8 Advanced Adjustments

When the diagnostic engine flags left truncation, informative censoring, or clustering — possibly in combination with other complexities — this module applies the appropriate corrections. These adjustments layer on top of the primary analysis module.

8.1 Script Pipeline

Script	Purpose
`01_left_truncation.R`	Delayed entry adjustment
`02_ipcw.R`	Inverse probability of censoring weights
`03_frailty_models.R`	Shared frailty for clustered data
`04_cluster_robust_se.R`	Cluster-robust sandwich standard errors
`05_combined_adjustments.R`	Apply multiple corrections simultaneously

8.2 Left Truncation

Subjects who enter observation after time zero (delayed entry) must be handled with left-truncated likelihoods. The standard Surv() call extends naturally:

# entry_time = time of study entry (left truncation point)
# time = event/censoring time
fit <- coxph(Surv(entry_time, time, status) ~ group + age, data = df)

When Does This Arise?

Prevalent cohort studies (enrolling patients who already have the condition), studies using calendar-time scales, and registries where subjects are only observable once they enter the system.

8.3 IPCW for Informative Censoring

When censoring is non-random (sicker patients are more likely to drop out), inverse probability of censoring weights (IPCW) correct for the selection bias:

# Step 1: Model the censoring mechanism
cens_model <- coxph(Surv(time, 1 - status) ~ age + comorbidity, data = df)

# Step 2: Compute weights
df$ipcw <- 1 / predict(cens_model, type = "survival")

# Step 3: Fit weighted Cox model
coxph(Surv(time, status) ~ group + age, data = df, weights = ipcw)

Weight Instability

IPCW weights can become extreme when the censoring probability approaches zero. Truncate or stabilise weights to avoid high-variance estimates.

8.4 Frailty Models vs Cluster-Robust SE

For clustered data (patients within hospitals, siblings within families), two approaches are available:

Approach	What It Does	When to Prefer
Shared frailty	Random effect per cluster	When heterogeneity is of interest
Cluster-robust SE	Sandwich variance estimator	When only adjusting for correlation

# Frailty
coxph(Surv(time, status) ~ group + age + frailty(hospital_id), data = df)

# Cluster-robust SE
coxph(Surv(time, status) ~ group + age + cluster(hospital_id), data = df)

8.5 Combining Adjustments

Script 05_combined_adjustments.R handles cases where multiple complications co-exist — for example, left truncation with clustering:

coxph(
  Surv(entry_time, time, status) ~ group + age + cluster(hospital_id),
  data = df
)

Layered Corrections

The diagnostic report specifies which adjustments are needed. The combined script reads this report and applies only the flagged corrections, avoiding unnecessary complexity.

8.6 Running the Module

make analyze-advanced PROJECT=my-study

8.7 Demo: Kitchen Sink (Scenario 6)

N=1000, 15 centers, 15.8% left truncated, informative censoring, ICC approximately 0.05.

8.7.1 Left Truncation

Adjusting for delayed entry had modest effects on the severity HR. The truncation-adjusted model estimated severity HR = 1.707, compared to the naive HR = 1.723 — a -0.89% change. Center and age estimates were essentially unchanged.

Table 8.1: Left truncation adjustment comparison

Variable	HR (truncated)	HR (naive)	% change
Center	1.010	1.012	-0.15%
Age	1.002	1.001	+0.02%
Severity	1.707	1.723	-0.89%

Figure 8.1: Truncation-adjusted vs naive KM curves

8.7.2 IPCW Weighting

Figure 8.2: IPCW-weighted vs unweighted KM curves

8.7.3 Frailty (Clustering)

The shared frailty model (coxme) estimated a center-level frailty variance of 0.187, corresponding to a moderate degree of between-center heterogeneity. Using robust sandwich SEs, the severity SE decreased from 0.152 (naive) to 0.126 (robust), a ratio of 0.83, indicating that ignoring clustering slightly inflated standard errors in this case.

Table 8.2: Fixed-effects vs frailty model comparison

Variable	HR (fixed)	HR (frailty)	Frailty variance
Age	1.002	1.001	0.187
Severity	1.726	1.819	0.187

Figure 8.3: Caterpillar plot of center-level random effects

Note

The frailty model had the largest impact: severity HR shifted from 1.726 (fixed-effects) to 1.819 (frailty-adjusted), a ~5.4% increase. Center-level random effects ranged from HR=0.60 to HR=2.26.