Best practices

Introduction

There are some rules of thumb that I follow when using the TreatmentPatterns package. These rules tend to work well in most situations, across databases and datasets.

TLDR

minPostCombinationWindow <= minEraDuration.
combinationWindow >= minEraDuration.
Small cohorts should not be considered.
Pathways with a low count should not be considered.
Possible number of pathways: \(n^n\)
Possible number of pathways, with no re-occurrence: \(n!\)
Total number of possible events (events + combinations): \(2^n - 1\)
Total number of possible combinations: \(2^n - (n + 1)\)

Cohorts

When creating cohorts, it is important to keep in mind that the subjects will be dived across pathways. Lets assume we have 10000 subjects in a fictitious cohort. Let’s also assume we have 5 event cohorts.

The total number of potential pathways, assuming only mono therapies equals to \(pathways_n = n^{n}\), assuming we do not allow for any re-occurring treatments it would still equal to \(pathways_n = n!\).

Assuming our 5 event cohorts this would equal to:

5^5

## [1] 3125

factorial(5)

## [1] 120

Combinations

Combinations add additional pathway possibilities. Each event can be uniquely combined with each other event. Each combination can combine with another singular event or any other combination. However, each event in a combination must be unique. So: \(AB = BA\). As an example it is irrelevant if a person receives penicillin and ibuprofen or ibuprofen and penicillin.

We can draw out all possible combinations in a graph for events \(A\) \(B\) \(C\).

The subscript of the nodes are the layers where the combination exists in. I.e. combination \(AB\) is in layer 2, and combinations \(ABCD\) is in layer 4. The layer coincides with the number of events in the combination.

We can count the number of nodes per layer, for each graph: \[ \begin{matrix} & l1 & l2 & l3 & l4 & sum\\ A & 1 & 3 & 3 & 1 & 8\\ B & 1 & 2 & 1 & 0 & 4\\ C & 1 & 1 & 0 & 0 & 2\\ D & 1 & 0 & 0 & 0 & 1 \end{matrix} \]

Our sums look suspiciously similar to \(2^n\).

2^1

## [1] 2

2^2

## [1] 4

2^3

## [1] 8

2^4

## [1] 16

We seem to overshoot by 1 \(n\), so we can try \(2^{n-1}\).

2^0

## [1] 1

2^1

## [1] 2

2^2

## [1] 4

2^3

## [1] 8

So our total number of events equals: \[ \sum^{n-1}_{i=0}2^{i} \]

Which we can define as a function \(f_1\).

sum(c(2^0, 2^1, 2^2, 2^3))

## [1] 15

# Or:
n <- 4
sum(2^(0:(n - 1)))

## [1] 15

f_1 <- function(n) {
  sum(2^(0:(n - 1)))
}

We can simulate our \(f_1\) function for 100 events.

n <- 1:25
f_1_events <- unlist(lapply(n, f_1))

data.frame(
  n = n,
  f_1 = f_1_events
)

##     n      f_1
## 1   1        1
## 2   2        3
## 3   3        7
## 4   4       15
## 5   5       31
## 6   6       63
## 7   7      127
## 8   8      255
## 9   9      511
## 10 10     1023
## 11 11     2047
## 12 12     4095
## 13 13     8191
## 14 14    16383
## 15 15    32767
## 16 16    65535
## 17 17   131071
## 18 18   262143
## 19 19   524287
## 20 20  1048575
## 21 21  2097151
## 22 22  4194303
## 23 23  8388607
## 24 24 16777215
## 25 25 33554431

Notice how the number of events increases with \(2^n-1\).

We define this as \(f_2\). We can compare \(f_1\) to \(f_2\).

f_2 <- function(n) {
  2^n - 1
}

n <- 1:25
f_1_events <- unlist(lapply(n, f_1))
f_2_events <- unlist(lapply(n, f_2))

data.frame(
  n = n,
  f_1 = f_1_events,
  f_2 = f_2_events
)

##     n      f_1      f_2
## 1   1        1        1
## 2   2        3        3
## 3   3        7        7
## 4   4       15       15
## 5   5       31       31
## 6   6       63       63
## 7   7      127      127
## 8   8      255      255
## 9   9      511      511
## 10 10     1023     1023
## 11 11     2047     2047
## 12 12     4095     4095
## 13 13     8191     8191
## 14 14    16383    16383
## 15 15    32767    32767
## 16 16    65535    65535
## 17 17   131071   131071
## 18 18   262143   262143
## 19 19   524287   524287
## 20 20  1048575  1048575
## 21 21  2097151  2097151
## 22 22  4194303  4194303
## 23 23  8388607  8388607
## 24 24 16777215 16777215
## 25 25 33554431 33554431

Now we can assert the following: \[ monoEvents = n \\ totalEvents = 2^n - 1 \\ combinationEvents = totalEvents - n \]

n <- 5
totalEvents <- 2^n - 1
combinationEvents <- totalEvents - n

sprintf("monoEvents: %s", n)

## [1] "monoEvents: 5"

sprintf("totalEvents: %s", totalEvents)

## [1] "totalEvents: 31"

sprintf("combinationEvents: %s", combinationEvents)

## [1] "combinationEvents: 26"

Settings

The minEraDuration, combinationWindow, and minPostCombinationWindow have significant effects on how the treatment pathways are built. Conciser the following example:

library(dplyr)

cohort_table <- tribble(
  ~cohort_definition_id, ~subject_id, ~cohort_start_date,    ~cohort_end_date,
  1,                     1,           as.Date("2020-01-01"), as.Date("2021-01-01"),
  2,                     1,           as.Date("2020-01-01"), as.Date("2020-01-20"),
  3,                     1,           as.Date("2020-01-22"), as.Date("2020-02-28"),
  4,                     1,           as.Date("2020-02-20"), as.Date("2020-03-3")
)

cohort_table

## # A tibble: 4 × 4
##   cohort_definition_id subject_id cohort_start_date cohort_end_date
##                  <dbl>      <dbl> <date>            <date>         
## 1                    1          1 2020-01-01        2021-01-01     
## 2                    2          1 2020-01-01        2020-01-20     
## 3                    3          1 2020-01-22        2020-02-28     
## 4                    4          1 2020-02-20        2020-03-03

Assume that the target cohort is cohort_definition_id: 1, the rest are event cohorts.

cohort_table <- cohort_table %>%
  mutate(duration = as.numeric(cohort_end_date - cohort_start_date))

cohort_table

## # A tibble: 4 × 5
##   cohort_definition_id subject_id cohort_start_date cohort_end_date duration
##                  <dbl>      <dbl> <date>            <date>             <dbl>
## 1                    1          1 2020-01-01        2021-01-01           366
## 2                    2          1 2020-01-01        2020-01-20            19
## 3                    3          1 2020-01-22        2020-02-28            37
## 4                    4          1 2020-02-20        2020-03-03            12

As you can see, the duration of the treatments are: 19, 37 and 12 days. Also cohort 3 overlaps with treatment 4 for 8 days.

We can compute the overlap as follows:

cohort_table <- cohort_table %>%
  # Filter out target cohort
  filter(cohort_definition_id != 1) %>%
  mutate(overlap = case_when(
    # If the result of the next cohort_end_date is NA, set 0
    is.na(lead(cohort_end_date)) ~ 0,
    # Compute duration of cohort_end_date - next cohort_start_date
    # 2020-02-28 - 2020-02-20 = -8
    .default = as.numeric(cohort_end_date - lead(cohort_start_date))))

cohort_table

## # A tibble: 3 × 6
##   cohort_definition_id subject_id cohort_start_date cohort_end_date duration
##                  <dbl>      <dbl> <date>            <date>             <dbl>
## 1                    2          1 2020-01-01        2020-01-20            19
## 2                    3          1 2020-01-22        2020-02-28            37
## 3                    4          1 2020-02-20        2020-03-03            12
## # ℹ 1 more variable: overlap <dbl>

We see that the overlap between treatment 2 and 3 is -2, so rather than an overlap there is a gap between these treatments. Between treatment 3 and 4 there is an 8 day overlap. There is no next treatment after treatment 4, so the overlap is 0, let’s assume our minEraDuration = 5.

We can draw it out like so:

2:   -------------------
3:                        -------------------------------------
4:                                                     ------------

If we set our minCombinationWindow = 5, the combination would be computed for cohort 3 and 4. This would leave us with the following treatments:

2:   -------------------
3:                        -----------------------------
3+4:                                                   --------
4:                                                             ----

Treatment 3 now lasts 11 days; Treatment 4 lasts 4 days; and combination treatment 3+4 lasts 8 days. If our minPostCombinationDuration is not set properly, we can filter out either too many, or too little treatments.

Assuming we would set minPostCombinationDuration = 10, we would lose treatment 4 and combination treatment 3+4. This would leave us with the following paths:

2:   -------------------
3:                        -----------------------------

Pathway: 2-3

As a rule of thumb the setting the minPostCombinationDuration <= minEraDuration seems to yield reasonable results. This would leave us with the following paths minPostCombinationDuration = 5:

2:   -------------------
3:                        -----------------------------
3+4:                                                   --------

Pathway: 2-3-3+4

Maarten van Kessel

2024-05-21

Introduction

TLDR

Cohorts

Combinations

Settings