Prediction and Simulation

Mathematical Derivations

Since residuals can be correlated, potentially existing observed outcomes of the same individual can be informative for predicting the unobserved valued of the same individual.

Assume that the data is sorted such that \(Y_{ij} = y_{ij}, j = k+1, k+2, \dots, p\) are observed and \(Y_{ij}, j = 1, 2, \dots, k\) are not. The special case of all outcomes being unobserved (new individual) is covered with \(k=p\).

Let further \[ \Sigma_i(X_i, \theta) = \begin{pmatrix} \Sigma_i^{new,new}(X_i,\theta) & \Sigma_i^{new,old}(X_i,\theta)\\ \Sigma_i^{old,new}(X_i,\theta) & \Sigma_i^{old,old}(X_i,\theta)\end{pmatrix} \]

be a block decomposition where \(\Sigma_i^{new,new}(X_i,\theta) = \Big(\big(\Sigma_i(X_i,\theta)\big)_{j,l}\Big)_{j = 1\dots k,\, l = 1\ldots k}\) and similarly for the other blocks.

Predictions can then be made based on the conditional distribution \[ Y_{i, 1\ldots k}\,|\,X_i,Y_{i,k+1\ldots p}=y_{i, k+1\ldots p}\sim\mathcal{N}(\mu_i, A_i) \]

with

\[ \mu_i(\beta,\theta) = (X_i \ \beta)_{1\ldots k} + \Sigma_i^{new,old}(X_i,\theta) \, \Big(\big(\Sigma_i^{old,old}(X_i,\theta)\big)^{-1} \big(y_i^{k+1\ldots p} - (X_i \ \beta)_{k+1\ldots p}\big)\Big) \] and

\[ A_i(\beta, \theta) = \Sigma_i^{new,new}(X_i,\theta) - \Sigma_i^{old,new}(X_i,\theta) \Big(\Sigma_i^{old,old}(X_i,\theta)\Big)^{-1} \Sigma_i^{new,old}(X_i,\theta) \ . \] Note that \(A_i\) does not depend on \(\beta\).

Implementation of predict

For implementing predict(), only \(\widehat{\mu}_i:=\mu_i(\widehat{\beta},\widehat{\theta})\) is required.

For predict(interval = "confidence") additionally standard errors are required. These could be derived using the delta methods since \(\mu_i\) is a function of the estimated model parameters \(\beta\) and \(\theta\). This would require the Jacobian \(\nabla\mu_i(\beta,\theta)|_{\big(\widehat{\beta},\widehat{\theta}\big)}\) in addition to the estimated variance covariance matrix of the parameter estimate \(\big(\widehat{\beta},\widehat{\theta}\big)\), \(\widehat{S}\). Standard errors for \(\widehat{\mu}^{\,(i)}\) are then given by the square root of the diagonal elements of \[ \Big(\nabla\mu_i(\beta,\theta)|_{\big(\widehat{\beta},\widehat{\theta}\big)}\Big)^\top\quad \widehat{S} \quad \Big(\nabla\mu_i(\beta,\theta)|_{\big(\widehat{\beta},\widehat{\theta}\big)}\Big) \] For predict(interval = "prediction") one would use the square root of the diagonal elements of \(A_i\big(\widehat{\beta},\widehat{\theta}\big)\) instead. The delta method could again be used to make upper and lower boundaries reflect parameter estimation uncertainty.

Alternatively, both intervals can be derived using a parametric bootstrap sample of the unrestricted parameters \(\theta\). This would probably also be easier for the interval = "prediction" case.

Please note that for these intervals, we assume that the distribution is approximately normal: we use \(\mu_{i,j}(\hat\beta, \hat\theta) \pm Z_{\alpha} * sqrt(A_{i, j, j}(\hat\beta, \hat\theta))\) to construct it, where \(\mu_{i,j}(\hat\beta, \hat\theta)\) is the \(j\)th element of \(\mu_i(\hat\beta, \hat\theta)\), and \(A_{i, j, j}(\hat\beta, \hat\theta)\) is the \(j,j\) element of \(A_i(\hat\beta, \hat\theta)\).

Parametric Sampling for Prediction Interval

With the conditional variance formula

\[ Var(Y_i) = Var(E(Y_i|\theta)) + E(Var(Y_i|\theta)) \]

and the conditional expectation \(E(Y_i|\theta)\) and the conditional variance \(Var(Y_i|\theta)\) being already described as

\[ E(Y_i|\theta) = \mu_i(\beta, \theta) \]

and

\[ Var(Y_i|\theta) = A_i(\beta, \theta), \]

we can sample on \(\theta\) and obtain \(\beta\), then calculate the variance of conditional mean and the mean of conditional variance.

Prediction of Conditional Mean for New Subjects

If there are no observations for a subject, then the prediction is quite simple:

\[ Y_i = X_i \hat\beta \]

Conditional Simulation

Under conditional simulation setting, the variance-covariance matrix, and the expectation of \(Y_i\) are already given in Mathematical Derivations.

Please note that in implementation of predict function, we only use the diagonal elements of \(A_i\), however, here we need to make use of the full matrix \(A_i\) to obtain correctly correlated simulated observations.

Marginal Simulation

To simulate marginally, we take the variance of \(\hat\theta\) and \(\hat\beta\) into consideration. For each simulation, we first generate \(\theta\) assuming it approximately follows a multivariate normal distribution. Then, conditional on the \(\theta\) we sampled, we generate \(\beta\) also assuming it approximately follows a multivariate normal distribution.

Now we have \(\theta\) and \(\beta\) estimates, and we just follow the conditional simulation.

Implementation of simulate

To implement simulate function, we first ensure that the expectation (\(\mu\)) and variance-covariance matrix (\(A\)) are generated in predict function, for each of the subjects.

For simulate(method = "conditional"), we use the estimated \(\theta\) and \(\beta\) to construct the \(\mu\) and \(A\) directly, and generate response with \(N(\mu, A)\) distribution.

For simulate(method = "marginal"), for each repetition of simulation, we generate \(\theta_{new}\) from the mmrm fit, where the estimate of \(\theta\) and variance-covariance matrix of \(\theta\) are provided. Using the generated \(\theta_{new}\), we then obtain the \(\beta_{new}\) and its variance-covariance matrix, with \(\theta_{new}\) and the data used in fit.

Then we sample \(\beta\) as follows. We note that on the C++ side we already have the robust Cholesky decomposition of the inverse of its asymptotic covariance matrix: \[ cov(\hat\beta) = (X^\top W X)^{-1} = (LDL^\top)^{-1} \] Hence we make sure to report the lower triangular matrix \(L\) and the diagonal matrix \(D\) back to the R side, and afterwards we can generate \(\beta\) samples as follows: \[ \beta_{sample} = \beta_{new} + L^{-\top}D^{-1/2}z_{sample} \] where \(z_{sample}\) is drawn from the standard multivariate normal distribution, since \[ cov(L^{-\top}D^{-1/2}z_{sample}) = L^{-\top}D^{-1/2} I_p D^{-1/2} L^{-1} = L^{-\top}D^{-1}L^{-1} = (LDL^\top)^{-1} = cov(\hat\beta) \] We note that calculating \(w = L^{-\top}D^{-1/2}z_{sample}\) is efficient via backwards solving \[ L^\top w = D^{-1/2}z_{sample} \] since \(L^\top\) is upper right triangular.

Then we simulate the observations once with simulate(method = "conditional", beta = beta_sample, theta = theta_new). We pool all the repetitions together and thus obtain the marginal simulation results.

predict options

1. predict(type = "confidence") gives the variance of the predictions conditional on the \(\theta\) estimate, taking into account the uncertainty of estimating \(\beta\). So here we ignore the uncertainty in estimating \(\theta\). It also does not add measurement error \(\epsilon\). We can use this prediction when we are only interested in predicting the mean of the unobserved \(y_i\), assuming the estimated \(\theta\) as the true variance parameters.
2. predict(type = "prediction") in contrast takes into account the full uncertainty, including the variance of \(\theta\) and the measurement error \(\epsilon\). We can use this prediction when we are interested in reliable confidence intervals for the unobserved \(y_i\) as well as the observed \(y_i\), assuming we would like to predict repeated observations from the same subjects and time points.

simulate options

1. simulate(type = "conditional") simulates observations while keeping the \(\theta\) and \(\beta\) estimates fixed. It adds measurement errors \(\epsilon\) when generating the simulated values. Hence the mean of the simulated values will be within the confidence intervals from predict(type = "conditional").
2. simulate(type = "marginal") simulates observations while taking into account the uncertainty of \(\beta\) and \(\theta\) through sampling from their asymptotic frequentist distributions. On top of that, it also adds measurement errors \(\epsilon\). This hence is using the same distribution as predict(type = "prediction").