rstanarm hierarchical model

\sigma_\alpha^2 & \rho\sigma_\alpha \sigma_\beta \\ In the above example, we use the SSlogis function, which is a lot like the logistic CDF, but with an additional Asym argument that need not be one and indicates what value the function approaches for large values of the first argument. stan_lmer decomposes this covariance matrix (up to a factor of \(\sigma_y\)) into (i) a correlation matrix \(R\) and (ii) a matrix of variances \(V\), and assigns them separate priors as shown below. Thus, the likelihood in a simple hierarchical model in rstarnarm is \[\mathbf{y} \thicksim \mathcal{N}\left(\alpha + \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{b}, \sigma^2 \mathbf{I}\right)\] and the observations are independent conditional on \(\mathbf{X}\) and \(\mathbf{Z}\). This model can be estimated by adding female to the formula in the lmer() function, which will allow only the intercept to vary by school, and while keeping the “slope” for being female constant across schools. The diagnostics which we use to assess whether the chains have converged to the posterior distribution are the statistics \(\hat{R}\) and \(N_{\text{eff}}\) (Gelman and Rubin 1992). The following example is based on Carpenter, Gabry, and Goodrich (2017) and the rstanarm vignette Hierarchical Partial Pooling for Repeated Binary Trials. This tutorial uses Stan version 2.17.3 and requires the following R packages. \[\mathbf{y} = \alpha + \mathbf{X} \boldsymbol{\beta} + \mathbf{Z} \mathbf{b} + \boldsymbol{\epsilon},\], \(\mathbf{Z}\mathbf{b} + \boldsymbol{\epsilon}\), \[\mathbf{y} \thicksim \mathcal{N}\left(\alpha + \mathbf{X}\boldsymbol{\beta}, \sigma^2 \mathbf{I}+\mathbf{Z}^\top \boldsymbol{\Sigma} \mathbf{Z} \right),\], \[\mathbf{y} \thicksim \mathcal{N}\left(\alpha + \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{b}, \sigma^2 \mathbf{I}\right)\], \(\mathbf{b} \thicksim \mathcal{N}\left(\mathbf{0},\boldsymbol{\Sigma}\right),\), \(\boldsymbol{\theta} = (\theta_1, \dots, \theta_J)\), \({\Sigma}_{jj} = \sigma^2_j = \text{var}(\theta_j)\), \(\boldsymbol{\mu} = g\left(\alpha + \mathbf{X} \boldsymbol{\beta} + \mathbf{Z}\mathbf{b}\right)\), deviations in the intercept(s) and / or coefficients that. See also W. J. Browne, Draper, and others (2006) for further discussion. The \(N_{\text{eff}}\) statistic should typically be at least 100 across parameters. One may start by quickly fitting many specifications in building a model using the lmer() function, and then take advantage of the flexibility of a fully Bayesian approach using rstanarm to obtain simulations summarizing uncertainty about coefficients, predictions, and other quantities of interest. The \(\hat{R}\) is essentially the ratio of between-chain variance to within-chain variance analogous to ANOVA. Evaluate how well the model fits the data and possibly revise the model. The above model can be fit using the stan_lmer() function in the rstanarm package as follows: Note that instead of the default priors in stan_lmer, \(\mu_{\alpha}\) and \(\beta\) are given normal prior distributions with mean 0 and standard deviation 100 by specifying the arguments prior and prior_intercept as normal(location = 0, scale = 100, autoscale = FALSE). In Linear Mixed Models, \(\mathbf{b}\) can be integrated out analytically, leaving a likelihood function that can be maximized over proposals for the parameters. \end{aligned} First, before accounting for the scale of the variables, \(\mu_{\alpha}\) is given normal prior distribution with mean 0 and standard deviation 10. Treating these estimates of \(\mu_\alpha\), \(\beta\), \(\sigma^2_{y}\), and \(\sigma^2_{\alpha}\) as the true parameter values, we can then obtain the Best Linear Unbiased Predictions (BLUPs) for the school-level errors \(\hat{u}_j = \hat{\alpha}_{j} - \hat{\mu}_{\alpha}\). Gelman, Andrew, and Jennifer Hill. We demonstrate how to do so in the context of making comparisons between individuals schools. The documentation of lme4 and gamm4 has various warnings that acknowledge that the estimated standard errors, confidence intervals, etc. \end{matrix} \right)\\ Unlike the latter, the 95% credible intervals have a 95% probability of containing the true value of the parameter given the data. A more direct approach to obtaining the posterior draws for specific parameters is to make use of the built in functionality of the as.matrix method for stanreg objects. Additionally, the regression coefficient \(\beta\) is given normal prior distributions with mean 0 and standard deviation 100. \[ 6.1: Posterior predictive checking of normal model for light data; 6.2: Posterior predictive checking for independence in binomial trials; 6.3: Posterior predictive checking of normal model with poor test statistic Furthermore, as in the previous two models, the Bayesian estimate for \(\sigma_{\alpha}\) (10.3) is larger than the ML estimate (10.15). By default, all rstanarm modeling functions will run 4 randomly initialized Markov chains, each for 2,000 iterations (including a warmup period of 1,000 iterations). \left(\begin{matrix} The vector of variances is set equal to the product of a simplex vector \(\boldsymbol{\pi}\) — which is non-negative and sums to 1 — and the scalar trace: \(J \tau^2 \boldsymbol{\pi}\). Note that here, we use the default priors which are mostly similar to what was done in Model 1. Input - rstanarm is able to take a data frame as input2. \sigma_y^2\left(\begin{matrix} Unlike stan_glmer, in stan_gamm4 it is necessary to specify group-specific terms as a one-sided formula that is passed to the random argument as in the lme function in the nlme package. The 95% credible interval is [-3.64, 13.66], so we are 95% certain that the true value of the difference between the two schools lies within the range, given the data. Half of these iterations in each chain are used as warm-up/burn-in (to allow the chain to converge to the posterior distribution), and hence we only use 1,000 samples per chain. \left(\begin{matrix} While we do not subset the data to only include complete cases to demonstrate that rstanarm automatically drops these observations, it is generally good practice to manually do so if required. 2\left(\frac{\sigma_\alpha^2/\sigma_y^2 + \sigma_\beta^2/\sigma_y^2}{2}\right)\left(\begin{matrix} Speed comparable to lme4 can be obtained with rstanarm using approximate Bayesian inference via the mean-field and full-rank variational algorithms (see help("rstanarm-package", "rstanarm") for details). The standard deviation of this prior distribution, 10, is five times as large as the standard deviation of the response if it were standardized. This is equivalent to assigning as a prior the exponential distribution with rate parameter set to 1 which is consistent with the prior assigned to \(\sigma_y\). In stan_nlmer, it is not necessary to supply starting values; however, in this case it was necessary to specify the init_r argument so that the randomly-chosen starting values were not more than \(0.5\) away from zero (in the unconstrained parameter space). \sigma_y^2\left(\begin{matrix} The hierarchical shrinkage (hs) prior in the rstanarm package instead utilizes a half Student t distribution for the standard deviation (with 3 degrees of freedom by default), scaled by a half Cauchy parameter, as described by Piironen and Vehtari (2015). This 95% credible interval is typically obtained by considering the 2.5th to 97.5th percentiles of the distribution of posterior draws. In rstanarm, these models can be estimated using the stan_lmer and stan_glmer functions, which are similar in syntax to the lmer and glmer functions in the lme4 package. When non-Bayesian methods are used, we can attempt to make such comparisons based on empirical Bayes (or Best Linear Unbiased) predictions of the varying intercepts, but it will generally be impossible to express the uncertainty for nonlinear function such as rankings. Identifiers - rstanarm does not require identifiers to be sequential4. For a small number of past roaches, the function is steep and then it appears to flatten out, although we become highly uncertain about the function in the rare cases where the number of past roaches is large. Using rstanarm we fit two models, one where there referee is a free parameter and one where it is a hierarchical parameter. While lme4 uses (restricted) maximum likelihood (RE)ML estimation, rstanarm enables full Bayesian inference via MCMC to be performed. The matrix of (scaled) variances \(V\) can first be collapsed into a vector of (scaled) variances, and then decomposed into three parts, \(J\), \(\tau^2\) and \(\pi\) as shown below. In the case where \(\boldsymbol{\Sigma}\) is \(1\times 1\), this shape parameter is the cross-group standard deviation in the parameters and its square is the variance. For each group, we assume the vector of varying slopes and intercepts is a zero-mean random vector following a multivariate Gaussian distribution with an unknown covariance matrix to be estimated. We can use these samples for predictions, summarizing uncertainty and estimating credible intervals for any function of the parameters. Missing Data - rstanarm automatically discards observations with NA values for any variable used in the model 3. Multilevel models1 are designed to model such within-cluster dependence. For example, a two-level model that allows for grouping of student outcomes within schools would include residuals at both the student and school level. stan_lmer assigns it an LKJ11 prior (Lewandowski, Kurowicka, and Joe 2009), with regularization parameter 1. The terminology for the unknowns in the model is diverse. Since, \(\mathbf{b}\) is considered part of the random error term, frequentists allow themselves to make distributional assumptions about \(\mathbf{b}\), invariably that it is distributed multivariate normal with mean vector zero and structured covariance matrix \(\boldsymbol{\Sigma}\). \end{matrix} \right)\\ &= In this section, we present how to fit and evaluate Model 1 using the rstanarm package. Of course, the same approach can be taken to generate 95% credible intervales for \(\sigma_y\) and \(\sigma_\alpha\). If the draws were independent \(N_{\text{eff}}\) would be the number of saved draws, 4,000 (4 chains \(\times\) (2,000 iterations - 1,000 iterations for warmup)), but \(N_{\text{eff}}\) tends to be smaller than 4,000 because Markov chain simulations tend to be autocorrelated. ... On the other hand, the rstanarm package makes this easy. The rstanarm package includes a stan_gamm4 function that is similar to the gamm4 function in the gamm4 package, which is in turn similar to the gamm function in the mgcv package. In the example below, so-called “thin-plate splines” are used to model counts of roaches where we might fear that the number of roaches in the current period is an exponentially increasing function of the number of roaches in the previous period. We can generate replicated datasets with a single line of code using the posterior_predict function: yrep <-posterior_predict ... Data Analysis Using Regression and Multilevel/Hierarchical Models. If we further assume that the student-level errors \(\epsilon_{ij}\) are normally distributed with mean 0 and variance \(\sigma_{y}^{2}\), and that the school-level varying intercepts \(\alpha_{j}\) are normally distributed with mean \(\mu_{\alpha}\) and variance \(\sigma_{\alpha}^{2}\), then the model can be expressed as, \[y_{ij} \sim N(\alpha_{j}, \sigma_{y}^{2}),\] \[\alpha_{j}\sim N(\mu_{\alpha}, \sigma_{\alpha}^{2}),\]. This tutorial is aimed primarily at educational researchers who have used lme4 in R to fit models to their data and who may be interested in learning how to fit Bayesian multilevel models. Alternatively, but not covered in this tutorial, one can also create a hand-written program in Stan and run it using the rstan package. However, rather than performing (restricted) maximum likelihood (RE)ML estimation, Bayesian estimation is performed via MCMC. For example, here is a plot of the link-level fit: Stan is a general purpose probabilistic programming language for Bayesian statistical inference. Gelman, Andrew, and Donald B Rubin. In a_sims, we have saved 4,000 posterior draws (from all 4 chains) for the varying intercepts \(\alpha_{j}\) of the 73 schools. \end{matrix} \right) Multiple populations can be modeled simultaneously with hierarchical models. This should be a close approximation to a noninformative prior over the range supported by the likelihood, which should give inferences similar to those obtained by maximum likelihood methods if similarly weak priors are used for the other parameters. Suppose you have a hierarchical ecological modeling problem with data clustered by space, time, and species, such as estimating the effect of ocean temperatures on coral growth. These statistics are automatically generated when using the rstanarm variants of these models using rstanarm we fit models! 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