Appendix
Grace windows
Grace period case example For working through concepts below, assume we are emulating a trial where eligible individuals are followed for up to 60 time periods from an arbitrary calendar date, and are randomly assigned to two treatment strategies:
- Do not receive treatment through the end of follow-up (60 time periods, e.g. 60 months ~ 5 years)
- Initiate treatment within 12 periods of index time zero
Individuals are followed from the index date through the end of follow-up as long as the observed data is consistent with their assigned treatment strategy. When the observed data becomes inconsistent, they are artificially censored at that point in time (i.e. our emulation will estimate per protocol effects from an RCT with perfect adherence).
It is possible to specify two or more interventions where a single person’s observed data is consistent with multiple strategies at a single point in time. Under both treatment strategies if the person does not get treatment for the first 5 periods, then up to period 12 they are faithfully adherent to both treatment strategies. At t=12, if they still are untreated they would continue to follow-up under the no treatment strategy but be non-adherent to the “treatment in 12 periods” strategy. This 12 period window we outline in our strategy is referred to as a “grace period” or “grace window”. Alternatively, if the person was assigned to “no treatment” and received treatment at any point in the 12 periods they would be censored upon initiation of treatment under the 1) “no treatment strategy” but follow-up continue under the 2) “treat within 12” strategy.
In a target trial emulation which employs these methods, each person’s observed data may be consistent with 1+ treatment strategies in the same time period, particularly the initial period of time. Because at baseline the person’s observed data is consistent with 1+ more strategies, the typical solution is to clone the person and assign each clone to a treatment strategy, e.g. duplicate or triplicate the person’ data and follow each clone as a unique observation.
The person could also be randomly assigned to one of the interventions (instead of cloned and assigned to both), but simulations suggest this is very statistically inefficient (much less precise estimates than cloning).
The cloning approach has an appealing effect of eliminating confounding at baseline (time zero). This is true because the observations in each treatment group are identical at baseline. However, as each clone is followed across time periods, their adherence to the assigned treatment strategy is monitored and each clone is censored (follow-up ends) when the observed data is no longer consistent with the assigned treatment strategy.
This censoring is artificial because we can still observe data for that person but we choose to ignore it because of their non-adherence to the assigned strategy. So although identical at baseline the groups observed characteristics (distributions of covariates) will diverge across time if the artificial censoring is informative and non-random (almost always true). Usually the artificial censoring is intrinsically linked to initiation of treatment, so the artificial censoring will introduce bias if the initiation of treatment also has common causes with the outcome of interest.
In order to adjust for this bias, inverse probability weighting is used. However, proper application of the weights requires some consideration of the probability of treatment initiation, grace periods and the artificial censoring mechanism.
Weight assignments in CCW design
Simple example: Probability weighting with a grace period
Assume three persons are assigned to a treatment strategy as follows: Get the treatment within 2 time periods, if you don’t get it by the second period you are censored for non-adherence. The goal of weighting is to create a pseudo-population that can estimate the effect in a counterfactual world where all those assigned to treatment take treatment by the end of the grace period, and all those assigned to not take treatment do not take it. In other words, we are estimating the causal effects of treatment in a trial where there is perfect adherence to study assignment, also known as a “per protocol” effect.
Observed treatments:
A: treated at last period
B: never treated
C: treated at first period
| Person | Period | Treat | Censor |
|---|---|---|---|
| A | 1 | 0 | 0 |
| A | 2 | 1 | 0 |
| B | 1 | 0 | 0 |
| B | 2 | 0 | 1 |
| C | 1 | 1 | 0 |
| C | 2 | 0 | 0 |
Three persons have observed data in our trial emulation. Persons A & C get the treatment and are not censored, Person B doesn’t get the treatment and is censored.
Now adding censoring probabilities Pr(C=1) to the table:
| Person | Period | Treat | Censor | Pr(C=1) |
|---|---|---|---|---|
| A | 1 | 0 | 0 | 0 |
| A | 2 | 1 | 0 | 1/2 |
| B | 1 | 0 | 0 | 0 |
| B | 2 | 0 | 1 | 1/2 |
| C | 1 | 1 | 0 | 0 |
| C | 2 | 0 | 0 | - |
Because people have 2 periods to get treatment, the Pr(C=1) in period 1 is 0 for all persons. In other words, people cannot be artificially censored in the first period, because by design we are allowing a grace of 2 periods. The Pr(C=1)= ½ for person A and B even though there are three people (i.e. ⅓)? Because person C already received the treatment in period 1, they are immutably uncensored from that point forward. So the only two who can censor are persons A and B, since person B censors in period 2, the Pr(C=1) = ½.
The probability of not censoring, Pr(C=0) is simply 1 - Pr(C=1.
| Person | Period | Treat | Censor | Pr(C=1) | Pr(C=0) |
|---|---|---|---|---|---|
| A | 1 | 0 | 0 | 0 | 1 |
| A | 2 | 1 | 0 | 1/2 | 1/2 |
| B | 1 | 0 | 0 | 0 | 1 |
| B | 2 | 0 | 1 | 1/2 | 1/2 |
| C | 1 | 1 | 0 | 0 | 1 |
| C | 2 | 0 | 0 | - | 1 |
The unstabilized inverse probability weight is 1 / Pr(C=0); in simple terms people who are likely to censor but dont are assigned higher weights and count for more in the analysis.
| Person | Period | Treat | Censor | Pr(C=1) | Pr(C=0) | IPW |
|---|---|---|---|---|---|---|
| A | 1 | 0 | 0 | 0 | 1 | 1 |
| A | 2 | 1 | 0 | 1/2 | 1/2 | 2 |
| B | 1 | 0 | 0 | 0 | 1 | 1 |
| B | 2 | 0 | 1 | 1/2 | 1/2 | 0 |
| C | 1 | 1 | 0 | 0 | 1 | 1 |
| C | 2 | 0 | 0 | - | 1 | 1 |
Person A is given a weight of 2, making up for the loss of censored person B. Person C is given a weight of 1 because they received treatment in the first period, and so couldn’t censor at the end of the grace period. Three person, and the sum of the weights in both periods is 3.
How does one carry the IPW weight forward after the grace period? In this simple scheme, assign a weight of 1 for each time period after grace, and compute a cumulative product of the weights for each timepoint up until the end of follow-up. So person B, IPW=2, person C, IPW=1, for each subsequent time-point. However, if another censoring mechanism was important (i.e. censor due to death or loss to follow-up), then you might compute that probability separately and take the product of the two.