trendtest.Gibbs.AR: Obtain Bayesian trend test and posterior distributions for...
In BayesSingleSub: Computation of Bayes factors for interrupted time-series designs

Description Usage Arguments Details Value Note Author(s) References See Also Examples

This function computes Bayes factors for the trend and intercept differences between two phases of a single subject data sequence, using Gibbs sampling. Posterior samples of parameters are also provided.

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trendtest.Gibbs.AR(before, after, iterations = 1000, intArea = c(-0.2,0.2), slpArea = c(-0.2, 0.2),  r.scaleInt = 1, r.scaleSlp = 1, 
			alphaTheta = 1, betaTheta = 5, sdMet = 0.3, progress = TRUE, return.chains = FALSE)

`before`	A vector of observations, in time order, taken in Phase 1 (e.g., before the treatment).
`after`	A vector of observations, in time order, taken in Phase 2 (e.g., after the treatment).
`iterations`	Number of Gibbs sampler iterations to perform.
`intArea`	Only used if `return.chains=TRUE`. Bounds for the interval null hypothesis for the intercept difference.
`slpArea`	Only used if `return.chains=TRUE`. Bounds for the interval null hypothesis for the trend difference.
`r.scaleInt`	Prior scale for the intercept difference (see Details below).
`r.scaleSlp`	Prior scale for the trend difference (see Details below).
`alphaTheta`	The alpha parameter of the beta prior on theta (see Details below).
`betaTheta`	The beta parameter of the beta prior on theta (see Details below).
`sdMet`	Scale for the Metropolis-Hastings sampling of theta (see Details below).
`progress`	Report progress with a text progress bar?
`return.chains`	Return posterior samples of parameters?.

This function computes Bayes factors for the differences in trend and intercept between two data sequences from a single subject, using Gibbs sampling. The Bayes factor for trend difference compares the null hypothesis of no true trend difference against the alternative hypothesis of a true trend difference. The Bayes factor for intercept difference compares the null hypothesis of no true intercept difference against the alternative hypothesis of a true intercept difference. Also, a joined Bayes factor for the trend and intercept combined is provided. Bayes factors larger than 1 support the null hypothesis, Bayes factors smaller than 1 support the alternative hypothesis. Auto-correlation of the errors is modeled by a first order auto-regressive process.

Posterior distributions of the model parameters can also be obtained. Model parameters of interest include mu0 (overall mean), sig*delta (difference between intercepts), beta0 (overall trend), sig*beta1 (difference between trends), sig2 (variance of the random shocks), and rho (auto-correlation).

Cauchy priors are placed on the standardized trend and intercept differences. The r.scaleInt and r.scaleSlp arguments control the scales of these Cauchy priors, with r.scaleInt = 1 and r.scaleSlp = 1 yielding standard Cauchy priors. A noninformative Jeffreys prior is placed on the variance of the random shocks of the auto-regressive process. A beta prior is placed on the auto-correlation theta. The alphaTheta and betaTheta arguments control the form of this beta prior.

Missing data are sampled from the likelihood function, conditioned at the observed data, at each iteration of the Gibbs sampler.

if return.chains=FALSE: a matrix containing MCMC estimates of the log Bayes factors, computed using the Savage-Dickey method (Morey, Rouder, Pratte, and Speckman, submitted) if return.chains=TRUE: a list containing the following:

`logbf`	MCMC estimates of the log Bayes factors, computed using the Savage-Dickey method (Morey, Rouder, Pratte, and Speckman, submitted).
`chains`	An object of type MCMC containing the chains for each parameter.
`acc`	The Metropolis-Hastings acceptance rate.
`logbfArea`	MCMC estimates of the log interval null Bayes factors.

For a more accurate method of computing the Bayes factor, see trendtest.MC.AR.

Richard D. Morey and Rivka de Vries

De Vries, R. M. \& Morey, R. D. (submitted). Bayesian hypothesis testing Single-Subject Data. Psychological Methods.

R code guide: http://drsmorey.org/research/rdmorey/

trendtest.MC.AR, ttest.Gibbs.AR, ttest.MCGQ.AR

## Define data
data = c(87.5, 82.5, 53.4, 72.3, 94.2, 96.6, 57.4, 78.1, 47.2,
 80.7, 82.1, 73.7, 49.3, 79.3, 73.3, 57.3, 31.7, 50.4, 77.8,
 67, 40.5, 1.6, 38.6, 3.2, 24.1)

## Obtain log Bayes factors
logBFs = trendtest.Gibbs.AR(data[1:10], data[11:25])

## Obtain log Bayes factors, chains, and log interval null Bayes factors
output = trendtest.Gibbs.AR(data[1:10], data[11:25], return.chains = TRUE, intArea = c(-0.2,0.2), slpArea = c(-0.2, 0.2))

## Look at the posterior distribution of the mean
plot(output$chains[,1])

## Obtain summary statistics of posterior distributions
summary(output$chains)