Table 10.

Illustration of stabilized IPTW and IPCW definitions with multiple time points

Stabilized WeightsTime 1Time 2Time 3
IPTWNumeratoraP(X1=a1)P(X2=a2|X1=a1) ×P(X1=a1)P(X3=a3|X1=a1, X2=a2) ×P(X2=a2|X1=a1) ×P(X1=a1)
DenominatoraP(X1=a1|Embedded Image)P(X2=a2|X1=a1, Embedded Image) ×P(X1=a1|Embedded Image)P(X3=a3|X1=a1, X2=a2,Embedded Image) ×P(X2=a2|X1=a1, Embedded Image) ×P(X1=a1|Embedded Image)
IPCWNumeratoraP(C1=0|Embedded Image)P(C2=0|C1=0, Embedded Image) ×P(C1=0|Embedded Image)P(C3=0| C1=0, C2=0, Embedded Image) × P(C2=0| C1=0, Embedded Image) ×P(C1=0|Embedded Image)
DenominatoraP(C1=0|Embedded Image, Embedded Image)P(C2=0|C1=0, Embedded Image, Embedded Image) ×P(C1=0|Embedded Image, Embedded Image)P(C3=0| C1=0, C2=0, Embedded Image, Embedded Image) × P(C2=0| C1=0, Embedded Image, Embedded Image) ×P(C1=0|Embedded Image, Embedded Image)
  • X1, X2, and X3 are the exposure; a1, a2, and a3 are the values of exposure; and the confounder history (i.e., confounder values since baseline to this time point) is Embedded Image,Embedded Image, and Embedded Image at time points 1, 2, and 3. C1, C2, and C3 are the censoring indicators at time points 1, 2, and 3 for one subject. They are defined as 1 if right-censored by that time point, and 0 otherwise. Embedded Image,Embedded Image, and Embedded Image are the exposure history at time points 1, 2, and 3. IPTW, inverse probability treatment weight; P, probability of; IPCW, inverse probability censoring weight.

  • a We can adjust for baseline covariates Embedded Image in all models.