Introduction

Panel data is common in various fields such as social sciences. These data consists of multiple subjects, followed over several time points, and there are often many observations per individual at each time, for example family status and income of each individual at each time point of interest. Such data can be analyzed in various ways, depending on the research questions and the characteristics of the data such as the number of individuals and time points, and the assumed distribution of the response variables. Popular, somewhat overlapping modeling approaches include dynamic panel models, fixed effect models, cross-lagged panel models (CLPM), CLPM with fixed or random effects Mulder and Hamaker (2021). Parameter estimation is typically based on ordinary least squares, generalized method of moments or maximum likelihood using structural equation models.

There are several R (R Core Team 2021) packages in CRAN focusing on panel data analysis. plm (Croissant and Millo 2008) provides various estimation methods and tests for linear panel data model, while fixest (Bergé 2018) supports multiple fixed effects and different distributions of response variables (based on stats::family). panelr (Long 2020) contains tools for panel data manipulation and estimation methods for “within-between” models which combine fixed effect and random effect models. This is done by using lme4 (Bates et al. 2015), geepack (Halekoh, Højsgaard, and Yan 2006), and brms (Bürkner 2018) packages as a backend. Finally, the lavaan (Rosseel 2012) package provides methods for general structural equation modeling, and thus can be used to estimate various panel data models such as cross-lagged panel models with fixed or random intercepts.

In traditional panel data models such as the ones provided by the aforementioned packages, the number of time points considered is often assumed to be relatively small, say less than 10, while the number of individuals can be for example hundreds or thousands. Due to the small number of time points, the effects of predictors are typically assumed to be time-invariant (although some extensions to time-varying effects have emerged (Hayakawa and Hou 2019)). On the other hand, when the number of time points is moderate or large, say hundreds or thousands, it can be reasonable to assume that the dynamics of the system change over time. There are various methods to model time-varying effects in (generalized) linear models based on state space models (Harvey and Phillips 1982; Durbin and Koopman 2012), with various R implementations (J. Helske 2022; Knaus et al. 2021; Brodersen et al. 2014). The state space modeling approaches are highly flexible but often computationally demanding due to assumed latent processes for the regression coefficients and an analytically intractable marginal likelihood, so they are most useful in settings with only few subjects. Methods based on the varying coefficients models (Hastie and Tibshirani 1993; Eubank et al. 2004) can be computationally more convenient, especially when the varying coefficients are based on splines. Some of the R implementations of time-varying coefficient models include (Casas and Fernandez-Casal 2019; Dziak et al. 2021).

The dynamite package provides an alternative approach to panel data inference which avoids some of the limitations and drawbacks of the aforementioned methods. First, dynamite supports estimating effects which vary smoothly over time according to splines, with penalization based on random walk priors. This allows modeling for example effects of interventions which increase or decrease over time. Second, dynamite supports wide variety of distributions for the response variables such as categorical distribution commonly encountered in life-course research (S. Helske, Helske, and Eerola 2018). Third, estimation is done in a fully Bayesian fashion using Stan (Stan Development Team 2022b, 2022a), which leads to transparent and interpretable quantification of parameter and predictive uncertainty. Finally, with dynamite we can model arbitrary number of measurements per individual (i.e. multiple channels following (S. Helske and Helske 2019)). Jointly modeling all endogenous variables allows us to consider long-term effects of interventions which take into account the interdependency of several variables of the model, and allows realistic simulation of complex counterfactual scenarios.

The dynamic multivariate panel model

Consider an individual \(i\) with observations \(y_{i,t} = (y^1_{i,t}, \ldots, y^c_{i,t},\ldots, y^C_{i,t})\), \(t=1,\ldots,T\), \(i = 1,\ldots,N\) i.e., at each time point \(t\) we have \(C\) observations from \(N\) individuals, where \(C\) is the number of channels, i.e., different response variables that have been measured. Assume that each element of \(y_{i,t}\) can depend on the past observations \(y_{i,t-\ell}\), \(\ell=1,\ldots\) (where the set of past values can be different for each channel) and also on additional exogenous covariates \(x_{i,t}\). In addition, \(y^c_{i,t}\) can depend on the elements of current \(y_{i,t}\), assuming that this does not lead to cyclic dependencies; this translates to a condition that the channels can be ordered so that \(y^c_{i,t}\) depends only on \(y^1_{i,t}, \ldots, y^{c-1}_{i,t}\), exogenous covariates, and past observations. For simplicity of the presentation, we now assume that the observations of each channel only depend on the previous time points, i.e., \(\ell = 1\) for all channels. We denote the set of all model parameters by \(\theta\). Now, assuming that the elements of \(y_{i,t}\) are independent given \(y_{i, t-1}\), \(x_{i,t}\), and \(\theta\) we have \[ \begin{aligned} y_{i,t} &\sim p_t(y_{i,t} | y_{i,t-1},x_{i,t},\theta) = \prod_{c=1}^C p^c_t(y^c_{i,t} | y^1_{i,t-1}, \ldots, y^C_{i,t-1}, x_{i,t}, \theta). \end{aligned} \] Importantly, the parameters of the conditional distributions \(p^c\) are time-dependent, enabling us to consider the evolution of the dynamics of our system over time.

Given a suitable link function depending on our distributional assumptions, we define a linear predictor \(\eta^c_{i,t}\) for the conditional distribution \(p^c_t\) of each channel \(c\) with the following general form: \[ \begin{aligned} \eta^c_{i,t} &= \alpha^c_{t} + X^{\beta, c}_{i,t} \beta^c + X^{\delta, c}_{i,t} \delta^c_{t} + X^{\nu, c}_{i,t} \nu_i^c + \lambda^c_i \psi^c_t, \end{aligned} \] where \(\alpha_t\) is the (possibly time-varying) common intercept term, \(X^{\beta, c}_{i,t}\) defines the predictors corresponding to the vector of time-invariant coefficients \(\beta^c\), and similarly for the time-varying term \(X^{\delta, c}_{i,t} \delta^c_{t}\). The term \(X^{\nu, c}_{i,t} \nu^c_{t}\) corresponds to individual-level random effects, where \(\nu^1_{i},\ldots, \nu^C_{i}\) are assumed to follow zero-mean Gaussian distribution, either with a diagonal or a full covariance matrix. Note that the predictors in \(X^{\beta, c}_{i,t}\), \(X^{\delta, c}_{i,t}\), and \(X^{\nu, c}_{i,t}\) may contain exogenous covariates or past observations of the response channels (or transformations of either).The final term \(\lambda^c_i \psi^c_t\) is a product of latent individual loadings \(\lambda^c_i\) and a univariate latent dynamic factor \(\psi^c_t\).

For time-varying coefficients \(\delta\) (and similarly for time-varying \(\alpha\) and latent factor \(\psi\)), we use Bayesian P-splines (penalized B-splines) (Lang and Brezger 2004) where \[ \delta^c_{k,t} = B_t \omega_k^c, \quad k=1,\ldots,K, \] where \(K\) is the number of covariates, \(B_t\) is a vector of B-spline basis function values at time \(t\) and \(\omega_k^c\) is a vector of corresponding spline coefficients. In general, the number of B-splines \(D\) used for constructing the splines for the study period \(1,\ldots,T\) can be chosen freely, but the actual value is not too important (as long as \(D \leq T\) and larger than the degree of the spline, e.g., three in case of cubic splines) (Wood 2020). Therefore, we use the same \(D\) basis functions for all time-varying effects. To mitigate overfitting due to too large a value of \(D\), we define a random walk prior for \(\omega^c_k\) as \[ \omega^c_{k,1} \sim p(\omega^c_{k,1}), \quad \omega^c_{k,d} \sim N(\omega^c_{k,d-1}, (\tau^c_k)^2), \quad d=2, \ldots, D. \] with a user defined prior \(p(\omega^c_{k,1})\) on the first coefficient, which due to the structure of \(B_1\) corresponds to the prior on the \(\delta^c_{k,1}\). Here, the parameter \(\tau^c_k\) controls the smoothness of the spline curves. While the different time-varying coefficients are modeled as independent a priori, the latent factors \(\psi\) can be modeled as correlated via correlated spline coefficients \(\omega\).

Defining the model

The models in the dynamite package are defined by combining the channel-specific formulas defined via the function dynamiteformula for which a shorthand alias obs is available. The function obs takes two arguments: formula and family, which define how the response variable of the channel depends on the predictor variables in the standard R formula syntax and the family of the response variable, respectively. These channel specific definitions are then combined into a single model with +. For example, the following formula

obs(y ~ lag(x), family = "gaussian") + 
obs(x ~ z, family = "poisson")

defines a model with two channels; first we declare that y is a gaussian variable depending on the previous value of x (lag(x)) and then we add a second channel declaring x as Poisson distributed depending on some exogenous variable z (for which we do not define any distribution). Note that the model formula can be defined without any reference to some external data, just as an R formula can. Currently the dynamite package supports the following distributions for the observations:

• Categorical: categorical (with a softmax link using the first category as reference). See the documentation of the categorical_logit_glm in the Stan function reference manual (https://mc-stan.org/users/documentation/).
• Gaussian: gaussian (identity link, parameterized using mean and standard deviation).
• Poisson: poisson (log-link, with an optional known offset variable).
• Negative-binomial: negbin (log-link, using mean and dispersion parameterization, with an optional known offset variable). See the documentation on NegBinomial2 in the Stan function reference manual.
• Bernoulli: bernoulli (logit-link).
• Binomial: binomial (logit-link).
• Exponential: exponential (log-link).
• Gamma: gamma (log-link, using mean and shape parameterization).
• Beta: beta (logit-link, using mean and precision parameterization).

obs(y ~ z, family = "gaussian") + 
obs(x ~ z, family = "poisson") +
lags(k = 1)

obs(y ~ z + lag(x, k = 1), family = "gaussian") + 
obs(x ~ z + lag(y, k = 1), family = "poisson")

obs(x ~ z + random(~1 + z), family = "gaussian")

obs(y ~ lag(log1x), family = "gaussian") + 
obs(x ~ z, family = "poisson") +
aux(numeric(log1x) ~ log(1 + x) | init(0))

obs(y ~ lag(log1x), family = "gaussian") + 
obs(x ~ z, family = "poisson") +
aux(numeric(log1x) ~ log(1 + x) | past(log(z)))

dynamite(dformula, data, time, group = NULL, 
  priors = NULL, backend = "rstan",
  verbose = TRUE, verbose_stan = FALSE, debug = NULL, ...)

dynamite(
  dformula = obs(x ~ varying(~ -1 + w), family = "poisson") + splines(df = 10), 
  data = d, 
  time = "year", 
  group = "id"
  chains = 2, 
  cores = 2
)

Example of Multichannel Model

As an example, consider the following simulated multichannel data:

library(dynamite)
head(multichannel_example)
#>   id time          g  p b
#> 1  1    1 -0.6264538  5 1
#> 2  1    2 -0.2660091 12 0
#> 3  1    3  0.4634939  9 1
#> 4  1    4  1.0451444 15 1
#> 5  1    5  1.7131026 10 1
#> 6  1    6  2.1382398  8 1

The data contains 50 unique groups (variable id), over 20 time points (time), a continuous variable g, a variable with non-negative integer values p, and a binary variable b. We define the following model (which actually matches the data generating process used in simulating the data):

set.seed(1)
f <- obs(g ~ lag(g) + lag(logp), family = "gaussian") +
  obs(p ~ lag(g) + lag(logp) + lag(b), family = "poisson") +
  obs(b ~ lag(b) * lag(logp) + lag(b) * lag(g), family = "bernoulli") +
  aux(numeric(logp) ~ log(p + 1))

As a directed acyclic graph (DAG), the model for the first three time points is

(The deterministic channel logp is omitted for clarity).

We fit the model using the dynamite function. In order to satisfy the package size requirements of CRAN, we have to use small number of iterations and additional thinning:

multichannel_example_fit <- dynamite(
  f, data = multichannel_example, 
  time = "time", group = "id"
  chains = 1, cores = 1, iter = 2000, warmup = 1000, init = 0, refresh = 0,
  thin = 5, save_warmup = FALSE)

You can update the estimated model which is included in the package as

multichannel_example_fit <- update(multichannel_example_fit,
  iter = 2000,
  warmup = 1000,
  chains = 4,
  cores = 4,
  refresh = 0,
  save_warmup = FALSE
)

Note that the dynamite call above produces a warning message related to the deterministic channel logp:

#> Warning: Deterministic channel `logp` has a maximum lag of 1 but you've supplied no
#> initial values:
#> ℹ This may result in NA values for `logp`.

This happens because the formulas of the other channels have lag(logp) as a predictor, whose value is undefined at the first time point without the initial value. However, in this case we can safely ignore the warning because the model contains lags of b and g as well meaning that the first time point in the model is fixed and does not enter into the model fitting process.

We can obtain posterior samples or summary statistics of the model using the as.data.frame, coef, and summary functions, but here we opt for visualizing the results using the plot_betas function:

plot_betas(multichannel_example_fit)

Note the naming of the model parameters; for example beta_b_g_lag1 corresponds to a fixed coefficient beta of lagged variable g of the channel b.

Assume now that we are interested in studying the causal effect of variable \(b_5\) on variable \(g_t\) at \(t = 6, \ldots, 20\). There is no direct effect from \(b_5\) to \(g_6\), but as \(g_t\) at affects \(b_{t+1}\) (and \(p_{t+1}\)), which in turn has an effect on all the variables at \(t+2\), we should see an indirect effect of \(b_5\) to \(g_t\) from time \(t=7\) onward.

For this task, we first create a new dataset where the values of our response variables after time 5 are set to NA:

d <- multichannel_example
d <- d |> 
  dplyr::mutate(
    g = ifelse(time > 5, NA, g),
    p = ifelse(time > 5, NA, p),
    b = ifelse(time > 5, NA, b))

We then predict time points 6 to 20 when \(b\) at time 5 is 0 or 1 for everyone:

newdata0 <- d |> 
  dplyr::mutate(
    b = ifelse(time == 5, 0, b))
pred0 <- predict(multichannel_example_fit, newdata = newdata0, type = "mean")
newdata1 <- d |> 
  dplyr::mutate(
    b = ifelse(time == 5, 1, b))
pred1 <- predict(multichannel_example_fit, newdata = newdata1, type = "mean")

By default, the output from predict is a single data frame containing the original new data and the samples from the posterior predictive distribution new observations. By defining type = "mean" we specify that we are interested in the posterior distribution of the expected values instead. In this case the the predicted values in a data frame are in the columns g_mean, p_mean, and b_mean where the NA values of newdata are replaced with the posterior samples from the model (the output also contains additional column corresponding to the auxiliary channel logp and index variable .draw). For more memory efficient format, by using argument expand = FALSE the output is returned as a list with two components, simulated and observed, with new samples and original newdata respectively.

head(pred0, n = 10) # first rows are NA as those were treated as fixed
#>      g_mean   p_mean    b_mean      logp id time .draw          g  p  b
#> 1        NA       NA        NA 1.7917595  1    1     1 -0.6264538  5  1
#> 2        NA       NA        NA 2.5649494  1    2     1 -0.2660091 12  0
#> 3        NA       NA        NA 2.3025851  1    3     1  0.4634939  9  1
#> 4        NA       NA        NA 2.7725887  1    4     1  1.0451444 15  1
#> 5        NA       NA        NA 2.3978953  1    5     1  1.7131026 10  0
#> 6  1.836878 3.658028 0.7359881 1.0986123  1    6     1         NA NA NA
#> 7  1.583641 4.057826 0.7063930 0.6931472  1    7     1         NA NA NA
#> 8  1.151348 3.476842 0.6524427 1.6094379  1    8     1         NA NA NA
#> 9  1.247886 2.469857 0.6889052 1.3862944  1    9     1         NA NA NA
#> 10 1.612244 5.014211 0.6963598 1.9459101  1   10     1         NA NA NA

We can now compute summary statistics over the individuals and then over the posterior samples to obtain posterior distribution of the causal effects as

sumr <- dplyr::bind_rows(
  list(b0 = pred0, b1 = pred1), .id = "case") |> 
  dplyr::group_by(case, .draw, time) |> 
  dplyr::summarise(mean_t = mean(g_mean)) |> 
  dplyr::group_by(case, time) |> 
  dplyr::summarise(
    mean = mean(mean_t),
    q5 = quantile(mean_t, 0.05, na.rm = TRUE), 
    q95 = quantile(mean_t, 0.95, na.rm = TRUE))
#> `summarise()` has grouped output by 'case', '.draw'. You can override using the
#> `.groups` argument.
#> `summarise()` has grouped output by 'case'. You can override using the
#> `.groups` argument.

It is also possible to do the marginalization over groups inside predict by using the argument funs which can be used to provide a named list of lists which contains functions to be applied for the corresponding channel. This approach can save considerably amount of memory in case of large number of groups. The names of the outermost list should be channel names. In our case we can write

pred0b <- predict(multichannel_example_fit, newdata = newdata0, type = "mean", 
  funs = list(g = list(mean_t = mean)))$simulated
pred1b <- predict(multichannel_example_fit, newdata = newdata1, type = "mean", 
  funs = list(g = list(mean_t = mean)))$simulated
sumrb <- dplyr::bind_rows(
  list(b0 = pred0b, b1 = pred1b), .id = "case") |> 
  dplyr::group_by(case, time) |> 
  dplyr::summarise(
    mean = mean(mean_t_g),
    q5 = quantile(mean_t_g, 0.05, na.rm = TRUE), 
    q95 = quantile(mean_t_g, 0.95, na.rm = TRUE))

The resulting tibble sumrb is equal to the previous sumr (apart from randomness due to simulation of new trajectories).

We can the visualize our predictions with the ggplot2 (Wickham 2016) as

library(ggplot2)
#> Warning: package 'ggplot2' was built under R version 4.2.2
ggplot(sumr, aes(time, mean)) + 
  geom_ribbon(aes(ymin = q5, ymax = q95), alpha = 0.5) +
  geom_line() + 
  scale_x_continuous(n.breaks = 10) + 
  facet_wrap(~ case) +
  theme_bw()
#> Warning: Removed 5 rows containing missing values (`geom_line()`).

Note that these estimates do indeed coincide with the causal effects (assuming of course that our model is correct), as we can find the backdoor adjustment formula (Pearl 1995) \[ E(g_t | do(b_5 = x)) = \sum_{g_5, p_5}E(g_t | b_5 = x, g_5, p_5) P(g_5, p_5), \] which is equal to mean_t in above codes.

We can also compute the difference \(P(g_t | do(b_5 = 1)) - P(g_t | do(b_5 = 0))\) as

sumr_diff <- dplyr::bind_rows(
  list(b0 = pred0, b1 = pred1), .id = "case") |> 
  dplyr::group_by(.draw, time) |>
  dplyr::summarise(
    mean_t = mean(g_mean[case == "b1"] - g_mean[case == "b0"])) |>
  dplyr::group_by(time) |>
  dplyr::summarise(
    mean = mean(mean_t),
    q5 = quantile(mean_t, 0.05, na.rm = TRUE),
    q95 = quantile(mean_t, 0.95, na.rm = TRUE))
#> `summarise()` has grouped output by '.draw'. You can override using the
#> `.groups` argument.

ggplot(sumr_diff, aes(time, mean)) +
  geom_ribbon(aes(ymin = q5, ymax = q95), alpha = 0.5) +
  geom_line() +
  scale_x_continuous(n.breaks = 10) +
  theme_bw()
#> Warning: Removed 5 rows containing missing values (`geom_line()`).

Which shows that there is a short term effect of \(b\) on \(g\), although the posterior uncertainty is quite large. Note that for multistep causal effects these estimates contain also Monte Carlo variation due to the simulation of the trajectories (e.g., there is also uncertainty in mean_t for fixed parameters given finite number of individuals we are marginalizing over).