idem: Inference in Randomized Controlled Trials with Death and Missingness

Chenguang Wang

2020-05-23

## Loading required package: idem
## Loading required package: Rcpp

Introduction

In randomized studies involving severely ill patients, functional outcomes are often unobserved due to missed clinic visits, premature withdrawal or death. It is well known that if these unobserved functional outcomes are not handled properly, biased treatment comparisons can be produced.

R package idem implement a procedure for comparing treatments that is based on the composite endpoint of both the functional outcome and survival. The procedure considers missing data imputation with a sensitivity analysis strategy to handle the unobserved functional outcomes not due to death.

Data accepted by idem

In dataset accepted by idem, each row should represent a subject with treatment assignment, baseline coveraites, baseline outcome, post-randomization outcomes and survival time.

The idem package provides dataset abc from ABC trial as an example data set.

head(abc);
##     AGE TRT SURV Y1 Y2
## 1 59.63   1  999 NA NA
## 2 66.89   0  999 49 52
## 3 59.70   1    1 NA NA
## 4 81.41   0   72 NA NA
## 5 66.52   1  999 51 45
## 6 40.27   0   65 NA NA

Basic steps

There are four major steps in conducting imputation and inference using idem. First, a class IDEMDATA object should be generated by the function imData. Second, the imputation models will be fit to the data observed from the completers by the function imFitModel. Third, imputation can be conducted by the function imImpAll. Lastly, treatment effect estimation and hypothesis testing can be performed by function imInfer.

Data generalization and visualization

In this step, the original dataset with specification parameters will be combined and checked. These parameters include variable names in the dataset, endpoint specification, duration of the study, etc.. If there is mis-specification, error messages will be generated. Otherwise, a class IDEMDATA object will be generated with certain data visulation functions implemented as its S3 methods.

rst.data <- imData(abc, trt="TRT", outcome=c("Y1","Y2"), y0=NULL,
                   endfml="Y3", bounds=c(10,20), duration=365,
                   err.terminate = FALSE);
print(rst.data);
## Model specification is invalid. Please check the following:
##     No survival time specified
##     Endpoint formula error: Error in eval(substitute(expr), data, enclos = parent.frame()) :   object 'Y3' not found
##     Upper bound is smaller than some observed outcomes
rst.data <- imData(abc, trt="TRT", surv="SURV", outcome=c("Y1","Y2"),
                   y0=NULL, endfml="Y2",
                   trt.label = c("UC+SBT", "SAT+SBT"),
                   cov=c("AGE"), duration=365, bounds=c(0,100));

The class IDEMDATA provides S3 plot and summary methods with multiple options for the visualization of the data.

Spaghetti plot for survivors

plot(rst.data, opt = "survivor");

Missing pattern frequency table

summary(rst.data, opt = "misstable");
##                       Y1       Y2   UC.SBT  SAT.SBT
## Deaths on study                   58 (62%) 38 (41%)
## S=1             Observed Observed 18 (19%) 32 (34%)
## S=2             Observed  Missing   8 (9%)   8 (9%)
## S=3              Missing Observed   1 (1%)   0 (0%)
## S=4              Missing  Missing  9 (10%) 15 (16%)
## Total                                   94       93

Missing pattern heatmap

plot(rst.data, opt = "missing", cols = c("blue", "gray"));

Kaplan-Meier curves

plot(rst.data, opt = "KM");

Missing data imputation

Model fitting

To fit the imputation model to data observed from the completers, i.e. the subjects who were alive at the end of the study without missing data, the class IDEMDATA object needs to be passed to the function imFitModel as parameters. The result has class name IDEMFIT, which will be passed to imputation functions.

rst.fit <- imFitModel(rst.data);

The goodness of fit diagnostics plots can be generated by the S3 plot method implemented for class IDEMFIT:

plot(rst.fit, mfrow=c(2,4));

Check convergence

The MCMC sampling is primarily done by rstan. It is suggested that the convergence of the MCMC chains should be checked. This can be done by the imImpSingle function which imputes missing data for an individual subject under the benchmark assumption.

rst.mixing <- imImpSingle(abc[1,], rst.fit, chains = 4, iter = 2000, warmup = 1000);
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 9e-06 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1: 
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## Chain 1:                0.026429 seconds (Sampling)
## Chain 1:                0.05045 seconds (Total)
## Chain 1: 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 3e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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## Chain 3:                0.021087 seconds (Sampling)
## Chain 3:                0.047508 seconds (Total)
## Chain 3: 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 3e-06 seconds
## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
## Chain 4: Adjust your expectations accordingly!
## Chain 4: 
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## Chain 4:                0.048136 seconds (Total)
## Chain 4:
plot(rst.mixing);

Imputation

The following code shows how to use imImpAll to get the imputed complete datasets under benchmark assmption delta=0 and for sensitivity analysis. We use 300 iterations to reduce the computation time.

rst.imp <- imImpAll(rst.fit, deltas=c(-0.25,0,0.25),
                    normal=TRUE, chains = 4, iter = 300, warmup = 100);
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
rst.imp

Plot denisity of the imputed data

The result from imIMPALL is class IDEMFIT. Density plots the imputed outcomes and the imputed functional endpoint can be generated by the S3 plot method associated with class IDEMFIT.

plot(rst.imp, opt = "imputed", deltas = c(-0.25,0,0.25), xlim=c(0,100), endp=FALSE);

plot(rst.imp, opt = "imputed", deltas = c(-0.25,0,0.25), xlim=c(0,100), endp=TRUE);

Composite endpoint analysis

Plot the cumulative distribution of the compositve endpoint

Treatment-specific cumulative distribution functions of the composite endpoint, where the values of the composite endpoint are labeled according to the survival time and functional endpoint among survivors, can be plotted by the S3 plot method of class IDEMFIT.

plot(rst.imp, delta=0);

Inference

The function imInfer implements bootstrap analysis for hypothesis testing, point estimation and confidence intervals of the treatment effects.

For illustration, we run 2 bootstrap samples by the following code:

rst.test <- imInfer(rst.imp, n.boot = 2);
## ---- Bootstrap 1 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: WARNING: There aren't enough warmup iterations to fit the
## Chain 1:          three stages of adaptation as currently configured.
## Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
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## Chain 1:            term_buffer = 10
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## Chain 2:          three stages of adaptation as currently configured.
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 3: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 2 : 1 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 5e-06 seconds
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
## Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
## Chain 1:          the given number of warmup iterations:
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 3: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
## Chain 4: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 3 : 2 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 3e-06 seconds
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 13 : 3 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 22 : 4 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 24 : 5 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 25 : 6 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 26 : 7 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 3: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 30 : 8 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 32 : 9 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 36 : 10 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 53 : 11 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: 
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## 
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## Chain 2: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 55 : 12 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 70 : 13 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 73 : 14 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 76 : 15 out of 43 
## 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 77 : 16 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 85 : 17 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 86 : 18 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 89 : 19 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 95 : 20 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 99 : 21 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 102 : 22 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 105 : 23 out of 43 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 107 : 24 out of 43 
## 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 113 : 26 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 115 : 27 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 118 : 28 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 4e-06 seconds
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## Chain 1: Adjust your expectations accordingly!
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## Chain 1:          three stages of adaptation as currently configured.
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 3: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
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## Chain 4:          Reducing each adaptation stage to 15%/75%/10% of
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## Chain 4:
## Warning: The largest R-hat is 1.08, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 122 : 29 out of 43 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
## Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## Chain 2: WARNING: There aren't enough warmup iterations to fit the
## Chain 2:          three stages of adaptation as currently configured.
## Chain 2:          Reducing each adaptation stage to 15%/75%/10% of
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 133 : 30 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
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## Chain 1:          the given number of warmup iterations:
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 134 : 31 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 5e-06 seconds
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## Chain 3: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 139 : 32 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 145 : 34 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 148 : 35 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 156 : 36 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 163 : 37 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 168 : 38 out of 43 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 172 : 39 out of 43 
## 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 175 : 40 out of 43 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 176 : 41 out of 43 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 177 : 42 out of 43 
## 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 183 : 43 out of 43 
## ---- Bootstrap 2 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4: Adjust your expectations accordingly!
## Chain 4: 
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## Chain 4: WARNING: There aren't enough warmup iterations to fit the
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 1 : 1 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## 
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 4 : 2 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: Adjust your expectations accordingly!
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## Chain 2: Adjust your expectations accordingly!
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 6 : 3 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
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## Chain 1:          three stages of adaptation as currently configured.
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 8 : 4 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## Chain 2: 
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 15 : 5 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 32 : 6 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 35 : 7 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
## Chain 1: 
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 48 : 8 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 54 : 9 out of 47 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 56 : 10 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 59 : 11 out of 47 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 63 : 12 out of 47 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 77 : 13 out of 47 
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 78 : 14 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'idem' NOW (CHAIN 4).
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## Chain 4:
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
## 83 : 15 out of 47 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 1).
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## Chain 1: WARNING: There aren't enough warmup iterations to fit the
## Chain 1:          three stages of adaptation as currently configured.
## Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
## Chain 1:          the given number of warmup iterations:
## Chain 1:            init_buffer = 15
## Chain 1:            adapt_window = 75
## Chain 1:            term_buffer = 10
## Chain 1: 
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## Chain 1: 
## Chain 1:  Elapsed Time: 0.00281 seconds (Warm-up)
## Chain 1:                0.005823 seconds (Sampling)
## Chain 1:                0.008633 seconds (Total)
## Chain 1: 
## 
## SAMPLING FOR MODEL 'idem' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 2e-06 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.
## Chain 2: Adjust your expectations accordingly!
## Chain 2: 
## Chain 2: 
## Chain 2: WARNING: There aren't enough warmup iterations to fit the
## Chain 2:          three stages of adaptation as currently configured.
## Chain 2:          Reducing each adaptation stage to 15%/75%/10% of
## Chain 2:          the given number of warmup iterations:
## Chain 2:            init_buffer = 15
## Chain 2:            adapt_window = 75
## Chain 2:            term_buffer = 10
## Chain 2: 
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## Chain 2: 
## Chain 2:  Elapsed Time: 0.002911 seconds (Warm-up)
## Chain 2:                0.005519 seconds (Sampling)
## Chain 2: