# multilevel modelling brms

The dotted lines intersect at the true values. Here was that formula. But this method has its limitations. The concept of partial pooling, however, took me some time to wrap my head around. B., & Bosker, R. J. With that in mind, the code for our first task of getting the posterior draws for the actor-level estimates from the b12.7 model looks like so. # how many simulated actors would you like? Let’s fit the intercepts-only model. But okay, now let’s do things by hand. Note the uncertainty in terms of both location $$\alpha$$ and scale $$\sigma$$. Age_j = 30 - \frac{\beta_{j1}}{2 \beta_{j2}}. Here’s the plot. \alpha_{\text{actor}} & \sim \text{Normal} (0, \sigma_{\text{actor}}) \\ Behold our two identical Gaussians in a tidy tibble. Part of the wrangling challenge is because coef() returns a list, rather than a data frame. \beta_1 & \sim \text{Normal} (0, 10) \\ brms allows users to specify models via the customary R commands, where. The syntax for the varying effects follows the lme4 style, ( | ). By average actor, McElreath referred to a chimp with an intercept exactly at the population mean $$\alpha$$. There is no effsamples() function. McElreath, R. (2016). This is what shrinkage does for us…, Second, “prediction” in the context of a multilevel model requires additional choices. If we would like to average out block, we simply drop it from the formula. The reason that the varying intercepts provides better estimates is that they do a better job trading off underfitting and overfitting. And. And you can get the data of a given brm() fit object like so. But here’s our analogue. \sigma_{\text{grouping variable}} & \sim \text{HalfCauchy} (0, 1) One of the primary examples they used in the paper was of 1970 batting average data. If you followed along closely, part of what made that a great exercise is that it forced you to consider what the various vectors in post meant with respect to the model formula. The extra data processing for dfline is how we get the values necessary for the horizontal summary lines. \text{pulled_left}_i & \sim \text{Binomial} (n_i = 1, p_i) \\ First, we should no longer expect the model to exactly retrodict the sample, because adaptive regularization has as its goal to trade off poorer fit in sample for better inference and hopefully better fit out of sample. For a full list of available vignettessee vignette(package = "brms"). But back on track, here’s our prep work for Figure 12.1. \sigma & \sim \text{HalfCauchy} (0, 1) \text{total_tools}_i & \sim \text{Poisson} (\mu_i) \\ By replacing the 1 with nrow(post), we do this nrow(post) times (i.e., 12,000). \sigma_{\text{block}} & \sim \text{HalfCauchy} (0, 1) This time we increased adapt_delta to 0.99 to avoid divergent transitions. I wrote a lot of code like this in my early days of working with these kinds of models, and I think the pedagogical insights were helpful. \log \left(\frac{p_{ij}}{1 - p_{ij}}\right) = \beta_{i0} + \beta_{i1} D_{ij} + \beta_{i2} D_{ij}^2 If you’re interested, pour yourself a calming adult beverage, execute the code below, and check out the Kfold(): “Error: New factor levels are not allowed” thread in the Stan forums. The dashed line is, the model-implied average survival proportion. Let’s get the chimpanzees data from rethinking. (p. 365). Introduction to brms (Journal of Statistical Software) Advanced multilevel modeling with brms (The R Journal) Website (Website of brms with documentation and vignettes) Blog posts (List of blog posts about brms) In addition to the model intercept and random effects for the individual chimps (i.e., actor), we also included fixed effects for the study conditions. \text{logit} (p_i) & = \alpha + \alpha_{\text{actor}_i} + (\beta_1 + \beta_2 \text{condition}_i) \text{prosoc_left}_i \\ Prior distributions Priors should be specified usi… We can still extract that information, though. But if we were to specify a value for block in the nd data, we would no longer be averaging over the levels of block anymore; we’d be selecting one of the levels of block in particular, which we don’t yet want to do. tidy-brms.Rmd. These, of course, are in the log-odds metric and simply tacking on inv_logit_scaled() isn’t going to fully get the job done. Every model is a merger of sense and nonsense. Let’s keep expanding our options. Note how we just peeked at the top and bottom two rows with the c(1:2, 59:60) part of the code, there. When you do that, you tell fitted() to ignore group-level effects (i.e., focus only on the fixed effects). \sigma_{\text{actor}} & \sim \text{HalfCauchy} (0, 1) For example, multilevel models are typically used to analyze data from the students’ performance at different tests. Smaller, ponds produce more error, but the partial pooling estimates are better, on average, especially in smaller ponds. But first, we’ll simulate new data. \alpha_{\text{grouping variable}} & \sim \text{Normal} (0, \sigma_{\text{grouping variable}}) \\ Theformula syntax is very similar to that of the package lme4 to provide afamiliar and simple interface for performing regression analyses. By the first argument, we that requested spead_draws() extract the posterior samples for the b_Intercept. – Installation of R packages brms for Bayesian (multilevel) generalised linear models (this tutorial uses version 2.9.0). Bayesian multilevel modelling using MCMC with brms So, now we are going to model the same curves, but using Markov Chain Monte Carlo (MCMC) instead of maximum likelihood. You might check out its structure via b12.3$fit %>% str(). They should be most useful for meta-analytic models, but can be produced from any brmsfit with one or more varying parameters. \alpha_{\text{actor}} & \sim \text{Normal} (0, \sigma_{\text{actor}}) \\ … The introduction of varying effects does introduce nuance, however. Somewhat discouragingly, coef() doesn’t return the ‘Eff.Sample’ or ‘Rhat’ columns as in McElreath’s output. Purpose Bayesian multilevel models are increasingly used to overcome the limitations of frequentist approaches in the analysis of complex structured data. When using brms::posterior_samples() output, this would mean working with columns beginning with the b_ prefix (i.e., b_Intercept, b_prosoc_left, and b_prosoc_left:condition). (p. 367). About half of them are lower than we might like, but none are in the embarrassing $$n_\text{eff} / N \leq .1$$ range. The initial solutions came with a few divergent transitions. So there’s no need to reload anything. Families and link functions Details of families supported by brms can be found inbrmsfamily. You can always take the mean out of a Gaussian distribution and treat that distribution as a constant plus a Gaussian distribution centered on zero. It’s common in multilevel software to model in the variance metric, instead. We’ve already had some practice with the first three, but I hope this section will make them even more clear. If you’d like the stanfit portion of your brm() object, subset with$fit. Here is our code for Figure 12.3. Let’s get the reedfrogs data from rethinking. The posterior is the solid orange, $$\alpha_{\text{tank}} \sim \text{Normal} (\alpha, \sigma)$$, \begin{align*} Go ahead and acquaint yourself with the reedfrogs. So if we simply leave out the r_block vectors, we are ignoring the specific block-level deviations, effectively averaging over them. The initial multilevel update from model b10.4 from the last chapter follows the statistical formula. \text{logit} (p_i) & = \alpha + \alpha_{\text{actor}_i} + \alpha_{\text{block}_i} + (\beta_1 + \beta_2 \text{condition}_i) \text{prosoc_left}_i \\ Let’s review what that returns. For a given player, define the peak age \end{align*}, # we could have included this step in the block of code below, if we wanted to, "The horizontal axis displays pond number. Purpose: Bayesian multilevel models are increasingly used to overcome the limitations of frequentist approaches in the analysis of complex structured data. Happily, brms::fitted() has a re_formula argument. Both are great. Each pond $$i$$ has $$n_i$$ potential survivors, and nature flips each tadpole’s coin, so to speak, with probability of survival $$p_i$$. This time we’ll be sticking with the default re_formula setting, which will accommodate the multilevel nature of the model. If we convert the $$\text{elpd}$$ difference to the WAIC metric, the message stays the same. This time, we no longer need that re_formula argument. Additionally, I’d like to do a three-way comparison between the empirical mean disaggregated model, the maximum likelihood estimated multilevel model, the full Bayesian model. Depending upon the variation among clusters, which is learned from the data as well, the model pools information across clusters. Here’s the formula for the un-pooled model in which each tank gets its own intercept. Here it is for model b12.2. Up till this point, we’ve really only used the tidybayes package for plotting (e.g., with geom_halfeyeh()) and summarizing (e.g., with median_qi()). \beta_2 & \sim \text{Normal} (0, 10) \\ (p. 364). Formula syntax of brms models Details of the formula syntax applied in brms can be found inbrmsformula. And because we made the density only using the r_actor[5,Intercept] values (i.e., we didn’t add b_Intercept to them), the density is in a deviance-score metric. This code is no more burdensome for 5 group levels than it is for 5000. If you’re new to multilevel models, it might not be clear what he meant by “population-level” or “fixed” effects. Now, notice we fed it two additional arguments. In the first block of code, below, we simulate a bundle of new intercepts defined by, $\alpha_\text{actor} \sim \text{Normal} (0, \sigma_\text{actor})$. So then, if we want to continue using our coef() method, we’ll need to augment it with ranef() to accomplish our last task. The brms package allows R users to easily specify a wide range of Bayesian single-level and multilevel models which are fit with the probabilistic programming language Stan behind the scenes. By default, the code returns the posterior samples for all the levels of actor. Increasing adapt_delta to 0.95 solved the problem. We can fit the model in a Bayesian framework using mildly informative priors and quantify uncertainty based on the posterior samples. \text{logit} (p_i) & = \alpha_{\text{tank}_i} \\ Consider an example from biology. \end{align*}\], \begin{align*} In the present vignette, we want to discuss how to specify multivariate multilevel models using brms. If it helps to keep track of which vector indexed what, consider this. Yep, you can use the exponential distribution for your priors in brms. “We can use and often should use more than one type of cluster in the same model” (p. 370). For more on the sentiment it should be the default, check out McElreath’s blog post, Multilevel Regression as Default. Recall we use brms::fitted() in place of rethinking::link(). But the contexts in which multilevel models are superior are much more numerous. Bayesian multilevel modelling using MCMC with brms. For our first step, we’ll introduce the models. They offer both the ability to model interactions (and deal with the dreaded collinearity of model parameters) and a built-in way to regularize our coefficient to minimize the impact of outliers and, thus, prevent overfitting. If we wish to validate a model against the specific clusters used to fit the model, that is one thing. (p. 376). The following graph shows the posterior distributions of the peak ages for all players. Here we add the actor-level deviations to the fixed intercept, the grand mean. Now we’ll fit this simple aggregated binomial model much like we practiced in Chapter 10. You specify the corresponding multilevel model like this. Consider trying both methods and comparing the results. \end{align*}, chimp 1's average probability of pulling left, chimp 2's average probability of pulling left, chimp 3's average probability of pulling left, chimp 4's average probability of pulling left, chimp 5's average probability of pulling left, chimp 6's average probability of pulling left, chimp 7's average probability of pulling left, $$\text{logit} (p_i) = \alpha + \alpha_{\text{actor}_i}$$, # we need an iteration index for spread() to work properly, $$\alpha_{\text{block}} \sim \text{Normal} (0, \sigma_{\text{block}})$$, # here we add in the block == 1 deviations from the grand mean, # within fitted(), this line does the same work that, # inv_logit_scaled() did with the other two methods, # you'll need this line to make the spread() line work properly, # this line allows us to average over the levels of block, Andrew MacDonald’s great blogpost on this very figure, improved estimates for repeated sampling (i.e., in longitudinal data), improved estimates when there are imbalances among subsamples, estimates of the variation across subsamples, avoiding simplistic averaging by retaining variation across subsamples, Partial pooling (i.e., the multilevel model for which. The former will allow us to marginalize across the specific actors in our data and the latter will instruct fitted() to use the multivariate normal distribution implied by the random effects. This is because we had 12,000 HMC iterations (i.e., execute nrow(post)). And, of course, we can retrieve the data from that model, too. A general overview is provided in thevignettes vignette("brms_overview") andvignette("brms_multilevel"). Enter detailed information about the course literature, books, essays, abstracts, articles. ", "The prior is the semitransparent ramp in the, background. # if you want to use geom_line() or geom_ribbon() with a factor on the x axis, # you need to code something like group = 1 in aes(), # our hand-made brms::fitted() alternative, # here we use the linear regression formula to get the log_odds for the 4 conditions, # with mutate_all() we can convert the estimates to probabilities in one fell swoop, # putting the data in the long format and grouping by condition (i.e., key), # here we get the summary values for the plot, # with the ., ., ., . syntax, we quadruple the previous line, # the fixed effects (i.e., the population parameters), # to simplify things, we'll reduce them to summaries. If you’re struggling with this, be patient and keep chipping away. Multilevel models… remember features of each cluster in the data as they learn about all of the clusters. The formula syntax is very similar to that of the package lme4 to provide a familiar and simple interface for performing regression analyses. Digressions aside, let’s get ready for the diagnostic plot of Figure 12.3. (p. 356). Introduction This document shows how you can replicate the popularity data multilevel models from the book Multilevel analysis: Techniques and applications, Chapter 2. This model formula follows the form, Now we’ve fit our two intercepts-only models, let’s get to the heart of this section. The brms package also allows fitting multivariate (i.e., with several outcomes) models by combining these outcomes with mvbind():rmvbind(Reaction, Memory) ~ Days + (1 + Days | Subject)--The right-hand side of the formula defines the *predictors* (i.e., what is used to predict the outcome.s). So this time we’ll only be working with the population parameters, or what are also sometimes called the fixed effects. \alpha & \sim \text{Normal} (0, 10) \\ \alpha_{\text{block}} & \sim \text{Normal} (0, \sigma_{\text{actor}}) \\ Don’t worry. The second-stage parameters $$\beta$$ and $$\Sigma$$ are independent with weakly informative priors. McElreath didn’t show what his R code 12.29 dens( posta_actor[,5] ) would look like. A wide range of distributions and link functions are supported, allowing users to fit -- among others -- linear, robust linear, count data, survival, response times, ordinal, zero-inflated, hurdle, and even self-defined mixture models all in a multilevel context. This probability $$p_i$$ is implied by the model definition, and is equal to: $p_i = \frac{\text{exp} (\alpha_i)}{1 + \text{exp} (\alpha_i)}$, The model uses a logit link, and so the probability is defined by the [inv_logit_scaled()] function. There are certainly contexts in which it would be better to use an old-fashioned single-level model. You learn one basic design and you get all of this for free. \] Partial pooling shown in black. \end{align*}\], \begin{align*} However, the summaries are in the deviance metric. So to be clear, our goal is to accomplish those three tasks with four methods, each of which should yield equivalent results. The two models yield nearly-equivalent information criteria values. However, we’ll also be adding allow_new_levels = T and sample_new_levels = "gaussian". Half of the brood were put into another fosternest, while the other h… One of the things I really like about this method is the b_Intercept + r_actor[i,Intercept] part of the code makes it very clear, to me, how the porterior_samples() columns correspond to the statistical model, $$\text{logit} (p_i) = \alpha + \alpha_{\text{actor}_i}$$. By default, we get the familiar summaries for mean performances for each of our seven chimps. We’ll get more language for this in the next chapter. To complete our first task, then, of getting the posterior draws for the actor-level estimates from the b12.7 model, we can do that in bulk. \sigma_{\text{culture}} & \sim \text{HalfCauchy} (0, 1) \\ However, we do have neff_ratio(). Our orange density, then, is the summary of that process. \log \left(\frac{p_{ij}}{1 - p_{ij}}\right) = \beta_{i0} + \beta_{i1} D_{ij} + \beta_{i2} D_{ij}^2 For our brms model with varying intercepts for actor but not block, we employ the pulled_left ~ 1 + ... + (1 | actor) syntax, specifically omitting a (1 | block) section. For situations where we have the brms::brm() model fit in hand, we’ve been playing with various ways to use the iterations, particularly with either the posterior_samples() method and the fitted()/predict() method. If we want to depict the variability across the chimps, we need to include sd_actor__Intercept into the calculations. Assume that $$y_{ij}$$ is binomial with sample size $$n_{ij}$$ and probability of success $$p_{ij}$$. We call a model multivariateif it contains multiple response variables, each being predicted by its own set of predictors. The method remains essentially the same for accomplishing our second task, getting the posterior draws for the actor-level estimates from the cross-classified b12.8 model, averaging over the levels of block. \end{align*}\], \[\begin{align*} I’m not aware of a way to do that directly, but we can extract the iter value (i.e., b12.2fit@sim$iter), the warmup value (i.e., b12.2$fit@sim$warmup), and the number of chains (i.e., b12.2$fit@sim$chains). With each of the four methods, we’ll practice three different model summaries. Introduction. As with our posterior_samples() method, this code was near identical to the block, above. This pooling tends to improve estimates about each cluster. The brms package implements Bayesian multilevel models in R using the probabilistic programming language Stan. Depending upon the variation among clusters, which is learned from the data as well, the model pools information across clusters. The coef() function, in contrast, yields the group-specific estimates in what you might call the natural metric. Let’s look at the summary of the main parameters. Multivariate models, in which each response variable can be predicted using the above mentioned op- tions, can be tted as well. the age at which the player achieves peak performance. Consider what coef() yields when working with a cross-classified model. \alpha & \sim \text{Normal} (0, 10) \\ To accomplish that, we’ll need to bring in ranef(). The trace plots look great. \alpha_{\text{block}} & \sim \text{Normal} (0, \sigma_{\text{block}}) \\ For the finale, we’ll stitch the three plots together. tidybayes, which is a general tool for tidying Bayesian package outputs. Age_j = 30 - \frac{\beta_{j1}}{2 \beta_{j2}}. Results should be very similar to results obtained with other software packages. If you’re willing to pay with a few more lines of wrangling code, this method is more general, but still scalable. This vignette describes how to use the tidybayes and ggdist packages to extract and visualize tidy data frames of draws from posterior distributions of model variables, fits, and predictions from brms::brm. By the second argument, r_actor[actor,], we instructed spead_draws() to extract all the random effects for the actor variable. That way, we’ll know the true per-pond survival probabilities. But that doesn’t really count.] In sum, taking aside the change from a frequentist to a bayesian perspective, what would be the proper way to specify this multivariate multilevel model in nlme with brms? The higher the point, the worse. So if we know the neff_ratio() values and the number of post-warmup iterations, the ‘Eff.Sample’ values are just a little algebra away. But as models get more complex, it is very difficult to impossible to understand them just by inspecting tables of posterior means and intervals. Within the brms workflow, we can reuse a compiled model with update(). To get a sense of how it worked, consider this: First, we took one random draw from a normal distribution with a mean of the first row in post$b_Intercept and a standard deviation of the value from the first row in post$sd_tank__Intercept, and passed it through the inv_logit_scaled() function. Fitting multilevel event history models in lme4 and brms; Fitting multilevel multinomial models with MCMCglmm; Fitting multilevel ordinal models with MCMCglmm and brms . Yep, those Gaussians look about the same. The function get_onbase_data() function collects on-base data for all players born in the year 1977 who have had at least 1000 career plate appearances. The se_diff is large relative to the elpd_diff. McElreath encouraged us to inspect the trace plots. \text{logit} (p_i) & = \alpha + \alpha_{\text{actor}_i} + \alpha_{\text{block}_i}\\ The brms package implements Bayesian multilevel models in R using the probabilistic programming language Stan. We are going to practice four methods for working with the posterior samples. Chapman & Hall/CRC Press. All we did was switch out b12.7 for b12.8. References Snijders, T. A. The reason we can still get away with this is because the grand mean in the b12.8 model is the grand mean across all levels of actor and block. The three plots together in order to keep track of which vector indexed what, consider this does introduce,. Basic design and you can use the brms::neff_ratio ( ) took model! 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