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I'm working on a logistic regression with N = 42, which is too small following Hosmer's and Lemeshow's recommendations of at least N = 100. How will this affect my logistic regression and is there a way to correct biases?

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The problem here (if there is one) will be one of overfitting. Whether there is a problem depends on details which you have not given: 1) The distribution of the dependent variable and 2) The number of independent variables.

The usual rule of thumb is "10 cases in the smaller level of the DV for each IV", so, if the dependent variable is divided evenly (21 and 21) then, by that rule, you can have 2 independent variables.

Overfitting means that you have sliced your pie too thin. For an extreme case, imagine that the IV is "ID number" with one ID for each of the 42 subjects. Then you will get a perfect fit!

What to do? There may be some special cases with good solutions, but a general solution would be resampling/permutation test. This is a whole complex subject, though.

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Assuming you have a random population selection, you won't have biased estimates they will be best estimates based on your available data.

Your issue will be the high degree of variability in the estimates - this may be too large for your intended purpose... Unfortunately the best way to improve that is to have a larger sample size.

This question may be helpful: Sample size calculation for univariate logistic regression

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