Impact of Informative Censoring �on Kaplan-Meier Estimate��

Charlie Ahn

September 30, 2023

Outline

• Lung Data
• Informative Censoring
• Slud-Rubinstein Method
• Efron’s Self-consistent Algorithm for KM estimate
• Link Method Based on Self-consistent Algorithm

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Lung data

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Leung, Elashoff, and Afifi (1997) reported survival times in weeks of 61 patients with inoperable adenocarcinoma of lung.

Survival probabilities from Kaplan-Meier (KM)

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Survival probabilities from Kaplan-Meier (KM)

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Leung, Elashoff, and Afifi (1997) stated “Observations were censored because the patients experienced metastasic disease or a significant increase in the size of their primary lesion. Such disease progression usually indicates a shortened residual survival time.”

In such situations, the KM method will overestimate the actual survival function, and we cannot apply the standard procedure (assuming non-informative censoring).

Survival probabilities with eventual survival times

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As expected, the KM rates are overestimated.

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Q: Can we correct it?

Informative Censoring

1. Patients remove themselves from study for reasons possibly related to therapy and thereby censor their survival time
2. Patients experiencing a specific critical event such as metastatic spread of disease are removed from study
3. Failure times from causes of secondary interest are recorded as censored observations of the failure times from the causes of primary interest

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Lagakos (1979, Biometrics) mentioned the following three situations in which the independence assumption (censoring times independent of survival times) is of questionable validity:

Models Proposed for Informative Censoring

• Efron (1967 Proceedings of the 5th Berkeley Symposium) introduced the self-consistent estimate that coincides with Kaplan and Meier's product limit estimate.
• Link (1989, JASA) used Efron's self-consistency algorithm and proposed a modified Kaplan-Meier estimator (MKME).
• Slud and Rubinstein (1983 Biometrika) modeled ρ(t) = h(t | C < t) / h(t | C ≥ t) that led to the generalized Kaplan-Meier estimate.
• Leung, et al (1997 Ann. Rev. Pub. Health) concluded that that assuming ignorable censoring can lead to a biased estimate of the survival function. They compared several models for informative censoring and concluded that Slud-Rubinstein’s estimate outperformed.

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Model for Censored Survival Analysis

• Observations X are of the form,

X = min(T, C),

where T and C are non-negative RVs representing lifetime and censoring time, respectively.

• Under the usual assumption of independence of T and C, the Kaplan-Meier estimator (KME) is the appropriate estimator of S(t) = Pr(T> t).
• The KME can lead to substantial overestimates of survival probabilities if the event of censoring indicates an unfavorable prognosis for future survival.
• We consider two alternative models based on (1) self consistent estimator of the survival function and (2) the ratios of two hazards- hazard of those who have been censored and hazard of those who are still in the study.

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Slud & Rubinstein (1983, Biometrika)

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Slud & Rubinstein generalized the KME with informative censoring situations. They assumed:

ρ(t) = h(t | C ≤ t ) / h(t | C > t ) measures the relative risk of failure at time T among previously censored subjects as compared with subjects not yet censored, where h(t | .) is the conditional hazard function of T.

Slud & Rubinstein

Numerator: P(t < T < t+δ | T > t, C ≤ t) for subjects previously censored

= P(t < T < t+δ, T > t, C ≤ t) / P(T > t, C ≤ t)

Applying the total probability: P(A) = P(AB) + P(AB^c) or, P(AB) =P(A) - P(AB^c)

P(t < T < t+δ, T > t, C ≤ t) = P(t < T < t+δ) - P(t < T < t+δ, C > t)

P(T > t, C ≤ t) = P(T > t) - P(T > t, C > t)

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🡪 f(t) - ψ(t)

🡪 S(t) - S_x(t)

Denominator: P(t < T < t+δ | T > t, C > t) for subjects not yet censored

= P(t < T < t+δ, T > t, C > t) / P(T > t, C > t) 🡪 ψ(t) /S_x(t)

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ρ(t) = { [f(t) - ψ(t)]/[S(t) - S_x(t)] } / {ψ(t) /S_x(t)} = [f(t)/ψ(t) – 1] [S(t)/S_x(t)-1]^-1

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Slud & Rubinstein

ρ > 1: + dependence between censoring and death

ρ < 1: − dependence between censoring and death

ρ = 1: S-R estimates = KM rates

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• The Kaplan-Meier estimator may overestimate the survival function if censoring and death are positively correlated, and underestimate the survival function if they are negatively correlated (Leung, et al, 1997)
• Let’s apply S&R method to Lung data with ρ > 1.

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Survival probabilities with eventual survival times

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ρ > 1: + dependence between

censoring and death

ρ < 1: − dependence between

censoring and death

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As ρ increases from 1, the S-R method produces the estimates closer to the true survival probabilities.

When ρ =4, the S-R estimates are very close to the true survival probabilities.

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Efron’s self-consistent estimator (Link, 1989 JASA)

Self-consistent algorithm to calculate S(t) = Pr( T > t )

• This alternative expression of KME is the sum of two parts:
• the percentage of observations (censored or uncensored) beyond time t
• the estimated percentage that would have survived beyond time t but were censored before t.

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• 1) has the form of the common right sided sample c.d.f.
• 2) is the portion of probabilities to KME from censored patients. S(t)/S(x_i) in (2) is the conditional probability that T > t given T > x_i. The closer to t, x_i is, the higher contribution is given to the probability of surviving past to t=15.

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Which subject would have higher chance of surviving past t=15?

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_Pr(T>15)

Subject A censored (x_A =6)

Subject B censored (x_B =10)

_t

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₆

₁₀

₁₅

Which subject would have higher chance of surviving past t=15?

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_Pr(T>15)

Subject A censored (x_A =6) P(T > 15 |T >6)

Subject B censored (x_B =10) P(T > 15 |T >10)

_t

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₁₀

₁₅

Self-consistent estimator

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Which subject would have higher chance of surviving past t=15?

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_Pr(T>15)

Subject A censored (x_A =6) P(T > 15 |T >6)=.5238/.8571=.6111

Subject B censored (x_B =10) P(T > 15 |T >10)=.5238/.7529=.6957

_t

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₁₀

₁₅

Self-consistent estimator

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Self-consistent estimator

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Modified Kaplan-Meier Estimator (MKME)

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S(t | Z = z ) is decreasing in z, so individuals with high frailties tend to have smaller lifetimes.

Link considered the survival function conditional on frailty Z = z.

A = { z | z ≥ a): KME overestimates S(t)

A = { z | z ≤ a): KME underestimates S(t)

Frailty a=0.333 provides the true S(t) in Link method

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iter #1

iter #2

iter #3

iter #6

converged

References

• Efron, B. (1967), "The Two Sample Problem With Censored Data," in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4), Berkeley: University of California Press, pp. 831-853.
• Kaplan, E. L., and Meier, P. (1958), "Nonparametric Estimation From Incomplete Observations," Journal of the American Statistical Association, 53, 457-481.
• Lagakos, S. W. (1979), "General Right Censoring and Its Impact on the Analysis of Survival Data," Biometrics, 35, 139-156.
• Lagakos, S. W., and Williams, J. S. (1978), "Models for Censored Survival Analysis: A Cone Class of Variable-Sum Models," Biometrika, 65, 181-189.
• Leung, K.M., Elashoff, R.M., Afifi, A.A. (1997). Censoring issues in survival analysis, Annu. Rev. Public Health 1997. 18:83–104.
• William A. Link (1989) A Model for Informative Censoring, JASA, 84:407, 749-752,
• Robertson, J. B., and Uppuluri, V. R. R. (1984), "A Generalized Kaplan-Meier Estimator," The Annals of Statistics, 12,366-371.
• Slud EV, Rubinstein LV. (1983). Dependent competing risks and summary survival curves. Biometrika, 70:643–49
• Williams, J. S., and Lagakos, S. W. (1977), "Models for Censored Survival Analysis: Constant-Sum and Variable-Sum Models," Biometrika, 64,215-224.

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