# Regarding log-normal specification

I'm working with a log-normal regression model. However, some of the dependent variable equal zero (not missing). Can I use an alternative specification like $log(y+1)$ ~ $X$ (most $y$s are really large)? Or should I just omit those observations? Do I have to do balance check every time I drop some observations and report all the result in appendix?

• Some zeros are not a problem with generalized linear models and logarithmic link. These work on the assumption that the mean is positive and do not use direct logarithmic transformation of the response. Nov 15, 2014 at 12:19
• Unless those observations are clearly erroneous (as determined by independent information), removing them would be mistake. The results would be biased.
– whuber
Nov 15, 2014 at 14:17
• Very similar questions have extensive discussions at stats.stackexchange.com/questions/30728 and stats.stackexchange.com/questions/41361.
– whuber
Nov 15, 2014 at 21:11

The answer to this question really depends on the meaning of $y=0$ in your application. You've said most $y$ are really large -- this means that those $y$'s that are zero are quite different from the other $y$'s. Are they really likely to be generated by the same process that generates the large $y$'s?

If $y=0$ is truly comparable with $y\gg0$ then you could use the approach you suggest of taking $\log(y+1)$. This would equate to a functional form of

$$y_i = e ^ {\beta_0} e^ {\beta_1 x_i,1} ... e^{\beta_n x_i,n}e^{\epsilon_i}+1$$

Note that the errors $\epsilon_i$ are exponential, not additive using this transformation.

If most $y$'s are really large, then $\log(y+1)$ will also be much larger when $y\neq0$, so including those observations for which $y=0$ will increase the error in your regression significantly. The distribution of error terms will be non-normal (and may effectively be a mixture of two different distributions). Coefficient estimates will potentially be biased if the process generating the $y=0$ terms is different to that generating the $y>0$ terms.

You could ignore the cases where $y=0$ entirely, if you considered that the aim of your model is to understand the relationship between the $y$'s and the $x$'s conditional of $y>0$.

Alternatively, and perhaps more satisfactorily if $y=0$ has meaning in your context, you could explicitly model the probability of $y=0$ vs $y>0$ using a binary model (ie a logitistic regression)and then have a sub-model that only considered the case where $y>0$. This would avoid omitting any data points, at the cost of having to consider what explanatory variables are suitable for explaining the probability of $y=0$.

Rob Hyndman's blog has some further discussion on transforming data with zeroes in it.

• Thanks for your answer. I think I will just omit those observations. Actually $y=0$ cases are quite unusual. Maybe there're some mistakes in the records. I also appreciate your solution of running two regressions. This will definitely be a useful method in some other cases. Nov 15, 2014 at 6:22
• Your characterization of leverage is erroneous. You also need to consider the effects of the $\log(y)+1$ transformation on the error terms, which have been neglected.
– whuber
Nov 15, 2014 at 14:21
• @whuber I've removed the erroneous leverage point, and added the error terms to the function. Nov 16, 2014 at 19:30