The Isotoma petraea example

If we put the data we've analyzed before into Hickory, we get the results shown in Table 1 $f$ is the within-population inbreeding coefficient, $F_{is}$, $\theta^{(II)}$ is the Bayesian analog of Weir and Cockerham's $\theta$,⁶ and $G_{st}^B$ is the Bayesian analog of Nei's $G_{st}$.

Table 1: Results from analyzing the Isotoma petraea data with Hickory.

		Credible Interval
Parameter	Mean (s.d.)	2.5%	97.5%
$f$	0.52 (0.10)	0.32	0.70
$\theta^{(II)}$	0.19 (0.12)	0.03	0.50
$G_{st}^B$	0.10 (0.07)	0.02	0.30

Hickory allows you to select from three models when running the data with codominant markers:

1. The full model estimates both $f$ and $\theta^{(II)}$.

2. The f $=$ 0 model constrains $f=0$ and estimates $\theta^{(II)}$.

3. The theta $=$ 0 model constrains $\theta^{(II)} = 0$ and estimates $f$.

As a result, we can use DIC comparisons among the models to determine whether we have evidence for inbreeding within populations ($f=0$ versus $f \ne 0$) or for genetic differentiation among populations ( $\theta^{(II)} = 0$ versus $\theta^{(II)} \ne 0$). If we do that with these data we get the results shown in Table 2.

Table 2: DIC analysis of the Isotoma petraea data.

Model	DIC
Full	331.2
$f=0$	355.4
$\theta^{(II)} = 0$	343.7

The $f=0$ has a much larger DIC than the full model, a difference of more than 20 units. Thus, we have strong evidence for inbreeding in these populations of Isotoma petraea.⁷ The $\theta^{(II)} = 0$ model also has a DIC substantially larger than the DIC for the full model, a difference of more than 10 units. Thus, we also have good evidence for genetic differentiation among these populations.⁸

If we ask Hickory to keep log files of our analyses, we can also compare the estimates of $f$ for the full and $\theta^{(II)} = 0$ models. The posterior means are 0.52 and 0.55, respectively, so they seem pretty similar. But we can do better than that. Since we have the full posterior distribution for $f$ in both models, we can pick points at random from each, take their difference, and construct a 95% credible interval. Doing that we find that the 95% credible interval for $f_{full} - f_{\theta^{(II)}=0}$ is (-0.29, 0.24), meaning we have no evidence that the estimates are different. That may be a little surprising, since we strong evidence that $\theta^{(II)} \ne 0$ in these data, but it's also good news. It means that our estimate of within-population inbreeding is not much affected by the amount of differentiation among populations.⁹ What may be more surprising is that the estimates for $\theta^{(II)}$ with and without inbreeding are also very similar: 0.19 versus 0.22, respectively. Moreover, the 95% credible interval for $\theta^{(II)}_{full} - \theta^{(II)}_{f=0}$ is (-0.38, 0.33). Thus, even though we have strong evidence that $f > 0$, our estimate of $\theta^{(II)}$ is not strongly affected by what we think about $f$ in these data.

Table 3: Comparison of $F_{is}$ and $F_{st}$ estimates calculated in different ways.

Method	$F_{is}$	$F_{st}$
Direct	0.14	0.21	0.32
Nei	0.31	0.24	0.47
Weir & Cockerham	0.54	0.04	0.56
Bayesian	0.52 (0.32, 0.70)	0.19 (0.03, 0.50)

One more thing we can do is to look at the posterior distribution of our parameter estimates and at the sample trace.¹⁰ (Figure 2). The result of an MCMC analysis, like the ones here or the ones in WinBUGS, is a large number of individual points. We can either fit a distribution to those points and display the results (the black lines in the figures), or we can use a non-parametric, kernel density estimate (the blue lines in the figures). The sample traces below show the values the chain took on at each point in the sampling process, and you can see that the values bounced around, which is good.

Figure 2: Posterior densities and traces for parameters of the full model applied to the Isotoma petraea data.

It's also useful to look back and think about the different ways we've used the data from Isotoma petraea (Table 3). Several things become apparent from looking at this table:

• The direct calculation is very misleading. A population that has only one individual sampled carries as much weight in determining $F_{st}$ and $F_{is}$ as populations with samples of 20-30 individuals.

• By failing to account for genetic sampling, Nei's statistics significantly underestimate $F_{is}$, while Weir & Cockerham's estimate is quite close to the Bayesian estimates.

• It's not illustrated here, but when a reasonable number of loci are sampled, say more than 8-10, the Weir & Cockerham estimates and the Bayesian estimates are quite similar. But the Bayesian estimates allow for more convenient comparisons of different estimates, and the credible intervals don't depend either on asymptotic approximations or on bootstrapping across a limited collection of loci. The Bayesian approach can also be extended more easily to complex situations.