# Can you compare two models with the same dataset but use DV vs LNDV?

**URL:** <https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151>\
**Category:** Concepts\
**Created:** [March 7, 2025, 9:44pm UTC](https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151 "2025-03-07T21:44:18Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![engie](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.pumas.ai/engie/32/531_2.png) [@engie](https://discourse.pumas.ai/u/engie)\
**Post date:** [March 7, 2025, 9:44pm UTC](https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151/1 "2025-03-07T21:44:18Z")

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This might be a very dumb question, but is it valid to compare a 2 compartment model using DV as an input versus a TMDD model using LNDV? As in, are the delta OFV values comparable to determine which one is better? Or would I need to compare the 2c LNDV vs TMDD LNDV or 2c DV vs TMDD DV model?

Thank you!

Engie

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**Author:** ![mohamed82008](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.pumas.ai/mohamed82008/32/127_2.png) [@mohamed82008](https://discourse.pumas.ai/u/mohamed82008)\
**Post date:** [March 8, 2025, 2:11pm UTC](https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151/2 "2025-03-08T14:11:46Z")

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Hi Engie,

You should compare models for the same response variable, either `DV` in both models or `LNDV` in both models (I am assuming that’s the natural logarithm of `DV`). A more detailed explanation is given below for why `DV` and `LNDV` models are not directly comparable.

When `LNDV` is assumed to be normally distributed, this is equivalent to assuming that `DV` is log-normally distributed with a caveat. When sampling/simulating, simulating from `DV ~ LogNormal(..)` is equivalent to simulating from `LNDV ~ Normal(..)` and then taking the `exp` of the output. However, when calculating the log probability of `LNDV` given its normal distribution, this log probability is not equal to the log probability of `DV` given its corresponding log-normal distribution.

Luckily, the difference between the 2 log probabilities is a constant in the parameters and only a function of the data `DV` which is constant during fitting. This means that fitting both models will give you the same parameter estimates at the end, assuming the rest of the model is the same obviously. In Pumas, `FOCE` supports `Normal` and `LogNormal` distributions so you can use either without a significant difference in performance.

However, the log likelihoods of the normal and log-normal models (using `LNDV` and `DV` as observations, respectively), and by extension their AICs and BICs, are not comparable. The following are the exact formulas for:

- The log probability of a single observation `log(x)` given a normal distribution with mean \mu and standard deviation \sigma, and

- \log (\sigma) - \frac{1}{2} \log (2 \pi) - \frac{(\log (x) - \mu)^2}{2 \sigma^2}

- The log probability of `x` directly given the equivalent log-normal distribution with log-scale mean \mu and log-scale standard deviation \sigma.

-\log (x) - \log (\sigma) - \frac{1}{2} \log (2 \pi) - \frac{(\log (x) - \mu)^2}{2 \sigma^2}

So the difference is -\log (x) for each observation. In other words, if you fit your model using `LNDV` and a normal distribution, you need to subtract the sum (across all observations) of `LNDV` from the log likelihood. The result is the log likelihood assuming the log-normal distribution model for `DV`. This means you can then compare it to any other model which models `DV` directly.

Note that the issue is not the normal vs log normal distribution. The issue is the response variable being different. It is fine to compare a normal distribution model for `DV` with a log-normal distribution model for `DV`. But a normal distribution model for `LNDV` is equivalent to a log-normal distribution model for `DV` which is why we need to do the above correction before comparing it to any other model for `DV`. I hope this helps.

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**Author:** ![mohamed82008](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.pumas.ai/mohamed82008/32/127_2.png) [@mohamed82008](https://discourse.pumas.ai/u/mohamed82008)\
**Post date:** [March 8, 2025, 2:14pm UTC](https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151/3 "2025-03-08T14:14:38Z")

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> [@mohamed82008](#):
>
> In Pumas, `FOCE` supports `Normal` and `LogNormal` distributions so you can use either without a significant difference in performance.

What I recommend is, if you can, to just make sure you use the same response variable in both models. If you want to model `LNDV` as a `Normal` distribution, use `DV ~ @. LogNormal(..)` in the derived block in Pumas instead. `FOCE` will still work and you won’t need any error-prone corrections.

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**Author:** ![engie](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.pumas.ai/engie/32/531_2.png) [@engie](https://discourse.pumas.ai/u/engie)\
**Post date:** [March 8, 2025, 8:44pm UTC](https://discourse.pumas.ai/t/can-you-compare-two-models-with-the-same-dataset-but-use-dv-vs-lndv/1151/4 "2025-03-08T20:44:36Z")

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Hi Mohamed, thank you for this very thorough and helpful explanation! I appreciate you explaining things in depth ☺ Your explanation makes perfect sense and the practical recommendation to use the same response variable in both models to make them directly comparable. Thank you!