Survival Analysis - Immortal Time Bias with Stata

I wrote a tutorial on how to handle immortal time bias with survival analysis using Stata. In the tutorial, I used a time-varying predictor for the grouping variable and assigned the period before exposure to the control group. This was inspired by the paper Redelmeier and Singh wrote on “Surival in Academy Award-Winner Actors and Actresses.” There was a lot of debate about the rigor of their analyses, and Sylvestre and colleagues re-analyzed the data with immortal time bias in mind. This tutorial uses data from Sylvestre and colleagues to re-create their results.

The tutorial is on my RPubs page. Data used for the tutorial is located on my GitHub page.

To load the data, you can use the Stata import command

import delimited "https://raw.githubusercontent.com/mbounthavong/Survival-analysis-and-immortal-time-bias/main/Data/data1.csv"

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Survival analysis in R

I wrote a tutorial on survival analysis using R, which is located on my RPubs page. The R Markdown code is located on my GitHub site.

I provide an introduction to survivor and hazard functions, Kaplan-Meier curves, and Cox proportional hazards model.

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Generating Survival Curves from Study Data: An Application for Markov Models (Part 2 of 2)

BACKGROUND

In a previous blog, we provided instructions on how to generate the Weibull curve parameters (λ and γ) from an existing Kaplan-Meier curve. The Weibull parameters will allow you to generate survival curves for cost-effectiveness analysis. In the second part of this tutorial, we will take you through the process of incorporating these Weibull parameters to simulate survival using a simple three-state Markov model. Finally, we’ll show how to extrapolate the survival curve to go beyond the time frame of the Kaplan-Meier curve so that you can perform cost-effectiveness analysis across a lifetime horizon.

In this tutorial, I will:

• Describe how to incorporate the Weibull parameters into a Markov model

• Compare the survival probability of the Markov model to the reference Kaplan-Meier curve to validate the method and catch any errors

• Extrapolate the survival curve across a lifetime horizon

Link to the Markov model used in this tutorial can be found here.

MOTIVATING EXAMPLE

We will use a three-state Markov model to illustrate how to incorporate the Weibull parameters and generate a survival curve (Figure 1).

Figure 1. Markov model.

To simulate a Markov model using 40-time units (e.g., months), you will need to think about the different transition probabilities. Figure 2 lists the different transition probabilities and their calculations. We made the assumption that the transition from the Healthy state to the Sick state was 20% across all time points.

Figure 2. Transition probabilities for all health states and associated calculations.

The variable pDeath(t) denotes the probability of mortality as a function of time. Since we have the lambda (λ=0.002433593) and gamma (γ=1.722273465) Weibull parameters, we can generate the Weibull curve using Excel. Figure 3 illustrates how I set up the Markov model in Excel. I used the following equation to estimate the pDeath(t):

where t_i  is some time point at i. This expression can be written in Excel as (assuming T=1 and T+1 = 2):

= 1 – EXP(lambda*(T^gamma – (T+1)^gamma))

Figure 3. Calculating the probability of mortality as a function of time in Excel.

You can estimate the probability of survival as a function of time S(t) by subtracting pDeath(t) from 1. Once you have these values, you can compare how well your Markov model was able to simulate the survival compared to the observed Kaplan-Meier curve (Figure 4).

Figure 4. Survival curve comparison between the Markov model and Kaplan-Meier curve.

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Once we are comfortable with the simulated survival curve, we can extrapolate the survival probability beyond the limits of the Kaplan-Meier curve. To do this, we will need to go beyond the reference Kaplan-Meier’s time period of 40 months. In Figure 5, I extended the time cycle (denoted as Time) from 40 to 100 (truncated at 59 months).

Figure 5 illustrates the Weibull distribution extrapolated out to an entire cohort’s life time in the Markov model (Figure 5 is truncated at 59 months to fit this into the tutorial).

Figure 6 provides an illustration of the lifetime survival of the cohort after extrapolating the time period from 40 months to 100 months. The survival curve does a relative good job of modeling the Kaplan-Meier curve. As the time period extends beyond 40 months, the Weibull curve will exponentially reach a point where all subjects will enter the Death state. This is reflected in the flat part of the Weibull curve at the late part of the time period.

Figure 6. Lifetime survival of the cohort using the Weibull extrapolation.

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SUMMARY

After extrapolating the survival curve beyond the reference Kaplan-Meier curve limit of 40 months, you can estimate the lifetime horizon for a cohort of patients using a Markov model. This method is very useful when simulating chronic diseases. However, it is always good practice to calibrate your survival curves with the most recent data on the population of interest.

The U.S. National Center for Health Statistics has life tables that you can use to estimate the life expectancy of the general population, which you can compare to your simulated cohort. Moreover, if you want to compare your simulated cohort’s survival performance to a reference specific to your chronic disease cohort, you can search the literature for previously published registry data or epidemiology studies. Using existing studies as a reference will allow you to make adjustments to your survival curves that will give them credibility and validation to your cost-effectiveness analysis.

CONCLUSIONS

Using the Kaplan-Meier curves from published sources can help you to generate your own time-varying survival curves for use in a Markov model. Using the Hoyle and Henley’s Excel template to generate the survival probabilities, which are then used in an R script to generate the lambda and gamma parameters, provides a powerful tool to integrate Weibull parameters into a Markov model. Moreover, we can take advantage of the Weibull distribution to extrapolate the survival probability over the cohort’s lifetime giving us the ability to model lifetime horizons.

The Excel template developed by Hoyle and Henley generates other parameters that can be used in probabilistic sensitivity analysis like the Cholesky decomposition matrix, which will be discuss in a later blog.

REFERENCES

Location of Excel spreadsheet developed by Hoyle and Henley (Update 02/17/2019: I learned that Martin Hoyle is not hosting this on his Exeter site due to a recent change in his academic appointment. For those interested in getting access to the Excel spreadsheet used in this blog, please download it at this link).

Location of the Markov model used in this exercise is available in the following link:

https://www.dropbox.com/sh/ztbifx3841xzfw9/AAAby7qYLjGn8ZfbduJmAsVva?dl=0

Symmetry Solutions. “Engauge Digitizer—Convert Images into Useable Data.” Available at the following url: https://www.youtube.com/watch?v=EZTlyXZcRxI

Engauge Digitizer: Mark Mitchell, Baurzhan Muftakhidinov and Tobias Winchen et al, "Engauge Digitizer Software." Webpage: http://markummitchell.github.io/engauge-digitizer [Last Accessed: February 3, 2018].

1. Hoyle MW, Henley W. Improved curve fits to summary survival data: application to economic evaluation of health technologies. BMC Med Res Methodol 2011;11:139.

 

ACKNOWLEDGMENTS

I want to thank Solomon J. Lubinga for helping me with my first attempt to use Weibull curves in a cost-effectiveness analysis. His deep understanding and patient tutelage are characteristics that I aspire to. I also want to thank Elizabeth D. Brouwer for her comments and edits, which have improved the readability and flow of this blog. Additionally, I want to thank my doctoral dissertation chair, Beth Devine, for her edits and mentorship.

Generating Survival Curves from Study Data: An Application for Markov Models (Part 1 of 2)

BACKGROUND

In cost-effectiveness analysis (CEA), a life-time horizon is commonly used to simulate a chronic disease. Data for mortality are normally derived from survival curves or Kaplan-Meier curves published in clinical trials. However, these Kaplan-Meier curves may only provide survival data up to a few months to a few years. Extrapolation to a lifetime horizon is possible using a series of methods based on parametric survival models (e.g., Weibull, exponential); but performing these projections can be challenging without the appropriate data and software.

This blog provides a practical, step-by-step tutorial to estimate a parameter method (Weibull) from a survival function for use in CEA models. Specifically, I will describe how to:

• Capture the coordinates of a published Kaplan-Meier curve and export the results into a *.CSV file

• Estimate the survival function based on the coordinates from the previous step using a pre-built template

• Generate a Weibull curve that closely resembles the survival function and whose parameters can be easily incorporated into a simple three-state Markov model

 

MOTIVATING EXAMPLE

We will use an example dataset from Stata’s data library. (You can use any published Kaplan-Meier curve. I use Stata's data library for convenience.) Open Stata and enter the following in the command line:

use http://www.stata-press.com/data/r15/drug2b
sts graph, by(drug) risktable

You should get a Kaplan-Meier curve that illustrates the survival probability of two different drugs (Figure 1). The Y-axis denotes the survival probability and the X-axis denotes the time in months. Below the figure is the number at risk for the two drug comparators. We will need this to generate our Weibull curves. (If possible, find a Kaplan-Meier curve with the number at risk. It will make the Weibull curve generation easier.) Alternative methods exist to use Kaplan-Meier curves without the number at risk, but they will not be discussed in this tutorial.

Figure 1. Kaplan-Meier curve.

You will need to download the “Engauge Digitizer” application to convert this Kaplan-Meier curve into a *.CSV file with the appropriate data points. This will help you to develop an accurate survival curve based on the Kaplan-Meier curve. You can download the “Engauge Digitizer” application here: https://markummitchell.github.io/engauge-digitizer/

After you download “Engauge Digitizer,” open it and import the Kaplan-Meier file. Your interface should look like the following:

Figure 2. Engauge Digitizer interface.

The right panel guides you in digitizing your Kaplan-Meier figure. Follow this guide carefully. I will not go into how to use “Engauge Digitizer;” however a YouTube video tutorial to use Engauge Digitizer was developed by Symmetry Solutions and is available here.

We will use the top Kaplan-Meier curve (which is highlighted with blue crosshairs in Figure 3) to generate our Weibull curves.

Figure 3. Select the top curve to digitize.

After you digitize the figure, you will export the data as a *.CSV file. The *.CSV file should have two columns corresponding to the X- and Y-axes of the Kaplan-Meier figure. Figure 4 has the X values end at row 20 to fit onto the page, but this extends till the end of the Kaplan-Meier time period, which is 40.

Figure 4. *.CSV file generated from the Kaplan-Meier curve (truncated to fit onto this page).

I usually superimpose the “Engauge Digitizer” results with the actual Kaplan-Meier figure to prove to myself (and others) that the curves are exactly the same (Figure 5). This is a good practice to convince yourself that your digitized data properly reflects the Kaplan-Meier curve from the study.

Figure 5. Kaplan-Meier curve superimposed on top of the Engauge Digitizer curve.

Now, that we have the digitized version of the Kaplan-Meier, we need to format the data to import into the Weibull curve generator. Hoyle and Henley wrote a paper that explains their methods for using the results from the digitizer to generate Weibull curves.[1] We will use the Excel template they developed in order to generate the relevant Weibull curve parameters. (The link to the Excel template is provided at the end of this tutorial.)

I always format the data to match the Excel template developed by Hoyle and Henley. The blue box indicates the number at risk at the time points denoted by Figure 1 and the red box highlights the evenly spaced time intervals that I estimated (Figure 6).

Figure 6. Setting up your data using the template from Hoyle and Henley.

In order to find the survival probability at each “Start time” listed in the Excel template by Hoyle and Henley, linear interpolation is used. [You can use other methods to estimate the survival probability between each time points given the data on Figure 3 (e.g., last observation carried forward); however, I prefer to use linear interpolation.] In Figure 7, the survival probabilities (Y) correspond to a time (X) that was generated by the digitizer. Now, we want to find the Y value corresponding to the X values on the Excel template.

Figure 7. Generating the Y-values using linear interpolation.

Figure 8 illustrates how we apply the linear interpolation to estimate the Y value that corresponds with the X values from the Excel template developed by Hoyle and Henley. For example, if you were interested in finding the Y value at X = 10, the you would need to input the following into the linear interpolation equation using the following expression:

This yields a Y value of 0.866993, which is approximately 0.87.

Figure 8. Y values are generated using linear interpolation.

After generating the Y values corresponding to the Start time from Figure 5, you can enter them into the Excel template by Hoyle and Henley (Figure 9). Figure 9 illustrates the inputted survival probabilities into the Excel template.

Figure 9. Survival probabilities are entered after estimating them from linear interpolation.

After the “Empirical survival probability S(t)” is populated, you will need to go to the “R Data” worksheet in the Excel template and save this data as a *.CSV file. In this example, I saved the data as “example_data.csv” (Figure 10).

Figure 10. Data is saved as “example_data.csv.”

Then I used the following R code to estimate the Weibull parameters. This R code is located in the “Curve fitting ‘R’ code” in the Excel templated developed by Hoyle and Henley. (I modified the R code written by Hoyle and Henley to allow for a *.CSV file import.)

rm(list=ls(all=TRUE))   
library(survival)   

#    Step 4.   Update directory name and text file name in line below   
setwd("insert the folder path where the data is stored")
data<- read.csv("example_data.csv")
attach(data)
data    

times_start <-c(  rep(start_time_censor, n_censors), rep(start_time_event, n_events) )  
times_end <-c(  rep(end_time_censor, n_censors), rep(end_time_event, n_events)  )   

#  adding times for patients at risk at last time point 
######code does not apply because 0 at risk at last time point  
######code does not apply because 0 at risk at last time point  

#   Step 5. choose one of these function forms   (WEIBULL was chosen for the example)
model <- survreg(Surv(times_start, times_end, type="interval2")~1, dist="exponential")   # Exponential function, interval censoring 
model <- survreg(Surv(times_start, times_end, type="interval2")~1, dist="weibull")   # Weibull function, interval censoring 
model <- survreg(Surv(times_start, times_end, type="interval2")~1, dist="logistic")   # Logistic function, interval censoring   
model <- survreg(Surv(times_start, times_end, type="interval2")~1, dist="lognormal")   # Lognormal function, interval censoring 
model <- survreg(Surv(times_start, times_end, type="interval2")~1, dist="loglogistic")   # Loglogistic function, interval censoring 

#   Compare AIC values  
n_patients <- sum(n_events) +  sum(n_censors)   
-2*summary(model)$loglik[1] + 1*2   #  AIC for exponential distribution 
-2*summary(model)$loglik[1] + 1*log(n_patients)   #  BIC exponential    
-2*summary(model)$loglik[1] + 2*2   #  AIC for 2-parameter distributions    
-2*summary(model)$loglik[1] + 2*log(n_patients)   #  BIC for 2-parameter distributions  


intercept <- summary(model)$table[1]   # intercept parameter    
log_scale <- summary(model)$table[2]   # log scale parameter    

#  output for the example of the Weibull distribution   
lambda <- 1/ (exp(intercept))^ (1/exp(log_scale))    # l for Weibull, where S(t) = exp(-lt^g)   
gamma <- 1/exp(log_scale)     # g for Weibull, where S(t) = exp(-lt^g)  
(1/lambda)^(1/gamma) * gamma(1+1/gamma)    # mean time for Weibull distrubtion  


#  For the Probabilistic Sensitivity Analysis, we need the Cholesky matrix, which captures the variance and covariance of parameters    
t(chol(summary(model)$var))    #  Cholesky matrix   

#  Simulate variability of mean for Weibull 
library(MASS)   

simulations <- 10000  # number of simulations for standard deviation of mean    
sim_parameters <- mvrnorm(n=simulations, summary(model)$table[,1],  summary(model)$var  )   # simulates simulations from multivariate normal    
intercepts <- sim_parameters[,1]   # intercept parameters   
log_scales <- sim_parameters[,2]   # log scale parameters   
lambdas <- 1/ (exp(intercepts))^ (1/exp(log_scales))    # l for Weibull, where S(t) = exp(-lt^g)    
gammas <- 1/exp(log_scales)     # g for Weibull, where S(t) = exp(-lt^g)    
means <- (1/lambdas)^(1/gammas) * gamma(1+1/gammas)    # mean times for Weibull distrubtion 
sd(means)   # standard deviation of mean survival   

# consider adding this (from Arman Oct 2016) to plot KM 
km <- survfit(Surv(times_start, times_end, type="interval2")~ 1)    
summary(km) 
plot(km, xmax=600, xlab="Time (Days)", ylab="Survival Probability") 

There are several elements generated by the above R code that you need to record, including the intercept and log-scale:  

> intercept
[1] 3.494443
> log_scale
[1] -0.5436452

Once you have this, input them into the Excel template sheet titled “Number events & censored,” which is the same sheet you used to generate the survival probabilities after entering the data from the “Engauge Digitizer.” Figure 11 illustrates where these parameters are entered (red square).

Figure 11. Enter the intercept and log scale parameters into the Excel template developed by Hoyle and Henley.

You can check the fit of the Weibull curve to the observed Kaplan-Meier curve in the tab “Kaplan-Meier.” Figure 12 illustrates the Weibull fit’s approximation of the observed Kaplan-Meier curve.

Figure 12. Weibull fit (red curve) of the observed Kaplan-Meier curve (blue line).

From Figure 11, we also have the lambda (λ=0.002433593) and gamma (γ=1.722273465) parameters which we’ll use to simulate survival using a Markov model.

 

SUMMARY

In the next blog, we will discuss how to use the Weibull parameters to generate a survival curve using a Markov model. Additionally, we will learn how to extrapolate the survival curve beyond the time period used to generate the Weibull parameters for cost-effectiveness studies that use a lifetime horizon.

 

REFERENCES

Location of Excel spreadsheet developed by Hoyle and Henley (Update 02/17/2019: I learned that Martin Hoyle is not hosting this on his Exeter site due to a recent change in his academic appointment. For those interested in getting access to the Excel spreadsheet used in this blog, please download it at this link).

Update: 12 January 2023 - The old link to the YouTube video on Engauge was broken. A new link was identified that provided the same content on how to use Engauge.

Location of the Markov model used in this exercise is available in the following link:

https://www.dropbox.com/sh/ztbifx3841xzfw9/AAAby7qYLjGn8ZfbduJmAsVva?dl=0

Design with Greg. “Engauge: A Free Tool for Engineering that pairs great with Excel.” Available at the following url: https://www.youtube.com/watch?v=i1bEFovvvbM

Engauge Digitizer: Mark Mitchell, Baurzhan Muftakhidinov and Tobias Winchen et al, "Engauge Digitizer Software." Webpage: http://markummitchell.github.io/engauge-digitizer [Last Accessed: February 3, 2018].

1. Hoyle MW, Henley W. Improved curve fits to summary survival data: application to economic evaluation of health technologies. BMC Med Res Methodol 2011;11:139.

ACKNOWLEDGMENTS

I want to thank Solomon J. Lubinga for helping me with my first attempt to use Weibull curves in a cost-effectiveness analysis. His deep understanding and patient tutelage are characteristics that I aspire to. I also want to thank Elizabeth D. Brouwer for her comments and edits, which have improved the readability and flow of this blog. Additionally, I want to thank my doctoral dissertation chair, Beth Devine, for her edits and mentorship.