# Journal of Statistical Software

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1 JSS Journal of Statistical Software January 2011, Volume 38, Issue 5. Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up Martyn Plummer International Agency for Research on Cancer Bendix Carstensen Steno Diabetes Center Abstract The Lexis class in the R package Epi provides an object-based framework for managing follow-up time on multiple time scales, which is an important feature of prospective epidemiological studies with long duration. Follow-up time may be split either into fixed time bands, or on individual event times and the split data may be used in Poisson regression models that account for the evolution of disease risk on multiple time scales. The summary and plot methods for Lexis objects allow inspection of the follow-up times. Keywords: epidemiology, survival analysis, R. 1. Introduction Prospective epidemiological studies follow a cohort of individuals until disease occurrence or death, or until the scheduled end of follow-up. The data for each participant in a cohort study must include three variables: time of entry, time of exit and status at exit. In the R language (R Development Core Team 2010), working with such data is made easy by the Surv class in the survival package (Therneau and Lumley 2010). The survival package also provides modelling functions that use Surv objects as outcome variables, and use the standard S syntax for model formulae. In epidemiological studies with long-term follow-up, there may be more than one time scale of interest. If follow-up time is measured in decades, for example, any analysis of disease risk must take account of the impact of the ageing of the population. In this case, calendar time and age are both time scales of interest. A time scale may also measure the time elapsed since an important event, such as study entry, first exposure to a risk factor or beginning of treatment.

2 2 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up Age Calendar time Figure 1: A simple example of a Lexis diagram showing schematically the follow-up of 13 individuals. The statistical problem of accounting for multiple time scales can be addressed using tools developed in the field of demography in the 19th century. Such tools are also used in descriptive epidemiology to separate time trends in chronic disease incidence rates into age, period and cohort effects. The increasing number of large population-based cohort studies with long-term follow-up has created a demand for these tools in analytical epidemiology. The Epi (Carstensen, Plummer, Laara, and Hills 2010) package contains functions and classes to facilitate the analysis of epidemiological studies in R. Among these, the Lexis class was designed to simplify the analysis of long term follow-up studies by tracking follow-up time on multiple time scales. It also accounts for many possible disease outcomes by having a status variable that is not a simple binary indicator (alive/dead or healthy/diseased) but may take multiple values. 2. Lexis diagrams and Lexis objects Figure 1 shows a simple example of a Lexis diagram, named after the demographer Wilhelm Lexis ( ). Each line in a Lexis diagram represents the follow-up of a single individual

3 Journal of Statistical Software 3 from entry to exit on two time scales: age and calendar time. Both time scales are measured in the same units (years) so that the follow-up traces a line at 45 degrees. Exit status is denoted by a circle for the 4 subjects who experienced a disease event. The other subjects are disease-free at the end of follow-up. The follow-up line of an individual in a Lexis diagram is defined by his or her entry time on the two time scales of interest (age and calendar time) and the duration of follow up. The Lexis class formalises this representation of follow-up in an R object. Lexis objects are not limited to 2 time scales, but allow follow-up time to be tracked on an arbitrary number of time scales. The only restriction is that time scales must be measured in the same units. To illustrate Lexis objects, we use a cohort of nickel smelting workers in South Wales (Doll, Mathews, and Morgan 1977), which was included as an example by Breslow and Day (1987). The data from this study are contained in the data set nickel in the Epi package. R> data("nickel") R> nicl <- Lexis(entry = list("period" = agein dob, "age" = agein, "tfe" = agein - age1st), exit = list("age" = ageout), exit.status = icd, id = id, data = nickel) Follow-up time is defined by entry, a named list of entry times on each time scale, and exit, another named list of exit times. Since duration of follow-up is the same on all time scales, it is only necessary to define the exit time on one of them, in this case age. The three time scales in this Lexis object are: ˆ period, representing calendar time. Date of entry is calculated as date of birth (dob) plus age at entry (agein). ˆ age, representing participant s age. ˆ tfe, representing time since first employment, which is used as a proxy for first exposure. Entry time on this scale is calculated as the difference between age at entry and age at first employment (age1st). The exit.status argument gives the individual s status at the end of follow-up. Since this is a study of mortality, the exit status is the cause of death according to the Seventh Revision of the International Classification of Diseases (ICD, World Health Organization 1957). For individuals who were still alive at the end of follow-up, the exit status is 0. The data argument gives a data frame that is the source of all the variables used to define entry time, exit time, status, and so on. The Lexis function transforms this data frame into a Lexis object Plotting Lexis objects Not surprisingly, the plot method for Lexis objects plots a Lexis diagram. Figure 2 shows a Lexis diagram for the nickel smelters cohort. The points method is used to annotate the Lexis diagram with the times of all deaths from lung cancer (ICD code 162 or 163): R> plot(nicl) R> case <- status(nicl) %in% c(162,163) R> points(subset(nicl, case), pch = "", col = "red")

4 4 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up period age Figure 2: Lexis diagram of the nickel smelters cohort. This example also illustrates the extractor function status, which returns the status at the beginning or (by default) end of each follow-up period. Other extractor functions dur, entry, and exit return respectively the duration of follow-up and the entry and exit times on any given time scale. By default, the plot method chooses the first two time scales of the Lexis object to plot. Other time scales may be chosen using the argument time.scale. A single time scale may be specified: R> plot(nicl, time.scale = "tfe", xlab = "Time since first employment (years)") This produces Figure 3, in which the y-axis is the unique id number and all history lines are horizontal. Such plots may reveal important features of the data. For example, Figure 3 shows that, on the tfe time scale, there are many late entries into the study with some participants entering over 20 years after first employment. Due to the method of selection for this cohort, no participant came under observation until 1934, even if they had been working many years in the smelting industry (Breslow and Day 1987).

5 Journal of Statistical Software 5 id number Time since first employment (years) Figure 3: Schematic representation of follow-up in the nickel smelters cohort Structure of a Lexis object Lexis objects inherit from the class data.frame. The Lexis object contains all the variables in the source data frame that was given as the data argument to the Lexis function. In addition, a variable is created for each time scale as well as a four variables with reserved names starting with lex. (lex.dur, lex.cst, lex.xst, and lex.id). R> head(nicl)[, 1:7] period age tfe lex.dur lex.cst lex.xst lex.id In this example, the first 3 variables (period, age, and tfe) show the entry times on the 3 time scales. The variable lex.dur shows the duration, lex.cst and lex.xst show the

6 6 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up current status and exit status respectively, and lex.id shows the unique identifier for each individual. 3. Splitting follow-up time The Cox proportional hazards model, which is the most commonly used model for timeto-event data in epidemiology, does not generalize to more than one time scale. A simpler parametric alternative is to use Poisson regression with a piecewise-constant hazard. In typical applications of Poisson regression the hazard is constant within time bands defined by 5-year periods of age or calendar year. A single individual may pass through several time bands as shown by Figure 4 which shows the follow-up of a single hypothetical individual and reproduces Figure 2.1 of Breslow and Day (1987). The individual represented in this Lexis diagram passes through 5 time bands before the end of follow-up. The total follow-up time is created by a call to the Lexis function: R> lx <- Lexis(entry = list(cal = , age = 43.71), exit = list(cal = ), exit.status = 1) R> lx cal age lex.dur lex.cst lex.xst lex.id This creates a simple Lexis object with only one row. The object may be split into separate time bands using the splitlexis function: R> lx <- splitlexis(lx, breaks=c(1955, 1960, 1965, 1970), time.scale = "cal") R> lx lex.id cal age lex.dur lex.cst lex.xst Splitting the follow-up time by 5-year calendar periods creates a new Lexis object with 3 rows. The total follow-up time of years is divided up into 3 periods of = years. A second call to splitlexis may be used to split the follow-up time along the age axis. R> lx <- splitlexis(lx, breaks=c(40, 45, 50, 55), time.scale = "age") R> lx lex.id cal age lex.dur lex.cst lex.xst

7 Journal of Statistical Software 7 Age at follow up (years) Calendar year of follow up Figure 4: Lexis diagram showing the follow-up of one person in a cohort study. The follow-up time is now divided into the 5 separate parts falling in different time bands defined by age and calendar time. Under the Poisson model, these separate follow-up periods make independent contributions to the likelihood and may therefore be treated as if they come from separate individuals (although, if needed, the lex.id variable keeps track of split follow-up times that come from the same individual). This simple example also shows what happens to the entry and exit status when follow-up time is split. It is assumed that an individual keeps their current status (entry status = 0) at each splitting time until the end of follow-up (exit status = 1) in the last interval. A call to the plot method for Lexis objects creates the plot shown in Figure 4. R> plot(lx, xlim = c(1955, 1971), ylim = c(40, 56), pty = "s", xlab = "Calendar year of follow-up", ylab = "Age at follow-up (years)") R> points(lx) When a split Lexis object is plotted, the break points are shown as a background grid. The points method annotates the end of each follow-up segment with a circle, showing how the follow-up line is split whenever it crosses either a horizontal or a vertical grid line.

8 8 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up 4. Modelling risk on multiple time scales Returning to the cohort of nickel smelters, we now show how time splitting may be combined with Poisson regression. R> nics1 <- splitlexis(nicl, "age", breaks = seq(40, 80, 10)) R> nics2 <- splitlexis(nics1, "tfe", breaks = c(20, 30)) The timeband function returns information about the time band on a given time scale. It can label the time bands in many different ways, according to the type argument. For Poisson regression, it is easiest to return a factor. R> nics2\$age.cat <- timeband(nics2, "age", type = "factor") R> nics2\$tfe.cat <- timeband(nics2, "tfe", type = "factor") R> subset(nics2, id == 8, select = c("age", "tfe", "lex.dur", "age.cat", "tfe.cat")) age tfe lex.dur age.cat tfe.cat (40,50] (20,30] (50,60] (20,30] (50,60] (30,Inf] (60,70] (30,Inf] Factors are labelled in the same way as for the cut function, as can be seen from the selected output for subject 8. These factors may then be used as predictor variables in a Poisson regression model that separates the effects of age from time since first employment: R> case <- status(nics2) %in% c(162,163) R> pyar <- dur(nics2) R> glm(case ~ age.cat tfe.cat offset(log(pyar)), family = poisson(), subset = (age >= 40), data = nics2) Call: glm(formula = case ~ age.cat tfe.cat offset(log(pyar)), family = poisson(), data = nics2, subset = (age >= 40)) Coefficients: (Intercept) age.cat(50,60] age.cat(60,70] age.cat(70,80] age.cat(80,inf] tfe.cat(20,30] tfe.cat(30,inf] Degrees of Freedom: 2757 Total (i.e. Null); Null Deviance: 999 Residual Deviance: 975 AIC: Residual Since no deaths occur from lung cancer before age 40 in this cohort, we have removed the lowest level of the age factor from the model using the subset argument to the glm function.

9 Journal of Statistical Software 9 5. Time splitting on an event Lexis objects also allow follow-up time to be split on an event. We illustrate this using data from a cohort of patients who were exposed to Thorotrast (Andersson, Vyberg, Visfeldt, Carstensen, and Storm 1994; Andersson, Carstensen, and Storm 1995), a contrast medium used for cerebral angiography in the 1930s and 1940s that was later found to cause liver cancer and leukaemia (IARC 2001). Data on the cohort are contained in the data set thoro in the Epi package. We convert the thoro data frame into a Lexis object using the data of injection of Thorotrast (injecdat) as the data of entry, and using time scales of calendar time ( cal ) and age ( age ). The cal.yr function from the Epi package is used to convert the Date variables to numeric calendar years. R> data("thoro") R> thorol <- Lexis(entry = list("cal" = cal.yr(injecdat), "age" = cal.yr(injecdat) - cal.yr(birthdat)), exit = list("cal" = cal.yr(exitdat)), entry.status = 2, exit.status = exitstat, id = id, data = thoro) For these data, the exit status may take three values (1 = dead, 2 = alive, 3 = lost to followup). We give all subjects an entry status of 2 (alive) in the call to the Lexis function. The summary method for Lexis objects prints a table showing the transitions between entry and exit states and a second table showing the transition rates. R> summary(thorol) Transitions: To From Records: Events: Risk time: Persons: Rates: To From Total In this study, 1966 out of 2470 participants (80%) died before the end of follow up in 1992, and the overall mortality rate was 4% per year. For participants in the cohort who developed liver cancer during follow-up, the variable liverdat contains the date of diagnosis of liver cancer. For other participants, this date is missing. We can use the cutlexis function to split follow-up time for liver cancer cases into pre-diagnosis and post-diagnosis: R> thorol2 <- cutlexis(thorol, cal.yr(thorol\$liverdat), timescale = "cal", new.state = 4) R> summary(thorol2) Transitions: To

10 10 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up age cal Figure 5: Survival from liver cancer in the Thorotrast study (35 cases). From Records: Events: Risk time: Persons: Sum Rates: To From Total Unlike the splitlexis function, the cutlexis function allows the status variable to be modified when follow-up time is split. The new.state = 4 argument means that the date we are splitting on is the date of transition to state 4 (liver cancer case). This is reflected in the updated transition table printed by the summary method, which shows 35 incident liver cancer cases (transitions 2 4), all of whom died during follow-up (transitions 4 1). The summary output also shows that the Lexis object has data on 2468 persons instead of

11 Journal of Statistical Software 11 the 2470 in the original object. In fact the 2 individuals who were dropped have no follow-up time in the study: their exit date is the same as the entry date. The cutlexis function automatically drops follow-up intervals with zero length. Survival from liver cancer can be analyzed by selecting only the rows of the Lexis object that represent follow-up after diagnosis, when the participant is in state 4. R> cases <- subset(thorol2, status(thorol2, at = "entry") == 4) R> plot(cases, col = "red") The results are shown in Figure 5. Survival from liver cancer in this cohort is very short, except for one case who survives more than 10 years after diagnosis. Such an anomalous result may prompt further checking of the data to ensure that the date of diagnosis or date of death had not been mis-coded. The same technique of splitting follow-up by event times can also be applied to multi-state disease models, in which arbitrarily complex transitions between disease states are possible. The additional machinery in the Epi package to handle this more complex situation is the subject of a companion paper (Carstensen and Plummer 2011). References Andersson M, Carstensen B, Storm HH (1995). Mortality and Cancer Incidence After Cerebral Angiography. Radiation Research, 142, Andersson M, Vyberg M, Visfeldt M, Carstensen B, Storm HH (1994). Primary Liver Tumours among Danish Patients Exposed to Thorotrast. Radiation Research, 137, Breslow NE, Day NE (1987). Statistical Methods in Cancer Research, volume II. International Agency for Research on Cancer, Lyon. Carstensen B, Plummer M (2011). Using Lexis Objects for Multistate Models in R. Journal of Statistical Software, 38(6), URL Carstensen B, Plummer M, Laara E, Hills M (2010). Epi: A Package for Statistical Analysis in Epidemiology. R package version , URL Doll R, Mathews JD, Morgan LG (1977). Cancers of the Lung and Nasal Sinuses in Nickel Workers: A Reassessment of the Period of Risk. British Journal of Industrial Medicine, 34, IARC (2001). Ionizing Radation Part 2: Some Internally Deposited Radionucludes. Number 78 in IARC Monographs on the Evaluation of Carcinogenic Risks to Humans. IARC- Press, Lyon, France. R Development Core Team (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN , URL http: // Therneau T, Lumley T (2010). survival: Survival Analysis Including Penalised Likelihood. R package version , URL

12 12 Lexis: An R Class for Epidemiological Studies with Long-Term Follow-Up World Health Organization (1957). International Classification of Diseases. WHO, Geneva. Affiliation: Martyn Plummer International Agency for Research on Cancer 150 Cours Albert-Thomas Lyon Cedex 08, France Bendix Carstensen Steno Diabetes Center Niels Steensens Vej Gentofte, Denmark & Department of Biostatistics, University of Copenhagen Web-site: Journal of Statistical Software published by the American Statistical Association Volume 38, Issue 5 Submitted: January 2011 Accepted:

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