Self-controlled designs

What this page answers

The instrumental variables page tackles the same predicament — unmeasured confounding — at a high price: two of three assumptions cannot be verified, and the confidence interval widens several-fold.

Self-controlled designs take an entirely different route, and the idea behind them is almost brutally simple:

If the two sides of the comparison are two stretches of time in the same person, then everything about that person that does not change over time disappears from the comparison.

Genes, constitution, sex, ancestry, childhood environment, socioeconomic position, the baseline burden of chronic disease, health-seeking habits — it makes no difference whether you measured any of them, because within one person they take the same value in both stretches of time and cancel out. This is not statistical adjustment; it is elimination by design.

The cost moves elsewhere. The design answers “why now?” and never “why me?”; it suits transient exposures paired with acute events with a definite date; and it carries three assumptions, at least one of which is routinely broken in pharmacoepidemiology.

Two kinds of anchor: SCCS and case-crossover

Every self-controlled design is defined by three things: an anchor (a fixed point on the timeline), a focal window (the stretch during which risk is hypothesised to be raised), and one or more referent windows (used to estimate how often things happen ordinarily). The analysis is always stratified by person — conditional Poisson or conditional logistic regression — and that step is where the time-invariant characteristics actually get eliminated.

The two families differ in where the anchor is pinned:

Those last two rows usually decide which design to use: when the outcome is often fatal, or when clinicians stop the drug after an event, case-crossover is the safer choice; when use of the exposure is changing quickly over calendar time — a vaccine rollout, for instance — SCCS is safer.

The simulation on this page

The setup matches the most typical pharmacoepidemiological question: a fourteen-day course of treatment and an acute event — admission for an arrhythmia, say — with a true incidence rate ratio of 3.0. Two things were built into it deliberately:

• Baseline event rates differ a great deal between people (the frailty term has a standard deviation of 1.0, so the highest- and lowest-risk people differ several-fold), and that frailty is never measured.
• Sicker people are more likely to be prescribed the drug — in the simulated data, exposed and unexposed people differ by 0.83 standard deviations of frailty.

The simulation follows 4000 people for 365 days each; 1614 of them used the drug at some point, and 4084 events occurred in total.

The SCCS conditional likelihood, written out

At the heart of SCCS is one line of algebra you can write by hand. Take a patient with $n_i$ events during observation, of which $n_{i,\text{risk}}$ fall inside the risk window, who spent $r_i$ days in the risk window and $c_i$ days in the referent window. Conditional on the total number of events that patient had, the probability of them landing in the risk window gives:

$e^\beta$ is the incidence rate ratio (IRR). The patient’s baseline risk does not appear in the expression — conditioning on how many events they had cancelled it out. That single line is the entire content of “every time-invariant characteristic is eliminated”.

It also explains two further facts: only people with both an event and an exposure contribute to the estimate (everyone else contributes a constant, which differentiates away), and this design cannot estimate absolute risk — the denominator was conditioned out, leaving only a ratio.

library(survival)
set.seed(20260822)

OBS <- 365L; RISK <- 14L; IRR <- 3.0; BASE <- 0.0016

# -- Simulation: frailty is unmeasured and drives both whether the drug is
#    prescribed and the event rate --------------------------------------
simulate_people <- function(n) {
  frailty <- rnorm(n)
  exposed <- rbinom(n, 1, plogis(-0.5 + 1.0 * frailty))   # confounding by indication
  start   <- ifelse(exposed == 1, sample(OBS - RISK, n, TRUE), NA)
  rate    <- BASE * exp(frailty)
  lapply(seq_len(n), function(i) {
    inrisk <- rep(FALSE, OBS)
    if (exposed[i] == 1) inrisk[start[i]:(start[i] + RISK - 1L)] <- TRUE
    days <- which(rbinom(OBS, 1, pmin(rate[i] * ifelse(inrisk, IRR, 1), 1)) == 1)
    list(exposed = exposed[i], inrisk = inrisk, eventDays = days,
         riskDays = sum(inrisk), controlDays = OBS - sum(inrisk),
         events = length(days),
         eventsInRisk = if (length(days)) sum(inrisk[days]) else 0L)
  })
}
rows <- simulate_people(4000)

# -- SCCS: optimise the conditional likelihood directly ------------------
# Only people with an event AND an exposure enter; the rest are a constant
use   <- Filter(function(r) r$events > 0 && r$exposed == 1, rows)
r_day <- sapply(use, `[[`, "riskDays")
c_day <- sapply(use, `[[`, "controlDays")
n_rsk <- sapply(use, `[[`, "eventsInRisk")
n_all <- sapply(use, `[[`, "events")

nll <- function(b) -sum(n_rsk * b - n_all * log(c_day + exp(b) * r_day))
o   <- optim(0, nll, method = "BFGS", hessian = TRUE)
se  <- sqrt(1 / o$hessian[1, 1])
exp(c(IRR = o$par, lcl = o$par - 1.96 * se, ucl = o$par + 1.96 * se))

# -- Case-crossover: one stratum per event, conditional logistic ---------
lags <- c(30, 60, 90)                     # three referent points, all pre-event
recs <- list()
for (k in seq_along(rows)) {
  r <- rows[[k]]
  for (ev in r$eventDays) {
    wins <- c(ev, ev - lags)
    if (any(wins < 1)) next
    recs[[length(recs) + 1]] <- data.frame(
      sid = paste0(k, "_", ev), case = c(1L, rep(0L, length(lags))),
      exposed = as.integer(r$inrisk[wins]))
  }
}
dd <- do.call(rbind, recs)
summary(clogit(case ~ exposed + strata(sid), data = dd))$conf.int

import numpy as np
from scipy.optimize import minimize
from scipy.special import expit

rng = np.random.default_rng(20260822)
OBS, RISK, IRR, BASE = 365, 14, 3.0, 0.0016

def simulate_people(n):
    frailty = rng.normal(size=n)
    exposed = rng.binomial(1, expit(-0.5 + 1.0 * frailty))
    start = rng.integers(1, OBS - RISK, size=n)
    rate = BASE * np.exp(frailty)
    out = []
    for i in range(n):
        inrisk = np.zeros(OBS, bool)
        if exposed[i]:
            inrisk[start[i]:start[i] + RISK] = True
        lam = rate[i] * np.where(inrisk, IRR, 1.0)
        days = np.flatnonzero(rng.binomial(1, np.minimum(lam, 1)))
        out.append(dict(exposed=exposed[i], inrisk=inrisk, days=days,
                        r=inrisk.sum(), c=OBS - inrisk.sum(),
                        n=len(days), nr=int(inrisk[days].sum()) if len(days) else 0))
    return out

rows = simulate_people(4000)
use = [r for r in rows if r["n"] > 0 and r["exposed"]]
rd = np.array([r["r"] for r in use]); cd = np.array([r["c"] for r in use])
nr = np.array([r["nr"] for r in use]); na = np.array([r["n"] for r in use])

nll = lambda b: -np.sum(nr * b[0] - na * np.log(cd + np.exp(b[0]) * rd))
fit = minimize(nll, [0.0], method="BFGS")
print(np.exp(fit.x[0]))

Five analyses, one true value

The first row is the point of the page: pool everyone’s exposed person-time, compare it with everyone’s unexposed person-time, and the estimate is 4.57 against a true value of 3.00. The bias is not an arithmetic error. It is that all 22596 exposed person-days came from the frailer part of the cohort — and a between-person comparison has no way to repair that, because frailty is not in the data.

SCCS, on the same data, gives 3.01 (2.66–3.41). It uses only the 1017 patients who had both an event and an exposure, 2555 events in total, of which 274 fell inside a risk window. No covariate was adjusted for, and no unexposed control group was used at all.

The three SCCS assumptions

What this design buys is substantial, so it is not cheap. Taking the assumptions one at a time:

Assumption one: events are independently recurrent, or rare enough

If one event changes the probability of later ones — after a first stroke, the probability of a second is permanently different — the derivation of the conditional likelihood no longer holds. The conventional response is to analyse only each person’s first event, and that is safe only when events are rare enough (an incidence below 10% over the study period). The simulation on this page uses an independent Poisson process, so every recurrence can be used.

Assumption two: the event must not change the probability of later exposure (event-independent exposure)

This is the one most often broken in pharmacoepidemiology. After a patient has an event, the clinician frequently stops prescribing the drug — so exposure in the period after the event is artificially suppressed, the referent window looks cleaner than it really is, and the IRR is overestimated.

The historical fix is to carve out a pre-exposure window before the exposure and drop it. Recent work shows that this repair is not dependable: how much bias remains depends not only on the length of the delay but on the length of the observation period and on where in that period exposure tends to occur. If you are worried about this assumption, the right response is a sensitivity analysis using an SCCS extension built for event-dependent exposure — not a pre-exposure window and a clear conscience.

Assumption three: the event must not change the length of the observation period (event-independent observation period)

SCCS is bidirectional precisely because observation does not end when an event occurs. When the outcome is death, that assumption fails outright — the patient dies and the rest of the observation period does not exist.

The last row of the simulation demonstrates it: change the data so that follow-up stops at the first event and leave everything else alone, and SCCS returns 3.87 (3.05–4.91), nearly thirty per cent above the true value of 3.00, with a confidence interval that does not cover the truth at all.

The core case-crossover assumption

CCO rests on a single sentence, but that sentence covers a lot of ground: under the null hypothesis that exposure has no effect, the probability of exposure in the referent window must represent the probability of exposure in the focal window.

Two consequences follow directly.

First, use of the exposure must not have a marked time trend. If prescriptions of this drug have been rising year on year, a patient is more likely to be on the drug “on the day of the event” than “thirty days earlier” — for reasons having nothing to do with the drug’s effects and everything to do with the calendar. The estimate is pushed up systematically. There are two responses: a bidirectional CCO, with referent windows on both sides of the event (which requires the extra assumption that the event does not change later exposure), or a case-time-control, which uses a group of people without events to estimate the odds ratio attributable to the time trend and divides it out.

Second, chronic medication produces an upward bias. Only people whose exposure status differs between windows carry any information in a CCO. If the drug is taken daily, the only people who can be “exposed on the day of the event, unexposed thirty days earlier” are those who had an event shortly after starting it. The reverse combination — unexposed at the event, exposed thirty days earlier — is almost nonexistent, because people who do not have events do not stop the drug. This is persistent user bias, a systematic problem for CCO with chronic medication; SCCS, being bidirectional, is much less affected.

The claim that only discordant people contribute can be counted directly in this simulation. The third row of the table above (every event used) has 3052 matched sets — one per event — of which only 445 (14.6%) are genuinely discordant between focal and referent windows and therefore enter the conditional likelihood. The fourth row (first event only) has 127 of 1168 (10.9%). In both rows, fewer than a sixth of the apparent matched sets are carrying the estimate — and a reader looking only at the table cannot see it. That is the entry in the misuse table below about not reporting how many people actually contributed information.

What this design does not eliminate

Self-controlled designs eliminate what does not change over time. So none of the following is touched:

• Time-varying confounders — age, season (influenza, temperature), the natural course of the disease itself. The longer the observation period, the worse this gets. The standard remedy is to cut age and calendar time into intervals and include them in the conditional model.
• Imprecise timing of exposure or outcome — focal and referent windows get drawn in the wrong place and events are assigned to the wrong period. This is why the design requires events whose date is unambiguous (acute events, precisely dated). Conditions with an insidious onset (endometriosis) and as-needed medication (an analgesic for migraine) do not fit.
• Selection bias — if events inside the focal window are more likely to be recorded than events outside it (post-vaccination symptoms are reported more readily), the association is overestimated.
• Absolute risk — this design cannot produce it. Converting to an attributable fraction or to cases per ten thousand people requires external denominator information, and requires great care.

When to reach for a self-controlled design

When all four of these hold, it is often the best available choice:

1. The exposure is transient (or the exposure is permanent but its effect on the outcome is transient — vaccines work this way).
2. The outcome is acute, with a clear date.
3. Unmeasured, time-invariant confounding is the main threat — exactly the grey nodes on a DAG.
4. No suitable comparison group exists, or the exposure spread through the whole population quickly (a vaccination campaign).

Conversely, when the exposure is chronic, the outcome’s onset is vague, or the dominant threat is time-varying confounding, a cohort study with weighting is the more practical route.

Why this page recommends no videos

Videos are listed on this site only after they have been verified. On this topic there is no teaching-level video in any language — nothing at all in Chinese, traditional or simplified, and in English only three recordings of seminars or lectures (view counts in the hundreds, thirty to sixty minutes long, mostly someone advancing slides).

Rather than pad the list, here is the situation as it stands. If you would rather hear it explained, the three closest are the Leicester RSS Hub’s Self-controlled case series methodology (32 minutes, methodological), Self controlled Design (60 minutes) given by Malcolm Maclure, who originated the case-crossover design, and OHDSI’s Vaccine safety evaluation using SCCS (58 minutes, applied). All three are pitched at people who already know the method, not at beginners.

The more efficient path is to read this page and then go straight to the CC BY review listed at the foot of it — it has a full glossary and an annotated bibliography.

Common misuses

Reproducing every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B6-05-self-controlled.R

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