- Overview of
`mcmcse`

Package - Diasorin Example
- Confidence Regions
- Effective Sample Size
- Graphical Diagnostics
- Homework

- Overview of
`mcmcse`

Package - Diasorin Example
- Confidence Regions
- Effective Sample Size
- Graphical Diagnostics
- Homework

`mcmcse`

provides estimates of Monte Carlo standard errors for MCMC.- Written by James M. Flegal, John Hughes, Dootika Vats, and Ning Dai.
- Useful when estimating means and quantiles of functions of the MCMC output. In addition to MCMC output, the package can be used for time series and other correlated processes.
- Predominantly useful after MCMC output has been obtained by the user.
- Also provides univariate and multivariate estimates of effective sample size and tools to determine whether enough Monte Carlo samples have been obtained.
- Graphical tools to ascertain the behavior of the Monte Carlo estimates.

- Renal osteodystrophy is a bone disease that occurs when the kidneys fail to maintain proper levels of calcium and phosphorus in the blood. Monitoring patients with loss of kidney function for lower than normal bone turnover aids in managing the disease. A commercially available diagnostic assay, Diasorin, claims to be able to tell patients apart who have low versus normal bone turnover.
- Cross-section of 34 kidney patients from the bone registry at the University of Kentucky were identified as low or normal turnover by other means and then given the commercial assay to determine whether it could correctly identify them. From boxplots a normal sampling model appears untenable due to marked skewness, but boxplots and quantile plots of the log transformed data seem reasonably normal.

set.seed(1) n1 <- 19 n2 <- 15 low <- c(91, 46, 95, 60, 33, 410, 105, 43, 189,1097, 54,178, 114, 137, 233, 101, 25,70,357) normal <- c(370, 267, 99,157, 75,1281, 48, 298, 268, 62,804,430,171,694,404) diasorin <- c(low,normal) group <- c(rep(1,n1),rep(2,n2)) group1 <- factor(group, 1:2, c('Low','Normal'))

- We elicit independence priors for each population.
- For the low bone turnover group \(\mu_L = \mu_1 \sim N(4.87,0.00288)\).
- For the normal bone turnover group \(\mu_N = \mu_2 \sim N(5.39,0.00280)\).
- We consider gamma priors with modes \(\tau_{0,1} = 37.60\) and \(\tau_{0,2} = 46.53\).
- Consider \(d = 0.001\) so \[ \tau_1 \sim \text{Gamma}(1.0376,0.001) \text{ and } \tau_2 \sim \text{Gamma}(1.04653,0.001). \]

- Use the Gibbs to obtain correlated samples from the posterior.

gibbs <- function(reps, mu, tau, y.bar, s2, n, a, b, c, d){ output <- matrix(0, nrow=reps, ncol=2) a.gamma <- c+n/2 for(i in 1:reps){ mu.hat <- n * tau / (n * tau + b) * y.bar + b / (n * tau + b) * a sd.norm <- 1/sqrt(n*tau + b) mu <- rnorm(1, mu.hat, sd.norm) b.gamma <- d + ((n-1)*s2 + (y.bar - mu)^2)/2 tau <- rgamma(1, a.gamma, b.gamma) output[i,] <- c(mu, tau) } return(output) }

- Simulate for 50,000 iterations.

N <- 50000 y.bar.l <- mean(log(low)) s2.l <- var(log(low)) y.bar.n <- mean(log(normal)) s2.n <- var(log(normal)) sample.l <- gibbs(N, mu=5, tau=1, y.bar = y.bar.l, s2 = s2.l, n = n1, a=4.87, b=0.00288, c=1.0376, d=0.001) sample.n <- gibbs(N, mu=5, tau=1, y.bar = y.bar.n, s2 = s2.n, n = n2, a=5.39, b=0.00280, c=1.04653, d=0.001)

- For using the
`mcmcse`

package the rows of the MCMC output should store each iteration of the algorithm. Thus the output should have \(N\) rows and \(p\) columns.

chain <- cbind(sample.l, sample.n) colnames(chain) <- c("mu1", "tau1", "mu2", "tau2") head(chain)

## mu1 tau1 mu2 tau2 ## [1,] 4.561362 1.3286643 5.209149 1.1472458 ## [2,] 4.538759 1.9654129 5.290550 1.3588665 ## [3,] 4.758976 0.9519949 5.562766 1.3602247 ## [4,] 4.635779 1.2538313 5.580864 1.6013328 ## [5,] 4.642499 1.9271081 5.333285 1.2905995 ## [6,] 4.769481 1.0219264 5.534604 0.9064198

- Discuss estimating two sets of features of interest.
- First, consider \(\nu = \left( E[\mu_1 | y], E[\tau_1 | y], E[\mu_1 | y], E[\tau_1 | y] \right)\) where the estimator is the Monte Carlo mean \[ \nu_n = \frac{1}{n} \sum_{i=1}^{n} y_i. \]
- In
`R`

can can obtain \(\nu_n\) using the usual`colMeans`

function. If \(p=1\), then we can use`mean`

instead of`colMeans`

.

colMeans(chain)

## mu1 tau1 mu2 tau2 ## 4.704665 1.315990 5.488848 1.295256

- Second, consider the ratio \(g(\tau_1, \tau_2) = \tau_2 / \tau_1\). This is defined in
`R`

by creating a function that takes a vector argument.

g <- function(chain){ chain[4]/chain[2] }

- Then the Monte Carlo estimate for \(g\) is \[ \nu_{g,n} = \frac{1}{n} \sum_{i=1}^{n} g(y_i). \]

gofy <- apply(chain, 1, g) mean(gofy)

## [1] 1.088283

- Thus, to obtain Monte Carlo estimates from MCMC output, the base package is sufficient (generally).
- However, Monte Carlo estimates must be reported with Monte Carlo standard error.
- That is, if the following central limit theorems hold \[ \sqrt{n} (\nu_n - \nu) \stackrel{d}{\rightarrow} N_p(0, \Sigma) \] and \[ \sqrt{n} (\nu_{g,n} - g) \stackrel{d}{\rightarrow} N(0, \Sigma_g). \]
- Then estimates of \(\Sigma\) and \(\Sigma_g\) must be reported.
- Since the samples obtained are correlated, these quantities require more sophisticated tools than usual sample estimators.

- In this package, the functions
`mcse`

,`mcse.mat`

,`mcse.multi`

, and`mcse.initseq`

estimate the Monte Carlo standard error of \(\nu_{n}\) or \(\nu_{g,n}\).`mcse`

gives consistent estimates of the square root of \(\Sigma/n\) (standard error) when \(\Sigma\) is \(1 \times 1\).`mcse.mat`

gives consistent estimates of the square root of the diagonals of \(\Sigma/n\).`mcse.multi`

gives consistent estimates of \(\Sigma\).`mcse.initseq`

gives asymptotically conservative estimates of \(\Sigma\) using initial sequence estimators.

`mcmcse`

Packagelibrary(mcmcse)

## mcmcse: Monte Carlo Standard Errors for MCMC ## Version 1.3-2 created on 2017-07-03. ## copyright (c) 2012, James M. Flegal, University of California, Riverside ## John Hughes, University of Colorado, Denver ## Dootika Vats, University of Warwick ## Ning Dai, University of Minnesota ## For citation information, type citation("mcmcse"). ## Type help("mcmcse-package") to get started.

`mcmcse`

Packagemcerror_bm <- mcse.multi(x = chain, method = "bm", size = "sqroot", g = NULL, level = .95, large = FALSE) mcerror_bart <- mcse.multi(x = chain, method = "bartlett", size = "cuberoot", g = NULL, level = .95, large = FALSE) mcerror_tuk <- mcse.multi(x = chain, method = "tukey", size = "sqroot", g = NULL, level = .95, large = FALSE) mcerror_is <- mcse.initseq(x = chain, g = NULL, level = .95, adjust = FALSE) mcerror_isadj <- mcse.initseq(x = chain, g = NULL, level = .95, adjust = TRUE)

`x`

takes the \(n \times p\) MCMC data.`x`

can take only numeric entries in the form of a matrix or data frame. The rows of`x`

are the iterations of the MCMC.`method = â€˜â€˜bmâ€™â€™, â€˜â€˜bartlettâ€™â€™, â€˜â€˜tukeyâ€™â€™`

calculates the estimate using the batch means method and spectral variance methods with the modified-Bartlett and Tukey-Hanning windows.`size`

is the batch size for the`bm`

method and the truncation point for`tukey`

and`bartlett`

methods.`size = â€˜â€˜sqrootâ€™â€™`

sets the`size`

as \(\lfloor \sqrt{n} \rfloor\) and`size = â€˜â€˜cuberootâ€™â€™`

sets it at \(\lfloor n^{1/3} \rfloor\). An integer value of size less than \(n\) is also valid.`g`

is a function that is applied to each row of`x`

and represents the features of interest of the process. Since here we are interested in only means,`g`

is`NULL`

.

`level`

is the confidence level of the resulting confidence region. This is required to calculate the volume of the confidence region.`large`

is a logical argument. If`large`

is`TRUE`

the volume of the confidence region is the large sample volume obtained using \(\chi^2\) critical values. By default, volume is calculated using \(F\) distribution critical values.`adjust`

is a logical argument only used for the`mcse.initseq`

function. If`adjust`

is`TRUE`

, the eigenvalues of the initial sequence estimator are increased slightly.

`mcse.multi`

and`mcse.initseq`

return a list with multiple components.`cov`

stores the estimate of \(\Sigma\) obtained using the method chosen,`vol`

returns the volume to the \(p\)th root of the resulting confidence region,`est`

stores the estimate of`g`

applied on the Markov chain and`nsim`

stores the arguments used to calculate \(\Sigma\).`mcse.multi`

also returns`size`

which indicates the size of batches/truncation,`method`

used, and whether a`large`

sample volume is returned.`mcse.initseq`

also returns`cov.adj`

,`vol.adj`

, and whether an adjusted estimator was used (`adjust`

).

mcerror_bm$cov

## [,1] [,2] [,3] [,4] ## [1,] 0.0394643598 0.001375493 0.002379704 -0.0008686777 ## [2,] 0.0013754926 0.185400330 -0.006510334 0.0272105676 ## [3,] 0.0023797040 -0.006510334 0.058530865 -0.0049212774 ## [4,] -0.0008686777 0.027210568 -0.004921277 0.1957994103

mcerror_bart$cov

## [,1] [,2] [,3] [,4] ## [1,] 0.0457547735 -0.002134084 0.001193809 -0.0003560083 ## [2,] -0.0021340836 0.162503174 -0.000597813 0.0028188956 ## [3,] 0.0011938092 -0.000597813 0.056796737 -0.0010611261 ## [4,] -0.0003560083 0.002818896 -0.001061126 0.1932724203

mcerror_tuk$cov

## [,1] [,2] [,3] [,4] ## [1,] 0.0421549392 0.002086938 0.0005128396 -0.003337617 ## [2,] 0.0020869382 0.176423621 -0.0035763980 0.012953344 ## [3,] 0.0005128396 -0.003576398 0.0532081937 -0.001401419 ## [4,] -0.0033376174 0.012953344 -0.0014014185 0.197610425

mcerror_is$cov

## [,1] [,2] [,3] [,4] ## [1,] 0.0445562278 -0.0006880102 0.0010955859 -0.0017663968 ## [2,] -0.0006880102 0.1653666228 -0.0008183728 0.0025448181 ## [3,] 0.0010955859 -0.0008183728 0.0564524841 0.0002125322 ## [4,] -0.0017663968 0.0025448181 0.0002125322 0.1947821982

mcerror_isadj$cov.adj

## [,1] [,2] [,3] [,4] ## [1,] 0.0448840031 -0.0009320214 0.0004870070 -0.0011130623 ## [2,] -0.0009320214 0.1656303983 -0.0005832360 0.0019029607 ## [3,] 0.0004870070 -0.0005832360 0.0581606915 -0.0005879199 ## [4,] -0.0011130623 0.0019029607 -0.0005879199 0.1963788341

rbind(mcerror_bm$est, mcerror_bart$est, mcerror_tuk$est, mcerror_is$est, mcerror_isadj$est)

## mu1 tau1 mu2 tau2 ## [1,] 4.704665 1.31599 5.488848 1.295256 ## [2,] 4.704665 1.31599 5.488848 1.295256 ## [3,] 4.704665 1.31599 5.488848 1.295256 ## [4,] 4.704665 1.31599 5.488848 1.295256 ## [5,] 4.704665 1.31599 5.488848 1.295256

c(mcerror_bm$vol, mcerror_bart$vol, mcerror_tuk$vol, mcerror_is$vol, mcerror_isadj$vol)

## [1] 0.006425778 0.006328959 0.006289714 0.006323004 0.006323004

- If the diagonals of \(\Sigma\) are \(\sigma^2_{ii}\), the function
`mcse`

and`mcse.mat`

return \(\sigma^2_{ii} / \sqrt{n}\). `mcse`

does it for one component and`mcse.mat`

does it for all diagonals.

mcse(x = chain[,1], method = "bm", g = NULL)

## $est ## [1] 4.704665 ## ## $se ## [1] 0.0008884184

mcse.mat(x = chain, method = "bm", g = NULL)

## est se ## mu1 4.704665 0.0008884184 ## tau1 1.315990 0.0019256185 ## mu2 5.488848 0.0010819507 ## tau2 1.295256 0.0019788856

- In order to estimate \(\nu_{g,n}\) and \(\Sigma_g\), we use the
`R`

function`g`

we had defined before. - Recall that
`g`

should be a function that takes vector inputs.

g

## function(chain){ ## chain[4]/chain[2] ## } ## <bytecode: 0x7fbe3ac4e3c8>

mcerror_g_bm <- mcse.multi(x = chain, g = g) mcerror_g_is <- mcse.initseq(x = chain, g = g)

- Initial Sequence error is larger than batch means, as expected.

c(mcerror_g_bm$cov, mcerror_g_is$cov)

## [1] 0.2822692 0.2912683

c(sqrt(mcerror_g_bm$cov/N), sqrt(mcerror_g_is$cov/N))

## [1] 0.002376002 0.002413580

- Using the function
`confRegion`

in the package, the user can create joint confidence regions for two parameters. - Input for this function is the output list from the
`mcse.multi`

or`mcse.initseq`

function. - Function uses the attributes
`cov`

,`est`

, and`nsim`

from the output list. - If the
`mcse.initseq`

is input and`adjust = TRUE`

had been used, then`cov.adj`

is used instead of`cov`

. `mcse.multi`

also uses the attribute`size`

.

plot(confRegion(mcerror_bm, which = c(1,2), level = .90), type = 'l', asp = 1, xlab="mu1", ylab="tau1") lines(confRegion(mcerror_bart, which = c(1,2), level = .90), col = "red")

- To determine the effect of the confidence level, we draw two regions with difference confidence levels. We use
`mcse.initseq`

this time.

plot(confRegion(mcerror_is, which = c(1,2), level = .95), type = 'l', asp = 1, xlab="mu1", ylab="tau1") lines(confRegion(mcerror_is, which = c(1,2), level = .90), col = "red")

- Reporting \(p \times p\) covariance matrix estimates is impractical and uninterpretable.
- The motivation of estimating Monte Carlo standard error is to ensure that said error is small.
- Essentially the idea behind estimating effective sample size and ensuring that the estimated effective sample size is larger than a prespecified lower bound.

- Before sampling the Markov chain, the user is advised to used the function
`minESS`

to ascertain what is the minimum effective sample size needed for stable analysis. `p`

is the dimension of the estimation problem.`alpha`

is the confidence level.`eps`

is the tolerance level. Default is .05. Reasonable levels are anywhere from .01 to .05. The smaller the tolerance, the larger the minimum effective samples.`eps`

represents a tolerance level relative to the variability in the target distribution. It is akin to the idea of margin-of-error.

minESS(p = 4, alpha = .05, eps = .05)

## minESS ## 8431

minESS(p = 1, alpha = .05, eps = .05)

## minESS ## 6146

`minESS`

is independent of the Markov chain or process, and is only a function of the`p`

,`alpha`

, and`eps`

.- User should find
`minESS`

and then sample their process until the required minimum samples are achieved. - Alternatively, we often donâ€™t have to luxury of obtaining a lot of samples, and reaching a minimum effective sample size os not possible. In such a scneario, it is useful to know the
`eps`

tolerance level the number of estimated effective samples correspond to.

- If we can only obtain 1000 effective samples,

minESS(p = 4, alpha = .05, ess = 1000)

## Epsilon ## 0.1451773

minESS(p = 1, alpha = .05, ess = 1000)

## Epsilon ## 0.123959

`multiESS`

and`ess`

are two functions that calculate the effective sample size of a correlated sample.`ess`

calculations are component-wise while`multiESS`

utilizes the multivariate nature of the problem.

ess(chain)

## mu1 tau1 mu2 tau2 ## 55801.49 44545.19 49721.07 50076.02

multiESS(chain)

## [1] 50237.53

multiESS(chain, covmat = mcerror_bart$cov)

## [1] 50233.54

- Since
`ess`

produces a different estimate for each component, conservative practice dictates choosing the smallest of the values. `multiESS`

returns one estimate of the effective sample size based on the whole sample.- Function calls
`mcse.multi`

function to obtain a batch means estimate of \(\Sigma\). User can provide another estimate of \(\Sigma\) using the`covmat`

argument. - Can compare effective sample size to minimum effective samples, and adjust our simulation length if necessary. For example, by looking at the ratio of the Monte Carlo samples size and
`multiESS`

.

- The function
`estvssamp`

plots the Monte Carlo estimates versus the sample size for a component of the MCMC output. This plot indicates whether the Monte Carlo estimate has stabilized.

estvssamp(chain[,1])

- Additionally, if \(p\) is not too small, due to the CLT and an estimate of \(\Sigma\) using the
`mcse.multi`

function, a QQ plot of the standardized estimates gives an idea of whether asymptopia has been achieved. - However, \(p=4\) in our example so we need another …

mu1 <- sample.l[,1] tau1 <- sample.l[,2] mu2 <- sample.n[,1] tau2 <- sample.n[,2] mu.diff <- mu2-mu1 tau.ratio <- tau2/tau1 post.sum <- function(vec){ return(c(mean(vec), sd(vec), quantile(vec, probs=c(0.25, .5, .975)))) } sum.table <- rbind(post.sum(mu1), post.sum(mu2), post.sum(mu.diff), post.sum(tau1), post.sum(tau2), post.sum(tau.ratio)) rownames(sum.table) <- c("mu1", "mu2", "mu2-mu1", "tau1", "tau2", "tau2/tau1") colnames(sum.table) <- c("mean", "sd", "2.5%", "50%", "97.5%") round(sum.table, 3)

## mean sd 2.5% 50% 97.5% ## mu1 4.705 0.210 4.566 4.705 5.117 ## mu2 5.489 0.241 5.332 5.489 5.968 ## mu2-mu1 0.784 0.319 0.572 0.784 1.415 ## tau1 1.316 0.406 1.024 1.274 2.227 ## tau2 1.295 0.443 0.976 1.246 2.299 ## tau2/tau1 1.088 0.543 0.711 0.975 2.439

- Use the
`mcmcse`

package to add MCSEs to the summary statistics in`sum.table`

. - Need to use the
`mcse.q`

function for quantiles. - What
`method`

and`size`

did you use? Why? - Create a plot using the
`mcmcse`

package of your choosing. Explain carefully what the plot is illustrating.