Fixed errors in a few references

Fixed error in a few references that prevented having more than one contiguous reference.

Author: feralaes

manuscript/cSTM_Tutorial_Intro.Rmd

@@ -199,12 +199,10 @@ knitr::opts_chunk$set(fig.path="figs/", fig.width=8, fig.height=6, fig.align="ce
 
 Policymakers are often tasked with allocating limited healthcare resources under constrained budgets and uncertainty about future outcomes. Health economic evaluations might inform their final decisions. These economic evaluations often rely on decision models to synthesize evidence from different sources and project long-term outcomes of various alternative strategies. A commonly used decision model is the discrete-time cohort state-transition model (cSTM), often referred to as a Markov model.[@Kuntz2017]
 
-A cSTM is a dynamic mathematical model in which a hypothetical cohort of individuals transition between different health states over time. A cSTM is most appropriate when the decision problem has a dynamic component (e.g., the disease process can vary over time) and can be described using a reasonable number of health states. cSTMs are often used because of their transparency, efficiency, ease of development, and debugging. cSTMs are usually computationally less demanding than iSTMs, providing the ability to conduct PSA and value-of-information (VOI) analyses that otherwise might not be computationally feasible with iSTMs.[@Siebert2012c] cSTMs have been used to evaluate screening and surveillance programs,[@Suijkerbuijk2018; @Sathianathen2018a] diagnostic procedures,[@Lu2018b] disease management programs,[@Djatche2018], interventions,[@Smith-Spangler2010] and policies.[@Pershing2014] 
+A cSTM is a dynamic mathematical model in which a hypothetical cohort of individuals transition between different health states over time. A cSTM is most appropriate when the decision problem has a dynamic component (e.g., the disease process can vary over time) and can be described using a reasonable number of health states. cSTMs are often used because of their transparency, efficiency, ease of development, and debugging. cSTMs are usually computationally less demanding than iSTMs, providing the ability to conduct PSA and value-of-information (VOI) analyses that otherwise might not be computationally feasible with iSTMs.[@Siebert2012c] cSTMs have been used to evaluate screening and surveillance programs,[@Suijkerbuijk2018; @Sathianathen2018a] diagnostic procedures,[@Lu2018b] disease management programs,[@Djatche2018] interventions,[@Smith-Spangler2010] and policies.[@Pershing2014] 
 
 In a recent review, we illustrated the increased use of R's statistical programming framework in health decision sciences. We provided a summary of available resources to apply to medical decision making.[@Jalal2017b] Many packages have been explicitly developed to estimate and construct cSTMs in R. For example, the `markovchain`[@Spedicato2017] and `heemod`[@Filipovic-Pierucci2017] packages are designed to build cSTMs using a pre-defined structure. The `markovchain` package simulates time-independent, and time-dependent Markov chains but is not designed to conduct economic evaluations. `heemod` is a well-structured R package for economic evaluations. However, these packages are necessarily stylized and require users to specify the structure and inputs of their cSTM in a particular way potentially without fully understanding how cSTMs work. Using these packages can be challenging if the desired cSTM does not fit within this structure.
 
-<!-- This introductory tutorial covers time-independent cSTMs and how to use them to conduct a full cost-effectiveness analysis (CEA) comparing multiple interventions and implementing probabilistic sensitivity analyses. The accompanying advanced tutorial covers cSTMs for which transition probabilities depend on simulation time and time spent in a state, calculation of various epidemiological outcomes, and incorporation of transition rewards.[@Alarid-Escudero2021b]  -->
-
 This tutorial demonstrates how to conduct a full cost-effectiveness analysis (CEA) comparing multiple interventions and implementing probabilistic sensitivity analysis (PSA) without needing a specialized cSTM package. We first describe each of the components of a time-independent cSTM. Then, we illustrate the implementation of these components with an example. Our general conceptualization should apply to other programming languages (e.g., MATLAB, Python, C++, and Julia). The reader can find the most up-to-date R code of the time-independent cSTM and the R code to create the tutorial graphs in the accompanying GitHub repository (https://github.com/DARTH-git/cohort-modeling-tutorial-intro) to replicate and modify the example to fit their needs. We assume that the reader is familiar with the basics of decision modeling and has a basic understanding of programming. Thus, a prior introduction to R, "for" loops, and linear algebra for decision modelers is recommended. The linear algebra concepts used throughout the code are explained in more detail in the Supplementary Material.
 
 This introductory tutorial aims to (1) conceptualize time-independent cSTMs for implementation in a programming language and (2) provide a template for implementing these cSTMs in _base_ R. We focus on using R _base_ packages, ensuring modelers understand the concept and structure of cSTMs and avoid the limitation of constructing cSTMs in a package-specific structure. We used previously developed R packages for visualizing CEA results and checking cSTMs are correctly specified.
@@ -630,7 +628,7 @@ v_cost_strAB <- m_M_strAB %*% v_c_strAB
 ```
 
 ## Within-cycle correction
-A discrete-time cSTM involves an approximation of continuous-time dynamics to discrete points in time. The discretization might introduce biases when estimating outcomes based on state occupancy.[@VanRosmalen2013] One approach to reducing these biases is to shorten the cycle length, requiring simulating the model for a larger number of cycles, which can be computationally burdensome. Another approach is to use within-cycle corrections (WCC).[@Siebert2012c, @Hunink2014] In this tutorial, we use Simpson's 1/3rd rule by multiplying the rewards (e.g., costs and effectiveness) by $1/3$ in the first and last cycles, by $4/3$ for the odd cycles, and by $2/3$ for the even cycles.[@Elbasha2016, @Elbasha2016a] We implement the WCC by generating a column vector $\mathbf{wcc}$ of size $n_T+1$ with values corresponding to the first, $t=0$, and last cycle, $t= n_T$, equal to $1/3$, and the entries corresponding to the even and odd cycles with $2/3$ and $4/3$, respectively.
+A discrete-time cSTM involves an approximation of continuous-time dynamics to discrete points in time. The discretization might introduce biases when estimating outcomes based on state occupancy.[@VanRosmalen2013] One approach to reducing these biases is to shorten the cycle length, requiring simulating the model for a larger number of cycles, which can be computationally burdensome. Another approach is to use within-cycle corrections (WCC).[@Siebert2012c;@Hunink2014] In this tutorial, we use Simpson's 1/3rd rule by multiplying the rewards (e.g., costs and effectiveness) by $1/3$ in the first and last cycles, by $4/3$ for the odd cycles, and by $2/3$ for the even cycles.[@Elbasha2016;@Elbasha2016a] We implement the WCC by generating a column vector $\mathbf{wcc}$ of size $n_T+1$ with values corresponding to the first, $t=0$, and last cycle, $t= n_T$, equal to $1/3$, and the entries corresponding to the even and odd cycles with $2/3$ and $4/3$, respectively.
 
 \[
   \mathbf{wcc} = \left[\frac{1}{3}, \frac{2}{3}, \frac{4}{3}, \cdots, \frac{1}{3}\right]
@@ -869,13 +867,13 @@ plot(elc_obj, log_y = FALSE,
 # Discussion
 In this introductory tutorial, we provided a step-by-step mathematical conceptualization of time-independent cSTMs and a walk-through of their implementation in R using a hypothetical disease example with accompanying code throughout the tutorial. While some of the presented implementation details are specific to the R programming language, much of the code structure shown in this tutorial would be similar in other programming languages. Thus, readers may use this tutorial as a template for coding cSTMs more generally in different programming languages.
 
-The parameterization of our example model assumes all parameters are known, or at least, the characterization of their uncertainty is known (i.e., we know their distributions). However, to construct a real-world cSTM, modelers must conduct a thorough synthesis of current evidence to determine these appropriate structures and inform all parameters based on the current evidence. For example, literature must be carefully considered when determining whether transitions between non-death health states are estimated conditional on being alive or are estimated as competing risks along with mortality risks.[@Briggs2012] Similarly, our PSA analysis simplifies reality where all model parameters are assumed to be independent of each other. However, parameters could be correlated or have a rank order, and appropriate statistical methods that simulate these correlations or rank order might be needed.[@Goldhaber-Fiebert2015] We encourage modelers to use appropriate statistical methods to accurately synthesize and quantify model parameters' uncertainty. For example, for the PSA of our case study, we used distributions based on the type of parameters following standard recommendations. For a more detailed description of how to choose distributions, we refer the reader to other literature.[@Briggs2002; @Briggs2003] In addition, modelers should appropriately specify all model parameters for the cycle length of the model.[@Hunink2014]
+The parameterization of our example model assumes all parameters are known, or at least, the characterization of their uncertainty is known (i.e., we know their distributions). However, to construct a real-world cSTM, modelers must conduct a thorough synthesis of current evidence to determine these appropriate structures and inform all parameters based on the current evidence. For example, literature must be carefully considered when determining whether transitions between non-death health states are estimated conditional on being alive or are estimated as competing risks along with mortality risks.[@Briggs2012] Similarly, our PSA analysis simplifies reality where all model parameters are assumed to be independent of each other. However, parameters could be correlated or have a rank order, and appropriate statistical methods that simulate these correlations or rank order might be needed.[@Goldhaber-Fiebert2015] We encourage modelers to use appropriate statistical methods to accurately synthesize and quantify model parameters' uncertainty. For example, for the PSA of our case study, we used distributions based on the type of parameters following standard recommendations. For a more detailed description of how to choose distributions, we refer the reader to other literature.[@Briggs2002;@Briggs2003] In addition, modelers should appropriately specify all model parameters for the cycle length of the model.[@Hunink2014]
 
 In general, cSTMs are recommended when the number of states is considered "not too large".[@Siebert2012c] This recommendation arises because it becomes more challenging to keep track of their construction as the number of states increases. It is possible to build reasonably complex cSTMs in R as long as the computer's RAM can store the size of the transition probability matrix and outputs of interest. For time-independent cSTMs, in general, this should not be a problem with the capacity of current RAM in personal computers. An alternative to reduce the explosion of disease states is individual-based state-transition model (iSTM), which is a type of STM where simulated individuals transition between health states over time.[@Siebert2012c] We have previously published a tutorial on the implementation of iSTM in R.[@Krijkamp2018]
 
 With increasing model complexity and interdependency of functions to conduct various analyses like PSA, it is essential to ensure all code and functions work as expected and all elements of the cSTM are valid. We can achieve this by creating functions that help with model debugging, validation, and thorough unit testing. In the accompanying GitHub repository, we provide functions to check that transition probability matrices and their elements are valid. These functions are an example of a broader standard practice in software development called unit testing that requires building functions to test and check that the model and model-based analysis perform as intended.[@Wickham2021] However, unit testing is beyond the scope of this tutorial. We refer the reader to a previously published manuscript that describes unit testing in more detail and provides accompanying code.[@Alarid-Escudero2019e]
 
-In this tutorial, we implemented a cSTM using a (discrete-time) transition matrix. However, cSTM can also be implemented via (discrete-time) difference equations or (continuous-time) differential equations in R.[@Grimmett2014; @Axler2005] We refer readers interested in learning more on continuous-time cSTMs to previously published manuscripts[@Cao2016;@VanRosmalen2013;@Begun2013;@Soares2012] and a tutorial using R.[@Frederix2013a] Finally, the variable names used in this paper reflect our coding style. While we provide standardized variable names, adopting these conventions is ultimately a personal preference.
+In this tutorial, we implemented a cSTM using a (discrete-time) transition matrix. However, cSTM can also be implemented via (discrete-time) difference equations or (continuous-time) differential equations in R.[@Grimmett2014;@Axler2005] We refer readers interested in learning more on continuous-time cSTMs to previously published manuscripts[@Cao2016;@VanRosmalen2013;@Begun2013;@Soares2012] and a tutorial using R.[@Frederix2013a] Finally, the variable names used in this paper reflect our coding style. While we provide standardized variable names, adopting these conventions is ultimately a personal preference.
 
 In summary, this tutorial provides a conceptualization of time-independent cSTMs and a step-by-step guide to implement them in R. We aim to add to the current body of literature and material on building this type of decision model so that health decision scientists and health economists can develop cSTMs in a more flexible, efficient, open-source manner and to encourage increased transparency and reproducibility. In the advanced cSTM tutorial, we explore generalizing this framework to time-dependent cSTM, generating epidemiological outcomes, and incorporating transition rewards.