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Someone wanted to know how to run a Bayesian data analysis for a two group longitudinal study. There are several ways you can do this, but I had to confess I did not have an immediate answer. So I took some time to figure out how to do this using jags software inside of R. I’ve done a fair amount of stuff in jags, but not anything close to a longitudinal design. The general principle is to start with something easy and work your way slowly up to the final analysis.

You can find the code and some supporting text files at

https://github.com/pmean/bayesian-longitudinal

I usually start with a very simple Bayesian analysis because if something is awry, I’ll notice it faster with a simple example. The code below implements a simple beta-binomial model.

Make sure that you have jags installed on your computer

http://sourceforge.net/projects/mcmc-jags/files/

as well as the rjags package

https://cran.r-project.org/web/packages/rjags/index.html

Assume you have a beta prior distribution with alpha=4 and beta=16. You collect 60 observations and observe 18 successes and 42 failures.

You don’t really need jags for this. The posterior distribution is beta with alpha=4+18=22 and beta=16+42=58. The posterior mean is 0.275 and the posterior standard deviation is 0.05.

`library("rjags")`

`## Loading required package: coda`

`## Linked to JAGS 4.2.0`

`## Loaded modules: basemod,bugs`

```
fnm <- "jags_beta_binomial.txt"
mon <- c("pi")
out <- list(description="jags output")
dat <- list(a=4, b=16, x=18, n=60)
mod <- jags.model(fnm, data=dat, quiet=TRUE)
out$bb1 <-
coda.samples(mod, variable.names=mon, n.iter=1000, by=1000, progress.bar=NULL)
summary(out$bb1)
```

```
##
## Iterations = 1:1000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 1000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## 0.273157 0.049814 0.001575 0.001575
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## 0.1811 0.2367 0.2718 0.3086 0.3698
```

Now, try to run one of the examples in the BUGS manuals that is reasonably close to your problem. For our problem, the very first example in volume 1 works well.

http://www.openbugs.net/Examples/Rats.html

I had to fix two things to get this example to run properly in jags. First, I had to remove some lines of code that used functions like post.p.value. Second, I have to transpose the matrix of data. But with these changes it ran just fine.

```
dat <- list(x = c(8.0, 15.0, 22.0, 29.0, 36.0), xbar = 22, N = 30, T = 5,
Y = structure(
.Data = c(151, 199, 246, 283, 320,
145, 199, 249, 293, 354,
147, 214, 263, 312, 328,
155, 200, 237, 272, 297,
135, 188, 230, 280, 323,
159, 210, 252, 298, 331,
141, 189, 231, 275, 305,
159, 201, 248, 297, 338,
177, 236, 285, 350, 376,
134, 182, 220, 260, 296,
160, 208, 261, 313, 352,
143, 188, 220, 273, 314,
154, 200, 244, 289, 325,
171, 221, 270, 326, 358,
163, 216, 242, 281, 312,
160, 207, 248, 288, 324,
142, 187, 234, 280, 316,
156, 203, 243, 283, 317,
157, 212, 259, 307, 336,
152, 203, 246, 286, 321,
154, 205, 253, 298, 334,
139, 190, 225, 267, 302,
146, 191, 229, 272, 302,
157, 211, 250, 285, 323,
132, 185, 237, 286, 331,
160, 207, 257, 303, 345,
169, 216, 261, 295, 333,
157, 205, 248, 289, 316,
137, 180, 219, 258, 291,
153, 200, 244, 286, 324),
.Dim = c(5,30)))
dat$Y <- t(dat$Y)
init <- list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250,
250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250),
beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6),
alpha.c = 150, beta.c = 10,
tau.c = 1, alpha.tau = 1, beta.tau = 1)
fnm <- "jags_rats.txt"
mon <- c("alpha0", "beta.c", "sigma")
mod <- jags.model(fnm, dat, init, quiet=TRUE)
out$rats <-
coda.samples(mod, variable.names=mon, n.iter=1000, by=1000, progress.bar=NULL)
summary(out$rats)
```

```
##
## Iterations = 1:1000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 1000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## alpha0 106.283 3.5884 0.113476 0.125482
## beta.c 6.196 0.1072 0.003391 0.004276
## sigma 6.108 0.6291 0.019893 0.033978
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## alpha0 99.495 104.008 106.248 108.670 113.344
## beta.c 5.996 6.128 6.193 6.268 6.399
## sigma 5.205 5.735 6.060 6.402 7.227
```

Now let’s see if we can modify this example to fit our particular setting.

Our longitudinal setting has two measurements: pre and post. So strip the last three columns from the matrix Y. There is one within subject factor, time, which is represents by X1. There is a between subjects factor, treatment group, which is represented by X2. Let’s pretend for now that that the first twenty observations represent your treatment group and the last ten observatitons represent your control group.

```
dat <- list(X1 = c(0, 1), X2 = rep(0:1, c(20, 10)), N = 30, T = 2,
Y = structure(
.Data = c(151, 199,
145, 199,
147, 214,
155, 200,
135, 188,
159, 210,
141, 189,
159, 201,
177, 236,
134, 182,
160, 208,
143, 188,
154, 200,
171, 221,
163, 216,
160, 207,
142, 187,
156, 203,
157, 212,
152, 203,
154, 205,
139, 190,
146, 191,
157, 211,
132, 185,
160, 207,
169, 216,
157, 205,
137, 180,
153, 200),
.Dim = c(2,30)))
dat$Y <- t(dat$Y)
init <- list(alpha = rep(150, 30), beta1 = rep(50, 30),
beta1.c = 50, beta1.tau=1,
beta2 = 20, beta3 = 5,
alpha.c = 150,
tau.c = 1, alpha.tau = 1)
fnm <- "jags_lon.txt"
mon <- c("alpha.c", "beta1.c", "beta2", "beta3")
mod <- jags.model(fnm, dat, init, quiet=TRUE)
out$lon <-
coda.samples(mod, variable.names=mon, n.iter=1000, by=1000, progress.bar=NULL)
summary(out$lon)
```

```
##
## Iterations = 1:1000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 1000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## alpha.c 154.095 2.576 0.08145 0.4494
## beta1.c 50.186 1.136 0.03593 0.1414
## beta2 -5.756 4.172 0.13193 1.4509
## beta3 -1.593 1.723 0.05447 0.5767
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## alpha.c 148.583 152.538 154.2840 155.7871 158.7248
## beta1.c 47.995 49.452 50.1588 50.9269 52.4221
## beta2 -8.877 -7.483 -7.1730 -6.8746 9.7413
## beta3 -5.556 -2.352 -0.9216 -0.6312 0.9299
```

Okay, so far, so good. Your next step would be to put your own data in place of the data from the example. I won’t show that here because I don’t have your data. But I hope you can try the general outline suggested here.

- Fit the simplest model possible, a beta-binomial.
- Fit an example taken from the BUGS software that is reasonably close to your example.
- Modify the example to match some of the structure of your data set, but also keep some of the original data.
- Substitute your data for the example data.

You might want to iterate step 3, making a small change to match your data a bit more closely, then another small change, and so forth.