Time unit sensitivity of the birth-death model?

[david....@gmail.com]

Hi everyone,

I have a question about this passage from p. 43 of the PAML manual:

While ideally one would want the biological results to be unchanged when one changes the time unit, we know that two components of the model are not invariant to the time scale: the log normal distribution for rates and the birth-death model for times.

It makes sense to me that the time unit of the birth-death prior has to match the time unit of the tree. If I want the mean of my speciation rate prior to be 0.1 events per lineage per Ma, but my time tree is in units of 100 Ma, the prior mean would need to be specified as "10" (i.e., 10 events per lineage per time unit). But does the passage really only refer to this need for adjustment / unit conversion when it speaks of the lack of invariance to the time unit? If I run the same analysis on the same data, only with a different time unit (and with all priors adjusted accordingly), I would expect the results to be the same (up to the unit conversion factor and some MCMC stochasticity). Is the passage trying to say that this expectation is wrong?

Many thanks in advance,

--

David Černý

[Sandra AC]

Hi David,

Perhaps you may want to take a look at the section "Tutorial 3: Changing the time scale" in the PDF file with the 2017 MCMCtree tutorials that is part of the PAML software (available on the PAML GitHub repo). If you access the first link above and go to pages 10-11, you will find an explanation regarding how the birth-death process is affected when the time unit is changed. The next pages in the PDF file show how the specific settings in the control file also need to change. Please note that now the `finetune` option is deprecated and there is no need to include this in the control file. It is also recommended to stop using the `RootAge` option in the control file and, instead, incorporate the node age constrain on the root in the tree file. Following the example in the tutorial, the tree file when changing the time scale from 100Myr to 1Myr would look like this:

```
7 1
((((human, (chimpanzee, bonobo)) '>6<8', gorilla), (orangutan, sumatran)) '>12<16', gibbon)'<1.0';
```

You can see that the root age is now constrained with an upper-bound calibration of 1 (time unit = 100 Myr). For more details about the notation that you can use to incorporate specific calibrations to constrain node ages, please check pages 49-52 in the PAML documentation (i.e., section "Fossil calibrations"). Hope the explanations in the tutorial and the documentation have the answer you are looking for!

All the best,
S.

[Ziheng]

Hi David

The paml manual is wrong to say that the birth death process was not invariant to a change of time scale.  If you change the time unit the calibrations change, and the rates are changed accordingly and the biological results should remain unchanged, exactly in the way you described.  Sandra has explained how such changes are specified.

I’ll try to remember to edit the manual to fix this.

Best, ziheng

[david....@gmail.com]

Hi Sandra, Ziheng,

Many thanks for your detailed and informative replies! I guess what confused me was in part the fact that the manual also references a paper that did consider the influence of the time unit to be something worth exploring: https://www.sciencedirect.com/science/article/pii/S1055790311002752 (see Suppl. Fig. 8). But in any case, it's great to have it confirmed that as long as all the priors are appropriately rescaled, the choice of the time unit does not affect the inference. Thanks again!

Best,

--

David Černý