MigrateTutorial
Tutorial Overview
How to get MIGRATE
Locally, the most recent copy of the software on the server is in the /class/molevol-software/migrate directory, you can either use the program that is already installed on the class server or use these files in the local_distribution directory to install MIGRATE on your own computer. if this fails, download MIGRATE from the Migrate download-website.
Comparison of gene flow models using Bayes Factors with MIGRATE
Most are familiar with the concept of likelihood ratio tests, or Akaike’s Information criterion for model comparison. This tutorial describes how to compare population models using Bayes Factors. These allow comparing nested and un-nested models, without assuming Normality, or large samples. Bayes factors are ratios of marginal likelihoods. In contrast to the maximum likelihood, the marginal likelihood is the integral of the likelihood function over the complete parameter range. MIGRATE can calculate such marginal likelihoods for a particular migration model (Beerli and Palczewski 2010).
Executive Summary
This tutorial steps through all necessary program runs to calculate Bayes factors for comparing different gene flow models. We need to do the following:
- Decide on the models that are interesting for a comparison. The method does not work well for a fishing expedition where one would try to evaluate all possible models; this is only possible only for population models with very few number of populations. It will be possible to enumerate all models for two or three populations but more will be very daunting.
- Run each model through MIGRATE. Use the same prior settings for each of them because the prior distribution has some influence on the Bayes factors. Use the heating menu to allow for at least four chains. The menu supplies a shortcut to specify the temperatures, it is #. It generates temperatures that are spaced in a particular way: they are spaced so that the inverse of the temperature are regularly spaced on the interval 0 to 1. For example, the 4 different chains have temperatures 1.0, 1.5, 3.0, 100,000.0, this results in the spacing 1.0, 0.666, 0.333, and 0.0.
- Compare the marginal likelihood of the different runs and calculate the Bayes factor and calculate the probability for each model.
More Details
The following pages detail all steps using a small example. We use a simulated dataset that was generated using parameters that force a direction of migration from the population Ascona (A) to the population Brisago (B). The Brissago population is larger than the Ascona population and no individual from Brissago ever goes to Ascona, but Brissago receives about 1 migrant every four generation from Ascona. The dataset name is twoswisstowns (if you are not at the workshop then download here) We will evaluate a total of 7 population models.
- a full migration model with two population sizes and two migration rates (from A to B and from B to A);
- a model with two population sizes and one migration rate to Brissago;
- a model with two population sizes and one migration rate to Ascona;
- a model where Ascona and Brissago are part of the same panmictic population.
- a model where Ascona is an ancient city and Brissago was built a new from people who left Ascona; but Brissago is still attractive and people migrate to there ever since.
- a model where Brissago is an ancient city and Ascona was built a new from people who left Brissago; but Ascona is still attractive and people migrate to there ever since.
- a model where Ascona is an ancient city and Brissago was built a new from people who left Ascona and they became hostile towards each other and nobody moves between the cities.
. We know the truth therefore we have some prejudice about the ranking of the models, model 5 should be best, models with the same migration pattern as the truth should work better than those with alternative migration patterns, or those with many parameters.
First we need to figure out how to run the dataset efficiently in MIGRATE. For that we pick a parameter-rich model 1 and experiment with run conditions until we are satisfied that the run converges and delivers posterior distributions that look acceptable. Here are now the detailed instructions how to rank population genetics models for a particular dataset.
Familiarize with MIGRATE [Tutorial Start]
- Make a new directory and download or copy the datafile
#if you are AT THE WORKSHOP use this rsync -avz /class/molevol-software/migrate/migrate_tutorial . cd migrate_tutorial
#if you are NOT at the workshop use this mkdir migrate_lab cd migrate_lab wget http://popgen.sc.fsu.edu/tutorials/BF_migrate_tutorial2/twoswisstowns
- Start the program: the regular distribution comes in two flavors the single cpu processor version called migrate-n and the parallel processing version that runs on cluster or computers with multiple cores is called migrate-n-mpi. We use a new version of MIGRATE: version 4.2.2. On the cluster you should be able to call migrate-n. (In this text I will call the program from now on simply MIGRATE).
We will run the exercise on the server. On the server type
migrate-n
On your laptop you may need to use "./migrate-n", if the program is in the same directory. The main menu will appear, looking like this
[pbeerli@class-02 migrate_lab]$ migrate-n ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + + + POPULATION SIZE, MIGRATION, DIVERGENCE, ASSIGNMENT, HISTORY + + Bayesian inference using the structured coalescent + + + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ PDF output enabled [Letter-size] Version 4.2.2a [July-18-2015] Program started at Tue Jul 21 15:03:15 2015 ============================================================= MAIN MENU ============================================================= D Data type currently set to: DNA sequence model I Input/Output formats and Event reporting P Parameters [start, migration model] S Search strategy W Write a parmfile Q Quit the program To change the settings type the letter for the menu to change Start the program with typing Yes or Y ===>
- Go to the Input/Output formats menu (press I and hit Enter), in the INPUT section change the Datafile name to twoswisstowns, Return to the main menu by typing Y.
- In the Search strategy menu: Change the Number of recorded steps in chain to 1000, and also change the Burn-in for each chain: to 1000. Do not worry about priors or other runtime options for the moment. Return to the main menu.
- Save the changes by using the menu item Write a parmfile. This will write a file named parmfile.
- Now run the program (pressing Y will start the run if you are in the main menu). For this dataset the runtime will be very short. On a modern computer this will take under a minute. If this takes more than 3 minute, something is not set up correctly! On the server this takes about 29.6 seconds (on my macOS 10.10.4: 8.8 seconds).
- The program writes considerable information during the run to the screen, that gives some information about the run. Most interesting are the acceptance ratio for the genealogy and the autocorrelations of the parameter and the genealogy. If the autocorrelation is high and the effective sample size is low (<500) then a longer run may be needed. If the priors boundaries are to tight, then you will see that the values reported are either very close or exactly at the upper prior boundary, in these cases you need to extend the prior range. See prior problems in the output, but for this dataset we will have no such problems.
- Look at the outfile.pdf, you will need to transfer the pdf file to your computer and use preview or acrobat or another PDF viewer. In the outputfile look at the figures labeled Bayesian analysis: Posterior distribution, you see histograms similar to the ones in Figure 1. We expect single peaks where the shading of the histogram shows one dark block in the center (50% credibility set), two light gray bars indicating the extent of the 95% credibility set, and two lighter gray bars indication the 99% credibility set. We expect a histrogram that looks smooth that usually has a single peak, commonly similar to an exponential or normal distribution with a heavy right tail.
- In your investigation of Figure 1 you recognize that the histogram has a few kinks because our run was too short. Now restart MIGRATE and set in the strategy menu the setting for change the number of recorded steps in chain from 1,000 to 10,000. This will lengthen the run by a factor of 10.
The next step would take 4x longer than before run the same way, but we could run this in parallel, by calculating the parameters for the loci in parallel.After saving your change to 10,000 steps, close down the program and restart, but this time instead of
migrate-n parmfile
execute this (these instructions will work on the MBL cluster and may most likely not work on your computer or other clusters).
module load openmpi mpirun -np 6 migrate-n-mpi parmfile
Run this way, it will set up a computer with 6 virtual nodes (one master and 5 workers); the workers will do most of the work and at the end report back to the master who then assembles and prints the results. On the cluster, this step takes 339 seconds (on my mac 149 seconds).
For the MBL tutorial: USE THE PARALLEL VERSION!
Because we want to use the thermodynamic integration method for our marginal likelihood calculations, we need to turn on heating. Once you start MIGRATE with the mpi command above, use the Strategy menu to turn on heating.
MIGRATE will tell what to do next, you will need to enter 4 chains sampling at every tenth (10) interval using the temperature scheme that is suggested with the character #. Save the parmfile, and run. Here the sequence of entries that you will need to enter:
...... 13 Heating: NO 16 Run analysis without data: NO Are the settings correct? (Type Y to go back to the main menu or the number for a menu to change)
===>13
Heating scheme? < NO | YES | STATIC | BOUNDED_ADAPTIVE >
===> Static
Enter the number of different "heated" chains. Minimum is 4
===> 4
Enter the interval between swapping trees Enter 0 (zero) for NO swapping [Current interval is 1]
===> 1
Enter 4 "temperatures" [The coldest temperature, which is the first, has to be 1 For example: 1.0 1.5 3.0 1000000.0] OR give a range of values [linear increase: 1 - 10] [exponential increase: 1 @ 10] or, most lazily, let me suggest a range [simply type a #] @@@@@ For model comparison, the range of temperatures @@@@@ @@@@@ MUST include a very hot chain (>100000.0) @@@@@
===> #
Chain Temperature ------------------------- 4 1000000.00000 3 3.00000 2 1.50000 1 1.00000 Is this correct? [YES or NO]
===> yes
Don't forget to write the parmfile to save the settings. It should give a better posterior distribution histogram and will add a full table of (natural) log marginal likelihoods is shown towards the end of the outfile.pdf, but you may still see double peaks.
With your own data you may want to do another round of refinements, but eventually, by comparing the medians and modes of the parameters in the table and the shape of the histograms you should see a good agreement on similar values, if the modes of the different runs are not within the 50% credibility intervals you certainly need to run longer.
Finally, let's work on the different models
We now start to work on the different models. Remember, we want to run 7 models, and this will take some time, I suggest that you work with your neighbor and use the parallel version.To make sure that we have examples of all different models for the discussion at the end of tonight, start with the model that is equivalent to the class computer name you use, for example I am on class07 and would start with model 7. If you are on class08, class09, class10 pick model 5, 6, or 7, respectively. Once you finished your designated model work on the other ones, the more models the better.
Create directories for all model
I have created a small script that assisst with the creation of all model directories, execute at the prompt:
create_migmodels
It will create the directories and will also copy the parmfile into all these directories because we will need to change each parmfile to accommodate the population model. The parmfile in the directories will be correctly set up, except for the populaiton model. You will also need to copy the data file 'twoswisstowns' into these new directories
Model 1: xxxx
This model allows migration between A and B, the populations are assumed to exist since a very long time. The xxxx in the title is a shortcut for this model it means that we estimate migration rates and population sizes. I use a particular notion to define the model, assume there is a migration matrix (or call it population interaction matrix), on the diagonal we have instructions for the population size, for example, estimate (x or *) or constant (c), there are several other possibilities which you could look up in the manual. The off-diagonal are the immigration rates, a * or 'x' means that this parameter is estimated. This particular model is the default and, in principle doe not need any further input and you can simply run MIGRATE, to report follow now the instructions in section Reporting values.
Model 2: x0xx
This model allows migration from A to B. The populations are assumed to exist since a very long time. The x0xx in the title is a shortcut for this model it means that we estimate an immigration rate into the second population and population sizes. I use a particular notion to define the model, read the instructions about the migration/divergence matrix completely before you start typing. Once you have changed the population model,
exit the parameter menu, write the parmfile, run MIGRATE to report follow now the instructions in section Reporting values.
Model 3: xx0x
This model allows migration from B to A. The populations are assumed to exist since a very long time. The xx0x in the title is a shortcut for this model; it means that we estimate an immigration rate into the first population and population sizes. I use a particular notion to define the model, read the instructions about the migration/divergence matrix completely before you start typing. Once you have changed the population model,
exit the parameter menu, write the parmfile, run MIGRATE to report follow now the instructions in section Reporting values.
Model 4: x
Start MIGRATE and choose the menu Parameter settings. Choose the entry about sampling locations. We want to use the data as if we would have sampled a single population, therefore we need to claim that the two locations Ascona and Brissago belong to the same panmictic population. MIGRATE's default is to assume that every location is a individual population. The dialog (figure on the right gives an example of this interaction with the menu) will ask first how many locations are in the dataset (for our example we have 2). After that, you will need to assign the locations to a population. For this model we need to assign each location to the same population, so location 1 (Ascona) belongs to population 1 and location 2 (Brissago) also belongs to population 1. You need to enter 1 1 (one space one). With multiple populations more complicated settings are possible. Save the parmfile and then run MIGRATE. Follow now the instructions in section Reporting values.
Model 5: x0Bx
This model allows divergence, B splits off from A, with migration from A to B after the split. The first population (A: Ascona) exists for a long time and the population B:Brissago splits off at the time we want to estimate. The x0Dx in the title is a shortcut for this model; it means that we estimate a divergence time and a immigration rate into the second population and estimate also population sizes. I use a particular notion to define the model, read the instructions about the migration/divergence matrix completely before you start typing. Once you have changed the population model,
exit the parameter menu, write the parmfile, run MIGRATE to report follow now the instructions in section Reporting values.
Model 6: xB0x
This model is the mirror image of Model 5. This model allows divergence, A splits off from B, with migration from B to A after the split. For details read model 6 (simply exchange A with B). Read the instructions about the migration/divergence matrix completely before you start typing. Once you have changed the population model, exit the parameter menu, write the parmfile, run MIGRATE to report follow now the instructions in section Reporting values.
Model 7: x0dx
This model allows divergence, B splits off from A. The first population (A: Ascona) exists for a long time and the population B:Brissago splits off at the time we want to estimate. After that split there is no interaction between the populations. This scenario is equivalent to a species split. The x0dx in the title is a shortcut for this model; it means that we estimate a divergence time and estimate also population sizes. I use a particular notion to define the model, read the instructions about the migration/divergence matrix completely before you start typing. Once you have changed the population model, exit the parameter menu, write the parmfile, run MIGRATE to report follow now the instructions in section Reporting values.
Report values
We want to compare models, but the tutorial time is too short to do a thorough job and thus our runs will be too short and the posterior distributions of the parameters may still be in bad shape. For our exercise we are mostly interested in the model selection/ordering using marginal likelihoods.
Come to the front and write down the log marginal likelihood into the spreadsheet (look at the example figure labeled Log-Probability of the data given model (marginal likelihood)) [the figure is from a different dataset, so be not alarmed that the marignal likelihoods are different. You will need to report a number from the row labeled All. There are three columns, report the values for the Bezier approximation column.
Migration/Divergence matrix modifications
- Start MIGRATE, choose the Parameter menu.
Choose the entry labeled Model is set to. MIGRATE will now show a dizzying list of options, don't panic, we will only use a few of them. MIGRATE will ask you how many populations are used: enter 2. For a 2-population model we can have 4 parameters. For example, two population sizes and two migration rates. A * or x means that that particular parameter will be unrestrictedly estimated, a zero (0) means that that particular parameter will not be estimated (is not used). Our goal is to set one of the migration/divergence parameters to 0. MIGRATE needs to know how to treat all connections between the populations. The connection matrix is square so we can label it like it is shown in Table 1.
- MIGRATE asks now that you input each row, this can be done by either specifying * 0 (see the lower subtable of Table1) and then return and then entering the next line * * return (second row in second table), or you can enter the whole matrix as * 0 * *.
- For 'Divergence models instead of the '*' you need to enter a 'd' for divergence without migration, and a 'D' for divergence with migration.
To\From | Ascona | Brissago |
Ascona | Θ | M_{B → A} |
Brissago | M_{A → B} | Θ |
To\From | Ascona | Brissago |
Ascona | * | 0 |
Brissago | * | * |
To\From | Ascona | Brissago |
Ascona | Θ | 0 |
Brissago | Δ_{A → B} | Θ |
To\From | Ascona | Brissago |
Ascona | * | 0 |
Brissago | D | * |
Comparing models
How to calculate Bayes factors? In the Table 3 I summarized all log marginal likelihoods, ln(mL), the Bayes factors are often calculated in very different ways. Here, I report the natural log Bayes factors where
Using the guidelines of Kass and Raftery (1995), values smaller than -2 suggest preference for 'model 2', values larger than 2 suggest preference for 'model 1'. We can use the log marginal likelihoods or the BF to order the models (see column Choice in the Table 3).
We also can calculate the model probability. It is calculated by dividing each marginal likelihood by the sum of the marginal likelihoods of all used models:
Note that for the above formula uses the marginal likelihoods, *not* the *log* marginal likelihoods (which is what the program reports). The calculation of model probabilities from the reported log likelihoods is easy with computer programs that have variable precision (for example Maple or Mathematica). Calculations on a desk calculator often fail, for example the likelihood of model 1 is a remarkable small number because the likelihood is exp(-4803.07)= 1.130323625060 x 10^{-2086}, my emulated HP sci 15C calculator delivers 0.0000. But you can calculate the above quantities using this recipe: (1) find the largest log likelihood (-4795.23), (2) subtract that number form each log likelihood in the list (result: -2.27, 0.0, 2.5, -26.67), (3) exponentiate each element in the new list (result: 0.1033, 1.0, 0.0821, 2.6144 x 10^{-12}) , (4) sum all elements in the list up (0.1033+1.0+0.0821+2.6144 x 10^{-12}), this is the denominator (1.1854). (5) now divide each element in the list by that sum and the numbers will look like the one in table 3 last column.
I have added a little python script that calculates the model probabilities from the text outfiles: If you want to see the results of the model exercise do:
grep " All " */outfile_* | sort -n -k 4,4 | migbf.py
Table 3: Showing log marginal likelihoods for all models tested (and three more, but without the panmictic model) and model probabilities
Model Log(mL) LBF Model-probability ----------------------------------------------------------- 1:xxxx: -9662.42 -23.73 0.0000 2:xBxx: -9661.98 -23.29 0.0000 3:xxBx: -9661.52 -22.83 0.0000 4:xd0x: -9656.51 -17.82 0.0000 5:xB0x: -9649.33 -10.64 0.0000 6:xx0x: -9648.93 -10.24 0.0000 7:x0dx: -9641.77 -3.08 0.0402 8:x0xx: -9641.01 -2.32 0.0859 9:x0Bx: [your model 5] -9638.69 0.00 0.8739 ----------------------------------------------------------- The MacOS file system is capitalization-blind, D and d are the same, for that you see capital B which are placeholders for capital D.
Looking at the model probabilities we can see that the “true” model has considerably higher support than the full model or the model that suggests a wrong direction of gene flow.
Summary of results
The best model (the one with the highest marginal likelihood) is model 5 (custom-migration={*0D*}). if we use the thermodynamic approximation of marginal likelihood. MIGRATE also reports the harmonic mean, but I suggest to ignore it and use thermodynamic integration (as we did) although it will be more costly to run. It was shown several times (for example, Beerli and Palczewski 2010, Xie et al. 2011 ) that the harmonic mean estimator is not a good estimator and may be misleading and prefer the more complex model. The picture below is a sample from the class tutorial done on August 1st 2011.
References
- Beerli, P. and M. Palczewski. 2010. Unified framework to evaluate panmixia and migration direction among multiple sampling locations. Genetics 185: 313–326.
- Kass, R. E. and A. E. Raftery. Bayes factors. 1995. Journal of the American Statistical Association 90(430): 773– 795.
- Xie, W., P. O. Lewis, Y. Fan, L. Kuo, and M.-H. Chen. 2011 Improving marginal likelihood estimation for Bayesian phylogenetic model selection, Systematic Biology, 60: 150–160.