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I am attempting to work out probability of sale (binary variable) via a logistic regression to deal with perfect predictors (separation). I am using a firthlogit command in Stata. When I run the command many of my coefficients are greater than 1, how can this be?

Some example coefficients would be: (coefficent, s.e)

WEEKEND |  -2.034167   1.231037
PRIME   |  -1.268926   1.233
FSQ1    |   6.531973   3.790429
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  • $\begingroup$ The coefficients are not the same as the fitted values. It is possible for all coefficients to be greater than $1$ in absolute value, yet all fitted values could lie between $0$ and $1$. For instance, in one observation of your example maybe WEEKEND has the value $1$, PRIME equals $1,$ and FSQ1 equals $0.6,$ whence the fitted response would be $0.62.$ This may provide a little more insight than just pointing out that these coefficients are used to fit the log odds of the probability of a response rather than fitting the probability itself. $\endgroup$
    – whuber ♦
    Commented Jul 30, 2018 at 13:54
  • $\begingroup$ What's the intercept term? It will be easier to explain with the intercept coefficient shown, as well. $\endgroup$
    – Mark White
    Commented Jul 30, 2018 at 13:56
  • $\begingroup$ the intercept is: -0.4733 (s.e. 3.5310) $\endgroup$
    – Tom Witten
    Commented Jul 30, 2018 at 14:02
  • $\begingroup$ i understand now that i am getting logistic rather than logit coefficients, i was confused by the command -firthlogit-. i need to know a way to conveniently convert to logit probabilities in stata. ideally i need coefficients bound between -1 and 1 $\endgroup$
    – Tom Witten
    Commented Jul 30, 2018 at 14:04

1 Answer 1

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firthlogit calculates a logistic regression model, for which the coefficients can take any value between -infinity and infinity, as the relation $$ \frac{log(\mathbb{P}[Y=1])}{1-log(\mathbb{P}[Y=1])} = logit(Y_{1/0}) = \alpha + \beta_1 x_1+ ... + \beta_p x_p + \varepsilon $$ applies here. Applying the logit transformation to the outcome is the same as applying the logarithm to the odds of outcome 1 vs. outcome 0 $Y_{1/0}$. In reverse, you apply the exponential function to your coefficients or intercept to receive the odds of the coefficient. For more detail, look here.

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  • $\begingroup$ Ah thats fantastic- thank you! i don't suppose you would know how to do this directly in stata? $\endgroup$
    – Tom Witten
    Commented Jul 30, 2018 at 13:57
  • $\begingroup$ I do not work with stata. But the function 'exp' or something along the line typically does the trick. $\endgroup$
    – Alex2006
    Commented Jul 30, 2018 at 14:21

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