After Fitting

Summary, visualization and evaluation of the results

2020-06-22

This vignette describes how the results from a JointAI model can be visualized, summarized and evaluated. We use the NHANES data for examples in cross-sectional data and the dataset simLong for examples in longitudinal data. For more info on these datasets, check out the vignette Visualizing Incomplete Data, in which the distributions of variables and missing values in both sets is explored.

The functions described in this section use, by default, the full MCMC sample and show only the parameters of the analysis model. A number of arguments are available to select a subset of the MCMC samples that is used to calculate the summary. The argument subset allows controlling which part of the MCMC sample is returned and follows the same logic as the argument monitor_params in *_imp(). The use of these arguments is further explained below.

Visualizing the posterior sample

The posterior sample can be visualized by two commonly used plots: a trace plot, showing samples across iterations, or a plot of the empirical density of the posterior sample.

Trace plot

A trace plot shows the sampled values per chain and node throughout iterations. It allows us to evaluate convergence and mixing of the chains visually, and can be obtained with the function traceplot():

mod13a <- lm_imp(SBP ~ gender + WC + alc + creat, data = NHANES, n.iter = 500,
                 seed = 2020)

traceplot(mod13a)

When the sampler has converged the chains show one horizontal band, as in the above figure. Consequently, when traces show a trend convergence has not been reached and more iterations are necessary (e.g., using add_samples()).

Graphical aspects of the traceplot can be controlled by specifying standard graphical arguments via the dots argument "...", which are passed to matplot(). This allows changing colour, line type and -width, limits, and so forth. Arguments nrow and/or ncol can be supplied to set specific numbers of rows and columns for the layout of the grid of plots.

With the argument use_ggplot it is possible to get a ggplot2 version of the traceplot. It can be extended using standard ggplot2 syntax.

library(ggplot2)
traceplot(mod13a, ncol = 3, use_ggplot = TRUE) +
  theme(legend.position = 'bottom') +
  scale_color_brewer(palette = 'Dark2')

Density plot

The posterior distributions can also be visualized using the function densplot(), which plots the empirical density per node per chain, or combining chains (when joined = TRUE).

The argument vlines takes a list of lists, containing specifications passed to abline, and allows us to add (vertical) lines to the plot, e.g., marking zero:

densplot(mod13a, ncol = 3, col = c("darkred", "darkblue", "darkgreen"),
         vlines = list(list(v = c(rep(0, length(coef(mod13a)$SBP)), NA),
                            col = grey(0.8))))

or marking the posterior mean and 2.5% and 97.5% quantiles:

res <- rbind(summary(mod13a)$res$SBP$regcoef[, c('Mean', '2.5%', '97.5%')],
             summary(mod13a)$res$SBP$sigma[, c('Mean', '2.5%', '97.5%')])

densplot(mod13a, ncol = 3,
         vlines = list(list(v = res[, 'Mean'], lty = 1,
                            lwd = 2),
                       list(v = res[, "2.5%"], lty = 2),
                       list(v = res[, "97.5%"], lty = 2)
         )
)

Like with traceplot() it is possible to use the ggplot2 version of densplot() when setting use_ggplot = TRUE. Here, vertical lines can be added as additional layers:

# fit the complete-case version of the model
mod13a_cc <- lm(formula(mod13a), data = NHANES, na.action = na.omit)


# make a dataset containing the quantiles of the posterior sample and
# confidence intervals from the complete case analysis:
quantDF <- rbind(data.frame(variable = rownames(summary(mod13a)$res$SBP$regcoef),
                            type = '2.5%',
                            model = 'JointAI',
                            value = summary(mod13a)$res$SBP$regcoef[, c('2.5%')]
                            ),
                 data.frame(variable = rownames(summary(mod13a)$res$SBP$regcoef),
                            type = '97.5%',
                            model = 'JointAI',
                            value = summary(mod13a)$res$SBP$regcoef[, c('97.5%')]
                 ),
                 data.frame(variable = names(coef(mod13a_cc)),
                            type = '2.5%',
                            model = 'cc',
                            value = confint(mod13a_cc)[, '2.5 %']
                 ),
                 data.frame(variable = names(coef(mod13a_cc)),
                            type = '97.5%',
                            model = 'cc',
                            value = confint(mod13a_cc)[, '97.5 %']
                 )
)


# ggplot version:
p13a <- densplot(mod13a, use_ggplot = TRUE, joined = TRUE) +
  theme(legend.position = 'bottom')


# add vertical lines for the:
# - confidence intervals from the complete case analysis
# - quantiles of the posterior distribution
p13a +
  geom_vline(data = quantDF, aes(xintercept = value, color = model),
             lty = 2) +
  scale_color_manual(name = 'CI from model: ',
                     limits = c('JointAI', 'cc'),
                     values = c('blue', 'red'),
                     labels = c('JointAI', 'compl.case'))

Model Summary

A summary of the posterior distribution estimated in a JointAI model can be obtained using the function summary().

The posterior summary consists of the mean, standard deviation and quantiles (by default, the 2.5% and 97.5% quantiles) of the MCMC samples from all chains combined. Moreover, the tail probability (see below), Gelman-Rubin criterion (see section below) and the ratio of the Monte Carlo error to the posterior standard deviation (see section below) are given.

Additionally, some essential characteristics of the MCMC samples on which the summary is based is given. This includes the range and number of iterations (“Sample size per chain”), thinning interval, and number of chains. Furthermore, the number of observations (number of rows in the data) is given.

summary(mod13a)
#> 
#> Bayesian linear model fitted with JointAI
#> 
#> Call:
#> lm_imp(formula = SBP ~ gender + WC + alc + creat, data = NHANES, 
#>     n.iter = 500, seed = 2020)
#> 
#> 
#> Posterior summary:
#>                Mean     SD  2.5%  97.5% tail-prob. GR-crit MCE/SD
#> (Intercept)  81.077 9.6921 61.66 99.602      0.000   1.011 0.0258
#> genderfemale  0.368 2.6138 -4.74  5.594      0.871   0.999 0.0258
#> WC            0.306 0.0736  0.16  0.448      0.000   1.012 0.0259
#> alc>=1        6.365 2.4692  1.38 10.897      0.016   1.006 0.0291
#> creat         7.747 7.5949 -7.19 22.496      0.299   1.003 0.0264
#> 
#> Posterior summary of residual std. deviation:
#>           Mean    SD 2.5% 97.5% GR-crit MCE/SD
#> sigma_SBP 14.4 0.779   13    16    1.02 0.0278
#> 
#> 
#> MCMC settings:
#> Iterations = 101:600
#> Sample size per chain = 500 
#> Thinning interval = 1 
#> Number of chains = 3 
#> 
#> Number of observations: 186

For mixed models, summary() also returns the posterior summary of the random effects covariance matrix D and the number of groups. The argument missinfo = TRUE will add information on the missing values in the variables involved in the model:

library(splines)
mod13b <- lme_imp(bmi ~ GESTBIR + ETHN + HEIGHT_M + ns(age, df = 3),
                  random = ~ 1 | ID,
                  data = subset(simLong, !is.na(bmi)),
                  n.iter = 500, no_model = 'age', seed = 2020)

summary(mod13b, missinfo = TRUE)
#> 
#> Bayesian linear mixed model fitted with JointAI
#> 
#> Call:
#> lme_imp(fixed = bmi ~ GESTBIR + ETHN + HEIGHT_M + ns(age, df = 3), 
#>     data = subset(simLong, !is.na(bmi)), random = ~1 | ID, n.iter = 500, 
#>     no_model = "age", seed = 2020)
#> 
#> 
#> Posterior summary:
#>                       Mean      SD    2.5%   97.5% tail-prob. GR-crit MCE/SD
#> (Intercept)      17.567738 2.30929 12.9758 21.9295      0.000    1.01 0.0271
#> GESTBIR          -0.049258 0.04749 -0.1427  0.0472      0.312    1.00 0.0258
#> ETHNother         0.022619 0.15328 -0.2763  0.3213      0.883    1.00 0.0269
#> HEIGHT_M          0.000925 0.00962 -0.0174  0.0195      0.936    1.02 0.0274
#> ns(age, df = 3)1 -0.245192 0.07574 -0.3947 -0.0950      0.000    1.00 0.0258
#> ns(age, df = 3)2  1.939673 0.11262  1.7100  2.1489      0.000    1.00 0.0258
#> ns(age, df = 3)3 -1.247972 0.05842 -1.3656 -1.1326      0.000    1.01 0.0258
#> 
#> 
#> Posterior summary of random effects covariance matrix:
#>                Mean     SD  2.5% 97.5% tail-prob. GR-crit MCE/SD
#> D_bmi_ID[1,1] 0.696 0.0781 0.552 0.858               1.02 0.0278
#> 
#> 
#> Posterior summary of residual std. deviation:
#>            Mean     SD  2.5% 97.5% GR-crit MCE/SD
#> sigma_bmi 0.684 0.0119 0.662 0.709       1 0.0269
#> 
#> 
#> MCMC settings:
#> Iterations = 101:600
#> Sample size per chain = 500 
#> Thinning interval = 1 
#> Number of chains = 3 
#> 
#> Number of observations: 1881 
#> Number of groups:
#>  - ID: 200
#> 
#> 
#> Number and proportion of complete cases:
#>         level    #   %
#> ID         ID  190  95
#> lvlone lvlone 1881 100
#> 
#> Number and proportion of missing values:
#>      level # NA % NA
#> bmi lvlone    0    0
#> age lvlone    0    0
#> 
#>          level # NA % NA
#> GESTBIR     ID    0    0
#> ID          ID    0    0
#> HEIGHT_M    ID    4    2
#> ETHN        ID    6    3

At the end of the summary output, a summary of the number and proportion of missing values per variable is printed

Tail probability

The tail probability, calculated as \(2\times\min\left\{Pr(\theta > 0), Pr(\theta < 0)\right\},\) where \(\theta\) is the parameter of interest, is a measure of how likely the value 0 is under the estimated posterior distribution. The figure visualizes three examples of posterior distributions and the corresponding minimum of \(Pr(\theta > 0)\) and \(Pr(\theta < 0)\) (shaded area):

Evaluation criteria

The convergence of the MCMC chains and precision of the posterior sample can also be evaluated more formally. Implemented in JointAI are the Gelman-Rubin criterion for convergence¹ and a comparison of the Monte Carlo Error with the posterior standard deviation.

Gelman-Rubin criterion for convergence

The Gelman-Rubin criterion evaluates convergence by comparing within and between chain variability and, thus, requires at least two MCMC chains to be calculated. It is implemented for JointAI objects in the function GR_crit(), which is based on the function gelman.diag() from the package coda. The upper limit of the confidence interval should not be much larger than 1.

GR_crit(mod13a)
#> Potential scale reduction factors:
#> 
#>              Point est. Upper C.I.
#> (Intercept)        1.01       1.02
#> genderfemale       1.00       1.01
#> WC                 1.00       1.01
#> alc>=1             1.00       1.01
#> creat              1.00       1.00
#> sigma_SBP          1.02       1.05
#> 
#> Multivariate psrf
#> 
#> 1.01

Besides the arguments start, end, thin, and subset, which are explained below, GR_crit() also takes the arguments of gelman.diag().

Monte Carlo Error

The precision of the MCMC sample can be checked with the function MC_error(). It uses the function mcmcse::mcmcse.mat() from the package mcmcse to calculate the Monte Carlo error (the error that is made since the sample is finite) and compares it to the standard deviation of the posterior sample. A rule of thumb is that the Monte Carlo error should not be more than 5% of the standard deviation². Besides the arguments explained below, MC_error() takes the arguments of mcmcse.mat().

MC_error(mod13a)
#>                est   MCSE    SD MCSE/SD
#> (Intercept)  81.08 0.2502 9.692   0.026
#> genderfemale  0.37 0.0675 2.614   0.026
#> WC            0.31 0.0019 0.074   0.026
#> alc>=1        6.37 0.0718 2.469   0.029
#> creat         7.75 0.2007 7.595   0.026
#> sigma_SBP    14.40 0.0217 0.779   0.028

MC_error() returns an object of class MCElist, which is a list containing matrices with the posterior mean, estimated Monte Carlo error, posterior standard deviation and the ratio of the Monte Carlo error to the posterior standard deviation, for the scaled (if they are part of the JointAI object) and unscaled (transformed back to the scale of the data) posterior samples. The associated print method prints only the latter.

To facilitate quick evaluation of the Monte Carlo error to posterior standard deviation ratio, plotting of an object of class MCElist using plot() shows this ratio for each (selected) node and automatically adds a vertical line at the desired cut-off (by default 5%).

par(mar = c(3, 5, 0.5, 0.5), mgp = c(2, 0.6, 0), mfrow = c(1, 2))
plot(MC_error(mod13a))  # left panel: all iterations 101:600
plot(MC_error(mod13a, end = 250))  # right panel: iterations 101:250

Subset of output

By default, the functions traceplot(), densplot(), summary(), GR_crit(), MC_Error() and predict() use all iterations of the MCMC sample and consider only the parameters of the analysis model (if they were monitored). In this section, we describe how the set of iterations and parameters to display can be changed using the arguments subset, start, end, thin and exclude_chains.

Subset of parameters

As long as the main parameters have been monitored in a JointAI object, only these parameters are returned in the model summary, plots and criteria shown above. When the main parameters of the analysis model were not monitored, i.e., monitor_params = c(analysis_main = FALSE), and the argument subset is not specified, all parameters that were monitored are displayed.

To display output for nodes other than the main parameters of the analysis model or for a subset of nodes, the argument subset needs to be specified.

Examples

To display only the parameters of the imputation models, we set subset = c(analysis_main = FALSE, other_models = TRUE) (after re-estimating the model with the monitor for these parameters switched on):

mod13c <- update(mod13a, monitor_params = c(other_models = TRUE))

summary(mod13c, subset = c(analysis_main = FALSE, other_models = TRUE))
#> 
#> Bayesian joint model fitted with JointAI 
#> 
#> Call:
#> lm_imp(formula = SBP ~ gender + WC + alc + creat, data = NHANES, 
#>     n.iter = 500, monitor_params = c(other_models = TRUE), seed = 2020)
#> 
#> 
#> # --------------------------------------------------------------------- #
#>   Bayesian binomial model for "alc"
#> # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - #
#> 
#> Posterior summary:
#>                  Mean     SD    2.5%   97.5% tail-prob. GR-crit MCE/SD
#> (Intercept)   0.51390 1.5325 -2.6068  3.4078     0.7080    1.01 0.0535
#> genderfemale -0.88236 0.3995 -1.6322 -0.0498     0.0373    1.04 0.0762
#> WC            0.00632 0.0115 -0.0169  0.0298     0.5627    1.01 0.0375
#> creat        -1.48151 1.2238 -3.9056  0.9785     0.2213    1.00 0.0603
#> 
#> 
#> # --------------------------------------------------------------------- #
#>   Bayesian linear model for "creat"
#> # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - #
#> 
#> Posterior summary:
#>                   Mean       SD      2.5%    97.5% tail-prob. GR-crit MCE/SD
#> (Intercept)   0.844704 0.076409  0.694938  0.99127      0.000       1 0.0258
#> genderfemale -0.178815 0.022122 -0.223627 -0.13699      0.000       1 0.0258
#> WC            0.000877 0.000772 -0.000612  0.00243      0.256       1 0.0258
#> 
#> Posterior summary of residual std. deviation:
#>              Mean      SD  2.5% 97.5% GR-crit MCE/SD
#> sigma_creat 0.145 0.00769 0.132 0.161    1.01 0.0277
#> 
#> 
#> # --------------------------------------------------------------------- #
#>   Bayesian linear model for "WC"
#> # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - #
#> 
#> Posterior summary:
#>               Mean   SD  2.5%   97.5% tail-prob. GR-crit MCE/SD
#> (Intercept)  97.41 1.52 94.48 100.469     0.0000    1.00 0.0258
#> genderfemale -5.16 2.21 -9.38  -0.971     0.0147    1.01 0.0258
#> 
#> Posterior summary of residual std. deviation:
#>          Mean    SD 2.5% 97.5% GR-crit MCE/SD
#> sigma_WC 14.5 0.785 13.2  16.3       1 0.0258
#> 
#> 
#> # ----------------------------------------------------------- #
#> 
#> MCMC settings:
#> Iterations = 101:600
#> Sample size per chain = 500 
#> Thinning interval = 1 
#> Number of chains = 3 
#> 
#> Number of observations: 186

To select only some of the parameters, they can be specified directly by name via the other element of subset.

densplot(mod13c, subset = list(analysis_main = FALSE,
                               other = c('beta[4]', 'beta[5]')), nrow = 1)

This also works when a subset of the imputed values should be displayed:

# re-fit the model and monitor the imputed values
mod13d <- update(mod13a, monitor_params = c(imps = TRUE))

# select all imputed values for 'WC' (4th column of M_lvlone)
sub3 <- grep('M_lvlone\\[[[:digit:]]+,4\\]', parameters(mod13d)$coef, value = TRUE)
sub3
#> [1] "M_lvlone[33,4]"  "M_lvlone[150,4]"

# pass "sub3" to "subset" via "other", for example in a traceplot:
traceplot(mod13d, subset = list(analysis_main = FALSE, other = sub3), ncol = 2)

When the number of imputed values is large or in order to check convergence of random effects, it may not be feasible to plot and inspect all trace plots. In that case a random subset of, for instance the random effects, can be selected (output not shown):

# re-fit the model monitoring the random effects
mod13e <- update(mod13b, monitor_params = c(ranef_main = TRUE))

# extract random intercepts
ri <- grep('^b_bmi_ID\\[[[:digit:]]+,1\\]$', colnames(mod13e$MCMC[[1]]), value = T)

# to plot the chains of 12 randomly selected random intercepts and slopes:
traceplot(mod13e, subset = list(analysis_main = FALSE,
                                other = sample(ri, size = 12)), ncol = 4)

Subset of MCMC samples

With the arguments start, end and thin it is possible to select which iterations from the MCMC sample are included in the summary. start and end specify the first and last iterations to be used, thin the thinning interval. Specification of start, thus, allows discarding a “burn-in”, i.e., the iterations before the MCMC chain had converged.

If a particular chain does not have converged it can be excluded from the summary or plot using the argument exclude_chains. This argument takes a numeric vector identifying chains to be excluded, for example:

mod14 <- lm_imp(SBP ~ gender + WC + alc + creat, data = NHANES, n.iter = 100,
                progress.bar = 'none', n.chains = 5)

densplot(mod14, ncol = 3)

densplot(mod14, exclude_chains = c(2,4), ncol = 3)

traceplot(mod14, thin = 10, ncol = 3)

traceplot(mod14, start = 150, ncol = 3)

traceplot(mod14, end = 120, ncol = 3)

Predicted values

Often, the aim of an analysis is not only to estimate the association between outcome and covariates but to predict future outcomes or outcomes for new subjects.

The function predict() allows us to obtain predicted values and corresponding credible intervals from JointAI objects. Note that for mixed models, currently, only marginal prediction but not prediction conditional on the random effects is implemented.

A dataset containing data which the prediction should be performed is specified via the argument newdata. If no newdata is given, the original data from the JointAI object are used. The argument quantiles allows the specification of the quantiles of the posterior sample that are used to obtain the credible interval (by default the 2.5% and 97.5% quantile). Arguments start, end, thin and exclude_chains control the subset of MCMC samples that is used.

predict(mod13a, newdata = NHANES[27, ])
#> $newdata
#>          SBP gender age             race   WC alc educ creat albu
#> 392 126.6667   male  32 Mexican American 94.1  <1  low  0.83  4.2
#>     uricacid bili occup  smoke      fit     2.5%    97.5%
#> 392      8.7    1  <NA> former 116.3273 112.4343 120.1817
#> 
#> $fitted
#>        fit     2.5%    97.5%
#> 1 116.3273 112.4343 120.1817

predict() returns a list with elements dat, fit and quantiles, containing newdata with the predicted values and quantiles appended, the predicted values and quantiles that form the credible interval.

Prediction to visualize non-linear effects

Another reason to obtain predicted values is the visualization of non-linear effects, i.e., to create an effect plot. To facilitate the generation of a dataset for such a prediction, the function predDF() can be used. It generates a data.frame that contains a sequence of values through the range of observed values for a covariate specified by the argument vars, and the median or reference value for all other continuous and categorical variables.

# create dataset for prediction
newDF <- predDF(mod13b, var = ~ age)

# obtain predicted values
pred <- predict(mod13b, newdata = newDF)
#> Warning: 
#> Prediction in multi-level settings currently only takes into
#> account the fixed effects, i.e., assumes that the random
#> effect realizations are equal to zero.

# plot predicted values and credible interval
matplot(pred$newdata$age, pred$fitted,
        lty = c(1,2,2), type = 'l', col = 1,
        xlab = 'age in months', ylab = 'predicted value')

It is possible to have multiple variables vary and to set values for these variables:

# create dataset for prediction
newDF <- predDF(mod13b, var = ~ age + HEIGHT_M + ETHN, HEIGHT_M = c(150, 180))

# obtain predicted values
pred <- predict(mod13b, newdata = newDF)
#> Warning: 
#> Prediction in multi-level settings currently only takes into
#> account the fixed effects, i.e., assumes that the random
#> effect realizations are equal to zero.


# plot predicted values and credible interval
library(ggplot2)
ggplot(pred$newdata, aes(x = age, y = fit, color = factor(HEIGHT_M),
                         fill = factor(HEIGHT_M))) +
  geom_ribbon(aes(ymin = `2.5%`, ymax = `97.5%`), alpha = 0.3) +
  geom_line(size = 1) +
  facet_wrap('ETHN') +
  theme(legend.position = 'top',
        panel.grid = element_blank()) +
  scale_fill_viridis_d(name = 'maternal height',
                       aesthetics = c('colour', 'fill')) +
  scale_y_continuous(name = 'Expected BMI', breaks = seq(15, 18, 0.5))

Export of imputed values

Imputed datasets can be extracted from a JointAI object (in which a monitor for the imputed values has been set, i.e., monitor_params = c(imps = TRUE)), with the function get_MIdat().

A completed dataset is created by taking the imputed values from a randomly chosen iteration of the MCMC sample, transforming them back to the original scale if scaling had been performed before the MCMC sampling, and filling them into the original incomplete data.

The argument m specifies the number of imputed datasets to be created, include controls whether the original data are included in the long format data.frame (default is include = TRUE), start specifies the first iteration that may be used and minspace is the minimum number of iterations between iterations eligible to be selected.

To make the selection of iterations to form the imputed data reproducible, a seed value can be specified via the argument seed.

When export_to_SPSS = TRUE the imputed data is exported to SPSS, i.e., a .txt file containing the data and a .sps file containing SPSS syntax to convert the data into an SPSS data file (with ending .sav) are written. Arguments filename and resdir allow the specification of the name of the .txt and .sps file and the directory they are written to.

get_MIdat() returns a long-format data.frame containing the imputed datasets (and by default the original data) stacked onto each other. The imputation number is given in the variable Imputation_, column .id contains a newly created id variable for each observation in cross-sectional data (multi-level data should already contain an id variable).

impDF <- get_MIdat(mod13d, m = 10, seed = 2018)

The function plot_imp_distr() allows us to visually compare the distribution of the observed and imputed values.

plot_imp_distr(impDF, nrow = 1)

---

1. Gelman, A. and Rubin, D.B. (1992). Inference from Iterative Simulation Using Multiple Sequences. Statistical Science 7(4), 457-472. doi: 10.1214/ss/1177011136.
   
   Brooks, S. P. and Gelman, A. (1998). General Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics 7(4), 434 - 455. doi: 10.1080/10618600.1998.10474787.↩︎

2. Lesaffre, E. M. and A. B. Lawson (2012). Bayesian Biostatistics. John Wiley & Sons. doi: 10.1002/9781119942412.↩︎