Choosing Between Methods

Administrative Censoring Only

The methods are mathematically equivalent when there is no censoring except for at the end of study follow-up (i.e. administrative censoring only) - that is, when you observe all participants until they either experience a competing risk or reach the end of study follow-up.

Synchronized Event-Interval Censoring

Even with censoring before the end of follow-up, the two methods yield identical results when censoring follows a very specific pattern. Censoring must occur after everyone remaining at risk has experienced their pth event, but before anyone has had a (p+1)th. This censoring pattern is observed when non-administrative censoring is only observed during time intervals where the event count sequence is synchronized at the cohort level. I like to call this pattern synchronized event-interval censoring.

Take the applied example given by Dong, et al.¹ - assume 5 participants enrolled in hypothetical study:

• Subject 1: alive at the end of follow-up and administratively censored at that time (t8)
• Subject 2: Lost to follow-up and censored at t1
• Subject 3: Died (competing-risk event) at t5
• Subject 4: Experience the event of interest at t2, t6, and t7 and was alive at the end of follow-up (administratively censored at t8)
• Subject 5: Experienced event of interest once at t3 and died (competing risk event) at the same time point

df <- data.frame(
  id = c(1, 2, 3, 4, 4, 4, 4, 5, 5),
  time = c(8, 1, 5, 2, 6, 7, 8, 3, 3),
  cause = c(0, 0, 2, 1, 1, 1, 0, 1, 2)
 )

# Calculating MCC using both estimators
mcc_eq <- mcc(
  df,
  id_var = "id",
  time_var = "time",
  cause_var = "cause",
  method = "equation"
)
#> ℹ Adjusted time points for events occurring simultaneously for the same subject.

mcc_sci <- mcc(
  df,
  id_var = "id",
  time_var = "time",
  cause_var = "cause",
  method = "sci"
)
#> ℹ Adjusted time points for events occurring simultaneously for the same subject.

In this example, both methods are equivalent because the only non-administrative censoring that happens is before everyone’s first event (i.e., synchronized event-interval censoring):

# Dong-Yasui estimator
mcc_details(mcc_eq)
#> # A tibble: 9 × 8
#>    time nrisk censor event cmprk overall_surv_previous ave_events   mcc
#>   <dbl> <dbl>  <dbl> <dbl> <dbl>                 <dbl>      <dbl> <dbl>
#> 1  0        5      0     0     0                  1          0     0   
#> 2  1        5      1     0     0                  1          0     0   
#> 3  2        4      0     1     0                  1          0.25  0.25
#> 4  3        4      0     1     0                  1          0.25  0.5 
#> 5  3.00     4      0     0     1                  1          0     0.5 
#> 6  5        3      0     0     1                  0.75       0     0.5 
#> 7  6        2      0     1     0                  0.5        0.25  0.75
#> 8  7        2      0     1     0                  0.5        0.25  1   
#> 9  8        2      2     0     0                  0.5        0     1

# Sum of cumulative incidences estimator
mcc_details(mcc_sci)
#> # A tibble: 9 × 5
#>    time   CI1   CI2   CI3 SumCIs
#>   <dbl> <dbl> <dbl> <dbl>  <dbl>
#> 1  0     0     0     0      0   
#> 2  1     0     0     0      0   
#> 3  2     0.25  0     0      0.25
#> 4  3     0.5   0     0      0.5 
#> 5  3.00  0.5   0     0      0.5 
#> 6  5     0.5   0     0      0.5 
#> 7  6     0.5   0.25  0      0.75
#> 8  7     0.5   0.25  0.25   1   
#> 9  8     0.5   0.25  0.25   1

When Events Are Exchangeable

Recurrent events in a healthcare context might be reasonably considered exchangeable if their ordering is irrelevant clinically or biologically because all events represent the same underlying process or a routine, maintenance-type activities where clinical meaning is unrelated to the order of the recurrent event count. Consider the following examples:

• Routine medication refills: Each refill of chronic medications (blood pressure pills, diabetes medications) might represent identical adherence behavior
• Preventive care visits: Annual physicals or routine dental cleanings represent the same preventive behavior whether it’s the 3rd or 10th visit
• Routine lab monitoring: Quarterly hemoglobin A1c tests for stable diabetes may represent identical surveillance activity
• Routine imaging surveillance: Follow-up mammograms or CT scans for stable conditions represent the same monitoring process

When Event Order Matters

Exchangeability is unreasonable (or naive) for some recurrent outcomes in health research. With some outcomes, the occurrence of each event may impact the probability of subsequent events (i.e. the events have different clinical meaning or biological mechanisms depending on where they fall in the count sequence). Consider the following examples:

• Cancer recurrences: First vs second recurrence may have different biology and prognosis
• Hospital readmissions: Early readmissions may indicate discharge planning issues, while multiple readmissions suggest complex care needs
• Medication adverse events: Early vs late adverse events may have different mechanisms and clinical significance
• Seizures: Early vs late seizures may indicate different disease progression patterns (depending on the underlying cause)