fisher_evaluation: Evaluation of the Fisher Information Matrix in Nonlinear...
In MIXFIM: Evaluation of the FIM in NLMEMs using MCMC

Description Usage Arguments Value Author(s) References Examples

fisher_evaluation is used to evaluate the Fisher information matrix for both continuous and discrete data in nonlinear mixed effect models using Markov Chains Monte Carlo.

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fisher_evaluation(t, y_ini=1, model, model2, model3, params, dim_b, 
set_seed=TRUE, seed=42, n_samp, n_rep=1, n_iter, n_burn, CV=FALSE, 
plot_graph=0, L_boot=1000, nb_patients=1)

`t`	Vector of sampling times (or doses).
`y_ini`	A possible value for the response y to initialize the MCMC process. The default value is set at 1 (which works for many types of outcomes: continuous, binary, ...).
`model`	Compiled STAN model describing the response model to sample in the conditionnal distribution of b given y.
`model2`	Compiled STAN model describing the response model for calculating the derivative of the log-likelihood with respoect to each parameter.
`model3`	Compiled STAN model describing the response model to sample in the marginal distribution of the response y.
`params`	Vector of parameters given as follows: fixed effetcs, variances of the random effets, standard deviations of the residual errors (if continuous data).
`dim_b`	Number of random effects.
`set_seed`	Boolean indicating if the seed shoud be fixed. The default value is set at TRUE.
`seed`	Integer for the fixed seed. Used only if `set_seed` is TRUE. The default value is set at 42.
`n_samp`	Integer representing the number of Monte Carlo (MC) samples, (i.e. number of samples for the outcome y).
`n_rep`	Integer representing the number of repeated measures at the same time (or dose) for each patient. The default value is set at 1 (for continuous data).
`n_iter`	Integer representing the number of Markov Chains Monte Carlo (MCMC) samples.
`n_burn`	Integer representing the number of burn-in samples for MCMC.
`CV`	Boolean indicating if some convergence information (variance of the determinant, mean of b, mean log-likelihood, ...) should be returned. The default value is set at FALSE.
`plot_graph`	An integer with value 0 (no graph should be plotted), 1 (graph of the determinant of the FIM), 2 (graph of the determinant of the FIM with confidence intervals assuming normal distribution), 3 (graph of the determinant of the FIM with bootstrap confidence intervals) or 4 (graph of the determinant of the FIM with both bootstrap confidence intervals and confidence intervals assuming normal distribution). The default value is set at 0.
`L_boot`	Number of samples for bootstrap estimation of the confidence intervals of the normalized determinant of the FIM. This argument is used/required only if plot_graph = 3 or 4. The default value is set at 1000.
`nb_patients`	Number of patients with the same elementary design for which the FIM is evaluated. The default value is set at 1.

An list is returned, composed of the following variables:

`FIM`	Expected Fisher information matrix (FIM). Of note, the FIM is an individual FIM and is calculated for `nb_patients` patients.
`FIM_covar`	Variance-covariance matrix of the FIM. (Of note, its dimension is of size 4 as the FIM is in dimension 2.)
`inv_FIM`	Inverse of the FIM.
`RSE`	Relative standard errors (square root of the diagonal elements of the inverse of the FIM).
`RSE_inf_boot`	Vector containing the lower bound of the bootstrap confidence interval of the RSEs.
`RSE_sup_boot`	Vector containing the upper bound of the bootstrap confidence interval of the RSEs.
`det_norm_FIM`	Normalized determinant of the FIM.
`det_IC_normal`	Vector containing the lower and upper bound of the confidence interval of the normalized determinant of the FIM assuming normal distribution.
`det_IC_boot`	Vector containing the lower and upper bound of the bootstrap confidence interval of the normalized determinant of the FIM.

If CV=TRUE:

`mean_dloglik1`	Mean of the partial derivatives of the log-likelihood according to the first MCMC sample and MC sample. Should be equal approximately to 0.
`mean_dloglik2`	Mean of the partial derivatives of the log-likelihood according to the second MCMC sample and MC sample. Should be equal approximately to 0.
`var_dloglik1`	Variance of the partial derivatives of the log-likelihood according to the first MCMC sample and MC sample.
`var_dloglik2`	Variance of the partial derivatives of the log-likelihood according to the second MCMC sample and MC sample.
`mean_b`	Mean of the samples in the conditionnal distribution of b given y. Should be equal approximately to 0.
`mat_A_k1`	Vector containing for each value sampled of the response y, the estimation of the integral of the partial derivatives of the log-likelihood over the random effects according to the first MCMC sample of the random effects b given y.
`mat_A_k2`	Vector containing for each value sampled of the response y, the estimation of the integral of the partial derivatives of the log-likelihood over the random effects according to the second MCMC sample of the random effects b given y.

In addition, plot_graph enables to plot a graph of the normalized determinant of the FIM with normal and bootstrap confidence intervals in function of the number of MC samples.

Marie-Karelle Riviere-Jourdan eldamjh@gmail.com

Riviere, M-K., Ueckert, S. and Mentre, F,. Evaluation of the Fisher information matrix in nonlinear mixed effect models using Markov Chains Monte Carlo.

############################
# PLEASE UNCOMMENT EXAMPLE #
############################
#times = c(0.5,1,2,6,24,36,72,120)
#params = c(1,8,0.15,0.6,0.02,0.07,0.1)

# Files cen be found in external data
#model = stan_model("model_b_given_y.stan")
#model2 = stan_model("model_derivatives.stan")
#model3 = stan_model("model_y.stan")

#model_Warfarin = fisher_evaluation(t=times, y_ini=0.5, model=model, 
#model2=model2, model3=model3, params=params, dim_b=3, set_seed=TRUE, seed=42, 
#n_samp=1000, n_rep=1, n_iter=200, n_burn=500, CV=TRUE, plot_graph=4, 
#nb_patients=32)
#model_Warfarin