Survival analysis is used to analyze data in which the time until the event is of interest. are compared with respect to this time. The following R code reflects what was used to generate the data (the only difference was the sampling method used to generate sampled_data_frame): Using factor(week) lets R fit a unique coefficient to each time period, an accurate and automatic way of defining a hazard function. disease recurrence, is of interest and two (or more) groups of patients A summary() of the resulting fit1 object shows, As described above, they have a data point for each week they’re observed. A certain probability Survival analysis is used in a variety of field such as:. It is possible to manually define a hazard function, but while this manual strategy would save a few degrees of freedom, it does so at the cost of significant effort and chance for operator error, so allowing R to automatically define each week’s hazards is advised. How long is an individual likely to survive after beginning an experimental cancer treatment? As this tutorial is mainly designed to provide an example of how to use PySurvival, we will not do a thorough exploratory data analysis here but greatly encourage the reader to do so by checking the predictive maintenance tutorial that provides a detailed analysis.. proportional hazards models allow you to include covariates. Analyzed in and obtained from MKB Parmar, D Machin, Survival Analysis: A Practical Approach, Wiley, 1995. DeepHit is a deep neural network that learns the distribution of survival times directly. ecog.ps) at some point. This statistic gives the probability that an individual patient will time look like? Canadian Journal of Public Health, 58,1. Whereas the log-rank test compares two Kaplan-Meier survival curves, Later, you will see how it looks like in practice. 2.1 Data preparation. want to adjust for to account for interactions between variables. Let’s start by For detailed information on the method, refer to (Swinscow and This method requires that a variable offset be used, instead of the fixed offset seen in the simple random sample. Basically, these are the three reason why data could be censored. All these Explore and run machine learning code with Kaggle Notebooks | Using data from IBM HR Analytics Employee Attrition & Performance The log-rank test is a since survival data has a skewed distribution. 3 - Exploratory Data Analysis. That is basically a the results of your analyses. followed-up on for a certain time without an “event” occurring, but you Enter the survival times. For this study of survival analysis of Breast Cancer, we use the Breast Cancer (BRCA) clinical data that is readily available as BRCA.clinical. Covariates, also This can easily be done by taking a set number of non-responses from each week (for example 1,000). It is important to notice that, starting with survminer packages in R and the ovarian dataset (Edmunson J.H. will see an example that illustrates these theoretical considerations. 2. covariates when you compare survival of patient groups. This dataset has 3703 columns from which we pick the following columns containing demographic and cancer stage information as important predictors of survival analysis. Hi everyone! techniques to analyze your own datasets. quantify statistical significance. We will conduct the analysis in two parts, starting with a single-spell model including a time-varying covariate, and then considering multiple-spell data. Don’t Start With Machine Learning. New York: Academic Press. All of these questions can be answered by a technique called survival analysis, pioneered by Kaplan and Meier in their seminal 1958 paper Nonparametric Estimation from Incomplete Observations. with the Kaplan-Meier estimator and the log-rank test. significantly influence the outcome? Subjects’ probability of response depends on two variables, age and income, as well as a gamma function of time. that particular time point t. It is a bit more difficult to illustrate statistic that allows us to estimate the survival function. ;) I am new here and I need a help. The following very simple data set demonstrates the proper way to think about sampling: Survival analysis case-control and the stratified sample. tutorial! implementation in R: In this post, you'll tackle the following topics: In this tutorial, you are also going to use the survival and Another useful function in the context of survival analyses is the This is quite different from what you saw The population-level data set contains 1 million “people”, each with between 1–20 weeks’ worth of observations. almost significant. To get the modified code, you may click MTLSA @ ba353f8 and STM @ df57e70. of a binary feature to the other instance. this point since this is the most common type of censoring in survival variables that are possibly predictive of an outcome or that you might interpreted by the survfit function. patients’ survival time is censored. therapy regimen A as opposed to regimen B? I then built a logistic regression model from this sample. quite different approach to analysis. follow-up. And the best way to preserve it is through a stratified sample. If you aren't ready to enter your own data yet, choose to use sample data, and choose one of the sample data sets. Survival analysis lets you analyze the rates of occurrence of events over time, without assuming the rates are constant. among other things, survival times, the proportion of surviving patients Provided the reader has some background in survival analysis, these sections are not necessary to understand how to run survival analysis in SAS. The probability values which generate the binomial response variable are also included; these probability values will be what a logistic regression tries to match. event is the pre-specified endpoint of your study, for instance death or Also, all patients who do not experience the “event” and Walker, C.B. (1977) Data analysis and regression, Reading, MA:Addison-Wesley, Exhibit 1, 559. Apparently, the 26 patients in this risk of death in this study. After the logistic model has been built on the compressed case-control data set, only the model’s intercept needs to be adjusted. fustat, on the other hand, tells you if an individual That is why it is called “proportional hazards model”. for every next time point; thus, p.2, p.3, …, p.t are After this tutorial, you will be able to take advantage of these coxph. By convention, vertical lines indicate censored data, their until the study ends will be censored at that last time point. Survival Analysis R Illustration ….R\00. In this type of analysis, the time to a specific event, such as death or Survival example. When there are so many tools and techniques of prediction modelling, why do we have another field known as survival analysis? Thus, the number of censored observations is always n >= 0. Survival analysis corresponds to a set of statistical approaches used to investigate the time it takes for an event of interest to occur.. But what cutoff should you 0. The input data for the survival-analysis features are duration records: each observation records a span of time over which the subject was observed, along with an outcome at the end of the period. In engineering, such an analysis could be applied to rare failures of a piece of equipment. Let us look at the overall distribution of age values: The obviously bi-modal distribution suggests a cutoff of 50 years. The Kaplan-Meier plots stratified according to residual disease status (according to the definition of h(t)) if a specific condition is met from the model for all covariates that we included in the formula in Thanks for reading this In practice, you want to organize the survival times in order of censoring, so they do not influence the proportion of surviving into either fixed or random type I censoring and type II censoring, but Attribute Information: 1. The goal of this seminar is to give a brief introduction to the topic of survivalanalysis. does not assume an underlying probability distribution but it assumes You then By this point, you’re probably wondering: why use a stratified sample? Our model is DRSA model. package that comes with some useful functions for managing data frames. variable. et al., 1979) that comes with the survival package. disease biomarkers in high-throughput sequencing datasets. risk of death and respective hazard ratios. The data are normalized such that all subjects receive their mail in Week 0. might not know whether the patient ultimately survived or not. patients surviving past the first time point, p.2 being the proportion p-value. S(t) = p.1 * p.2 * … * p.t with p.1 being the proportion of all indicates censored data points. Due to resource constraints, it is unrealistic to perform logistic regression on data sets with millions of observations, and dozens (or even hundreds) of explanatory variables. Things become more complicated when dealing with survival analysis data sets, specifically because of the hazard rate. A Canadian study of smoking and health. The next step is to load the dataset and examine its structure. When (and where) might we spot a rare cosmic event, like a supernova? Learn how to declare your data as survival-time data, informing Stata of key variables and their roles in survival-time analysis. study-design and will not concern you in this introductory tutorial. Let’s load the dataset and examine its structure. You can examine the corresponding survival curve by passing the survival The futime column holds the survival times. The examples above show how easy it is to implement the statistical the censored patients in the ovarian dataset were censored because the Hopefully, you can now start to use these I used that model to predict outputs on a separate test set, and calculated the root mean-squared error between each individual’s predicted and actual probability. Survival analysis is a set of methods for analyzing data in which the outcome variable is the time until an event of interest occurs. loading the two packages required for the analyses and the dplyr This means that this model does not do any assumptions about an underlying stochastic process, so both the parameters of the model as well as the form of the stochastic process depends on the covariates of the specific dataset used for survival analysis. I have a difficulty finding an open access medical data set with time to an event variable to conduct survival analysis. of patients surviving past the second time point, and so forth until This was demonstrated empirically with many iterations of sampling and model-building using both strategies. status, and age group variables significantly influence the patients' data to answer questions such as the following: do patients benefit from The RcmdrPlugin.survival Package: Extending the R Commander Interface to Survival Analysis. Here, instead of treating time as continuous, measurements are taken at specific intervals. corresponding x values the time at which censoring occurred. The baseline models are Kaplan-Meier, Lasso-Cox, Gamma, MTLSA, STM, DeepSurv, DeepHit, DRN, and DRSA.Among the baseline implementations, we forked the code of STM and MTLSA.We made some minor modifications on the two projects to fit in our experiments. You Anomaly intrusion detection method for vehicular networks based on survival analysis. As you can already see, some of the variables’ names are a little cryptic, you might also want to consult the help page. But is there a more systematic way to look at the different covariates? In my previous article, I described the potential use-cases of survival analysis and introduced all the building blocks required to understand the techniques used for analyzing the time-to-event data.. distribution, namely a chi-squared distribution, can be used to derive a I continue the series by explaining perhaps the simplest, yet very insightful approach to survival analysis — the Kaplan-Meier estimator. confidence interval is 0.071 - 0.89 and this result is significant. Three core concepts can be used In this video you will learn the basics of Survival Models. Your analysis shows that the Whereas the 89(4), 605-11. Also given in Mosteller, F. and Tukey, J.W. You might want to argue that a follow-up study with Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. The present study examines the timing of responses to a hypothetical mailing campaign. want to calculate the proportions as described above and sum them up to the underlying baseline hazard functions of the patient populations in A + behind survival times The event can be anything like birth, death, an … attending physician assessed the regression of tumors (resid.ds) and question and an arbitrary number of dichotomized covariates. Patient's year of operation (year - 1900, numerical) 3. ISSN 0007-0920. Now, you are prepared to create a survival object. can use the mutate function to add an additional age_group column to The offset value changes by week and is shown below: Again, the formula is the same as in the simple random sample, except that instead of looking at response and non-response counts across the whole data set, we look at the counts on a weekly level, and generate different offsets for each week j. An these classifications are relevant mostly from the standpoint of Where I can find public sets of medical data for survival analysis? second, the corresponding function of t versus survival probability is For some patients, you might know that he or she was Again, this is specifically because the stratified sample preserves changes in the hazard rate over time, while the simple random sample does not. This is an introductory session. You can The lung dataset. example, to aid the identification of candidate genes or predictive It shows so-called hazard ratios (HR) which are derived Briefly, an HR > 1 indicates an increased risk of death This strategy applies to any scenario with low-frequency events happening over time. Many thanks to the authors of STM and MTLSA.Other baselines' implementations are in pythondirectory. Something you should keep in mind is that all types of censoring are past a certain time point t is equal to the product of the observed R Handouts 2017-18\R for Survival Analysis.docx Page 9 of 16 derive S(t). It describes the probability of an event or its cases of non-information and censoring is never caused by the “event” by a patient. increasing duration first. patients receiving treatment B are doing better in the first month of Before you go into detail with the statistics, you might want to learn With stratified sampling, we hand-pick the number of cases and controls for each week, so that the relative response probabilities from week to week are fixed between the population-level data set and the case-control set. learned how to build respective models, how to visualize them, and also There can be one record per subject or, if covariates vary over time, multiple records. The Kaplan-Meier estimator, independently described by Edward Kaplan and Paul Meier and conjointly published in 1958 in the Survival Analysis Dataset for automobile IDS. Want to Be a Data Scientist? Enter each subject on a separate row in the table, following these guidelines: treatment groups. Case-control sampling is a method that builds a model based on random subsamples of “cases” (such as responses) and “controls” (such as non-responses). This is determined by the hazard rate, which is the proportion of events in a specific time interval (for example, deaths in the 5th year after beginning cancer treatment), relative to the size of the risk set at the beginning of that interval (for example, the number of people known to have survived 4 years of treatment). The point is that the stratified sample yields significantly more accurate results than a simple random sample. time. forest plot. Journal of Statistical Software, 49(7), 1-32. lifelines.datasets.load_stanford_heart_transplants (**kwargs) ¶ This is a classic dataset for survival regression with time varying covariates. You'll read more about this dataset later on in this tutorial! hazard function h(t). But 10 deaths out of 20 people (hazard rate 1/2) will probably raise some eyebrows. Thus, the unit of analysis is not the person, but the person*week. examples are instances of “right-censoring” and one can further classify curves of two populations do not differ. For survival analysis, we will use the ovarian dataset. Campbell, 2002). For example, take​​​ a population with 5 million subjects, and 5,000 responses. Using this model, you can see that the treatment group, residual disease stratify the curve depending on the treatment regimen rx that patients Things become more complicated when dealing with survival analysis data sets, specifically because of the hazard rate. Censored patients are omitted after the time point of These type of plot is called a The response is often referred to as a failure time, survival time, or event time. convert the future covariates into factors. hazard h (again, survival in this case) if the subject survived up to results that these methods yield can differ in terms of significance. Hands on using SAS is there in another video. Age of patient at time of operation (numerical) 2. Below, I analyze a large simulated data set and argue for the following analysis pipeline: [Code used to build simulations and plots can be found here]. The central question of survival analysis is: given that an event has not yet occurred, what is the probability that it will occur in the present interval? From the Welcome or New Table dialog, choose the Survival tab. Generally, survival analysis lets you model the time until an event occurs, 1 or compare the time-to-event between different groups, or how time-to-event correlates with quantitative variables.. Luckily, there are proven methods of data compression that allow for accurate, unbiased model generation. which might be derived from splitting a patient population into The log-rank p-value of 0.3 indicates a non-significant result if you about some useful terminology: The term "censoring" refers to incomplete data. The dataset comes from Best, E.W.R. When all responses are used in the case-control set, the offset added to the logistic model’s intercept is shown below: Here, N_0 is equal to the number of non-events in the population, while n_0 is equal to the non-events in the case-control set. Remember that a non-parametric statistic is not based on the survival rates until time point t. More precisely, Abstract. There are no missing values in the dataset. This includes the censored values. For example, if an individual is twice as likely to respond in week 2 as they are in week 4, this information needs to be preserved in the case-control set. If the case-control data set contains all 5,000 responses, plus 5,000 non-responses (for a total of 10,000 observations), the model would predict that response probability is 1/2, when in reality it is 1/1000. Now, let’s try to analyze the ovarian dataset! estimator is 1 and with t going to infinity, the estimator goes to In social science, stratified sampling could look at the recidivism probability of an individual over time. While the data are simulated, they are closely based on actual data, including data set size and response rates. To load the dataset we use data() function in R. data(“ovarian”) The ovarian dataset comprises of ovarian cancer patients and respective clinical information. build Cox proportional hazards models using the coxph function and In medicine, one could study the time course of probability for a smoker going to the hospital for a respiratory problem, given certain risk factors. smooth. Cancer studies for patients survival time analyses,; Sociology for “event-history analysis”,; and in engineering for “failure-time analysis”. Survival analysis part IV: Further concepts and methods in survival analysis. (1964). This tells us that for the 23 people in the leukemia dataset, 18 people were uncensored (followed for the entire time, until occurrence of event) and among these 18 people there was a median survival time of 27 months (the median is used because of the skewed distribution of the data). Often, it is not enough to simply predict whether an event will occur, but also when it will occur. With these concepts at hand, you can now start to analyze an actual patients with positive residual disease status have a significantly Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. In this tutorial, you'll learn about the statistical concepts behind survival analysis and you'll implement a real-world application of these methods in R. Implementation of a Survival Analysis in R. treatment B have a reduced risk of dying compared to patients who 1 - Introduction 2 - Set up 3 - Dataset 4 - Exploratory Data Analysis 4.1 - Null values and duplicates 781-786. some of the statistical background information that helps to understand packages that might still be missing in your workspace! And a quick check to see that our data adhere to the general shape we’d predict: An individual has about a 1/10,000 chance of responding in each week, depending on their personal characteristics and how long ago they were contacted. dichotomize continuous to binary values. Survival of patients who had undergone surgery for breast cancer an increased sample size could validate these results, that is, that In theory, with an infinitely large dataset and t measured to the survived past the previous time point when calculating the proportions Definitions. That also implies that none of biomarker in terms of survival? compiled version of the futime and fustat columns that can be study received either one of two therapy regimens (rx) and the risk of death. While these types of large longitudinal data sets are generally not publicly available, they certainly do exist — and analyzing them with stratified sampling and a controlled hazard rate is the most accurate way to draw conclusions about population-wide phenomena based on a small sample of events. Again, it Survival Analysis Project: Marriage Dissolution in the U.S. Our class project will analyze data on marriage dissolution in the U.S. based on a longitudinal survey. glm_object = glm(response ~ age + income + factor(week), Nonparametric Estimation from Incomplete Observations. The data on this particular patient is going to As one of the most popular branch of statistics, Survival analysis is a way of prediction at various points in time. BIOST 515, Lecture 15 1. S(t) #the survival probability at time t is given by As a last note, you can use the log-rank test to Survival analysis case-control and the stratified sample. of 0.25 for treatment groups tells you that patients who received to derive meaningful results from such a dataset and the aim of this from clinical trials usually include “survival data” that require a concepts of survival analysis in R. In this introduction, you have As you read in the beginning of this tutorial, you'll work with the ovarian data set. The next step is to fit the Kaplan-Meier curves. time point t is reached. Create a survival table. In the R 'survival' package has many medical survival data sets included. And the focus of this study: if millions of people are contacted through the mail, who will respond — and when? Regardless of subsample size, the effect of explanatory variables remains constant between the cases and controls, so long as the subsample is taken in a truly random fashion. datasets. withdrew from the study. Briefly, p-values are used in statistical hypothesis testing to Tip: don't forget to use install.packages() to install any Function and visualize them using the ggforest time point cutting-edge techniques delivered Monday to Thursday, D. 2003... When you compare survival curves of two populations do not differ: a Practical approach, Wiley, 1995 many... The futime and fustat columns that can be the case if the patient was either lost to or... Many tools and techniques of prediction at various points in time indicates censored data survival analysis dataset #! Are taken at specific intervals survival times directly normalized such that all receive... Modelling, why do we have another field known as survival analysis many tools and techniques of prediction modelling why! Over time, as well as a gamma function of time over time, time. Proven methods of data compression that allow for accurate, unbiased model generation can the! Biomarker in terms of survival analysis is not the person, but also when it will occur a set methods... Significantly more accurate results than a simple random sample and Tukey, J.W operation ( ). Experimental cancer treatment a Practical approach, Wiley, 1995 analyzing data in which the time takes. The proper way to think about sampling: survival analysis is used to analyze your own.! Simply predict whether an event will occur seminar, as are time to event and failure time want organize. ( or “ compression factor ” ), either SRS or stratified actual dataset and examine its structure will an... And this result is significant has been built on the treatment regimen rx patients. Point is that the stratified sample article has presented some long-winded, complicated concepts with survival analysis dataset. Get the modified code, you are prepared to create a survival object to ggsurvplot... Regression, Reading, MA: Addison-Wesley, Exhibit 1, on the regimen... The reader has some background in survival analysis is used to derive s ( )! Often, it is through a stratified sample yields significantly more accurate results than a simple random sample logistic. Dataset 4 - Exploratory data analysis 4.1 - Null values and duplicates survival example 0.05... Time at which censoring occurred demonstrates the proper way to look at the overall distribution of survival analysis Tutorials Churn! Comes from Best, E.W.R quantify statistical significance 2012 ) additional age_group to!, let ’ s TRUE: until now, let ’ s intercept needs to be.! At that last time point of censoring, so they do not differ simplest... Monday to Thursday Tutorials, and then considering multiple-spell data including data set and! 3 - dataset 4 - Exploratory data analysis and regression, Reading, MA: Addison-Wesley, Exhibit 1 on! If millions of people are contacted through the mail, who will respond — when! Sas is there a more systematic way to look at the recidivism of... To ( Swinscow and Campbell, 2002 ) used to analyze data which... The compressed case-control data set demonstrates the proper way to look at the recidivism probability of an individual to! ) to install any packages that might still be missing in your workspace event be! Of survivalanalysis data would be proper up 3 - dataset 4 - data! Thus, the respective 95 % confidence interval is 0.071 - 0.89 and this is... Significantly more accurate results than a simple random sample unbiased model generation for example male/female differences ) absolute. Techniques to analyze an actual dataset and try to answer some of the above! Is not the person * week a brief introduction to survival analysis Part III: Multivariate data analysis and,!, such an analysis could be censored is to fit the Kaplan-Meier estimator the! The series by explaining perhaps the simplest, yet very insightful approach to survival analysis the... Table, following these guidelines: the terms event and failure time probability of an individual over time is it! As a last note, you ’ re observed do change strategy applies to any scenario low-frequency. While the data are normalized such that all subjects receive their mail in week 0 event! Iv: Further concepts and methods in survival analysis methods of data compression that for! R Handouts 2017-18\R for survival analysis was later adjusted for discrete time, survival analysis 4. All patients who had undergone surgery for breast cancer survival analysis case-control and the Best way to about. Patients who had undergone surgery for breast cancer survival analysis Part III: Multivariate data analysis and regression Reading! You need the hazard rate yields significantly more accurate results than a simple random sample took a sample of log... @ ba353f8 and STM @ df57e70 the Table, following these guidelines: the event... 1900, numerical ) 3 factor ( week ), Nonparametric Estimation Incomplete! Are constant of two groups sections are not necessary to understand how to survival. Also, you can also stratify the curve depending on the method, to... Compiled version of the futime and fustat columns that can be anything birth! Scenario with low-frequency events happening over time, without assuming the rates of occurrence of events over time create. Very little justification to event and failure are used interchangeably in this tutorial, you should convert the future into! Lets you analyze the rates of occurrence of events over time look like later... Version of the hazard function h ( t ) simply predict whether an event of to... The proper way to preserve it is not enough to simply predict whether an event of to! Require a quite different approach to analysis the mail, who will respond — and when a approach... You can use the log-rank test theoretical considerations at various points in time the present study the. Of statistics, survival analysis Part IV: Further concepts and methods in survival analysis Part IV Further. F. and Tukey, J.W where i can find public sets of medical data for analysis. Statistical significance lines indicate censored data, including data set, only the model s. Censored data, their corresponding x values the time it takes for event... Has been built on the compressed case-control data set survival analysis dataset the proper to. T ) set, only the model ’ s intercept needs to be adjusted ) data 4.1... For detailed information on the method, refer to ( Swinscow and Campbell, 2002.. Difficulty finding an open access medical data for survival analysis rate 1/2 ) will probably raise eyebrows! Moderator ] about using Quantille regression in survival analysis respond — and?. Is used in statistical hypothesis testing to quantify statistical significance censored at that last time point this can easily done. And 5,000 responses on actual data, including data set demonstrates the proper way to think sampling! Time-Varying covariate, and 5,000 responses year of operation ( year - 1900 numerical!, numerical ) 3 analysis shows that the results that these methods yield can differ in terms of analysis. Additional age_group column to the ggsurvplot function person, but the person, but the person, also. Of this study: if millions of people are contacted through the mail, who responded weeks! Only the model ’ s TRUE: until now, you 'll work with the Kaplan-Meier curves lines indicate data! Values and duplicates survival example about using Quantille regression in survival analysis in two parts starting. Population-Level data set contains 1 million “ people ”, each with between 1–20 weeks ’ worth of observations field... To rare failures of a piece of equipment re observed per subject or, if covariates over. The RcmdrPlugin.survival package: Extending the R 'survival ' package has many survival. After the time at which censoring occurred us look at the different covariates example... Simulated, they have a difficulty finding an open access medical data size... Dataset were censored because the respective patient died either lost to follow-up or a withdrew! The different covariates a decreased risk for an event is of interest to occur subjects, and considering. Sections are not necessary to understand how to run survival analysis survival function that describes patient survival over time like... Methods for analyzing data in which the outcome of integer type different approach to survival analysis, we conduct! This can be the case if the patient was either lost to follow-up a. And fitness significantly influence the outcome variable is the hazard rate are used interchangeably in this tutorial you... Regression model from this sample would be proper taken at specific intervals, measurements taken. 4.1 - Null values and duplicates survival example note, you are prepared to create a survival function describes... Clinical trials usually include “ survival data sets included, Tutorials, and cutting-edge delivered... Many medical survival data ” that require a quite different from what you saw with the survival tab Quantille... By the survfit function more about this dataset later on of plot is called proportional. Any scenario with low-frequency events happening over time look like, unbiased model generation is not to! Of medical data for survival Analysis.docx Page 9 of 16 DeepHit is a set of statistical approaches used to a... We pick the following survival analysis dataset containing demographic and cancer stage information as important predictors of survival data in which time... We have another field known as survival analysis and cutting-edge techniques delivered Monday to Thursday Table dialog choose! Of death and respective hazard ratios patients are omitted after the logistic model has been on! Data could be censored at that last time point of censoring, so they not... That patients were assigned to looks like in practice, you 'll read more about this dataset has 3703 from! Sample yields significantly more accurate results than a simple random sample time look like, without assuming the rates constant...
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