Introduction to intervalaverage
In intervalaverage: Time-Weighted Averaging for Interval Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(intervalaverage)
set.seed(1)

This package and vignette makes extensive use of data.table. If you're unfamiliar with the data.table syntax, a brief review of that package's introductory vignette may be useful.

Averaging values measured over intervals

Consider the following dataset which represents average (predicted) pm2.5 exposure and no2 exposure at some location over four sequential 7-day periods at the beginning of the year 2000:

exposure_dataset <- data.table(location_id=1, start=seq(as.IDate("2000-01-01"),by=7,length=4),
                end=seq(as.IDate("2000-01-07"),by=7,length=4),pm25=rnorm(4,mean=15), no2=rnorm(4,mean=25))
exposure_dataset

Note that the above intervals are stored as a column for the start of the interval and a column for the end of the interval. For the purpose of this package, intervals are ALWAYS treated as closed (i.e. inclusive of start and end values) and the variables storing interval starts and ends must be discrete (e.g., class integer or IDate).

If we wanted to calculate the average of the first two weeks of pm25 data, this would simply be the average of the two pm25 values for those weeks:

exposure_dataset[start %in% as.IDate(c("2000-01-01","2000-01-08")),mean(pm25)]

But we wanted the average of the first 10 days of that pm25 data, we would need to take a weighted average since the period from Jan 1 to Jan 10 doesn't align perfectly with the intervals over which the pm2.5 data is recorded:

exposure_dataset[start %in% as.IDate(c("2000-01-01","2000-01-08")),weighted.mean(pm25,w=c(7/10,3/10))]

The intervalaverage package and specifically the intervalaverage function was written to facilitate this sort averaging operation. In order to use this the package function, we'll need a dataset containing data that's stored over intervals (such as in exposure_dataset) as well as a dataset containing the periods you'd like to average over.

Let's create a dataset containing some periods we'd like averages over:

averaging_periods <- data.table(start=seq(as.IDate("2000-01-01"),by=10,length=3),
                end=seq(as.IDate("2000-01-10"),by=10,length=3))
averaging_periods

Now that we have defined intervals to average over, let's use the intervalaverage function to calculate the averages:

Note that in order for the intervalaverage function to work, the start and end columns need to have the same column names in x and in y. These column names are specified via the interval_vars argument. And the variables in x that you want averages calculated for are specified via value_vars.

averaged_exposures <- intervalaverage(x=exposure_dataset,y=averaging_periods,
                                      interval_vars=c("start","end"),value_vars=c("pm25","no2")
                                      )
averaged_exposures[, list(start,end,pm25,no2)]

The return value of the intervalaverage function is a data.table. Just the first four columns of that return data.table are printed. The return data.table always contains the exact intervals specified in y (and, as such, the number of rows of the return is always the number of rows in y). The return also contains a column for each value_var specified from x. These columns contain the values of the variables from x averaged over periods in y. Note that the value of the pm25 in the first row is what we calculated manually above.

Note that the third entry for both the pm25 and the no2 column is NA or missing. This makes sense because the x ( exposure_dataset) didn't have measurements for every day in the interval in y (averaging_periods) from Jan 21, 2000 to Jan 30, 2000.

Displaying the full data.table returned by the function gives us some more information:

averaged_exposures

The xduration column tells us the number of days that were present in x for each interval specified in y. The first two averaging_periods intervals were fully represented in x, whereas x only contained data for 8 of the 10 days in the third y interval. The xmaxend column shows us that the last day in the interval from Jan 21,2000 to Jan 30, 2000 that was present in y was Jan 28, 2000.

These supplementary columns are useful for diagnosing incomplete data in exposure_dataset.

If we're ok with calculating an average based on incomplete data, we can set the the tolerance for missingness lower. Let's say we're ok with calculating a non-missing average if 75% or more of the period is observed:

intervalaverage(x=exposure_dataset,
                y=averaging_periods,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                required_percentage = 75)

The results are the same for the first two rows but now we have nonmissing values in the third which are calculated based on the available data in exposure_dataset. If there had been a period with less than 75% of the data present, the function would still return NA for those value variables.

Averaging within values of a grouping variable (e.g. at multiple locations)

Often, we might have interval data at more than one location or identifier at a time. Let's create a data.table similar to exposure_dataset but with several (three) locations:

exposure_dataset2 <- rbindlist(lapply(1:3, function(z){
  data.table(location_id=z, 
             start=seq(as.IDate("2000-01-01"),by=7,length=4),
             end=seq(as.IDate("2000-01-07"),by=7,length=4),pm25=rnorm(4,mean=15),
             no2=rnorm(4,mean=25))} ))
exposure_dataset2

If you want to use the intervalaverage function to calculate averages of values in x over a set of averaging periods separately for each level of an identifier variable, that identifier variable needs to be crossed with every averaging period in y. It takes an extra step to cross the identifier with the averaging periods to create y, but in creating y this way you explicitly define the form of the return value of intervalaverage, since intervalaverage always returns one row for each row in y.

Let's cross the previous averaging periods table with every unique value of the identifier in the new exposure dataset:

#unexpanded:
averaging_periods
#expanded to every location_id:
rbindlist(lapply(1:3, function(z)copy(averaging_periods)[,location_id:=z][]))

The above code is a bit esoteric so the intervalaverage package contains function to simplify and generalize this process of repeating/expanding a set of intervals (or more generally, a set of rows in a table) for every location_id (or more generally, for every row in another table). To use this CJ.dt function, just create a data.table with a column containing unique ids, then call CJ.dt on the two tables:

exposure_dataset2_unique_locs <- data.table(location_id=unique(exposure_dataset2$location_id))
averaging_periods2 <- CJ.dt(averaging_periods, exposure_dataset2_unique_locs)
averaging_periods2

#or, more concisely:
averaging_periods2 <- CJ.dt(averaging_periods, unique(exposure_dataset2[,list(location_id)]))

Now, to take averages of x values over intervals in y within groups, all we have to do is use the same call as in the first example to intervalaverage while specifying one more argument: group_vars="location_id".

intervalaverage(x=exposure_dataset2,
                y=averaging_periods2,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars="location_id",
                required_percentage = 75)[, list(location_id, start,end, pm25,no2)]

The group_vars argument tells the intervalaverage function to calculate averages separately within each group.

Of course, we could have completed the above by calling intervalaverage repeatedly for each value of location_id in x and y using a for loop. The reason to prefer using the group_vars approach is that the intervalaverage function is written to be faster than looping when with dealing with grouping. It also saves you the trouble of writing a loop and combining the results.

Additionally, group_vars accepts a vector of character column names, meaning that you can calculate averages within combinations of groups without writing nested for loops.

Values in overlapping periods are not allowed

Note that all the intervals used in this package are treated as inclusive. So far, we've dealt with data which have intervals which do not overlap. However, consider the following dataset where the end day of a previous interval is the start day of the next interval:

exposure_dataset_overlapping <- rbindlist(lapply(1:3, function(z){
  data.table(location_id=z, 
             start=seq(as.IDate("2000-01-01"),by=7,length=4),
             end=seq(as.IDate("2000-01-08"),by=7,length=4),
             pm25=rnorm(4,mean=15),
             no2=rnorm(4,mean=25) )
} ))
exposure_dataset_overlapping

If we try to average this exposure dataset, we get an error:

intervalaverage(exposure_dataset_overlapping,averaging_periods2,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars="location_id",
                required_percentage = 75)

That's because the intervalaverage function is written to throw an error if there are overlaps in within groups. This is to encourage the user to explicitly and consciously deal with overlaps prior to averaging.

Note that we can also check whether there are overlapping intervals (within specified groups) using is.overlapping

is.overlapping(exposure_dataset_overlapping, interval_vars=c('start','end'),group_vars="location_id")

In order to deal with partially overlapping intervals, we need to split intervals into areas of exact overlap and non-overlap with the isolateoverlaps function:

exposure_dataset_isolated <- isolateoverlaps(exposure_dataset_overlapping, 
                                             interval_vars=c("start","end"), 
                                             group_vars="location_id",
                                             interval_vars_out=c("start2","end2"))
exposure_dataset_isolated[1:15] #only show the first 15 rows

Inspect the above table and compare it to exposure_dataset. start2 and end2 are the new intervals and start and end are the original intervals. Note how there are two rows for every overlapping period (ie in the start2 and end2 columns), but the pm25 and no2 values differ within these rows since one value comes from the first overlapping period and the second value comes from the second overlapping period.

We can then average exposure values within periods of exact overlap:

exposure_dataset_overlaps_averaged <-
  exposure_dataset_isolated[, list(pm25=mean(pm25),no2=mean(no2)),by=c("location_id","start2","end2")]

setnames(exposure_dataset_overlaps_averaged, c("start2","end2"),c("start","end"))
exposure_dataset_overlaps_averaged

This version of the dataset where values in overlapping periods have already been averaged can now be averaged to the times in averaging_periods2 using the intervalaverage function:

intervalaverage(exposure_dataset_overlaps_averaged,
                averaging_periods2,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars="location_id",
                required_percentage = 75)[,list(location_id, start,end,pm25,no2)]

Overlapping averaging periods

While overlapping periods in x are not allowed, there's nothing wrong with averaging to multiple partially overlapping periods in y at the same time:

overlapping_averaging_periods <- data.table(start=as.IDate(c("2000-01-01","2000-01-01")),
                                            end=as.IDate(c("2000-01-10","2000-01-20"))
                                            )
overlapping_averaging_periods
overlapping_averaging_periods_expanded <-
  CJ.dt(overlapping_averaging_periods,unique(exposure_dataset2[,list(location_id)]))

overlapping_averaging_periods_expanded


intervalaverage(exposure_dataset_overlaps_averaged,
                overlapping_averaging_periods_expanded,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars="location_id",
                required_percentage = 75)[,list(location_id, start,end,pm25,no2)]

Note that if you specify identical intervals (within groups defined by group_vars), duplicate intervals in y will be dropped with a warning resulting in a return data.table with fewer rows than in y.

Averaging values measured over intervals: different averaging periods for different groups

Often we're interested in calculating averages over different periods for different locations. First to make this more realistic, let's generate ~20 years of data at 2000 locations:

n_locs <- 2000
n_weeks <- 1000
exposure_dataset3 <- rbindlist(
  lapply(1:n_locs, function(id) {
    data.table(
      location_id = id,
      start = seq(as.IDate("2000-01-01"), by = 7, length = n_weeks),
      end = seq(as.IDate("2000-01-07"), by = 7, length = n_weeks),
      pm25 = rnorm(n_weeks, mean = 15),
      no2 = rnorm(n_weeks, mean = 25)
    )
  })
)

exposure_dataset3

Now let's pick a different random start and end date for each location's averaging period. We'll pick start dates at random and define the end the date as 3 years after each start date, thus creating different three-year intervals for every location.

averaging_periods3 <- data.table(location_id=1:n_locs,
                                 start=sample(
                                   x=seq(as.IDate("2000-01-01"),as.IDate("2019-12-31"),by=1),
                                   size=n_locs
                                 )
)
averaging_periods3[,end:=start+round(3*365.25)]
averaging_periods3

Because we've already generated y (averaging_period3) to contain the desired averaging interval for each value of the grouping variable location_id, it's ready to be used as an argument to intervalaverag:

intervalaverage(exposure_dataset3,
                averaging_periods3,
                interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars="location_id")[,list(location_id,start,end,pm25,no2)]

You shouldn't be surprised to see some missingness since the earliest possible latest possible averaging period start date is Dec 31, 2019 but the exposures stop in early 2019. The required_percentage argument could be set here to something less than the default of 100 to compute partial averages and get fewer missing values.

Finally, a quick trick if you'd like to calculate 1-year, 2-year, and 3-year averages all at once, starting with a fixed set of end dates:

averaging_periods3[, avg3yr:=end-round(3*365.25)]
averaging_periods3[, avg2yr:=end-round(2*365.25)]
averaging_periods3[, avg1yr:=end-round(1*365.25)]
#reshape the data.table:
averaging_periods4 <- melt(averaging_periods3,id.vars=c("location_id","end"),
                           measure.vars = c("avg3yr","avg2yr","avg1yr")) 
setnames(averaging_periods4, "value","start")
setnames(averaging_periods4, "variable","averaging_period")
averaging_periods4

intervalaverage(exposure_dataset3,averaging_periods4,interval_vars=c("start","end"),
                value_vars=c("pm25","no2"),
                group_vars=c("location_id"),
                required_percentage = 75)[,list(location_id,start,end,pm25,no2)]

Interval intersects: averaging with an address history

So far we've fully covered the functionality of the intervalaverage function and how to use it when we want to average over time at specific locations.

The above examples also cover the approach we'd use if we wanted to average over a cohort of study participants for whom we only have a single address (and we are ok assuming that participants never move).

However, in cohort studies, each participants shares their past locations/addresses and indicated the time periods over which they lived at each of those addresses. Typically this information is represented through a table we refer to as an "address history."

We'll start with a very simple example to demonstrate what an address history looks like and how we might use this in exposure averaging. Consider the following address history and exposure datasets:

address_history0 <- data.table(addr_id=c(1,2,2,3,5),
                 ppt_id=c(1,1,1,2,2),
                 addr_start=c(1L,10L,12L,1L,13L),
                 addr_end=c(9L,11L,14L,12L,15L)) 
exposure_dataset5 <- data.table(addr_id=rep(1:4,each=3),
  exp_start=rep(c(1L,8L,15L),times=4),
  exp_end=rep(c(7L,14L,21L),times=4),
  exp_value=c(rnorm(12))
 )

exposure_dataset5

Note that exposure_dataset5 has two regular (length-7) intervals for each address and corresponding measurements for those periods.

Here is a sample address history table:

address_history0

The first thing to note is that the address intervals (addr_start and addr_end) are non-overlapping, which is good because intervalaverage requires non-overlapping intervals as we saw previously. (If the addresses were overlapping we might consider using the isolateoverlaps function on it to identify overlapping periods in the address history and make decisions about which address to use in each overlapping period).

The address_history0 table has one participant with three rows and two addresses (addr_ids 1 and 2). In practice this would be the same data if the two intervals where ppt_id==1 & addr_id==2 were stored as a single row corresponding to the interval [10,14], but the dataset has been created like this to demonstrate that a single address represented over non-overlapping intervals doesn't cause problems. The second participant also has two addresses (addr_idss 3 and 5).

The goal here is to get exposures merged and clipped to the address intervals, but the problem is that the address intervals don't line up nicely with the exposure intervals. Participant 1 lived add address 1 from [1,9] but exposure is measured over [1,7] and [8,14]. The solution is to create two rows for that participant, one row for [1,7] and a second row from [8,9]. This can be accomplished using the intervalintersect function:

exposure_addresss_table <- intervalintersect(exposure_dataset5,
                                             address_history0,
                                             interval_vars=c(exp_start="addr_start",
                                                             exp_end="addr_end"),
                                             "addr_id")
exposure_addresss_table

intervalintersect takes every possible combination of overlapping intervals within group_vars (in this sense it is a cartesian join. More on this below).

intervalintersect is also an inner join because rows from either table that are not joined are not included in the output. For example, rows where addr_id==4 in exposure_dataset5 is not included since there are no rows in address_history0 where addr_id==4. Additionally, none of the exposure periods from exposure_dataset5 measured over exposure_start==15 to exposure_start==21 are included in the result, because none of the address history intervals overlap with those periods. Finally, there is an address (addr_id==5 from ppt_id==2) in the addr_history table that isn't in the exposure_dataset5 table. This address is also excluded from the result since exposure estimates do not exist for that participant.

It's worth doing some checks after completion of the intersection to identify what information has been dropped by the inner join:

setdiff(address_history0$addr_id,exposure_addresss_table$addr_id)
setdiff(exposure_dataset5$addr_id,exposure_addresss_table$addr_id)

Finally, note that the syntax of interval_vars allows those columns to be named differently in x and y via a named vector: interval_vars=c(exposure_start="addr_start",exposure_end="addr_end"). This is useful for keeping track of interval names since the return data.table has three sets of intervals: those from x, those from y, and their intersections.

Interval intersects: averaging with a larger address history

starting with the unique set of locations extracted from exposure_dataset3, let's generate 300 participants and a random number of addresses each participant lived at

n_ppt <- 300
addr_history <- data.table(ppt_id=paste0("ppt",sprintf("%03d", 1:n_ppt)))
addr_history[, n_addr := rbinom(.N,size=length(unique(exposure_dataset3$location_id)),prob=.001)]
addr_history[n_addr <1L, n_addr := 1L] 

#repl=TRUE because it's possible for an address to be lived at multiple different time intervals:
addr_history <- addr_history[, 
                             list(location_id=sample(exposure_dataset3$location_id,n_addr,replace=TRUE)),
                             by="ppt_id"
                             ]   
addr_history

#note that not all of these 2000 locations in exposure_dataset3 were "lived at" in this cohort:
length(unique(addr_history$location_id)) 

#also note that it's possible for different participants to live at the same address.
addr_history[,list(loc_with_more_than_one_ppt=length(unique(ppt_id))>1),
             by=location_id][,
                             sum(loc_with_more_than_one_ppt)]
#Because of the way I generated this data, it's way more common than you'd expect in a real cohort 
#but it does happen especially in cohorts with familial recruitment or people living in nursing home complexes.

I've generated intervals which are non-overlapping representing the participant address history (the code to achieve this is hidden because it's complicated and not the point of this vignette):

#generate a vector from which dates will be sampled

sample_dates <- function(n){
  stopifnot(n%%2==0)
  dateseq <- seq(as.IDate("1960-01-01"),as.IDate("2015-01-01"),by=1)
  dates <- sort(sample(dateseq,n))
  #90% of the time, make the last date "9999-01-01" which represents that the currently
   #lives at that location and we're carrying that assumption forward
  if(runif(1)>.1){
    dates[length(dates)] <- as.IDate("9999-01-01") 
  }
  dates
}


addr_history_dates <- addr_history[,list(date=sample_dates(.N*2)) ,by="ppt_id"] #for every address, ppt needs two dates: start and end
addr_history_dates_wide <- addr_history_dates[, list(start=date[(1:.N)%%2==1],end=date[(1:.N)%%2==0]),by="ppt_id"]
addr_history <- cbind(addr_history,addr_history_dates_wide[, list(start,end)])
setnames(addr_history, c("start","end"),c("addr_start","addr_end"))
#addr_history[,any(end=="9999-01-01"),by="ppt_id"][, sum(V1)]
setkey(addr_history, ppt_id, addr_start)
#here i'm using a trick to map distinct values of location_id to integers (within ppt) by coercing to factor then back to numeric.
addr_history[, addr_id:=paste0(ppt_id, "_", as.numeric(as.factor(location_id))),by=ppt_id]

addr_history

Oftentimes participant addresses are given their own keys that are distinct from location_id and that's represented in the above table. This means that a single location_id may map to multiple addr_ids.

It's important for these address tables to be non-overlapping within ppt. As shown previously, there's a function in the intervalaverage package for that check:

is.overlapping(addr_history,interval_vars=c("addr_start","addr_end"),
               group_vars="ppt_id") #FALSE is good--it means there's no overlap of dates within ppt.

This table passes that check because I've generated the data to be non-overlapping, but often times people report overlapping address histories and analytic decisions need to be made to de-overlap them (again, the isolateoverlap function would be useful for isolating sections of overlapping address intervals).

interval intersects is a cartesian join

So far we've seen exposure datasets stored by location_id but it's possible also to store exposures stored by addr_id (such that the series of exposure estimates for a single locations may be repeated multiple time if that location_id maps to multiple addr_ids )

exposure_dataset3_addr <- unique(addr_history[, list(location_id,addr_id)])[exposure_dataset3, 
                                                                            on=c("location_id"),
                                                                            allow.cartesian=TRUE,
                                                                            nomatch=NULL]
exposure_dataset3
exposure_dataset3_addr

Note that exposure_dataset3_addr contains repeat locations whereas exposure_dataset3 contains exactly one location per time point:

exposure_dataset3[, sum(duplicated(location_id)),by=c("start")][,max(V1)] #no duplicate locations at any date
exposure_dataset3_addr[, sum(duplicated(location_id)),by=c("start")][,max(V1)]

exposure_dataset3_addr has duplicate locations since multiple ppts may live at the same location or because a single participant lives at the same location multiple times.

(Storing exposure data according to addr_id rather than location_id takes up more space but may be beneficial for constraining exposure model revisions to be the same within cohorts)

location_id may not even be present in the address table if it's stored by address_id:

exposure_dataset3_addr[, location_id:=NULL]

Even if this distinction between how exposures are stored doesn't seem relevant, this section will demonstrate how intervalintersect is actually a cartesian join (in addition to being an inner interval join).

Whether the exposure dataset is stored by address or location, the intervalintersect will result in values from the exposure dataset merged to every address. In the case of the exposure table being stored by addr_id, this is a simple one to one merge (since for every address in the address history there's a set of exposures in the exposure table).

But in the case of the exposure table being stored by location_id, this becomes a one to many merge (since a single location id may merge to multiple locations in the address history table).

This works because the function that intervalintersect relies on (data.table::foverlaps) is performing an inner cartesian merge: that is--it only takes rows which match on the keying variables but also does a cartesian expansion if there are multiple matches in both tables.

z <- intervalintersect(x=exposure_dataset3,
               y=addr_history,
               interval_vars=c(
                 start="addr_start",
                 end="addr_end"
                 ),
               group_vars=c("location_id"),
               interval_vars_out=c("start2","end2")
) 

z_addr <- intervalintersect(x=exposure_dataset3_addr,
               y=addr_history,
               interval_vars=c(
                 start="addr_start",
                 end="addr_end"
                 ),
               group_vars=c("addr_id"),
               interval_vars_out=c("start2","end2")
) 

setkey(z,ppt_id,start2,end2)
setkey(z_addr,ppt_id,start2,end2)
all.equal(z,z_addr)

z

(note that this example of a one to many join maybe isn't a true "cartesian" join, but intervalintersect is capable of doing a true many-to-many cartesian expansion if provided the right datasets, although I'm not sure in context that would actually make any sense.)

In any case, the morale here is that whether the exposures are stored by location or address, using intervalintersect in combination will result in a dataset containing relevant exposures clipped to each address period.

This dataset can then be used to calculate averages over where a participant lived in a given period:

final_averaging_periods <- data.table(ppt_id=sort(unique(addr_history$ppt_id)))
final_averaging_periods[, end2:=sample(seq(as.IDate("2003-01-01"),as.IDate("2015-01-01"),by=1),.N)]
final_averaging_periods[,start2:=as.IDate(floor(as.numeric(end2-3*365.25)))]
final_averaging_periods

intervalaverage(z,final_averaging_periods, interval_vars=c("start2","end2"),
                value_vars=c("pm25","no2"),group_vars="ppt_id",required_percentage = 95
                )

There's lots of missingness because the address history I generated is nowhere near complete, but this demonstrates how important it is to have a good address history!