5 when all predictor variables are zero. This is rarely ever a reasonable assumption, and as such, the intercept term is almost always kept in the logistic regression model. Let’s look at the model for logistic regression, and explore some of the math behind this issue:

Ln(Odds) = β0 + β1*X1 + …

If we set the value of Xi to zero in the equation above, it’s reduced to:

Ln(Odds) = β0

Thus, the intercept β0 is equal to the log odds when all other predictors are zero. Excluding the intercept from the model is mathematically equivalent to setting the intercept term equal to zero. If we set eβ0 to zero and exponentiate the equation above, we obtain:

eLn(Odds) = eβ0

Odds = eβ0

Odds = e0

Odds = 1

If we now use the fact that the odds can be written as the ratio of the “probability of Y=1” and the “1 - probability of Y=1”.

Odds = 1

P(Y=1)/[1 - P(Y=1)] = 1

Solving for P(Y=1), we obtain:

P(Y=1) = (1)*[1 - P(Y=1)]

P(Y=1) = 1 - P(Y=1)

2*P(Y=1) = 1

P(Y=1) = 0.5

Thus, by excluding the intercept from the logistic regression model, you have made the assumption that when all predictors are zero, the probability of observing a success is 50%. This assumption is rarely applicable, and so the intercept term is almost always included in the logistic regression model.