## One-way ANOVA with fixed and random effects from a Bayesian perspective

This blog post is derived from a computer practical session that I ran as part of my new course on Statistics for Big Data, previously discussed. This course covered a lot of material very quickly. In particular, I deferred introducing notions of hierarchical modelling until the Bayesian part of the course, where I feel it is more natural and powerful. However, some of the terminology associated with hierarchical statistical modelling probably seems a bit mysterious to those without a strong background in classical statistical modelling, and so this practical session was intended to clear up some potential confusion. I will analyse a simple one-way Analysis of Variance (ANOVA) model from a Bayesian perspective, making sure to highlight the difference between fixed and random effects in a Bayesian context where everything is random, as well as emphasising the associated identifiability issues. R code is used to illustrate the ideas.

### Example scenario

We will consider the body mass index (BMI) of new male undergraduate students at a selection of UK Universities. Let us suppose that our data consist of measurements of (log) BMI for a random sample of 1,000 males at each of 8 Universities. We are interested to know if there are any differences between the Universities. Again, we want to model the process as we would simulate it, so thinking about how we would simulate such data is instructive. We start by assuming that the log BMI is a normal random quantity, and that the variance is common across the Universities in question (this is quite a big assumption, and it is easy to relax). We assume that the mean of this normal distribution is University-specific, but that we do not have strong prior opinions regarding the way in which the Universities differ. That said, we expect that the Universities would not be very different from one another.

### Simulating data

A simple simulation of the data with some plausible parameters can be carried out as follows.

set.seed(1)
Z=matrix(rnorm(1000*8,3.1,0.1),nrow=8)
RE=rnorm(8,0,0.01)
X=t(Z+RE)
colnames(X)=paste("Uni",1:8,sep="")
Data=stack(data.frame(X))
boxplot(exp(values)~ind,data=Data,notch=TRUE)


Make sure that you understand exactly what this code is doing before proceeding. The boxplot showing the simulated data is given below.

### Frequentist analysis

We will start with a frequentist analysis of the data. The model we would like to fit is

$y_{ij} = \mu + \theta_i + \varepsilon_{ij}$

where i is an indicator for the University and j for the individual within a particular University. The “effect”, $\theta_i$ represents how the ith University differs from the overall mean. We know that this model is not actually identifiable when the model parameters are all treated as “fixed effects”, but R will handle this for us.

> mod=lm(values~ind,data=Data)
> summary(mod)

Call:
lm(formula = values ~ ind, data = Data)

Residuals:
Min       1Q   Median       3Q      Max
-0.36846 -0.06778 -0.00069  0.06910  0.38219

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  3.101068   0.003223 962.244  < 2e-16 ***
indUni2     -0.006516   0.004558  -1.430 0.152826
indUni3     -0.017168   0.004558  -3.767 0.000166 ***
indUni4      0.017916   0.004558   3.931 8.53e-05 ***
indUni5     -0.022838   0.004558  -5.011 5.53e-07 ***
indUni6     -0.001651   0.004558  -0.362 0.717143
indUni7      0.007935   0.004558   1.741 0.081707 .
indUni8      0.003373   0.004558   0.740 0.459300
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1019 on 7992 degrees of freedom
Multiple R-squared:  0.01439,	Adjusted R-squared:  0.01353
F-statistic: 16.67 on 7 and 7992 DF,  p-value: < 2.2e-16


We see that R has handled the identifiability problem using “treatment contrasts”, dropping the fixed effect for the first university, so that the intercept actually represents the mean value for the first University, and the effects for the other Univeristies represent the differences from the first University. If we would prefer to impose a sum constraint, then we can switch to sum contrasts with

options(contrasts=rep("contr.sum",2))


and then re-fit the model.

> mods=lm(values~ind,data=Data)
> summary(mods)

Call:
lm(formula = values ~ ind, data = Data)

Residuals:
Min       1Q   Median       3Q      Max
-0.36846 -0.06778 -0.00069  0.06910  0.38219

Coefficients:
Estimate Std. Error  t value Pr(>|t|)
(Intercept)  3.0986991  0.0011394 2719.558  < 2e-16 ***
ind1         0.0023687  0.0030146    0.786 0.432048
ind2        -0.0041477  0.0030146   -1.376 0.168905
ind3        -0.0147997  0.0030146   -4.909 9.32e-07 ***
ind4         0.0202851  0.0030146    6.729 1.83e-11 ***
ind5        -0.0204693  0.0030146   -6.790 1.20e-11 ***
ind6         0.0007175  0.0030146    0.238 0.811889
ind7         0.0103039  0.0030146    3.418 0.000634 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1019 on 7992 degrees of freedom
Multiple R-squared:  0.01439,	Adjusted R-squared:  0.01353
F-statistic: 16.67 on 7 and 7992 DF,  p-value: < 2.2e-16


This has 7 degrees of freedom for the effects, as before, but ensures that the 8 effects sum to precisely zero. This is arguably more interpretable in this case.

### Bayesian analysis

We will now analyse the simulated data from a Bayesian perspective, using JAGS.

#### Fixed effects

All parameters in Bayesian models are uncertain, and therefore random, so there is much confusion regarding the difference between “fixed” and “random” effects in a Bayesian context. For “fixed” effects, our prior captures the idea that we sample the effects independently from a “fixed” (typically vague) prior distribution. We could simply code this up and fit it in JAGS as follows.

require(rjags)
n=dim(X)[1]
p=dim(X)[2]
data=list(X=X,n=n,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (j in 1:p) {
theta[j]~dnorm(0,0.0001)
for (i in 1:n) {
X[i,j]~dnorm(mu+theta[j],tau)
}
}
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)
autocorr.plot(output)
pairs(as.matrix(output))
crosscorr.plot(output)


On running the code we can clearly see that this naive approach leads to high posterior correlation between the mean and the effects, due to the fundamental lack of identifiability of the model. This also leads to MCMC mixing problems, but it is important to understand that this computational issue is conceptually entirely separate from the fundamental statisticial identifiability issue. Even if we could avoid MCMC entirely, the identifiability issue would remain.

A quick fix for the identifiability issue is to use “treatment contrasts”, just as for the frequentist model. We can implement that as follows.

data=list(X=X,n=n,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (j in 1:p) {
for (i in 1:n) {
X[i,j]~dnorm(mu+theta[j],tau)
}
}
theta[1]<-0
for (j in 2:p) {
theta[j]~dnorm(0,0.0001)
}
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)
autocorr.plot(output)
pairs(as.matrix(output))
crosscorr.plot(output)


Running this we see that the model now works perfectly well, mixes nicely, and gives sensible inferences for the treatment effects.

Another source of confusion for models of this type is data formating and indexing in JAGS models. For our balanced data there was not problem passing in data to JAGS as a matrix and specifying the model using nested loops. However, for unbalanced designs this is not necessarily so convenient, and so then it can be helpful to specify the model based on two-column data, as we would use for fitting using lm(). This is illustrated with the following model specification, which is exactly equivalent to the previous model, and should give identical (up to Monte Carlo error) results.

N=n*p
data=list(y=Data$values,g=Data$ind,N=N,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (i in 1:N) {
y[i]~dnorm(mu+theta[g[i]],tau)
}
theta[1]<-0
for (j in 2:p) {
theta[j]~dnorm(0,0.0001)
}
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)


As suggested above, this indexing scheme is much more convenient for unbalanced data, and hence widely used. However, since our data is balanced here, we will revert to the matrix approach for the remainder of the post.

One final thing to consider before moving on to random effects is the sum-contrast model. We can implement this in various ways, but I’ve tried to encode it for maximum clarity below, imposing the sum-to-zero constraint via the final effect.

data=list(X=X,n=n,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (j in 1:p) {
for (i in 1:n) {
X[i,j]~dnorm(mu+theta[j],tau)
}
}
for (j in 1:(p-1)) {
theta[j]~dnorm(0,0.0001)
}
theta[p] <- -sum(theta[1:(p-1)])
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)


Again, this works perfectly well and gives similar results to the frequentist analysis.

#### Random effects

The key difference between fixed and random effects in a Bayesian framework is that random effects are not independent, being drawn from a distribution with parameters which are not fixed. Essentially, there is another level of hierarchy involved in the specification of the random effects. This is best illustrated by example. A random effects model for this problem is given below.

data=list(X=X,n=n,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (j in 1:p) {
theta[j]~dnorm(0,taut)
for (i in 1:n) {
X[i,j]~dnorm(mu+theta[j],tau)
}
}
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
taut~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","taut","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)


The only difference between this and our first naive attempt at a Bayesian fixed effects model is that we have put a gamma prior on the precision of the effect. Note that this model now runs and fits perfectly well, with reasonable mixing, and gives sensible parameter inferences. Although the effects here are not constrained to sum-to-zero, like in the case of sum contrasts for a fixed effects model, the prior encourages shrinkage towards zero, and so the random effect distribution can be thought of as a kind of soft version of a hard sum-to-zero constraint. From a predictive perspective, this model is much more powerful. In particular, using a random effects model, we can make strong predictions for unobserved groups (eg. a ninth University), with sensible prediction intervals based on our inferred understanding of how similar different universities are. Using a fixed effects model this isn’t really possible. Even for a Bayesian version of a fixed effects model using proper (but vague) priors, prediction intervals for unobserved groups are not really sensible.

Since we have used simulated data here, we can compare the estimated random effects with the true effects generated during the simulation.

> apply(as.matrix(output),2,mean)
mu           tau          taut      theta[1]      theta[2]
3.098813e+00  9.627110e+01  7.015976e+03  2.086581e-03 -3.935511e-03
theta[3]      theta[4]      theta[5]      theta[6]      theta[7]
-1.389099e-02  1.881528e-02 -1.921854e-02  5.640306e-04  9.529532e-03
theta[8]
5.227518e-03
> RE
[1]  0.002637034 -0.008294518 -0.014616348  0.016839902 -0.015443243
[6] -0.001908871  0.010162117  0.005471262


We see that the Bayesian random effects model has done an excellent job of estimation. If we wished, we could relax the assumption of common variance across the groups by making tau a vector indexed by j, though there is not much point in persuing this here, since we know that the groups do all have the same variance.

#### Strong subjective priors

The above is the usual story regarding fixed and random effects in Bayesian inference. I hope this is reasonably clear, so really I should quit while I’m ahead… However, the issues are really a bit more subtle than I’ve suggested. The inferred precision of the random effects was around 7,000, so now lets re-run the original, naive, “fixed effects” model with a strong subjective Bayesian prior on the distribution of the effects.

data=list(X=X,n=n,p=p)
init=list(mu=2,tau=1)
modelstring="
model {
for (j in 1:p) {
theta[j]~dnorm(0,7000)
for (i in 1:n) {
X[i,j]~dnorm(mu+theta[j],tau)
}
}
mu~dnorm(0,0.0001)
tau~dgamma(1,0.0001)
}
"
model=jags.model(textConnection(modelstring),data=data,inits=init)
update(model,n.iter=1000)
output=coda.samples(model=model,variable.names=c("mu","tau","theta"),n.iter=100000,thin=10)
print(summary(output))
plot(output)


This model also runs perfectly well and gives sensible inferences, despite the fact that the effects are iid from a fixed distribution and there is no hard constraint on the effects. Similarly, we can make sensible predictions, together with appropriate prediction intervals, for an unobserved group. So it isn’t so much the fact that the effects are coupled via an extra level of hierarchy that makes things work. It’s really the fact that the effects are sensibly distributed and not just sampled directly from a vague prior. So for “real” subjective Bayesians the line between fixed and random effects is actually very blurred indeed…

## Introduction to the particle Gibbs sampler

### Introduction

Particle MCMC (the use of approximate SMC proposals within exact MCMC algorithms) is arguably one of the most important developments in computational Bayesian inference of the 21st Century. The key concepts underlying these methods are described in a famously impenetrable “read paper” by Andrieu et al (2010). Probably the most generally useful method outlined in that paper is the particle marginal Metropolis-Hastings (PMMH) algorithm that I have described previously – that post is required preparatory reading for this one.

In this post I want to discuss some of the other topics covered in the pMCMC paper, leading up to a description of the particle Gibbs sampler. The basic particle Gibbs algorithm is arguably less powerful than PMMH for a few reasons, some of which I will elaborate on. But there is still a lot of active research concerning particle Gibbs-type algorithms, which are attempting to address some of the deficiencies of the basic approach. Clearly, in order to understand and appreciate the recent developments it is first necessary to understand the basic principles, and so that is what I will concentrate on here. I’ll then finish with some pointers to more recent work in this area.

### PIMH

I will adopt the same approach and notation as for my post on the PMMH algorithm, using a simple bootstrap particle filter for a state space model as the SMC proposal. It is simplest to understand particle Gibbs first in the context of known static parameters, and so it is helpful to first reconsider the special case of the PMMH algorithm where there are no unknown parameters and only the state path, $x$ of the process is being updated. That is, we target $p(x|y)$ (for known, fixed, $\theta$) rather than $p(\theta,x|y)$. This special case is known as the particle independent Metropolis-Hastings (PIMH) sampler.

Here we envisage proposing a new path $x_{0:T}^\star$ using a bootstrap filter, and then accepting the proposal with probability $\min\{1,A\}$, where $A$ is the Metropolis-Hastings ratio

$\displaystyle A = \frac{\hat{p}(y_{1:T})^\star}{\hat{p}(y_{1:T})},$

where $\hat{p}(y_{1:T})^\star$ is the bootstrap filter’s estimate of marginal likelihood for the new path, and $\hat{p}(y_{1:T})$ is the estimate associated with the current path. Again using notation from the previous post it is clear that this ratio targets a distribution on the joint space of all simulated random variables proportional to

$\displaystyle \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

and that in this case the marginal distribution of the accepted path is exactly $p(x_{0:T}|y_{1:T})$. Again, be sure to see the previous post for the explanation.

### Conditional SMC update

So far we have just recapped the previous post in the case of known parameters, but it gives us insight in how to proceed. A general issue with Metropolis independence samplers in high dimensions is that they often exhibit “sticky” behaviour, whereby an unusually “good” accepted path is hard to displace. This motivates consideration of a block-Gibbs-style algorithm where updates are used that are always accepted. It is clear that simply running a bootstrap filter will target the particle filter distribution

$\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

and so the marginal distribution of the accepted path will be the approximate $\hat{p}(x_{0:T}|y_{1:T})$ rather than the exact conditional distribution $p(x_{0:T}|y_{1:T})$. However, we know from consideration of the PIMH algorithm that what we really want to do is target the slightly modified distribution proportional to

$\displaystyle \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$,

as this will lead to accepted paths with the exact marginal distribution. For the PIMH this modification is achieved using a Metropolis-Hastings correction, but we now try to avoid this by instead conditioning on the previously accepted path. For this target the accepted paths have exactly the required marginal distribution, so we now write the target as the product of the marginal for the current path times a conditional for all of the remaining variables.

$\displaystyle \frac{p(x_{0:T}^k|y_{1:T})}{M^T} \times \frac{M^T}{p(x_{0:T}^k|y_{1:T})} \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

where in addition to the correct marginal for $x$ we assume iid uniform ancestor indices. The important thing to note here is that the conditional distribution of the remaining variables simplifies to

$\displaystyle \frac{\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})} {\displaystyle p(x_0^{b_0^k})\left[\prod_{t=0}^{T-1} \pi_t^{b_t^k}p\left(x_{t+1}^{b_{t+1}^k}|x_t^{b_t^k}\right)\right]}$.

The terms in the denominator are precisely the terms in the numerator corresponding to the current path, and hence “cancel out” the current path terms in the numerator. It is therefore clear that we can sample directly from this conditional distribution by running a bootstrap particle filter that includes the current path and which leaves the current path fixed. This is the conditional SMC (CSMC) update, which here is just a conditional bootstrap particle filter update. It is clear from the form of the conditional density how this filter must be constructed, but for completeness it is described below.

The bootstrap filter is run conditional on one trajectory. This is usually the trajectory sampled at the last run of the particle filter. The idea is that you do not sample new state or ancestor values for that one trajectory. Note that this guarantees that the conditioned on trajectory survives the filter right through to the final sweep of the filter at which point a new trajectory is picked from the current selection of $M$ paths, of which the conditioned-on trajectory is one.

Let $x_{1:T} = (x_1^{b_1},x_2^{b_2},\ldots,x_T^{b_T})$ be the path that is to be conditioned on, with ancestral lineage $b_{1:T}$. Then, for $k\not= b_1$, sample $x_0^k \sim p(x_0)$ and set $\pi_0^k=1/M$. Now suppose that at time $t$ we have a weighted sample from $p(x_t|y_{1:t})$. First resample by sampling $a_t^k\sim \mathcal{F}(a_t^k|\boldsymbol{\pi}_t),\ \forall k\not= b_t$. Next sample $x_{t+1}^k\sim p(x_{t+1}^k|x_t^{a_t^k}),\ \forall k\not=b_t$. Then for all $k$ set $w_{t+1}^k=p(y_{t+1}|x_{t+1}^k)$ and normalise with $\pi_{t+1}^k=w_{t+1}^k/\sum_{i=1}^M w_{t+1}^i$. Propagate this weighted set of particles to the next time point. At time $T$ select a single trajectory by sampling $k'\sim \mathcal{F}(k'|\boldsymbol{\pi}_T)$.

This defines a block Gibbs sampler which updates $2(M-1)T+1$ of the $2MT+1$ random variables in the augmented state space at each iteration. Since the block of variables to be updated is random, this defines an ergodic sampler for $M\geq2$ particles, and we have explained why the marginal distribution of the selected trajectory is the exact conditional distribution.

Before going on to consider the introduction of unknown parameters, it is worth considering the limitations of this method. One of the main motivations for considering a Gibbs-style update was concern about the “stickiness” of a Metropolis independence sampler. However, it is clear that conditional SMC updates also have the potential to stick. For a large number of time points, particle filter genealogies coalesce, or degenerate, to a single path. Since here we are conditioning on the current path, if there is coalescence, it is guaranteed to be to the previous path. So although the conditional SMC updates are always accepted, it is likely that much of the new path will be identical to the previous path, which is just another kind of “sticking” of the sampler. This problem with conditional SMC and particle Gibbs more generally is well recognised, and quite a bit of recent research activity in this area is directed at alleviating this sticking problem. The most obvious strategy to use is “backward sampling” (Godsill et al, 2004), which has been used in this context by Lindsten and Schon (2012), Whiteley et al (2010), and Chopin and Singh (2013), among others. Another related idea is “ancestor sampling” (Lindsten et al, 2014), which can be done in a single forward pass. Both of these techniques work well, but both rely on the tractability of the transition kernel of the state space model, which can be problematic in certain applications.

### Particle Gibbs sampling

As we are working in the context of Gibbs-style updates, the introduction of static parameters, $\theta$, into the problem is relatively straightforward. It turns out to be correct to do the obvious thing, which is to alternate between sampling $\theta$ given $y$ and the currently sampled path, $x$, and sampling a new path using a conditional SMC update, conditional on the previous path in addition to $\theta$ and $y$. Although this is the obvious thing to do, understanding exactly why it works is a little delicate, due to the augmented state space and conditional SMC update. However, it is reasonably clear that this strategy defines a “collapsed Gibbs sampler” (Lui, 1994), and so actually everything is fine. This particular collapsed Gibbs sampler is relatively easy to understand as a marginal sampler which integrates out the augmented variables, but then nevertheless samples the augmented variables at each iteration conditional on everything else.

Note that the Gibbs update of $\theta$ may be problematic in the context of a state space model with intractable transition kernel.

In a subsequent post I’ll show how to code up the particle Gibbs and other pMCMC algorithms in a reasonably efficient way.