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Why is the pseudo-R2 for tobit negative or greater than one?

Title   Pseudo-R2 for tobit
Author William Sribney, StataCorp
Date June 1997

Concerning the pseudo-R2, we use the formula

        pseudo-R2 = 1 − L1/L0 

where L0 and L1 are the constant-only and full model log-likelihoods, respectively.

For discrete distributions, the log likelihood is the log of a probability, so it is always negative (or zero). Thus 0 ≥ L1L0, and so 0 ≤ L1/L0 ≤ 1, and so 0 ≤ pseudo-R2 ≤1 for DISCRETE distributions.

For continuous distributions, the log likelihood is the log of a density. Since density functions can be greater than 1 (cf. the normal density at 0), the log likelihood can be positive or negative. Similarly, mixed continuous/discrete likelihoods like tobit can also have a positive log likelihood.

If L1 > 0 and L0 < 0, then L1/L0 < 0, and 1 − L1/L0 > 1.

If L1 > L0 > 0 and then L1/L0 > 1, and 1 − L1/L0 < 0.

Hence, this formula for pseudo-R2 can give answers > 1 or < 0 for continuous or mixed continuous/discrete likelihoods like tobit. So, it makes no sense.

For many models, including tobit, the pseudo-R2 has no real meaning.

This formula for pseudo-R2 is nothing more than a reworking of the model chi-squared, which is 2(L1L0). Thus even for discrete distributions where 0 ≤ pseudo R2 ≤ 1, it is still better to report the model chi-squared and its p-value—not the pseudo-R2.

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