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## Interpreting the Output File

The output file is divided into four main sections, some of which will appear only if you ask for them (by setting output to "verbose"):

- ML Estimates of Parameters
- Profile Likelihoods
- User Specified Options
- Run Reports

The output report is best viewed and printed in a monospaced font such as Courier.

### Maximum Likelihood Estimates (MLEs) of Parameters

(or)

### Most Probable Estimates (MPEs) of Parameters

This section presents the actual results: maximum likelihood estimates (MLEs) in a likelihood run or most probable estimates (MPEs) in a bayesian run of whichever parameters were specified. Each evolutionary force (coalescence, migration, recombination, growth, etc.) used in the analysis has its own column or set of columns in the output.

If profiling has been turned on, each parameter will be presented with some information about its possible error. This information is calculated during profiling.

If you specified "percentile" profiling, you will be given approximate confidence intervals around the estimate of each parameter. These are only asymptotically correct, so take them with a grain of salt; in too-short runs, particularly, they are likely to be much too narrow. For the most accurate confidence intervals you will need to run multiple replicates (see the "Search Strategy" article). If you selected "concise" output with percentile profiling, only the 95% confidence intervals will be shown (the .025 and .975 percentiles). If "normal" or "verbose" output was selected, a full range of percentiles will be shown, from .005 to .995.

If you specified "fixed" profiling, no information about the support intervals is given in this section, but can be deduced from the information in the next section (Profile Likelihoods).

#### Theta ("coalescence" force)

The "Theta" data presents estimates of Theta for each population. Within each population an estimate is given for each region, along with an overall estimate combining information from all regions.

While most papers describe Theta as 4Nmu, this is true only for diploids. If you put haploid data into LAMARC (like that from mtDNA) the Theta estimates will be estimates of 2Nf(mu) instead. It is best to think of Theta as "number of heritable copies in population * 2 * mutation rate" since this definition works no matter what the ploidy is. (The "2" comes from the fact that two sequences that have diverged for time T are different by 2 * mu * T mutations, since both diverging lineages accumulate mutations.)

If you have used the "multiple mutation rates" option of the data model, then the mu in Theta is relative to the mean mutation rate across all your categories, weighted by the probabilities of each category. The categories reported in the output will be normalized to have a mean of 1.0. They may therefore appear different from the values you put in.

You can combine regions with different N or different mu, but you must know the relative N or mu of each region or segment, and must inform the program of this. If you believe your mu rates to vary over regions, you can tell LAMARC these rates were drawn from a gamma distribution (see the 'alpha' section, below).

#### r (recombination force)

The "Recombination Rate" information presents estimates of r for each region. There are no per-population estimates; we model only the case where the recombination rate is constant within a phylogenetic tree.

The parameter r is C/mu, with C being the per-site recombination rate and mu the per-site mutation rate. Thus r=1 describes a situation where the risk of recombination at a site is the same as the risk of mutation at that site. (Values of r as high as this will be difficult to estimate, and the program will tend to bog down.)

#### g (growth force)

The "Growth Rate" information presents estimates of the exponential growth or shrinkage rate for each population.

The parameter g shows the relationship between Theta, which is now the estimate of modern-day population size, and population size in the past through the equation Theta(t) = Theta(now) exp(-gt) where t is a time in the past. Positive values of g indicate that the population has been growing, and negative values indicate that it has been shrinking.

The units of t in this equation are mutational units; one unit of time is the expected time for one mutation to occur. To interpret the magnitude of g (in contrast to its sign, which has a straightforward meaning) you will need information about the mutation rate. When such information is unavailable you have two options: (1) use the values of g only to compare among organisms which presumably have the same mutation rate, or (2) consider a range of possible values.

If you have information about the mutation rate mu, you can solve for values of Theta a given number of generations in the past using the relationship:

Theta(T generations in the past) = Theta(now) exp(-gT(mu))

#### M (migration force)

The "Migration Rate" data are more complex, since we estimate the immigration rate into each population from every other population. There is an estimate for each migration rate parameter. For example if there are three populations we present immigration from 1 to 2, from 1 to 3, from 2 to 1, from 2 to 3, from 3 to 1 and from 3 to 2, a total of 6 parameters. If multiple genetic regions are present, then each parameter will have a separate estimate for each region and a joint "overall" estimate involving all the regions together.

The parameter M is m, the per-generation migration rate, divided by mu, the per-site mutation rate. Be careful in comparing results with other studies; there are two common ways to report migration rates, and many studies use 4Nm (where N is the population size of the receiving population) instead. To convert the M value into 4Nm, multiply it by the Theta value of the recipient population. For example, to convert a migration rate (M) from population 1 into population 3 to 4Nm, multiply by the Theta of population 3.

Please bear in mind that we always estimate immigration rates--rates at which migrants enter a population. This may seem backwards if one thinks in terms of the fate of individuals, but to a population as a whole the individuals entering it are much more significant to its future than the individuals leaving it.

#### alpha (scaled shape parameter of the best-fit gamma distribution of mutation rate over unlinked regions)

For data collected from multiple unlinked genomic regions, if you enable the gamma "force," you can have LAMARC distribute the unknown relative single-region mutation rates according to the gamma distribution which best fits your data. The general gamma distribution has two parameters, the "shape parameter" alpha (α) and the "scale parameter" beta (β); to avoid overparameterization, LAMARC internally sets β = 1/α so that the mean of the distribution is the product αβ = 1. The value of α that LAMARC estimates is a pure, positive number which best fits the landscape of rate variation among genomic regions in your data set. α = 1 corresponds to exponentially-distributed relative mutation rates; smaller α values imply most of your regions are nearly invariant and one or two are highly variable (data that are completely invariant everywhere would have α = 0). Large values of α imply your regions are mutating at similar relative rates that are approximately distributed according to a normal distribution (data in which each region mutates at exactly the same rate would have an infinite value for α). Some more information is available here.

Because there is very little power available to distinguish between very high values of α, LAMARC might, during the course of its analysis, decide to hold α constant at an arbitrarily high value such as 100, to avoid being pulled too close to infinity, where the likelihood calculations would become invalid.

### Profile Likelihoods

This section gives more detailed information about the possible error of each estimate, and the relationships between the parameters.

A profile likelihood table is a way of visualizing how change in one parameter affects the estimates of the other parameters. For each table, one parameter is set to several interesting values and held constant at those values while all other parameters are maximized. For example, we may hold Theta1 constant at 10x its MLE and see how that affects the best values of the other parameters.

If varying one parameter causes another to vary wildly, the two are correlated. If varying one parameter leaves another nearly constant, the two are uncorrelated. An example of correlated parameters would be the migration rates from North to South America and South to North America. If there are considerable similarities between the North and South American populations, then lowering the N->S migration rate will force a compensating raising of the S->N rate. This would be visible in profile likelihood tables as a marked curve in the S->N rate when the N->S rate is being profiled, and vice versa.

Profiles can be done in two ways, "percentile" or "fixed", but only one of those can be used per force. You can also turn off profiling of any individual parameter, perhaps because you already know it cannot be sensibly estimated.

"Percentile" profiles are estimated at percentiles of the likelihood curve; for example, they may hold Theta1 fixed at the value which could just be rejected at the 95% level, and see what happens to the other parameters. These are the most informative kind of profiles, but they are expensive to calculate because the program must find the correct percentile.

"Fixed" profiles are estimated at multiples of the MLE parameter value; for example, they may hold Theta1 fixed at 1/10 of its MLE value and at 10x its MLE value. These are not always as informative (the chosen values may be well off the edges of the curve) but they are quick to compute.

The "verbose" form of the output report gives profiles for every region as well as for the overall results. Lesser levels of output reporting give only the overall profiles. If you have many parameters and regions, you will probably want to avoid "verbose" as the output can be overwhelming. The "normal" form of the output report gives only the overall profiles. The "concise" form of the output report gives only the overall profiles, and also only calculates the two percentiles that correspond to a 95% support intervals, or, in the case of fixed profiles, only the 1/10X and 10X multipliers.

Profiles are very time-consuming, and if you don't want them it's best to turn them off. Be aware, however, that if you don't do any profiling there will be no confidence-interval information either. (The confidence intervals presented with the MLEs are, in fact, slices through the profile likelihood tables.)

If speed is an issue (and profiling can take up the majority of a LAMARC run), one option is to turn on summary file reading and writing (see Input and Output related tasks). Once a summary file has been written, you can change the profiling options, read it in again, and get mathematically identical results (to a certain degree of precision), but with percentile profiling instead of fixed, or normal output instead of concise.

### User Specified Options

This section lists the settings under which the program was run. It is useful as a record of what you are doing, and to verify that your instructions were interpreted correctly

#### Data summary

This section summarizes details of the input data. If there are multiple linked segments in the data, it provides a table listing all of the segments grouped by region. Each segment shows its type of data and the relative mutation rate used for that segment. The next section is a table showing for each region the number of variable markers found in that region, the relative Ne and mu of the region, and a pairwise estimate of Theta based on the method of Watterson or the FST estimator. It also gives the number of individuals sampled for that region in that population.

This section may provide a useful warning. If the values of the pairwise-estimator Thetas are widely variable and their support intervals do not overlap, you could be combining regions that should not be combined, or you have mistaken (or omitted) your relative Ne or mu values. The per-region estimates are still valid, but the combined estimate should be regarded with suspicion.

You may also see that, for example, it is hopeless to estimate recombination in a region because there are no variable sites.

Following the region summary is a summary of the data model(s) used and their parameters. For DNA, the Felsenstein '84 model reports four base frequencies and the transition/transversion ratio. The GTR model reports the four base frequencies and the six base-base mutation parameters. For microsatellite data, the type of model is listed (Brownian, Stepwise or K-Allele). For Stepwise and K-Allele the number of allele bins is reported, and for the Mixed-KS model the value of the percent_stepwise parameter is also reported.

We also report on the use of multiple mutation rate categories (method of Churchill and Felsenstein). If multiple categories are in use, we report the number of categories, the (normalized) rate and probability of each, and the mean length of autocorrelated regions.

#### Input Genetic Data

If you request normal or verbose output, this section will contain a copy of your input data, formatted like PHYLIP's "interleaved" format with 60 bases per line. This is useful to check that your data are properly aligned. It can also be cut and pasted into other programs.

### Run Reports by Region

If you requested verbose or normal output, the reports normally printed to screen during the program run will be repeated here (even if their screen printing was suppressed). This is useful in diagnosing problems such as too-short chains.

The reports are organized by region, and within regions by chain. They give the following:

"Accepted" indicates the proportion of proposed changes that were accepted. The search is in trouble if this dips below 5% and probably not working if it dips below 1%. Consider heating to remedy this.

"Prior lnL" compares how well the genealogies of this chain fit their ending parameter estimates as opposed to their starting parameter estimates. DO NOT use this number in likelihood ratio tests; it is a relative likelihood and has no meaning outside of context. It is provided only because very high prior lnLs are a symptom of having too few chains, or chains which are too short. As a rule of thumb, by the final chains the prior lnL should be no greater than 2x-3x the number of parameters being estimated.

"Data lnL" indicates the likelihood of the genetic data on the last genealogy in this chain. It is in the same units that DNAMLK from PHYLIP would produce. If the Data lnL is improving rapidly from chain to chain all the way to the end of the program, you are not running the program long enough--it is still finding much better trees than it ever found before.

If you are using the Brownian motion approximation of the microsatellite likelihood methods, "Data lnL" of zero indicates that your population is small enough for the approximation to break down. One zero, in an initial chain, may not be cause for concern but multiple zeros or zeros in the final chains suggest that the Brownian method should not be used.

For more information on fine-tuning your search, see the documentation article on "Search Strategies."

If genealogies were discarded due to too many migrations or recombinations, a line will be printed giving the number of bad genealogies. If this number remains high into the final chains, it is cause for concern.

If more than one arrangement strategy was in use (for example, you were searching over genotype reconstructions) there will be a summary of acceptance rates for each strategy. It is important to keep an eye on these and not simply look at the overall acceptance rates (see "Genotypic Data.").

If multiple temperatures were in use, there will be a table showing the rates of swapping between adjacent temperatures. We feel that optimal swapping rates are between 10% and 40%; if your rates are not in this range you may wish to adjust the number of temperatures or the difference between adjacent temperatures.

When adaptive heating is in use, the temperatures shown are averages over the course of the chain.

Finally, there is a summary of the parameter estimates for this chain.

The output file runtime reports differ from the ongoing runtime reports in that they omit prognosis of the ending time.

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