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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Interrupted time series analysis (ITSA) with Stata

Interrupted time series analysis (ITSA) is a study design used to study the effects of an intervention across time. An important feature of the ITSA is the time when the intevention occurs. The time before and after the intervention are of interest because we want to visualize if the trends are similar or different. Additionally, we want to visualize the change immediately after the intervention is implemenated. I call this period the index date.

In this article, I’ll review the single-group ITSA and multiple groups ITSA. Then I’ll review how to perform an ITSA in Stata.

You can view the complete tutorial on my RPubs site.

Stata tutorial: Adding the 95% Confidence Interval to a Two-way Line Plot

I created a tutorial on how to add the 95% CI to a two-way line plot in Stata. I use the “connected” command to generate a line plot in Stata, and then I added the 95% CI to each value. Surprisingly, Stata does not have a native feature to allow users to generate these 95% CI on a two-way line plot.

I used the AHRQ Medical Expenditure Panel Survey (MEPS) database for the motivating example. In this tutorial, we plotted the average total healthcare expenditure from 2008 to 2019.

I build this tutorial on Stata, but I used R Markdown to write the tutorial. The R Markdown code is located in my GitHub site (Stata - Line plot with 95% CI tutorial).

You can find the tutorial on my Github site and RPubs page.

I used Stata SE 17 to build this.

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Using Stata’s bysort command for panel data in time series analysis

BACKGROUND

Sorting information in panel data is crucial for time series analysis. For example, sorting by the time for time series analysis requires you to use the sort or bysort command to ensure that the panel is ordered correctly. However, when it comes to panel data where you may have to distinguish a patient located at two different sites or a patient with multiple events (e.g., deaths), it’s important to organize the data properly.

You can download the sample data and Stata code at the following links:

Data

Code

MOTIVATING EXAMPLE

In this example, we have a data set with time (months) in the column and patients in the rows (this is called a wide format data set). For each month, there are different numbers of observations. For instance, in Month 1, there were 5 observations. But in Month 7 there were only three.

The highlighted boxes indicate a patient was observed at two different sites. There are two ways to approach this: (1) remove the patient from Site B or (2) keep the patient by distinguishing it at each sight. Removing the patient will result in a loss of information for Site B, but keeping the patient complicates the panel data when we convert from wide to long format.

Converting this from wide to long format would result in the following panel data. Review each patient, in particular, the months of observations reported for the months. Notice that not all patients have observations for all the months (Months 1 to 7). Some patients have observations for scattered months (e.g., Patient 3). Of note is Patient 2 who has observations at Sites A and B. Since we opted to keep Patient 2 data for Sites A and B, we need to distinguish a method to ensure that the panel data is ordered correctly. Interestingly, Patient 8 has an observed event  (Death) three times at Months 5, 6, and 7. Since a patient should experience death only once, this may be a coding error and should be removed. Using the Stata sort and bysort command will allow us to fix this problem.

The bysort command has the following syntax:

bysort varlist1 (varlist2): stata_cmd

Stata orders the data according to varlist1 and varlist2, but the stata_cmd only acts upon the values in varlist1. This is a handy way to make sure that your ordering involves multiple variables, but Stata will only perform the command on the first set of variables.

REMOVE REPEATED DEATHS FROM PATIENT 8

First, we want to make sure we eliminate the repeated deaths from Patient 8. We can do this using the bysort command and summing the values of Death. Since Death == 1, we can sum up the total Deaths a patient experiences and drop those values that are greater than 1—because a patient can only die once.

***** Identify patients with repated death events. 
bysort id site (month death): gen byte repeat_deaths = sum(death==1)
drop if repeat_deaths > 1 

The alternative methods use the sort command:

* Alternative Method 1:
by id site (month death), sort: gen byte repeat_deaths = sum(death==1)
drop if repeat_deaths > 1

* Alternative Method 2:
sort id site (month death)
by id site (month death): gen byte repeat_deaths = sum(death ==1)
drop if repeat_deaths > 1

Now we have a data set without the unnecessary death values for Patient 8. Therefore, Patient 8 will not be counted in months 6 and 7 because they are no longer contributing to the denominator.

COUNT THE NUMBER OF DEATHS PER MONTH

Suppose we want to perform a single group time series analysis. We would want to sum up the number of deaths across the months. We can do this using the bysort command.

First, we have to think about how we want to count death. Since Death == 1, we want to add up the number of Death for each month. Initially, we were worried that Death would be counted two more times for Patient 8, but we solved this problem by removing these events from Patient 8.

The following command will yield the above results in a long format.

bysort month: egen byte total_deaths = total(death)

We use the egen command because we are using a more complex function. Detailers on when to use gen versus the egen commands are located at this site.

DETERMINING THE DENOMINATOR—COUNTING THE NUMBER OF PATIENTS CONTRIBUTION INFORMATION

Next, we want to determine that number of patient observations that are contributed to each month. To do this, we can use the bysort command again.

***** Determine the denominator -- using bysort and counter variable
gen counter = 1
bysort month: egen byte total_obs = total(counter)

This should yield the following results:

CHANGING FROM PATIENT-LEVEL DATA TO SINGLE-GROUP DATA

Currently, the data is set up using the patient-level. We want to change this to the single-group level or the aggregate monthly level. To do this, we have to eliminate the repeated month measurements for our total deaths (numerator) and total observations (denominator).

***** Drop duplicate months
bysort month: gen dup = cond(_N==1, 0, _n)
drop if dup > 1

We can visualize this by plotting two separated lines connected at the values for each month.

****** Plot the total number of deaths and total number of observations
graph twoway (connected total_deaths month, lcol(navy)) ///
             (connected total_obs month, lcol(cranberry) ytitle("Number") ///
	      xtitle("Months") ylab(, nogrid) graphregion(color(white)))

We can take this a step further and calculate the prevalence.

***** Estimate the prevalence (per 100 population) and plot
gen prev = (total_deaths / total_obs ) * 100	

graph twoway connected prev month, ytitle("Prevalence of Death" "per 100 population") ///
	     xtitle("Months") ylab(, nogrid) graphregion(color(white))

CONCLUSIONS

Using the bysort command can help us fix a variety of data issues with time series analysis. In this example, we have patient-level data that contained deaths for one patient and a patient who was observed at different sites. Using the bysort command to distinguish between sites allowed us to properly identify the patient as unique to the site. Additionally, we used the bysort to identify the patient with multiple deaths and eliminated these values from the aggregate monthly values. Then we finalized out single-group data set by summing the total deaths and observations per month and removing the duplicates.

You can download the Stata code from my Github site.

REFERENCES

I used the following references to write this blog.

Stata commands: bysort:

https://stats.idre.ucla.edu/stata/faq/can-i-do-by-and-sort-in-one-command/

Stata commands: gen versus egen:

https://stats.idre.ucla.edu/stata/seminars/notes/stata-class-notesmodifying-data/

Estimating marginal effects using Stata Part 1 – Linear models

BACKGROUND

Regression models provide unique opportunities to examine the impact of certain predictors on a specific outcome. These predictors’ effects are usually isolated using the model coefficients adjusting for all other predictors or covariates. A simple linear regression model with a single predictor x_i is represented as

where y_i denotes the outcome (dependent) variable for subject i, beta0 denotes the intercept, beta1 is the model coefficient that denotes the change in y due to a 1-unit change in x, and epsilon_i is the error term for subject i.

A 1-unit increase in x is associated with some change in the outcome y. This finding may explain predictor variable x’s impact on outcome variable y, but it doesn’t not tell us the impact of a representative or prototypical case.

The marginal effect allows us to examine the impact of variable x on outcome y for representative or prototypical cases. For example, Stata’s margins command can tell us the marginal effect of body mass index (BMI) between a 50-year old versus a 25-year old subject.

There are three types of marginal effects of interest:

1.       Marginal effect at the means (MEM)

2.       Average marginal effect (AME)

3.       Marginal effect at representative values (MER)

Each of these marginal effects have unique interpretations that will impact how you examine the regression results. (We will focus on the first two, since the third one is an extension of the AME.) The objective of this tutorial is to review these marginal effects and understand their interpretations through examples using Stata.

 

MOTIVATING EXAMPLE

We will use the Second National Health and Nutrition Examination Survey (NHANES) data from the 1980s, which can be found in Stata’s library using the following command:

use http://www.stata-press.com/data/r15/nhanes2.dta

Table 1 summarizes the characteristics of the NHANES population.

 

ADJUSTED PREDICTIONS

Adjusted prediction for a regression model provides the expected value of an outcome y conditioned on x assuming all other things are equal. In other words, this is the effect of the predictor variable x regressed to outcome variable y adjusting or controlling for other covariates. Therefore, if you were comparing the effect of a 1-unit increase in age to the BMI, then you could compare this across all patients who are equally White, Black, or Others.

Example 1

A simple linear regression model can capture the incremental effect of age on body mass index. For example, the impact of age on body mass index (BMI) can be represented as a linear regression:

where BMI_1  is the body mass index for individual i, beta0 denotes the intercept (or BMI when AGE=0), beta1 denotes the change in BMI for each 1-unit increase in Age for individual i, and episilon_i denotes the error term for individual i. (The unit of BMI is kg/m^2).

The Stata command to perform a simple linear regression:

regress bmi age

The corresponding regression output is:

In this regression output example, the predictor of interest is AGE. The _cons parameter denotes the coefficient beta0 otherwise known as the intercept; therefore, a subject with AGE = 0 has a BMI that is 23.2 kg/m^2. (Although this is unrealistic, we will ignore this for now.) The impact AGE has on the BMI is denoted by the slope parameter beta1, which is the change in BMI due to a 1-unit change in Age. In this example, the a 1-unit increase in Age is associated with a 0.05 kg/m^2 increase in BMI.

If we wanted to know the difference in BMI between a 50-year old and 25-year old, we need to estimate the adjusted prediction, which estimates the difference in the outcome based on some user-defined values for the x variables.

To estimate the adjusted predicted BMI for a 50-year old, we used the following equation:

which is 25.7 kg/m^2. We can do this using the following Stata command:

di _b[_cons] + 50*_b[age]
25.655896

Similarly, we can estimate the adjusted predicted BMI for a 25-year old:

which is 24.4 kg/m^2.

The difference between these two is:

25.655896 - 24.433991 = 1.2 kg/m^2. 

Therefore, the difference in BMI between a 50-year old and 25-year old is on average 1.2 kg/m^2. This seems like a tedious process, but let’s see how we can make this exercise simpler using Stata’s margins command.

We can use Stata’s margins command to estimate the adjusted predicted BMI for a 50-year old and 25-year old:

margins, at(age=(25 50))

Figure 2. Stata’s margins command output for adjusted prediction of BMI for a 50-year old and 25-year old.

Example 2

We use a linear regression with other independent variables to illustrate the complexity of having other covariates adjusted in the model.

The regression model has the structure:

where  is the body mass index for individual i, beta0 is the intercept (or BMI when AGE=0), beta1 is the change in BMI for each 1-unit increase in Age for individual i, beta2 denotes the change in BMI for a female relative to a male, beta3 denotes the change in BMI due to contrasts in race categories (White, Black, and Other), and  is the error term for individual i. (The unit of BMI is kg/m^2).

For this example, RACE will be included into the regression model as a dummy variable using the following Stata command:

regress bmi age i.race i.sex

The corresponding regression output is:

The following are interpretations of the regression output.

A 1-unit increase in age is associated with a BMI increase of 0.05 kg/m^2 adjusting for race and sex or all things being equal.

Blacks are associated with a BMI increase of 1.4 kg/m^2 adjusting for age and sex compared to Whites.

Others are associated with a BMI decrease of 1.2 kg/m^2 adjusting for age and sex compared to Whites.

Females are associated with a BMI increase of 0.03 kg/m^2 adjusting for age and race.

If we wanted to know the adjusted prediction for a 50-year old and 25-year old, we can use the margins command:

margins, at(age=(25 50)) atmeans vsquish

The output is similar to Example 1 but there are some differences.

The atmeans option captures the “average” sample covariates. In our example, the mean proportion of females is 0.525, males is 0.475, Whites is 0.876, Blacks is 0.105, and Others is 0.019. Therefore, the adjusted predictions for 50-year old and 25-year old’s BMI is conditioned on the “average” values of the covariates in the model. This may not make sense because an individual subject can’t be 0.525 female and 0.475 male. Fortunately, we have other ways to address this with the marginal effect.

 

MARGINAL EFFECT

Marginal effect with the margins command generates the change in the conditional mean of outcome y with respect to a single predictor variable x. In other words, this is the partial effect of x on the outcome y for some representative or prototypical case. Usually this is obtained by performing a first-order derivative of the regression expression:

where the partial effect of the expected value of y condition on x is the first order derivative of the expected value of y condition on x with respect to x.

The representative or prototypical case can be the mean, observed, or a user defined case.

MARGINAL EFFECT OF THE MEAN (MEM)

MEM is the partial effect of on the dependent variable (y) conditioned on a regressor (x) after setting all the other covariates (w) at their means. In other words, MEM is the difference in x’s effect on y when all other covariates (RACE and FEMALE) are at their mean.

Let’s revisit the linear regression model but with the dummy variables included:

In the output the beta1 = 0.0493881.

Getting the partial effect with respect to Age at the means for the other covariates, we use the following command:

regress bmi age i.race i.sex

margins, dydx(age) atmeans vsquish

Interpretation: For a subject who is average on all characteristics, the marginal change of a 1-unit increase in age is a 0.049 increase in the BMI.

We can also look at the MEM at different ages (e.g., 25 and 50 years):

margins, dydx(age) at(age=(25 50)) atmeans vsquish

This command performs the MEM for 25- and 50-year old subjects with their covariates set at the population mean. We interpret the results as the effect of age at different values of age at the average values of the other covariates.

The MEM should be:

The effect of age at 25 and 50 years old is an increase of 0.05 years. Notice that the MEM for 25- and 50-year olds are the same (MEM = 0.0493881). This is because the model is a linear regression. For every incremental increase in age, the incremental increase in the BMI is 0.0493881 given the other covariates are set at the mean.

To illustrate, we can manually perform this operation using the information above. Recall that the linear regression model with the dummy variables is represented as:

BMI for a 25-year old subject at the mean = intercept + 25*(beta1) + (mean of Black)*(beta2) + (mean of Other)*(beta3) + (mean of Female)*(beta4) = 24.42243 kg/m^2.

BMI for a 25-year old subject at the mean = 23.0528 + 25*(0.0493881) + .1049174*(1.382849) + .0193218*(-1.2243) + .5251667*(.025702) = 24.42243 kg/m^2, which is the same as the value presented in the adjusted prediction output.

Why are these the same? The linear regression is predictable in terms of the slope coefficients. Therefore, an incremental increase in predictor variable x will have the same incremental marginal increase in outcome variable y. When you apply the MEM to non-linear models, the slopes are no longer linear and will change based on varying levels of the continuous predictor x.

AVERAGE MARGINAL EFFECT (AME)

Unlike the MEM the average marginal effect (AME) doesn’t use the mean for the covariates when estimating the partial effect of the predictor variable x on the outcome variable y. Rather, the AME estimates the partial effect of the variable x on the outcome variable y for using the observed values for the covariates and then the average of that partial effect is estimated. In other words, the partial derivative is estimated with respect to x using the observed values for the other covariates (RACE and FEMALE), and then the average of that first-order derivative are averaged over the entire population to yield the AME. This is represented as:

where the partial derivative of the estimated value of the outcome variable y with respect to x is conditioned on the values of covariates (w) for subject i over the entire population (N) and multiplied by beta_k (or the parameters of interest) .

Getting the partial effect with respect to Age at the observed values for the other covariates, we use the following command:

regress bmi age i.race i.sex

margins, dydx(age) asobserved vsquish

Interpretation: The average marginal effect of a 1-unit increase in age is a 0.049 increase in the BMI.

We can also look at the AME at different ages (e.g., 25 and 50 years):

margins, dydx(age) at(age=(25 50)) asobserved vsquish

This command performs the MEM for 25- and 50-year old subjects with their covariates set at the observed values. We interpret the results as the effect of age at different values of age at the observed values of the other covariates.

The AME should be:

The effect of age at 25 and 50 years old is an increase of 0.05 years. Notice that the AME for 25- and 50-year olds are the same (MEM = 0.0493881). Similar to the MEM, this is because the model is a linear regression. For every incremental increase in age, the incremental increase in the BMI is 0.0493881 given the other covariates are set at the observed values.

CONCLUSIONS

We see that the MEM and AME are exactly the same because of the linear model. The marginal effect of an increase in 1-unit of age is an increase in 0.05 kg/m^2 of the BMI. In the next part, non-linear models will be used to demonstrate that the MEM and AME are not equal.

REFERENCES

I used the following websites to help create this tutorial:

https://thomasleeper.com/margins/articles/Introduction.html

https://support.sas.com/rnd/app/ets/examples/margeff/index.html

https://www.ssc.wisc.edu/sscc/pubs/stata_margins.htm

I also used the following paper by Richard Williams:

Using the margins command to estimate and interpret adjusted predictions and marginal effects. The Stata Journal. 2012;12(2):308-331.

https://www.stata-journal.com/article.html?article=st0260

Counting and Data Manipulation for an ITSA

BACKGROUND

In time series analysis, we are interested on the impact of some exposure over a time period. Exposure can be coded as event==1. If this is time-varying, then the event can occur at any time across a time period. Time series analysis requires us to identify the time when the event first occurred. In most cases this is also considered the post period. In this example, we will label the exposure of interest as event.

Longitudinal data can come in either a wide or long format. However, it is easier to perform longitudinal data analysis in the long format. This assumes that you declare either the xtset or tsset as a panel or time-series data set, respectively.

MOTIVATING EXAMPLE

Let’s assume that we have two subjects (A and B), who can experience an event at any time between some time variable 1 and 5, time(1:5). This is a longitudinal data set in the long format with id as the unique subject-level identifier, the exposure variable of interest event as the exposure, and time as an arbitrary time variable ranging from 1 to 5. The event for subject A occurs at time==3.

Suppose you want to create a variable that counts the number of times the subject has the event. We will call this variable duration.

The following Stata code will generate the duration variable.

by id (time), sort: gen byte duration = sum(event==1)

Sorting by the id and then time will nest the time sequence for each id. The sum() will add all event that is coded as 1.

It’s critical that you put time in parentheses (); otherwise, you can generate incorrect values. For instance, if you make the mistake of typing the Stata code as follows, you will generate a dataset which doesn’t provide the cumulative duration of having the event. Notice how the duration variable only has 1 instead of 1, 2, and 3.

by id time, sort: gen byte duration = sum(event==1)

Similarly, if you use the following code, you will generate the incorrect values. The sum(event)==1 syntax should be sum(event==1). However, this will “flag” the time when the event first occurred, which may be useful in some cases. 

by id (time), sort: gen byte duration = sum(event)==1

Let’s take our example further and generate a variable column that takes into consideration the period before the subject experiences an event. Suppose subject A experiences an event at time==3, but we want to center this as 0 and previous months as -1, -2, and so on. We need to first identify the time when the event occurs and populate that as a new variable, which we will call firstevent. We can use the following State code to generate firstevent based on the condition that the event==1 and the variable it occurs is time==3.

egen firstevent = min(cond(event == 1, time, .)), by(id)

There will be missing values since not all subjects experience the event. To populate the missing values for the subjects with no events (event==0), we need to replace firstevent by identifying the max time of the entire study period using the summary command. Once we have the max time of the study period, we add 1 to this and replace the missing values from the firstevent variable.

replace firstevent = max(time) if firstevent == .
summ time
global maxtime = r(max)

replace firstevent = $maxtime + 1 if firstevent == .

We can subtract the time from the first event to generate a new variable (its) that will capture the negative time before the event occurs and the positive time after the event occurs, centered on when event==1.

by id (time), sort: gen byte its = _n – firstevent

The new variable its is short for interrupted time series analysis. An investigator can use the its variable to plan any interrupted times series analysis without having to go through the ordeal of generating this variable using other software.

Here is a summary of the entire Stata code, which you can also find on my Github page:

***** Declare XTSET panel dataset.
* Variable list: id event time 
* id        =   subject identifier
* event     =   exposure of interest
* time      =   time interval

**** Create the duration variable to capture time after event.
by id (time), sort: gen byte duration = sum(event==1)

**** Create a variable for the time before the event.
egen firstevent = min(cond(event == 1, time, .)), by(id)

**** Identify the maxtime.
summ time
global maxtime = r(max)

**** Replace missing data with the maxtime + 1.
replace firstevent = $maxtime + 1 if firstevent == .

**** Create its to capture time before and after event. 
by id (time), sort: gen byte its = _n - firstevent

ACKNOWLEDGEMENTS

I used several online references to develop this tutorial for Stata. Nicholas J. Cox has some excellent tutorials that was influential in developing this piece.

Cox N. First and last occurrences in panel data. From https://www.stata.com/support/faqs/data-management/first-and-last-occurrences/

The Statlist forum was also helpful; in particular, the following discussion threads.

https://www.statalist.org/forums/forum/general-stata-discussion/general/965910-how-generate-variable-that-indicates-current-and-prior-event-occurrence-for-id-in-panel-data

https://www.statalist.org/forums/forum/general-stata-discussion/general/1297707-creating-duration-variable-for-panel-data

https://www.stata.com/statalist/archive/2010-12/msg00193.html

https://www.stata.com/statalist/archive/2012-09/msg00286.html

The UCLA Institute for Digital Research and Education has a tutorial on using _N and _n to count in Stata.

Counting from _N to _N. UCLA: Statistical Consulting Group. From https://stats.idre.ucla.edu/stata/seminars/notes/counting-from-_n-to-_n/