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France Mentré - One of the best experts on this subject based on the ideXlab platform.
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A new method for evaluation of the Fisher Information Matrix for discrete mixed effect models using Monte Carlo sampling and adaptive Gaussian quadrature
2016Co-Authors: Sebastian Ueckert, France MentréAbstract:The design of experiments for discrete mixed effect models is challenging due to the unavailability of a closed-form expression for the Fisher Information Matrix (FIM), on which most optimality criteria depend. Existing approaches for the computation of the FIM for those models are all based on approximations of the likelihood. A new method is presented which is based on derivatives of the exact conditional likelihood and which uses Monte Carlo (MC) simulations as well as adaptive Gaussian quadrature (AGQ) to integrate those derivatives over the data and random effects. The method is implemented in R and evaluated with respect to the influence of the tuning parameter, the accuracy of the FIM approximation, and computational complexity. The accuracy evaluation is performed by comparing the expected relative standard errors (RSE) from the MC/AGQ FIM with RSE obtained in a simulation study with four different discrete data models (two binary, one count and one repeated time-to-event model) and three different estimation algorithms. Additionally, the results from the MC/AGQ FIM are compared with expected RSE from a marginal quasi-likelihood (MQL) approximated FIM. The comparison resulted in close agreement between the MC/AGQ-based RSE and empirical RSE for all models investigated, and better performance of MC/AGQ than the MQL approximated FIM for variance parameters. The MC/AGQ method also proved to be well suited to calculate the expected power to detect a group effect for a model with binary outcomes.
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Evaluation of the Fisher Information Matrix in nonlinear mixed effect models using adaptive Gaussian quadrature
2014Co-Authors: Thu Thuy Nguyen, France MentréAbstract:Nonlinear mixed effect models (NLMEM) are used in model-based drug development to analyse longitudinal data. To design these studies, the use of the expected Fisher Information Matrix (MF) is a good alternative to clinical trial simulation. Presently, MF in NLMEM is mostly evaluated with first-order linearisation. The adequacy of this approximation is, however, influenced by model nonlinearity. Alternatives for the evaluation of MF without linearisation are proposed, based on Gaussian quadratures. The MF, expressed as the expectation of the derivatives of the log-likelihood, can be obtained by stochastic integration. The likelihood for each simulated vector of observations is approximated by Gaussian quadrature centred at 0 (standard quadrature) or at the simulated random effects (adaptive quadrature). These approaches have been implemented in R. Their relevance was compared with clinical trial simulation and linearisation, using dose-response models, with various nonlinearity levels and different number of doses per patient. When the nonlinearity was mild, three approaches based on MF gave correct predictions of standard errors, when compared with the simulation. When the nonlinearity increased, linearisation correctly predicted standard errors of fixed effects, but over-predicted, with sparse designs, standard errors of some variability terms. Meanwhile, quadrature approaches gave correct predictions of standard errors overall, but standard Gaussian quadrature was very time-consuming when there were more than two random effects. To conclude, adaptive Gaussian quadrature is a relevant alternative for the evaluation of MF for models with stronger nonlinearity, while being more computationally efficient than standard quadrature.
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Fisher Information Matrix for nonlinear mixed effects multiple response models: evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model.
2009Co-Authors: Caroline Bazzoli, Sylvie Retout, France MentréAbstract:We focus on the Fisher Information Matrix used for design evaluation and optimization in nonlinear mixed effects multiple response models. We evaluate the appropriateness of its expression computed by linearization as proposed for a single response model. Using a pharmacokinetic-pharmacodynamic (PKPD) example, we first compare the computation of the Fisher Information Matrix with approximation to one derived from the observed Matrix on a large simulation using the stochastic approximation expectation-maximization algorithm (SAEM). The expression of the Fisher Information Matrix for multiple responses is also evaluated by comparison with the empirical Information obtained through a replicated simulation study using the first-order linearization estimation methods implemented in the NONMEM software (first-order (FO), first-order conditional estimate (FOCE)) and the SAEM algorithm in the MONOLIX software. The predicted errors given by the approximated Information Matrix are close to those given by the Information Matrix obtained without linearization using SAEM and to the empirical ones obtained with FOCE and SAEM. The simulation study also illustrates the accuracy of both FOCE and SAEM estimation algorithms when jointly modelling multiple responses and the major limitations of the FO method. This study highlights the appropriateness of the approximated Fisher Information Matrix for multiple responses, which is implemented in PFIM 3.0, an extension of the R function PFIM dedicated to design evaluation and optimization. It also emphasizes the use of this computing tool for designing population multiple response studies, as for instance in PKPD studies or in PK studies including the modelling of the PK of a drug and its active metabolite.
Guy Melard - One of the best experts on this subject based on the ideXlab platform.
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invertibility condition of the Fisher Information Matrix of a varmax process and the tensor sylvester Matrix
2020Co-Authors: Andre Klein, Guy MelardAbstract:In this paper the invertibility condition of the asymptotic Fisher Information Matrix of a controlled vector autoregressive moving average stationary process, VARMAX, is displayed in a theorem. It is shown that the Fisher Information Matrix of a VARMAX process becomes invertible if the VARMAX Matrix polynomials have no common eigenvalue. Contrarily to what was mentioned previously in a VARMA framework, the reciprocal property is untrue. We make use of tensor Sylvester matrices since checking equality of the eigenvalues of Matrix polynomials is most easily done in that way. A tensor Sylvester Matrix is a block Sylvester Matrix with blocks obtained by Kronecker products of the polynomial coefficients by an identity Matrix, on the left for one polynomial and on the right for the other one. The results are illustrated by numerical computations.
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an algorithm for the exact Fisher Information Matrix of vector armax time series
2014Co-Authors: Andre Klein, Guy MelardAbstract:Abstract In this paper an algorithm is developed for the exact Fisher Information Matrix of a Gaussian vector ARMAX or VARMAX process. The algorithm proposed in this paper is composed by Chandrasekhar recursion equations at a vector–Matrix level, and some of these recursions consist of derivatives based on appropriate differential rules applied to a state space model for a vector process. The chosen representation is such that the recursions extracted from the state space model are given in terms of expectations of derivatives of innovations, and not the process and observation disturbances. The algorithm will be illustrated by an example. On that example, a comparison is made with results from E4, a toolbox for Matlab, and with the asymptotic Information Matrix.
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on the resultant property of the Fisher Information Matrix of a vector arma process
2005Co-Authors: Andre A Klein, Guy Melard, Peter SpreijAbstract:A Matrix is called a multiple resultant Matrix associated to two Matrix polynomials when it becomes singular if and only if the two Matrix polynomials have at least one common eigenvalue. In this paper a new multiple resultant Matrix is introduced. It concerns the Fisher Information Matrix (FIM) of a stationary vector autoregressive and moving average time series process (VARMA). The two Matrix polynomials are the autoregressive and the moving average Matrix polynomials of the VARMA process. In order to show that the FIM is a multiple resultant Matrix two new representations of the FIM are derived. To construct such representations appropriate Matrix differential rules are applied. The newly obtained representations are expressed in terms of the multiple Sylvester Matrix and the tensor Sylvester Matrix. The representation of the FIM expressed by the tensor Sylvester Matrix is used to prove that the FIM becomes singular if and only if the autoregressive and moving average Matrix polynomials have at least one common eigenvalue. It then follows that the FIM and the tensor Sylvester Matrix have equivalent singularity conditions. In a simple numerical example it is shown however that the FIM fails to detect common eigenvalues due to some kind of numerical instability. Whereas the tensor Sylvester Matrix reveals it clearly, proving the usefulness of the results derived in this paper.
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an algorithm for computing the asymptotic Fisher Information Matrix for seasonal siso models
2004Co-Authors: Andre Klein, Guy MelardAbstract:The paper presents an algorithm for computing the asymptotic Fisher Information Matrix of a possibly seasonal single input single output (SISO) time series model. That Matrix is a block Matrix whose elements are basically integrals over the oriented unit circle of rational functions. The procedure makes use of the autocovariance function of one or the cross-covariance function of two autoregressive processes based on the same noise. The algorithm also works when the input variable is omitted, the case of a seasonal ARMA model.
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a program for computing the exact Fisher Information Matrix of a gaussian varma model
2004Co-Authors: Andre Klein, Guy Melard, Jerzy Niemczyk, Toufik ZahafAbstract:A program in the MATLAB environment is described for computing the Fisher Information Matrix of the exact Information Matrix of a Gaussian vector autoregressive moving average (VARMA) model. A computationally efficient procedure is used on the basis of a state space representation. It relies heavily on Matrix operations. An illustration of the procedure is given for simple VARMA models and an example of output from a more realistic application is discussed.
Andre Klein - One of the best experts on this subject based on the ideXlab platform.
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invertibility condition of the Fisher Information Matrix of a varmax process and the tensor sylvester Matrix
2020Co-Authors: Andre Klein, Guy MelardAbstract:In this paper the invertibility condition of the asymptotic Fisher Information Matrix of a controlled vector autoregressive moving average stationary process, VARMAX, is displayed in a theorem. It is shown that the Fisher Information Matrix of a VARMAX process becomes invertible if the VARMAX Matrix polynomials have no common eigenvalue. Contrarily to what was mentioned previously in a VARMA framework, the reciprocal property is untrue. We make use of tensor Sylvester matrices since checking equality of the eigenvalues of Matrix polynomials is most easily done in that way. A tensor Sylvester Matrix is a block Sylvester Matrix with blocks obtained by Kronecker products of the polynomial coefficients by an identity Matrix, on the left for one polynomial and on the right for the other one. The results are illustrated by numerical computations.
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Matrix algebraic properties of the Fisher Information Matrix of stationary processes
2014Co-Authors: Andre KleinAbstract:In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on Matrix algebraic properties of the Fisher Information Matrix (FIM) of stationary processes. The FIM is an ingredient of the Cram´er-Rao inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant Matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum Information, a different statistical distance measure is set forth. It is related to the Fisher Information but where the Information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized.
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an algorithm for the exact Fisher Information Matrix of vector armax time series
2014Co-Authors: Andre Klein, Guy MelardAbstract:Abstract In this paper an algorithm is developed for the exact Fisher Information Matrix of a Gaussian vector ARMAX or VARMAX process. The algorithm proposed in this paper is composed by Chandrasekhar recursion equations at a vector–Matrix level, and some of these recursions consist of derivatives based on appropriate differential rules applied to a state space model for a vector process. The chosen representation is such that the recursions extracted from the state space model are given in terms of expectations of derivatives of innovations, and not the process and observation disturbances. The algorithm will be illustrated by an example. On that example, a comparison is made with results from E4, a toolbox for Matlab, and with the asymptotic Information Matrix.
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transformed statistical distance measures and the Fisher Information Matrix
2012Co-Authors: Andre Klein, Peter SpreijAbstract:Abstract Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, q × q say, in which case the i-th column is given by u ( z i ) , where we write u ( z ) = ( 1 , z , … , z q − 1 ) ⊤ . If all the z i ( i = 1 , … , q ) are different, the Vandermonde Matrix is non-singular, otherwise not. The latter case obviously takes place when all z i are the same, z say, in which case one could speak of a confluent Vandermonde Matrix. Non-singularity is obtained if one considers the Matrix V ( z ) whose i-th column ( i = 1 , … , q ) is given by the ( i − 1 ) -th derivative u ( i − 1 ) ( z ) ⊤ . We will consider generalizations of the confluent Vandermonde Matrix V ( z ) by considering matrices obtained by using as building blocks the matrices M ( z ) = u ( z ) w ( z ) , with u ( z ) as above and w ( z ) = ( 1 , z , … , z r − 1 ) , together with its derivatives M ( k ) ( z ) . Specifically, we will look at matrices whose ij-th block is given by M ( i + j ) ( z ) , where the indices i , j by convention have initial value zero. These in general non-square matrices exhibit a block-Hankel structure. We will answer a number of elementary questions for this Matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on z? What are left or right inverses? It turns out that answers can be obtained by factorizing the Matrix into a product of other Matrix polynomials having a simple structure. The answers depend on the size of the Matrix M ( z ) and the number of derivatives M ( k ) ( z ) that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of Matrix polynomials is needed.
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tensor sylvester matrices and the Fisher Information Matrix of varmax processes
2010Co-Authors: Andre Klein, Peter SpreijAbstract:The purpose of this paper is to develop compact expressions for the Fisher Information Matrix (FIM) of a Gaussian stationary vector autoregressive and moving average process with exogenous or input variables, a vector ARMAX or VARMAX process. We develop a representation of the FIM based on multiple Sylvester matrices. An extension of this representation yields another one but in terms of tensor Sylvester matrices. In order to obtain the results presented in this paper, the approach used in [A. Klein, G. Melard, P. Spreij, On the resultant property of the Fisher Information Matrix of a vector ARMA process, Linear Algebra Appl. 403 (2005) 291-313] is extended.
Sylvie Retout - One of the best experts on this subject based on the ideXlab platform.
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Fisher Information Matrix for nonlinear mixed effects multiple response models: evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model.
2009Co-Authors: Caroline Bazzoli, Sylvie Retout, France MentréAbstract:We focus on the Fisher Information Matrix used for design evaluation and optimization in nonlinear mixed effects multiple response models. We evaluate the appropriateness of its expression computed by linearization as proposed for a single response model. Using a pharmacokinetic-pharmacodynamic (PKPD) example, we first compare the computation of the Fisher Information Matrix with approximation to one derived from the observed Matrix on a large simulation using the stochastic approximation expectation-maximization algorithm (SAEM). The expression of the Fisher Information Matrix for multiple responses is also evaluated by comparison with the empirical Information obtained through a replicated simulation study using the first-order linearization estimation methods implemented in the NONMEM software (first-order (FO), first-order conditional estimate (FOCE)) and the SAEM algorithm in the MONOLIX software. The predicted errors given by the approximated Information Matrix are close to those given by the Information Matrix obtained without linearization using SAEM and to the empirical ones obtained with FOCE and SAEM. The simulation study also illustrates the accuracy of both FOCE and SAEM estimation algorithms when jointly modelling multiple responses and the major limitations of the FO method. This study highlights the appropriateness of the approximated Fisher Information Matrix for multiple responses, which is implemented in PFIM 3.0, an extension of the R function PFIM dedicated to design evaluation and optimization. It also emphasizes the use of this computing tool for designing population multiple response studies, as for instance in PKPD studies or in PK studies including the modelling of the PK of a drug and its active metabolite.
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further developments of the Fisher Information Matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics
2003Co-Authors: Sylvie RetoutAbstract:We extend the development of the expression of the Fisher Information Matrix in nonlinear mixed effects models for designs evaluation. We consider the dependence of the marginal variance of the observations with the mean parameters and assume an heteroscedastic variance error model. Complex models with interoccasions variability and parameters quantifying the influence of covariates are introduced. Two methods using a Taylor expansion of the model around the expectation of the random effects or a simulated value, using then Monte Carlo integration, are proposed and compared. Relevance of the resulting standard errors is investigated in a simulation study with NONMEM.
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Fisher Information Matrix for non linear mixed effects models evaluation and application for optimal design of enoxaparin population pharmacokinetics
2002Co-Authors: Sylvie Retout, Rene BrunoAbstract:We address the problem of the choice and the evaluation of designs in population pharmacokinetic studies that use non-linear mixed-effects models. Criteria, based on the Fisher Information Matrix, have been developed to optimize designs and adapted to such models. We optimize designs under different constraints and evaluate them for a population pharmacokinetics study, within a new phase III trial of enoxaparin, a low molecular weight heparin. To do this, we approximate the expression of the Fisher Information Matrix for non-linear mixed-effects models including the residual error variance as a parameter to be estimated. We use the Fedorov-Wynn algorithm to minimize the inverse of the determinant of this Matrix as required by the D-optimality criterion. Two optimal designs, as well as a design defined by pharmacologists, are evaluated by the simulation of 30 replicated data sets with NONMEM; all designs involve 220 patients with four measurements per patient. We also evaluate the relevance of the standard errors of estimation given from the Fisher Information Matrix by comparison with those given by NONMEM. The three designs provide more precise population parameter estimates; the optimal design gives the best precision and offers a simple clinical implementation. The expected standard errors given by the Information Matrix are close to those obtained by NONMEM on the simulation. Moreover, the proposed criterion of D-optimality appears to be a good measure to compare designs for population studies.
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development and implementation of the population Fisher Information Matrix for the evaluation of population pharmacokinetic designs
2001Co-Authors: Sylvie Retout, Stephen B DuffullAbstract:In population pharmacokinetic studies, the precision of parameter estimates is dependent on the population design. Methods based on the Fisher Information Matrix have been developed and extended to population studies to evaluate and optimize designs. In this paper we propose simple programming tools to evaluate population pharmacokinetic designs. This involved the development of an expression for the Fisher Information Matrix for nonlinear mixed-effects models, including estimation of the variance of the residual error. We implemented this expression as a generic function for two software applications: S-PLUS and MATLAB. The evaluation of population designs based on two pharmacokinetic examples from the literature is shown to illustrate the efficiency and the simplicity of this theoretic approach. Although no optimization method of the design is provided, these functions can be used to select and compare population designs among a large set of possible designs, avoiding a lot of simulations.
Brian Munsky - One of the best experts on this subject based on the ideXlab platform.
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the finite state projection based Fisher Information Matrix approach to estimate Information and optimize single cell experiments
2019Co-Authors: Zachary R Fox, Brian MunskyAbstract:Modern optical imaging experiments not only measure single-cell and single-molecule dynamics with high precision, but they can also perturb the cellular environment in myriad controlled and novel settings. Techniques, such as single-molecule fluorescence in-situ hybridization, microfluidics, and optogenetics, have opened the door to a large number of potential experiments, which begs the question of how to choose the best possible experiment. The Fisher Information Matrix (FIM) estimates how well potential experiments will constrain model parameters and can be used to design optimal experiments. Here, we introduce the finite state projection (FSP) based FIM, which uses the formalism of the chemical master equation to derive and compute the FIM. The FSP-FIM makes no assumptions about the distribution shapes of single-cell data, and it does not require precise measurements of higher order moments of such distributions. We validate the FSP-FIM against well-known Fisher Information results for the simple case of constitutive gene expression. We then use numerical simulations to demonstrate the use of the FSP-FIM to optimize the timing of single-cell experiments with more complex, non-Gaussian fluctuations. We validate optimal simulated experiments determined using the FSP-FIM with Monte-Carlo approaches and contrast these to experiment designs chosen by traditional analyses that assume Gaussian fluctuations or use the central limit theorem. By systematically designing experiments to use all of the measurable fluctuations, our method enables a key step to improve co-design of experiments and quantitative models.
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the finite state projection based Fisher Information Matrix approach to estimate Information and optimize single cell experiments
2018Co-Authors: Zachary R Fox, Brian MunskyAbstract:Abstract Modern optical imaging experiments not only measure single-cell and single-molecule dynamics with high precision, but they can also perturb the cellular environment in myriad controlled and novel settings. Techniques, such as single-molecule fluorescence in-situ hybridization, microfluidics, and optogenetics, have opened the door to a large number of potential experiments, which begs the question of how best to choose the best possible experiment. The Fisher Information Matrix (FIM) estimates how well potential experiments will constrain model parameters and can be used to design optimal experiments. Here, we introduce the finite state projection (FSP) based FIM, which uses the formalism of the chemical master equation to derive and compute the FIM. The FSP-FIM makes no assumptions about the distribution shapes of single-cell data, and it does not require precise measurements of higher order moments of such distributions. We validate the FSP-FIM against well-known Fisher Information results for the simple case of constitutive gene expression. We then use numerical simulations to demonstrate the use of the FSP-FIM to optimize the timing of single-cell experiments with more complex, non-Gaussian fluctuations. We validate optimal simulated experiments determined using the FSP-FIM with Monte-Carlo approaches and contrast these to experiment designs chosen by traditional analyses that assume Gaussian fluctuations or use the central limit theorem. By systematically designing experiments to use all of the measurable fluctuations, our method enables a key step to improve co-design of experiments and quantitative models. Author summary A main objective of quantitative modeling is to predict the behaviors of complex systems under varying conditions. In a biological context, stochastic fluctuations in expression levels among isogenic cell populations have required modeling efforts to incorporate and even rely upon stochasticity. At the same time, new experimental variables such as chemical induction and optogenetic control have created vast opportunities to probe and understand gene expression, even at single-molecule and single-cell precision. With many possible measurements or perturbations to choose from, researchers require sophisticated approaches to choose which experiment to perform next. In this work, we provide a new tool, the finite state projection based Fisher Information Matrix (FSP-FIM), which considers all cell-to-cell fluctuations measured in modern data sets, and can design optimal experiments under these conditions. Unlike previous approaches, the FSP-FIM does not make any assumptions about the shape of the distribution being measured. This new tool will allow experimentalists to optimally perturb systems to learn as much as possible about single-cell processes with a minimum of experimental cost or effort.
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the finite state projection based Fisher Information Matrix approach to estimate and maximize the Information in single cell experiments
2018Co-Authors: Zachary R Fox, Brian MunskyAbstract:Modern experiments not only measure single-cell and single-molecule dynamics with high precision, but they can also perturb the cellular environment in myriad controlled and novel settings. Such techniques have opened the door to an infinite number of potential experiments, which begs the question of how best to choose the next experiment. The Fisher Information Matrix (FIM) estimates how well potential experiments will constrain model parameters and can be used to design optimal experiments. Here, we introduce the finite state projection (FSP) based FIM, which uses the formalism of the chemical master equation to derive and compute the FIM. The FSP-FIM makes no assumptions about the distribution shapes of single-cell data, and it does not require precise measurements of higher order moments of such distributions. We validate the FSP-FIM against well-known Fisher Information results for the simple case of constitutive gene expression. We then demonstrate the use of the FSP-FIM to optimize the timing of single-cell experiments with more complex, non-Gaussian fluctuations. We validate optimal experiments determined using the FSP-FIM with Monte-Carlo approaches and contrast these to experiments chosen by traditional analyses that assume Gaussian fluctuations or use the central limit theorem. By systematically designing experiments to use all of the measurable fluctuation Information, our method enables a key step to improve co-design of experiments and quantitative models.