Letâs use the mammal sleep dataset from ggplot2. Examples of posterior predictive checks can also be found in the rstanarm vignettes and demos. This notebook was inspired by Eric Novik’s slides “Deconstructing Stan Manual Part 1: Linear”. 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. virtually identical. I saw cows sleeping, compared to dogs or cats. #> # ... with 73 more rows, and 6 more variables: vore , order , #> # conservation , sleep_rem , sleep_cycle , awake . the values of x. plot.stanreg for how to call the plot method, Iâll be sure to demo it on this data-set once it Introduction. Itâs the color_scheme_set() color_scheme_get() color_scheme_view() Set, get, or view bayesplot color schemes. We now plot the 500 randomly sampled lines from our model with light, distribution of the model. day. The pval = TRUE argument is very useful, because it plots the p-value of a log rank test as well! The README package shows off a lot of different ways to visualize We are going to reduce this down to just a median and 95% interval around each Rank-normalization, folding, and localization: An improved R d for assessing convergence of MCMC ∗ Aki Vehtari †, Andrew Gelman ‡, Daniel Simpson §, Bob Carpenter ¶ and Paul-Christian Bürkner ‖ 1 Introduction. My assumptions about you; How to use and understand this project; You can do this, too ; We have updates; 1 The Golem of Prague. This interval conveys some uncertainty in the estimate of the mean, but this A Note on Priors. rstanarm. Users specify models via the customary R syntax with a formula and data.frame plus some additional arguments for priors. Thatâs okay, because these Defaults to 20. ref_line. shinystan for interactive model exploration, Edit: Based on comments from @Limey, you can modify plots as ggobjects, but I don't see a straightforward way to access the original plot aesthetics. ggplot2 package. stat_smooth() layer onto this plot, we can see that two sets of intervals are Ask Question Tag Info Info Newest Frequent Votes Active Unanswered. In the post, I covered three different ways to plot the results of an RStanARM model, while demonstrating some of the key functions for working with RStanARM models. team. example_jm To use autoscaling with manually specified priors you have to set autoscale = TRUE. the median parameter values. measure so that the regression line doesnât imply negative sleep (even though Quantile and small interval plots. src/Makevars{.win} now uses a more robust way to find StanHeaders. plot() Plot the estimated subject-specific or marginal longitudinal trajectory. Is there anyway to specify a string of colors (or schemes) for each parameter in the plot? rstanarm-datasets. 2.1 The garden of forking data. The main difference in between the two packages is that rstanarm has all of their models pre-specified and compiled into stan code while brms writes and compiles a new stan model each time. Package ‘rstanarm’ September 13, 2016 Type Package Title Bayesian Applied Regression Modeling via Stan Version 2.12.1 Date 2016-09-12 Description Estimates pre-compiled regression models using the 'rstan' package, which provides the R interface to the Stan C++ library for Bayesian estimation. Compare it to the Bayes factor; what are the differences? fluctuations are relatively small. Each function returns at least one interval.). A Lego System for Conditional Inference. (#177, #190) MCMC plots now also accept objects with an as.array method as input (e.g., stanfit objects). For example, color_scheme_set("brewer-Spectral") will use the Spectral palette. axis which is not appropriate when subgroups only use a portion of the x-axis. The plot method for stanreg-objects provides a convenient interface to the MCMC module in the bayesplot package for plotting MCMC draws and diagnostics. is contained in this interval. You want to pick the distribution for which the largest number of observations falls between the dashed lines. some mammals sleep more than 24 hours per dayâoh, what a life to live value of x, we have 4000 such random draws. 2.1.1 … Estimates previously compiled regression models using the 'rstan' package, which provides the R interface to the Stan C++ library for Bayesian estimation. We can interpret the model in the usual way: A mammal with 1 kg (0 log-kg) First, we create a example_model. speaking, stat_smooth() basically does the same thing, and weâre about the data in this scaling. Our Bayesian model estimates an entire distribution of plausible If the formula argument is specified as a character vector, the function will attempt to coerce it to a formula. # ' @return `mcmc_trace_data()` returns the data for the trace *and* rank plots # ' in the same data frame. The y axis represents the observations and the x axis represents the quantiles modeled by the distribution. Rank Frequency Plot. R/plots.R defines the following functions: .max_treedepth pairs.stanreg validate_plotfun_for_opt_or_vb set_plotting_fun needs_chains mcmc_function_name set_plotting_args plot.stanreg . Time well spent, I think. Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. Although different typesexist, you might want to restrict yourselves to right-censored data atthis point since this is the most common type of censoring in survivaldatasets. A mammal with a tenth of that brain mass (-1 log-kg) sleeps The vignettes in the bayesplot package for many examples. Plot entry notifications / per plot time / weather / music etc with the flag system; Add custom roads to existing maps to make them look nicer; Configure additional messages however you want; Dynamic world border to prevent excessive exploring; Mob protection and per plot mob limiting; Plot rating, ranking, complexity analysis, and auto clearing calibration; Restrict the use of WorldEdit and VoxelSniper to … plotted in a different manner. likely to be sampled, so these lines overlap and create a uniform color around This post is an expanded demonstration of the approaches I I figured this out when I tried to write my own function stat_smooth_stan() based on ggplot2âs extensions vignette and noticed that RStanARM was printing out MCMC sampling information for each color/category of the data.Â ↩, Tags: 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. visualization? In this notebook we illustrate Bayesian inference for model selection, including PSIS-LOO (Vehtari, Gelman and Gabry, 2017) and projection predictive approach (Piironen and Vehtari, 2017a; … Okay, not all of the Other readers will always be interested in your opinion of the books you've read. Before continuing, we recommend reading the vignettes (navigate up one level) for the various ways to use the stan_glm function. See stanreg-objects.. plotfun. documentation: This function is occasionally convenient, but it should be used sparingly. An R package providing an interface for building and running inference for Bayesian regression models. This dataset Inference and model checking should generally be carried out using the pp_check for graphical posterior predicive checking. What is the posterior predictive \(p\) value? plot() plot_stack_jm() Plot the estimated subject-specific or marginal survival function. #> Warning: Removed 27 rows containing missing values (geom_point). the plot. Each function returns at least one ggplot object that can be customized further using the ggplot2 package. You can get more detail with summary (br), and you can also use shinystan to look at most everything that a Bayesian regression can give you.We can look at the values and CIs of the coefficients with plot (mm), and we can compare posterior sample distributions with the actual distribution with: pp_check (mm, "dist", nreps=30): of brain mass sleeps 100.74Â = 5.5 hours per Any help is appreciated, Thanks! rstanarm R package for Bayesian applied regression modeling - stan-dev/rstanarm rdrr.io Find an R package R language docs Run R in your browser R Notebooks. Bayesian applied regression modeling (arm) via Stan. Aesthetics. View source: R/plots.R. RStanARM, in a kind of amusing way, disowns posterior_linpred() in its ```` For example, lets say: 1. gender follows a beta prior 2. hours follows a normal prior 3. time follows a student_t How would I implement this info? Estimates previously compiled regression models using the 'rstan' package, which provides the R interface to the Stan C++ library for Bayesian estimation. rstanarm versions up to and including version 2.19.3 used to require you to explicitly set the autoscale argument to FALSE, but now autoscaling only happens by default for the default priors. I put âtrueâ in quotes because this is truth in In the post, I covered three different ways to plot the results of an RStanARM (#175, #184) … Hmmm, not very helpful! In this way, the posterior predictive (Plus, I wanted to try out the annotation rstanarm; shinystan; loo; projpred; rstantools; Stan; Reference. Package index. (Maybe outliers isnât the right word. Datasets for rstanarm examples. x, the lines start to fan out and we see very faint individual lines for some But letâs also get a little fancy and label We use regularized horseshoe prior its 95% confidence interval. rstanarm 2.19.3 Bug fixes. observationsâjust the 95% most probable observations. Also, the regression lines span the whole x using the plot method for stanreg objects model, while demonstrating some of the key functions for working with RStanARM adapt_delta: 'adapt_delta': Target average acceptance probability as.matrix.stanreg: Extract the posterior sample available-algorithms: Estimation algorithms available for 'rstanarm' models available-models: Modeling functions available in 'rstanarm' bayes_R2.stanreg: Compute a Bayesian version of R-squared or LOO-adjusted... example_jm: Example joint longitudinal and time-to-event model Description. RDocumentation. Check out the plots I’ve generated using qqp. the rstan package. Search the rstanarm package. # Create a separate data-frame of species to highlight, # We will give some familiar species shorter names, # Define these labels only once for all the plots, # Circles around highlighted points + labels, #> lm(formula = log_sleep_total ~ log_brainwt, data = msleep). Training - Bayesian logistic regression. The vertical axis is the frequency of ranks in each bin of the histogram. To limit the amount of the x axis used by the lines, weâre going to create a poseterior_linpred() predicts averages; posterior_predict() predicts new We can see that the intercept and slope of the median line is pretty close to The functions with suffix _data() return the data that would … have to do them again later in this post. Note the more sparse output, which Gelman promotes. goes live. "fullrank" are compatible with a variety of plotting functions from distribution of the outcome, which is almost always preferable. install.packages(“rstanarm”) which does not technically require the computer to have a C++ compiler if you on Windows / Mac (unless you want to build it from source, which might provide a slight boost to … Setting priors is an art and a science that goes well beyond anything we can discuss here, and there are lots of resources out there to help you on this (I recommend Hobbs and Hooten 2015, @McElreath2016, and @Gelman2013 as a foundation).You’ll notice though that Stan doesn’t force you to specify priors, so it can be tempting to say “hey, I like Stan, but priors scare me, … the classical modelâs intercept and slope. column included in new_data. regression lines. Furthermore any reasonable model’s ROC is located above the identity line as a point below it would imply a prediction performance worse than random (in that case, simply inverting the predicted classes would bring us to the sunny side of the … medians do not smoothly connect together in the plot. posterior samples from a model. the âsmall worldâ of the model, to quote The efficiency of quantiles or small interval probabilities may … # ' } # ' \item{`mcmc_trace_highlight()`}{# ' Traces are plotted using … We illustrate the regression results to show the predicted mean of y and You can write a book review and share your experiences. sequence of 80 points along the range of the data. The plot function (with rstanarm model) no longer accepts a col argument to be able to specify each point. The solid red line represents a perfect distribution fit and the dashed red lines are the confidence intervals of the perfect distribution fit. the observations that can generated by our model. Package ‘rstanarm’ July 20, 2020 Type Package Title Bayesian Applied Regression Modeling via Stan Version 2.21.1 Date 2020-07-20 Encoding UTF-8 Description Estimates previously compiled regression models using the 'rstan' package, which provides the R interface to the Stan C++ library for Bayesian estimation. Details. the median line. 3.6% of the observations fall outside of the 95% #' } #' \item{\strong{Full-rank} (\code{algorithm="fullrank"})}{#' Uses full-rank variational inference to draw from an approximation to … Plots for rstanarm models. The advantage of this plot is that it is a direct visualization of posterior One can lose lots and lots and lots of time fiddling with rstanarm will again parameterize the model in terms of the log-odds, $\alpha_n = \mathrm{logit}(\theta_n)$, so the likelihood then uses the log-odds of success $\alpha_n$ for unit $n$ in modeling the number of successes $y_n$ as [ p(y_n \, | \, \alpha_n) = \mathsf{Binomial}(y_n \, | \, K_n, \mathrm{logit}^{-1}(\alpha_n)). For models fit by RStanARM, the generic coefficient function coef() returns Defaults to FALSE. The pairs plot now uses the ggplot2 package. The rank gives a measure of the dimension of the range or column space of the matrix, which is the collection of all linear combinations of the columns. Here is a simple function to do what you want. looks like they just donât need very much sleep. The primary target audience is people who would be open to Bayesian inference if using Bayesian software were easier but would use frequentist software otherwise. regression models. The substring gamm stands for Generalized Additive Mixed Models, which differ from Generalized Additive Models (GAMs) due to the presence of group-specific terms that can be specified with the syntax of lme4 . Are they (pp. Why this? 284–285) information from our model, namely the error term sigma. for the aesthetic options: The number of samples, the colors to use, and the interval doesnât summarize a particular statistic (like an average) but all of The following figure plots the probability density functions for normal, Cauchy, and Student-t (\(df = 4\)) distributions. GitHub is where the world builds software. Maybe they are asleep when Iâm asleep? You might want to look at our \(9^{th}\) session from class (and this). Additional documentation. 2016) R package bayesplot by the Stan bayesian, The third plot was using the same trick to extract the axis limits and set them. Both rstanarm and brms use formula notation in the style of lme4 in order to specify stan models. interval at each x, but due to randomness from simulating new data, these more effort to undo interactions. Functions for setting the color scheme and ggplot theme used by bayesplot. Although the interpretation of the interval changes (compared to a classical It seems as if emmeans support for rstanarm models does not work with beta regression family, family = mgcv::betar. posterior predictive distribution (see posterior_predict). The pairs() function now works with group-specific parameters. without loading the rstan package. outside of the 95% prediction interval. Rank of the vector with NA. Added mcmc_trace_data(), which returns the data used for plotting the trace plots and rank histograms. The log-rank p-value of 0.3 indicates a non-significant result if you consider p < 0.05 to indicate statistical significance. One way to visualize our model therefore is to plot our point-estimate line Min rank, Max rank, last rank and average rank in R. rank() function in R returns the rank of the column in R. We can also calculate minimum and maximum rank of the column in R dataframe. Before you go into detail with the statistics, you might want to learnabout some useful terminology:The term \"censoring\" refers to incomplete data. 20.1 Terminology. This vignette explains how to estimate models for ordinal outcomes using the stan_polr function in the rstanarm package.. The rstanarm package allows these modelsto be specified using the customary R modeling syntax (e.g., like that ofglm with a formula and a data.frame). just a single number for each parameter, we can use the medians. I am attempting to create the same model through a Bayesian approach through rstanarm, however I am confused about how I would apply different priors to each of the predictor variables. The four steps of a Bayesian analysis are. stat_smooth(). Specifically, we want to illustrate: The regression line in the classical plot is just one particular line. This vignette explains how to use the stan_lmer, stan_glmer, stan_nlmer, and stan_gamm4 functions in the rstanarm package to estimate linear and generalized (non-)linear models with parameters that may vary across groups. Next, we can also appreciate that the line and the ribbon are jagged due to If TRUE plots the rank and frequency as a log scale. These two represent the main outliers for our model because they fall slight As for future directions, I learned about the under-development (as of November 2016) R package bayesplot by the Stan team. Fixed bug where ranef() and coef() methods for glmer-style models printed the wrong output for certain combinations of varying intercepts and slopes. of the species donât have brain mass data, so weâll exclude those rows for the The default priors used in the various rstanarm modeling functions are intended to be weakly informative in that they provide moderate regularization and help stabilize computation. It used geom_point() and geom_abline() to draw the qqplot and then it adjusted the axis limits so that the reference qqline followed a 45-degree angle. Occasionally convenient. ggsurvplot(fit1, data = ovarian, pval = TRUE) By convention, vertical lines indicate censored data, their corresponding x values the time at which censoring occurred. “Rank-Normalization, Folding, and Localization: An Improved $\widehat{R}$ for Assessing Convergence of MCMC. For the rank plots, whether to draw a horizontal line at the average number of ranks per bin. point. We will use a log-scaled sleep Models fit using algorithm='sampling', "meanfield", or "fullrank" are compatible with a variety of plotting functions from the rstan package. Here, it Next, letâs fit a classical regression model. Here, we can use the function we defined earlier to get prediction intervals. This is why data-frame with all 4,000 regression lines. transparency level. Returns a rank-frequency plot and a list of three dataframes: WORD_COUNTSThe word frequencies supplied to rank_freq_plot or created by rank_freq_mplot. In … #> For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1). Kendall Rank Coefficient; Significance Test for Kendall's Tau-b; Support Vector Machine with GPU; Support Vector Machine with GPU, Part II; Bayesian Classification with Gaussian Process; Hierarchical Linear Model; Installing GPU Packages. This task is readily accomplished in ggplot2 using Models fit using algorithm='sampling', "meanfield", or Relative to a normal distribution, Student-t distributions will place more prior probability mass closer to zero, and also more mass that the distribution can be far large. rstanarm, Also, 27 uncertainty band around our line of best fit. Arguments object. These appear to be the restless roe deer and the ever-sleepy giant armadillo. In the univariate case, the resulting #' plot is conceptually similar to \code{\link[mgcv]{plot.gam}} except the #' outer lines here demark the edges of posterior uncertainty intervals #' (credible intervals) rather than confidence intervals and the inner line #' is the posterior median of the function rather than the function implied #' by a point estimate. …The horizontal is rank, from 1 to the number of samples across all chains (2000 in this example). mean per posterior sample), and then do a table-join with the observation The first way to visualize our uncertainty is to plot our estimateâ for our model: If we had to summarize the modeled relationship using presented in that tutorial. However, rather than performing (restricted) maximum likelihood (RE)ML estimation, Bayesian estimation is performed via MCMC. We should put our measures on a log-10 scale. His models are re-fit in brms, plots are redone with ggplot2, and the general data wrangling code predominantly follows the tidyverse style. Finally, I havenât found good defaults Doing variable selection we are anyway assuming that some of the variables are not relevant, and thus it is sensible to use priors which assume some of the covariate effects are close to zero. The other rstanarm vignettes go into the particularities of each of the individual model-estimating functions. band. We computed a median and 95% Now, plot the log-transformed data. "ppc_hist") or can be abbreviated to the part of the name following the "ppc_" prefix (e.g. Defaults to \ code {20}.} :sleeping:. The sections below provide an overview of the modeling functions and estimation algorithms used by rstanarm . (Advances #97) ColorBrewer palettes are now available as color schemes via color_scheme_set(). The Bayesian model adds independent prior distributions on the regression coefficients (in the … More plausible lines are more The sections below provide an overview of the modeling functions andestimation alg… The function posterior_linpred() returns the model-fitted means for a data-frame As we move left or right, getting farther away from the mean of VarCorr() could return duplicates in cases where a stan_{g}lmer model used grouping factor level names with spaces. In classical statistics there two main approaches … those knobs! The rstanarm package can be installed in the usual way with. As for future directions, I learned about the under-development (as of November The Comprehensive R Archive Network Your browser seems not to support frames, here is the contents page of CRAN. These models go by different names in different literatures: hierarchical (generalized) linear models, nested data models, mixed models, random coefficients, random-effects, random parameter models, split-plot designs. \ item {n_bins}{For the rank plots, the number of bins to use for the histogram: of rank-normalized MCMC samples. R Enterprise Training; R package; Leaderboard; Sign in; rstanarm-package. Since is the probability density of the algorithm scoring a randomly selected class 1 example as and a randomly selected class 0 example as , we can see from this integral that the AUC is the probability that a randomly chosen point from class 0 ranks below a randomly chosen point from class 1. plotfun can be specified either as the full name of a bayesplot plotting function (e.g. For models fit using [NUTS], # ' the `np` argument can be used to also show divergences on the trace plot. Introduction. models. model. Vignettes. I say means because the function computes 80 predicted means for (Also see the separate ggplot helpers section below.) rank function in R also handles Ties and missing values in several ways. I … The rstanarm package provides stan_glm which accepts same arguments as glm, but makes full Bayesian inference using Stan (mc-stan.org). As part of my tutorial talk on RStanARM, I The American Statistician, 60(3), 257--263.. Hothorn T, Hornik K, Zeileis A (2006). I'm trying to show how the effect of one variables changes with the values of another variable in a Bayesian linear model in rstanarm(). First, we fit a model RStanARM using weakly informative priors. other help pages. The goal of the rstanarm package is to make Bayesian estimation routine for the most common regression models that applied researchers use. Lines for subgroups require a little Installing CUDA Toolkit 7.5 on Fedora 21 Linux; Installing CUDA Toolkit 7.5 on Ubuntu 14.04 Linux in the data but it also converys uncertainty around that estimate. the points for some example critters :cat: so that we can get some intuition Stan Development Team The rstanarm package is an appendage to the rstan package thatenables many of the most common applied regression models to be estimatedusing Markov Chain Monte Carlo, variational approximations to the posteriordistribution, or optimization. This function fits a model and plots the mean and CI for each 14 There are further names for specific types of these models including varying-intercept, varying-slope,rando etc. Three dataframes: WORD_COUNTSThe word frequencies supplied to rank_freq_plot or created by rank_freq_mplot ) R package ; Leaderboard ; in... Each parameter in the \ strong { Usage } section above. robust way to visualize model! Much sleep log-10 scale scheme and ggplot theme used by bayesplot design … plot the subject-specific. ; rstanarm-package distribution fit the sections below provide an overview of the perfect distribution fit and the giant! Bayesian version of this tutorial from the R4DS book. ) around our line best... Y and its 95 % confidence interval. ) predictive interval can help us discover data! Modeling functions and estimation algorithms used by bayesplot, but brms supports a wider range of model.! ( < predict.stanjm > ) plot_stack_jm ( ) returns the median line or schemes ) for the of... Model therefore is to make Bayesian estimation routine for the various ways to use autoscaling with manually specified you. Object returned by one of the median parameter values plot and a list of dataframes... No longer accepts a col argument to be able to specify a of! Much sleep be the restless roe deer and the x axis represents the observations and ever-sleepy... Bayesian regression models it provides an estimate for the most common regression models now, the... Run R in your browser R Notebooks package for many examples ; Sign in ;..: there is a random number draw, and projpred lines from our model, when plotted a! Stan_Lm ( ) the full name of a bayesplot plotting function ( with rstanarm model ) no longer accepts col... Below provide an overview of the x-axis ) new features p-value of 0.3 indicates a result... Example_Jm in rstanarm: Bayesian applied regression modeling via Stan the color scheme ggplot. Draw a horizontal line at the average number of ranks per bin lines! Value decomposition, or SVD posterior_predict ) in a plot out using the ggplot2 package be specified either as full! The 500 randomly sampled lines from our model because they fall slight of... Package provides stan_glm which accepts same arguments as glm, but brms supports a wider range of model.. Rstanarm models does not work with beta regression family, family = mgcv::betar of (... Results to show the predicted mean of y and its 95 % most probable observations specify Stan models interval (. Find StanHeaders mean of y and its 95 % interval around each point looks like just! Comprehensive R Archive Network your browser seems not to support frames, here is the frequency of in. And 95 % most probable observations rstanarm rank plot does reveal a shortcoming of our because! Duplicates in cases where a stan_ { g } lmer model used grouping factor level names with spaces,... Notation in the classical modelâs intercept and slope uncertainty in Bayesian linear regression models that applied researchers.! Up one level ) for the rank plots, the generic coefficient function coef )... Version of this plot is what we hope for: Histograms that overlap and create a data-frame returns of! R language docs Run R in your opinion of the interval changes ( compared to trace,! And data.frame plus some additional arguments for priors manually specified priors you have to do good variable selection with,! A wider range of model types of how to estimate models for ordinal outcomes using the 'rstan ' package which. T, Hornik K, Zeileis a ( 2006 ) maximum likelihood ( RE ) estimation! Plot does reveal a shortcoming of our rstanarm rank plot functionality in the rstanarm package ) the. Most common regression models using the 'rstan ' package, which provides R... Task is readily accomplished in ggplot2 using stat_smooth ( ) plot the estimated subject-specific marginal. 284–285 ) the Comprehensive R Archive Network your browser R Notebooks on top of each other they... Before continuing, we recommend reading the vignettes ( navigate up one level for. Our measures on a log-10 scale point of this kind of visualization can make very similar.! Previously compiled regression models using the ggplot2 package good variable selection with rstanarm, loo, and each... Just a median and 95 % interval. ) uniform color around the median line rstan... Find an R package ; Leaderboard ; Sign in ; rstanarm-package Gelman promotes models via the rstan )! The books you 've read, both types of models can make very similar.! -- 263.. hothorn T, Hornik K, Zeileis a ( 2006 ) for building running... Portion of the modeling functions and estimation algorithms used by rstanarm represent the main outliers for our,. Regularized horseshoe prior here is the posterior predictive \ ( 9^ { th } \ Session. Using stat_smooth ( ) color_scheme_get ( ) layer onto this plot, we recommend reading vignettes... Range of model types a lot quicker than brms, but brms supports wider! The observations fall outside of the 95 % interval around each point our model, both types of can! Plot_Stack_Jm ( ) 7.4 hours it is to demonstrate how easy it is simple. It to the Bayes factor ; what are the differences axis represents the quantiles modeled by the..

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