# Space and SEMs

One question that comes up time and time again when I teach my SEM class is, “What do I do if I have spatially structured data?” Maybe you have data that was sampled on a grid, and you know there are spatial gradients. Maybe your samples are clustered across a landscape. Or at separate sites. A lot of it boils down to worrying about the hidden spatial wee beasties lurk in the background.

I’m going to stop for a moment and suggest that before we go any further you read Brad Hawkins’s excellent Eight (and a half) deadly sins of spatial analysis where he warns of the danger of throwing out the baby with the bathwater. Remember, in any modeling technique, you want to ensure that you’re capturing as much biological signal as is there, and then adjust for remaining spatial correlation. Maybe your drivers vary in a spatial pattern. That’s OK! They’re still your drivers.

That said, ignoring residual spatial autocorrelation essentially causes you to think you have a larger sample size than you think you do (remember the assumption of independent data points) and as such your standard errors are too tight, and you may well produce overconfident results.

To deal with this in a multivariate Structural Equation Modeling context, we have a few options. First, use something like Jon Lefcheck’s excellent piecewiseSEM package and fit your models with mixed model or generalized least squares tools that can accomodate spatial correlation matrices as part of the model. If you have non-spatial information about structure, I’ve started digging into the lavaan.survey package, which has been fun (and is teaching me a lot about survey statistics).

But, what if you just want to go with a model you’ve fit using covariance matrices and maximum likelihood, like you do, using lavaan in R? It should be simple, right?

Well, I’ve kind of tossed this out as a suggestion in the ‘advanced topics’ portion of my class for years, but never implemented it. This year, I got off of my duff, and have been working this up, and have both a solid example, and a function that should make your lives easier – all wrapped up over at github. And I’d love any comments or thoughts on this, as, to be honest, spatial statistics is not where I spend a lot of time. Although I seem to be spending more and more time there these days… silly spatially structured observational datasets…that I seem to keep creating.

Anyway, let’s use as an example the Boreal Vegetation dataset from Zuur et al.’s Mixed Effects Models and Extensions in Ecology with R. The data shows vegetation NDVI from satellite data, as well as a number of other covariates – information on climate (days where the temperature passed some threshold, I believe), wetness, and species richness. And space. Here’s what the data look like, for example:

# Boreality data from http://www.highstat.com/book2.htm
# Mixed Effects Models and Extensions in Ecology with R (2009).
# Zuur, Ieno, Walker, Saveliev and Smith. Springer

#For later
source("./lavSpatialCorrect.R")

#Let's look at the spatial structure
library(ggplot2)

qplot(x, y, data=boreal, size=Wet, color=NDVI) +
theme_bw(base_size=18) +
scale_size_continuous("Index of Wetness", range=c(0,10)) +


So, there are both clear associations of variables, but also a good bit of spatial structure. Ruh roh! Well, maybe it’s all in the drivers. Let’s build a model where NDVI is affected by species richness (nTot), wetness (Wet), and climate (T61) and richness is itself also affected by climate.

library(lavaan)

## This is lavaan 0.5-17
## lavaan is BETA software! Please report any bugs.

# A simple model where NDVI is determined
# by nTot, temperature, and Wetness
# and nTot is related to temperature
borModel <- '
NDVI ~ nTot + T61 + Wet
nTot ~ T61
'

#note meanstructure=T to obtain intercepts
borFit <- sem(borModel, data=boreal, meanstructure=T)


OK, great, we have a fit model – but we fear that the SEs may be too small! Is there any spatial structure in the residuals? Let’s look.

# residuals are key for the analysis
borRes <- as.data.frame(residuals(borFit, "casewise"))

#raw visualization of NDVI residuals
qplot(x, y, data=boreal, color=borRes$NDVI, size=I(5)) + theme_bw(base_size=17) + scale_color_gradient("NDVI Residual", low="blue", high="yellow")  Well…sort of. A clearer way to see this that I like is just to see signs of residuals. #raw visualization of sign of residuals qplot(x, y, data=boreal, color=borRes$NDVI>0, size=I(5)) +
theme_bw(base_size=17) +
scale_color_manual("NDVI Residual >0", values=c("blue", "red"))


OK, we can clearly see the positive residuals clustering on the corners, and negatives ones more prevalent in the middle. Sort of. Are they really? Well, we can correct for them one we know the degree of spatial autocorrelation, Moran’s I. To do this, there are a few steps. First, calculate the spatial weight matrix – essentially, the inverse of the distance between any pair of points. Close points should have a lower weight on the resulting analyses than nearer points.

#Evaluate Spatial Residuals
#First create a distance matrix
library(ape)
distMat <- as.matrix(dist(cbind(boreal$x, boreal$y)))

#invert this matrix for weights
distsInv <- 1/distMat
diag(distsInv) <- 0


OK, that done, we can determine whether there was any spatial autocorrelation in the residuals. Let’s just focus on NDVI.

#calculate Moran's I just for NDVI
mi.ndvi <- Moran.I(borRes$NDVI, distsInv) mi.ndvi ##$observed
## [1] 0.08265236
##
## $expected ## [1] -0.001879699 ## ##$sd
## [1] 0.003985846
##
## $p.value ## [1] 0  Yup, it’s there. We can then use this correlation to calculate a spatially corrected sample size, which will be smaller than our initial sample size. #What is our corrected sample size? n.ndvi <- nrow(boreal)*(1-mi.ndvi$observed)/(1+mi.ndvi$observed)  And given that we can get parameter variances and covariances from the vcov matrix, it’s a snap to calculate new SEs, remembering that the variance of a parameter has the sample size in the denominator. #Where did we get the SE from? sqrt(diag(vcov(borFit))) ## NDVI~nTot NDVI~T61 NDVI~Wet nTot~T61 NDVI~~NDVI ## 1.701878e-04 2.254616e-03 1.322207e-01 5.459496e-01 1.059631e-04 ## nTot~~nTot NDVI~1 nTot~1 ## 6.863893e+00 6.690902e-01 1.617903e+02 #New SE ndvi.var <- diag(vcov(borFit))[1:3] ndvi.se <- sqrt(ndvi.var*nrow(boreal)/n.ndvi) ndvi.se ## NDVI~nTot NDVI~T61 NDVI~Wet ## 0.0001848868 0.0024493462 0.1436405689 #compare to old SE sqrt(diag(vcov(borFit)))[1:3] ## NDVI~nTot NDVI~T61 NDVI~Wet ## 0.0001701878 0.0022546163 0.1322207383  Excellent. From there, it’s a hop, skip, and a jump to calculating a z-score and ensuring that this parameter is still different from zero (or not!) #new z values z <- coef(borFit)[1:3]/ndvi.se 2*pnorm(abs(z), lower.tail=F) ## NDVI~nTot NDVI~T61 NDVI~Wet ## 5.366259e-02 1.517587e-47 3.404230e-194 summary(borFit, standardized=T) ## lavaan (0.5-17) converged normally after 62 iterations ## ## Number of observations 533 ## ## Estimator ML ## Minimum Function Test Statistic 1.091 ## Degrees of freedom 1 ## P-value (Chi-square) 0.296 ## ## Parameter estimates: ## ## Information Expected ## Standard Errors Standard ## ## Estimate Std.err Z-value P(>|z|) Std.lv Std.all ## Regressions: ## NDVI ~ ## nTot -0.000 0.000 -2.096 0.036 -0.000 -0.044 ## T61 -0.035 0.002 -15.736 0.000 -0.035 -0.345 ## Wet -4.270 0.132 -32.295 0.000 -4.270 -0.706 ## nTot ~ ## T61 1.171 0.546 2.144 0.032 1.171 0.092 ## ## Intercepts: ## NDVI 10.870 0.669 16.245 0.000 10.870 125.928 ## nTot -322.937 161.790 -1.996 0.046 -322.937 -30.377 ## ## Variances: ## NDVI 0.002 0.000 0.002 0.232 ## nTot 112.052 6.864 112.052 0.991  See! Just a few simple steps! Easy-peasy! And a few changes – the effect of species richness is no longer so clear, for example OK, I lied. That’s a lot of steps. But, they’re repetative. So, I whipped up a function that should automate this, and produce useful output for each endogenous variable. I need to work on it a bit, and I’m sure issues will come up with latents, composites, etc. But, just keep your eyes peeled on the github for the latest update. lavSpatialCorrect(borFit, boreal$x, boreal$y) ##$Morans_I
## $Morans_I$NDVI
##     observed     expected          sd p.value    n.eff
## 1 0.08265236 -0.001879699 0.003985846       0 451.6189
##
## $Morans_I$nTot
##     observed     expected          sd p.value    n.eff
## 1 0.03853411 -0.001879699 0.003998414       0 493.4468
##
##

#get x
x <- rq_obj$x[,2] #assumes no intercept xx <- seq(min(x, na.rm=T),max(x, na.rm=T),1) #calculate y over all taus f <- coef(rq_obj) yy <- cbind(1,xx)%*%f if(length(lincol)==1) lincol=rep(lincol, length(taus)) #plot all lines for(i in 1:length(taus)){ lines(xx,yy[,i], col=lincol[i], ...) } }  And an example use. data(engel) attach(engel) taus <- c(.05,.1,.25,.75,.9,.95) plot(income,foodexp,xlab="Household Income", ylab="Food Expenditure", pch=19, col=alpha("black", 0.5)) rq_fit <- rq((foodexp)~(income),tau=taus) quantRegLines(rq_fit)  Oh, and I set it up to make pretty colors in plots, too. plot(income, foodexp, xlab = "Household Income", ylab = "Food Expenditure", pch = 19, col = alpha("black", 0.5)) quantRegLines(rq_fit, rainbow(6)) legend(4000, 1000, taus, rainbow(6), title = "Tau")  All of this is in a repo over at github (natch), so, fork and play. # More on Bacteria and Groups Continuing with bacterial group-a-palooza I followed Ed’s suggestions and tried both a binomial distribution and a Poisson distribution for abundance such that the probability of a density of one species s in one group g in one plot r where there are S_g species in group gis $A_rgs ~ Poisson(\frac{A_rg}{S_g})$ In the analysis I’m doing, interesting, the results do change a bit such that the original network only results are confirmed. I am having one funny thing, though, which I can’t lock down. Namely, the no-group option always has the lowest AIC once I include abundances – and this is true both for binomial and Poisson distributions. Not sure what that is about. I’ve put the code for all of this here and made a sample script below. This doesn’t reproduce the behavior, but, still. Not quite sure what this blip is about. For the sample script, we have five species and three possible grouping structures. It looks like this, where red nodes are species or groups and blue nodes are sites: And the data looks like this  low med high 1 2 3 1 1 1 1 50 0 0 2 2 1 1 45 0 0 3 3 2 2 0 100 1 4 4 2 2 0 112 7 5 5 3 2 0 12 110  So, here’s the code: And the results: > aicdf k LLNet LLBinomNet LLPoisNet AICpois AICbinom AICnet low 5 0 0.00000 -20.54409 71.08818 60.00000 30 med 3 0 -18.68966 -23.54655 65.09310 73.37931 18 high 2 0 -253.52264 -170.73361 353.46723 531.04527 12  We see that the two different estimations disagree, with the binomial favorint disaggregation and poisson favoring moderate aggregation. Interesting. Also, the naive network only approach favors complete aggregation. Interesting. Thoughts? # Filtering Out Exogenous Pairs of Variables from a Basis Set Sometimes in an SEM for which you're calculating a test of D-Separation, you want all exogenous variables to covary. If you have a large model with a number of exogenous variables, coding that into your basis set can be a pain, and hence, you can spend a lot of time filtering out elements that aren't part of your basis set, particularly with the ggm library. Here's a solution – a function I'm calling filterExoFromBasiSet  #Takes a basis set list from basiSet in ggm and a vector of variable names filterExoFromBasiSet <- function(set, exo) { pairSet <- t(sapply(set, function(alist) cbind(alist[1], alist[2]))) colA <- which(pairSet[, 1] %in% exo) colB <- which(pairSet[, 2] %in% exo) both <- c(colA, colB) both <- unique(both[which(duplicated(both))]) set[-both] }  How does it work? Let's say we have the following model: y1 <- x1 + x2 Now, we should have no basis set. But… library(ggm) modA <- DAG(y1 ~ x1 + x2) basiSet(modA)  ## [[1]] ## [1] "x2" "x1"  Oops – there's a basis set! Now, instead, let's filter it basisA <- basiSet(modA) filterExoFromBasiSet(basisA, c("x1", "x2"))  ## list()  Yup, we get back an empty list. This function can come in handy. For example, let's say we're testing a model with an exogenous variable that does not connect to an endogenous variable, such as y1 <- x1 x2 (which is exogenous) Now –  modB <- DAG(y ~ x1, x2 ~ x2) basisB <- basiSet(modB) filterExoFromBasiSet(basisB, c("x1", "x2"))  ## [[1]] ## [1] "x2" "y" "x1"  So, we have the correct basis set with only one element. What about if we also have an endogenous variable that has no paths to it?  modC <- DAG(y1 ~ x1, x2 ~ x2, y2 ~ y2) basisC <- basiSet(modC) filterExoFromBasiSet(basisC, c("x1", "x2"))  ## [[1]] ## [1] "y2" "x2" ## ## [[2]] ## [1] "y2" "x1" ## ## [[3]] ## [1] "y2" "y1" "x1" ## ## [[4]] ## [1] "x2" "y1" "x1"  This yields the correct 4 element basis set. # Extracting p-values from different fit R objects Let's say you want to extract a p-value and save it as a variable for future use from a linear or generalized linear model – mixed or non! This is something you might want to do if, say, you were calculating Fisher's C from an equation-level Structural Equation Model. Here's how to extract the effect of a variable from multiple different fit models. We'll start with a data set with x, y, z, and a block effect (we'll see who in a moment).  x <- rep(1:10, 2) y <- rnorm(20, x, 3) block <- c(rep("a", 10), rep("b", 10)) mydata <- data.frame(x = x, y = y, block = block, z = rnorm(20))  Now, how would you extract the p-value for the parameter fit for z from a linear model object? Simply put, use the t-table from the lm object's summary alm <- lm(y ~ x + z, data = mydata) summary(alm)$coefficients

##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   1.1833     1.3496  0.8768 0.392840
## x             0.7416     0.2190  3.3869 0.003506
## z            -0.4021     0.8376 -0.4801 0.637251


# Note that this is a matrix.
# The third row, fourth column is the p value
# you want, so...

p.lm <- summary(alm)$coefficients[3, 4] p.lm  ## [1] 0.6373  That's a linear model, what about a generalized linear model? aglm <- glm(y ~ x + z, data = mydata) summary(aglm)$coefficients

##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   1.1833     1.3496  0.8768 0.392840
## x             0.7416     0.2190  3.3869 0.003506
## z            -0.4021     0.8376 -0.4801 0.637251


# Again, is a matrix.
# The third row, fourth column is the p value you
# want, so...

p.glm <- summary(aglm)$coefficients[3, 4] p.glm  ## [1] 0.6373  That's a linear model, what about a generalized linear model?  anls <- nls(y ~ a * x + b * z, data = mydata, start = list(a = 1, b = 1)) summary(anls)$coefficients

##   Estimate Std. Error t value  Pr(>|t|)
## a   0.9118     0.1007   9.050 4.055e-08
## b  -0.4651     0.8291  -0.561 5.817e-01


# Again, is a matrix.
# The second row, fourth column is the p value you
# want, so...

p.nls <- summary(anls)$coefficients[2, 4] p.nls  ## [1] 0.5817  Great. Now, what if we were running a mixed model? First, let's look at the nlme package. Here, the relevant part of the summary object is the tTable library(nlme) alme <- lme(y ~ x + z, random = ~1 | block, data = mydata) summary(alme)$tTable

##               Value Std.Error DF t-value  p-value
## (Intercept)  1.1833    1.3496 16  0.8768 0.393592
## x            0.7416    0.2190 16  3.3869 0.003763
## z           -0.4021    0.8376 16 -0.4801 0.637630


# Again, is a matrix.
# But now the third row, fifth column is the p value
# you want, so...

p.lme <- summary(alme)$tTable[3, 5] p.lme  ## [1] 0.6376  Last, what about lme4? Now, for a linear lmer object, you cannot get a p value. But, if this is a generalizes linear mixed model, you are good to go (as in Shipley 2009). Let's try that here. library(lme4) almer <- lmer(y ~ x + z + 1 | block, data = mydata) # no p-value! summary(almer)@coefs  ## Estimate Std. Error t value ## (Intercept) 4.792 0.5823 8.231   # but, for a genearlined linear mixed model # and yes, I know this is a # bad model but, you know, demonstration! aglmer <- lmer(y + 5 ~ x + z + (1 | block), data = mydata, family = poisson(link = "log")) summary(aglmer)@coefs  ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 1.90813 0.16542 11.535 8.812e-31 ## x 0.07247 0.02471 2.933 3.362e-03 ## z -0.03193 0.09046 -0.353 7.241e-01   # matrix again! Third row, fourth column p.glmer <- summary(aglmer)@coefs[3, 4] p.glmer  ## [1] 0.7241  # A Quick Note in Weighting with nlme I’ve been doing a lot of meta-analytic things lately. More on that anon. But one quick thing that came up was variance weighting with mixed models in R, and after a few web searches, I wanted to post this, more as a note-to-self and others than anything. Now, in a simple linear model, weighting by variance or sample size is straightforward. #variance lm(y ~ x, data = dat, weights = 1/v) #sample size lm(y ~ x, data = dat, weights = n)  You can use the same sort of weights argument with lmer. But, what about if you’re using nlme? There are reasons to do so. Things change a bit, as nlme uses a wide array of weighting functions for the variance to give it some wonderful flexibility – indeed, it’s a reason to use nlme in the first place! But, for such a simple case, to get the equivalent of the above, here’s the tricky little difference. I’m using gls, generalized least squares, but this should work for lme as well. #variance gls(y ~ x, data=dat, weights = ~v) #sample size gls(y ~ x, data = dat, weights = ~1/n)  OK, end note to self. Thanks to John Griffin for prompting this. # Why I’m Teaching Computational Data Analysis for Biology This is a x-post from the blog I’ve setup for my course blog. As my first class at UMB, I’m teaching An Introduction to Computational Data Analysis for Biology – basically mixing teaching statistics and basic programming. It’s something I’ve thought a long time about teaching – although the rubber meeting the road has been fascinating. As part of the course, I’m adapting an exercise that I learned while taking English courses – in particular from a course on Dante’s Divine Comedy. I ask that students write 1 page weekly to demonstrate that they are having a meaningful interaction with the material. I give them a few pages from this book as a prompt, but really they can write about anything. One student will post on the blog per week (and I’m encouraging them to use the blog for posting other materials as well – we shall see, it’s an experiment). After they post, I hope that it will start a conversation, at least amongst participants in the class. I also think this post might pair well with some of Brian McGill’s comments on statistical machismo to show you a brief sketch of my own evolution as a data analyst. I’ll be honest, I’m excited. I’m excited to be teaching Computational Data Analysis to a fresh crop of graduate students. I’m excited to try and take what I have learned over the past decade of work in science, and share that knowledge. I am excited to share lessons learned and help others benefit from the strange explorations I’ve had into the wild world of data. I’m ready to get beyond the cookbook approach to data. When I began learning data analysis, way back in an undergraduate field course, it was all ANOVA all the time (with brief diversions to regression or ANCOVA). There was some point and click software that made it easy, so long as you knew the right recipe for the shape of your data. The more complex the situation, the more creative you had to be in getting an accurate sample, and then in determining what was the right incantation of sums of squares to get a meaningful test statistic. And woe be it if your p value from your research was 0.051. I think I enjoyed this because it was fitting a puzzle together. That, and I love to cook, so, who doesn’t want to follow a good recipe? Still, there was something that always nagged me. This approach – which I encountered again and again – seemed stale. The body of analysis was beautiful, but it seemed divorced from the data sets that I saw starting to arrive on the horizon – data sets that were so large, or chocked full of so many different variables, that something seemed amiss. The answer rippled over me in waves. First, comments from an editor – Ram Meyers – for a paper of mine began to lift the veil. I had done all of my analyses as taught (and indeed even used for a class) using ANOVA and regression, multiple comparison, etc. etc. in the classic vein. Ram asked why, particularly given that the Biological processes that generated my data should in no way generate something with a normal – or even log-normal – distribution. While he agreed that the approximation was good enough, he made me go back, and jump off the cliff into the world of generalized linear models. It was bracing. But he walked me through it – over the phone even. So, a new recipe, yes? But it seemed like something more was looming. Then, an expiration of a JMP site license with one week left on a paper revision left me bereft. The only free tool I could turn to that seemed to do what I wanted it to do was R. Wonderful, messy, idiosyncratic R. I jumped in and learned the bare minimum of what I needed to know to do my analysis…and lost myself. I had taken computer science in college, and even written the backend of a number of websites in PERL (also wonderful, messy, and idiosyncratic). What I enjoyed most about programming was that you could not hide from how you manipulated information. Programming has a functional aspect at the core where an input must be translated into a meaningful output according to the rules that you craft. Working with R, I was crafting rules to generate meaningful statistical output. But what were those rules but my assumptions about how nature worked. The fundamentals of what I was doing all along – fitting a line to data with an error distribution – that should be based in biology, not arbitrary assumptions – was laid all the more bare. Some grumblingly lovely help from statistical denizens on the R help boards helped to bring this in sharp focus. So, I was ready when, for whatever reason, fate thrust me into a series of workshops on Bayesian statistics, AIC analysis, hierarchical modeling, time series analysis, data visualization, meta-analysis, and last – Structural Equation Modeling. I was delighted to learn more and more of how statistical analysis had grown beyond what I had been taught. I drank deeply of it. I know, that’s pretty nerdy, but, there you have it. The new techniques all shared a common core – that they were engines of inference about biological processes. How I, as the analyst, made assumptions about how the world worked was up to me. Once I had a model of how my system worked in mind – sketched out, filled with notes on error distributions, interactions, and more, I could sit back and think about what inferential tools would give me the clearest answers I needed. I had moved instead of finding the one right recipe in a giant cookbook to choosing the right tools out of a toolbox. And then using the tools of computer science – optimizing algorithms, thinking about efficient data storage, etc – to let my tools work bring data and biological models together. It’s exciting. And that’s the core philosophy I’m trying to convey in this semester. (N.B. the spellchecker tried to change convey to convert – there’s something there). Think about biology. Think about a line. Think about a probability distribution. Put them together, and find out what stories your data can tell you about the world. # Missing my Statsy Goodness? Check out #SciFund! I know, I know, I have been kinda lame about posting here lately. But that’s because my posting muscle has been focused on the new analyses for what makes a succesful #SciFund proposal. I’ve been posting them at the #SciFund blog under the Analysis tag – so check it out. There’s some fun stats, and you get to watch me be a social scientist for a minute. Viva la interdisciplinarity! # Running R2WinBUGS on a Mac Running OSX I have long used JAGS to do all of my Bayesian work on my mac. Early on, I tried to figure out how to install WinBUGS and OpenBUGS and their accompanying R libraries on my mac, but, to no avail. I just had too hard of a time getting them running and gave up. But, it would seem that some things have changed with Wine lately, and it is now possible to not only get WinBUGS itself running nicely on a mac, but to also get R2WinBUGS to run as well. Or at least, so I have discovered after an absolutely heroic (if I do say so myself) effort to get it all running (this was to help out some students I’m teaching who wanted to be able to do the same exercises as their windows colleagues). So, I present the steps that I’ve worked out. I do not promise this will work for everyone – and in fact, if it fails at some point, I want to know about it so that perhaps we can fix it so that more people can get WinBUGS up and running. Or just run JAGS (step 1} install the latest version, step 2} install rjags in R. Modify your code slightly. Run it. Be happy.) So, this tutorial works to get the whole WinBUGS shebang running. Note that it hinges on installing the latest development version of Wine, not the stable version (at least as of 1/17/12). If you have previously installed wine using macports, good on you. Now uninstall it with “sudo port uninstall wine”. Otherwise, you will not be able to do this. Away we go! 1) Have the free version of XCode Installed from http://developer.apple.com/xcode/. You may have to sign up for an apple developer account. Whee! You’re a developer now! 2) Have X11 Installed from your system install disc. 3) Install http://www.macports.org/install.php and install from the package installer. See also here for more information. Afterwards, open the terminal and type  echo export PATH=/opt/local/bin:/opt/local/sbin:$PATH$'n'export MANPATH=/opt/local/man:$MANPATH | sudo tee -a /etc/profile

4) Open your terminal and type

 sudo port install wine-devel

5) Go have a cup of coffe, check facebook, or whatever you do while the install chugs away.

7) Open your terminal, and type

 cd Downloads wine WinBUGS14.exe 

8 ) Follow the instructions to install WinBUGS into c:Program Files.

9) Run WinBUGS via the terminal as follows:

 wine ~/.wine/drive_c/Program Files/WinBUGS14/WinBUGS14

10) After first running WinBUGS, install the immortality key. Close WinBUGS. Open it again as above and install the patch. Close it. Open it again and WinBUGS away!

11) To now use R2WinBugs fire up R and install the R2WinBUGS library.

12) R2WinBugs should now work normally with one exception. When you use the bugs function, you will need to supply the following additional argument:

 bugs.directory='/Users/YOURUSERNAME/.wine/drive_c/Program Files/WinBUGS14'

filling in your username where indicated. If you don’t know it, in the terminal type

 ls /Users

No, ~ will not work for those of you used to it. Don’t ask me why.