Fitting a linear regression model
Fitting a linear regression model with JointAI is straightforward with the function lm_imp():
lm1 <- lm_imp(SBP ~ gender + age + race + WC + alc + educ + albu + bili,
data = NHANES, n.iter = 500, progress.bar = 'none')
The specification of lm_imp() is similar to the specification of a linear regression model for complete data using lm(). In this minimal example the only difference is that for lm_imp() the number of iterations n.iter has to be specified. Of course there are many more parameters that can or should be specified. In the vignette Model Specification many of these parameters are explained in detail.
n.iter specifies the length of the Markov Chain, i.e., the number of draws from the posterior distribution of the parameter or unobserved value. How many iterations are necessary depends on the data and complexity of the model and can vary from as few as 100 up to thousands or millions.
One important criterion is that the Markov chains need to have converged. This can be evaluated visually with a traceplot.
Traceplot
The function traceplot() produces a plot of the sampled values across iterations per parameter. By default, three Markov chains are produced for each parameter and represented by different colors.
When the sampler has converged the chains show a 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 the function add_samples()).
When convergence has been achieved, we can obtain the result of the model from the model summary.
Model Summary
Results from a model fitted with JointAI can be printed using summary():
summary(lm1)
#>
#> Linear model fitted with JointAI
#>
#> Call:
#> lm_imp(formula = SBP ~ gender + age + race + WC + alc + educ +
#> albu + bili, data = NHANES, n.iter = 500, progress.bar = "none")
#>
#> Posterior summary:
#> Mean SD 2.5% 97.5% tail-prob. GR-crit
#> (Intercept) 61.411 23.4312 14.4040 108.363 0.00933 1.005
#> genderfemale -3.218 2.2840 -7.6393 1.267 0.16400 1.003
#> age 0.364 0.0731 0.2182 0.503 0.00000 1.011
#> raceOther Hispanic 0.882 4.9074 -9.0399 10.469 0.85200 1.000
#> raceNon-Hispanic White -1.368 3.1853 -7.4884 4.923 0.64000 0.999
#> raceNon-Hispanic Black 9.053 3.6267 2.0569 16.194 0.01733 0.999
#> raceother 3.849 3.6480 -3.6682 10.906 0.29333 1.005
#> WC 0.235 0.0805 0.0791 0.399 0.00400 1.003
#> alc>=1 7.166 2.3004 2.6582 11.587 0.00267 0.999
#> educhigh -3.397 2.2360 -7.8988 0.954 0.12133 0.999
#> albu 5.279 4.2311 -3.1135 13.546 0.22400 1.006
#> bili -5.826 5.0408 -15.6047 4.216 0.26667 1.000
#>
#> Posterior summary of residual std. deviation:
#> Mean SD 2.5% 97.5% GR-crit
#> sigma_SBP 13.2 0.742 11.9 14.8 1.02
#>
#>
#> MCMC settings:
#> Iterations = 101:600
#> Sample size per chain = 500
#> Thinning interval = 1
#> Number of chains = 3
#>
#> Number of observations: 186
The output gives the posterior summary, i.e., the summary of the MCMC (Markov Chain Monte Carlo) sample (which consists of all chains combined).
By default, summary() will only print the posterior summary for the main model parameters of the analysis model. How to select which parameters are shown is described in the vignette Selecting Parameters.
The summary consists of the posterior mean, the standard deviation and the 2.5% and 97.5% quantiles of the MCMC sample, the tail probability and the Gelman-Rubin criterion for convergence.
The tail probability is a measure of how likely the value 0 is under the estimated posterior distribution, and is calculated as \[2\times\min\left\{Pr(\theta > 0), Pr(\theta < 0)\right\}\] (where \(\theta\) is the parameter of interest).
In the following graphics, the shaded areas represent the minimum of \(Pr(\theta > 0)\) and \(Pr(\theta < 0)\): 
The Gelman-Rubin criterion, also available via the function GR_crit(), compares the within and between chain variation. When it is close enough to 1, the chains can be assumed to have converged.
In the model summary, additionally, some important characteristics of the MCMC samples on which the summary is based, are given. This includes the range and number of iterations (= Sample size per chain), thinning interval and number of chains.
Furthermore, the number of observations (the sample size of the data) is given.
With the arguments start, end and thin it is possible to select which iterations from the MCMC sample are included in the summary.
For example:
When the traceplot shows that the chains only converged after 1500 iterations, start = 1500 should be specified in summary().
Plot of the posterior distributions
The posterior distributions can be visualized using the function densplot():
By default, densplot() plots the empirical distribution of each of the chains separately. When joined = TRUE the distributions of the combined chains are plotted.