A Simple Example: Multilevel Manifest AR(1) Model
In mlts: Multilevel Latent Time Series Models with 'R' and 'Stan'

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
options(rmarkdown.html_vignette.check_title = FALSE)

library(mlts)

One of the simplest models we can fit with mlts is a multilevel first order autoregressive model with only one observed variable. We start by specifying the model with mlts_model(). The argument q controls the number of time-series constructs. For this simple model, the following call is sufficient:

ar1_model <- mlts_model(q = 1)

We can check the parameters present in the model by just calling the object:

ar1_model

When mlts_model() sets up this model, all model parameters are free (i.e., not constrained) by default. On the within level, there are three fixed effects: the grand mean (mu_1) of the outcome, the autoregressive effect (phi(1)_11), and the natural log of the innovation variance (ln.sigma2_1). The (1) in brackets for the autoregressive effect parameter indicates a lag of first order, and the _11 subscript denotes that the first construct is as well predicted by the first construct. Note that for the innovation variance, the natural log is used to prevent the variance from dropping below zero. For each of these effects, random effects are also estimated on the between level, which are drawn from a multivariate normal distribution with zero mean. Standard deviations of random effects are indicated with a sigma_ prefix, and random effect correlations are indicated with an r_ prefix.

A TeX formula for the above model can be obtained by calling the mlts_model_formula() function on the model object. By default, the function produces an RMarkdown file and renders it to a pdf file using knitr. However, the TeX file can also be kept by calling keep_tex = TRUE within the function call.

mlts_model_formula(ar1_model)

[\text{Decomposition.}] [ \begin{gathered} \begin{bmatrix} y_{1, it} \ \end{bmatrix} = \begin{bmatrix} \mu_{1,i} \ \end{bmatrix} + \begin{bmatrix} y_{1, it}^w \ \end{bmatrix} \end{gathered} ] \vspace{1em} [\text{Within-level model.}] [ \begin{gathered} \begin{bmatrix} y_{1, it}^w \ \end{bmatrix} = \begin{bmatrix} \phi_{(1)11,i} \ \end{bmatrix} \begin{bmatrix} y_{1,i(t - 1)}^w \ \end{bmatrix} + \begin{bmatrix} \zeta_{1, it} \ \end{bmatrix} ,\ \text{with}~\zeta_{1,it} \sim \mathit{N}(0, \sigma^2_{\zeta_{1},i}) \end{gathered} ] \vspace{1em} [\text{Between-level model.}] [ \begin{gathered} \begin{bmatrix} \mu_{1,i}\ \phi_{(1)11,i}\ \ln(\sigma^2_{\zeta_{1},i})\ \end{bmatrix} = \begin{bmatrix} \gamma_{0,\mu_{1}}\ \gamma_{0,\phi_{(1)11}}\ \gamma_{0,\ln(\sigma^2_{\zeta_{1}})}\ \end{bmatrix} + \begin{bmatrix} \upsilon_{\mu_{1},i}\ \upsilon_{\phi_{(1)11},i}\ \upsilon_{\ln(\sigma^2_{\zeta_{1}}),i}\ \end{bmatrix} ,\ \text{with}~ \upsilon_{i} \sim \mathit{MVN}(\mathbf{0}, \mathbf{\Omega}) \end{gathered} ] \vspace{1em}

Furthermore, a path model can also be produced with the function mlts_model_paths(). Again, the function produces an RMarkdown and pdf file by default. A png file for each level can also be produced by calling add_png = TRUE and the TeX code for the path model can be kept with keep_tex = TRUE.

mlts_model_paths(ar1_model)

knitr::include_graphics(c("../vignettes/pathmodel_ar1.png"))

By path model convention, rectangles represent observed variables and circles represent latent (i.e., unobserved) variables. A single-headed arrow represents a directed path (i.e., a regression), a double-headed arrow represents (co-)variation, and a dot on a path indicates that this parameter can vary across between-level units. To fit the above model, we pass it together with the data set to mlts_fit(). The data set for this example is an artificial data set simulated from an autoregressive model:

load("../data/ar1_data.rda")

head(ar1_data)

We need to specify the variable in data that contains the time-series process in the ts argument and the variable that contains the unit identifier in the id argument. With the argument tinterval, the time interval for approximation of a continuous time process can be specified @Asparouhov2018. We don't specify it here, but see the Vignette Approximation of a Continuous Time Model for more details.

ar1_fit <- mlts_fit(
  model = ar1_model,
  data = ar1_data,
  id = "ID",
  ts = "Y1",
  iter = 4000
)

ar1_fit <- readRDS("../vignettes/ar1_fit.rds")

The model summary() shows general information about the model and data:

summary(ar1_fit)

The line Time series variables as indicated by parameter subscripts: 1 --> Y1 shows that model parameters indexed by a 1 refer to the variable Y1 in the data set. The Model convergence criteria provide an overview across convergence diagnostics for all model parameters (i.e., also parameters which are not printed in the summary() by default). For the simple AR1-model, all parameters converged well after 4,000 iterations.

The section Fixed Effects provides information about the fixed effects in the model, i.e., $\gamma_{0, \mu_1}$, $\gamma_{0, \phi_{(1)11}}$, and $\gamma_{0,\ln(\sigma^2_{\zeta_{1}})}$ in the above formula. For example, the posterior mean of the autoregressive effect parameter phi(1)_11 is estimated at .275 with 95%-credible interval [.218, .331]. The log variance of the innovations $\zeta_{1t}$ is estimated at -.304. The estimate needs to be exponentiated to be on the original scale: exp(-.304) = r round(exp(-.304), 3).

The section Random Effects SDs shows standard deviations of the random effects $\upsilon_{\mu_{1},i}$, $\upsilon_{\phi_{(1)11},i}$, and $\upsilon_{\ln(\sigma^2_{\zeta_{1}}),i}$. The section Random Effects Correlations shows correlations between random effects. For example, while random effects of the person mean $\upsilon_{\mu_{1},i}$ and the autoregressive effect $\upsilon_{\phi_{(1)11},i}$ display nearly no correlation, r round(ar1_fit$pop.pars.summary[ar1_fit$pop.pars.summary$Param == "r_mu_1.phi(1)_11", "mean"], 3), there is a positive correlation between the person mean and log innovation variance, r round(ar1_fit$pop.pars.summary[ar1_fit$pop.pars.summary$Param == "r_mu_1.ln.sigma2_1", "mean"], 3). This indicates that individuals with a higher person mean in the variable Y1 also tend to have a higher innovation variance.