Other Survival Models (1) Non-PH models We briefly discussed the non-proportional hazards (non-ph) model λ(t Z) = λ 0 (t) exp{β(t) Z}, where β(t) can be estimated by: piecewise constants (recall how); splines, which give a more smooth ˆβ(t) (Gray, 1992); other methods. More broadly, non-ph can mean any survival model that is not the Cox model. 1
(2) Parametric regression models First we will discuss some parametric regression models, some are in fact PH models. We have (X i, δ i, Z i ), i = 1,..., n, i.i.d., where X i = min(t i, C i ), Z i = (Z i1, Z i2,..., Z ip ). If we assume that T i follows an exponential distribution with a parameter λ i = exp(β 0 + β Z i ), then we have a exponential regression model: T i Z i Exp(e β 0+β Z i ). Since the hazard functions are constants, these are PH models. Q: what is the baseline hazard? How do we estimate the unknown parameters? (hint: see separate notes on parametric regressions) 2
We may also assume that T i Z i follows a Weibull distribution. Recall that the hazard function for a Weibull distribution is: λ(t) = κ λ t (κ 1). The Weibull regression model assumes that given covariates Z i, λ i = exp(β 0 +β Z i ) as in the exponential regression. The conditional hazard function is then λ i (t) = κ exp(β 0 + β Z i ) t (κ 1). Note that the shape parameter κ is the same for all subjects; Is this a PH model? (what is the baseline hazard?) How do we estimate the unknown parameters κ, β 0 and β? (hint: see separate notes on parametric regressions) 3
Comparison of Exponential with Kaplan-Meier We can see how well the Exponential model fits by comparing the survival estimates for males and females under the exponential model, i.e., P (T t) = e ( ˆλ z t), to the Kaplan- Meier survival estimates: 1. 0 0. 9 0. 8 S 0. 7 u r 0. 6 v i 0. 5 v 0. 4 a l 0. 3 0. 2 0. 1 0. 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 L e n g t h o f S t a y ( d a y s ) 4
Comparison of Weibull with Kaplan-Meier We can see how well the Weibull model fits by comparing the survival estimates, P (T t) = e ( ˆλ z tˆκ), to the Kaplan- Meier survival estimates. 1. 0 0. 9 0. 8 S 0. 7 u r 0. 6 v i 0. 5 v 0. 4 a l 0. 3 0. 2 0. 1 0. 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 L e n g t h o f S t a y ( d a y s ) Which do you think fits better? 5
Comparison of Models Exponential Regression: λ(t Z) = exp(β 0 + β 1 Z 1 +... + β p Z p ) Weibull Regression: RR = exp(β 1 Z 1 +... + β p Z p ) λ(t Z) = κ t κ 1 exp(β 0 + β 1 Z 1 +... + β p Z p ) RR = exp(β 1 Z 1 +... + β p Z p ) Proportional Hazards Model: λ(t Z) = λ 0 (t) exp(β 1 Z 1 +... + β p Z p ) RR = exp(β 1 Z 1 +... + β p Z p ) 6
Remarks Exponential model is a special case of the Weibull model with κ = 1 Exponential and Weibull models are both special cases of the Cox PH model. If either the exponential model or the Weibull model is valid, then these models should be more efficient (somewhat smaller variances of the parameter estimates) than the semiparametric PH model. This is because they assume a particular form for λ 0 (t), with only one or two unknown parameters, rather than leaving it as an infinite dimensional parameter and estimating it at each distinct failure time. Note however, that the Cox partial likelihood estimator is semiparametrically efficient, meaning: as more and more parameters are used to model λ 0 (t), the limit of the (asymptotic) variance of the estimated β equals the variance of the (asymptotic) variance of the partial likelihood estimator. 7
(3) Accelerated Failure Time Model The accelerated failure time (AFT) model is a linear regression model with log(t ) as the response: where log(t i ) = β Z i + ɛ i log(t i ) is the log of the survival time; (why take log?) β is the vector of regression parameters including intercept (and Z includes a 1 with a slight change of notation here); ɛ i is a random error term. Note that the AFT model can be also written T i = T 0 exp(β Z i ), where T 0 has the same distribution as e ɛ. 8
Write φ = exp( β Z). It can be shown: S(t Z) = S 0 (φ t) That is, the effect of covariates is to accelerate (or decelerate) the time-scale. If S i (M i ) = 0.5, then S 0 (φ i M i ) = 0.5. This means M 0 = φ i M i, or: M i = M 0 /φ = M 0 exp(β Z i ) 9
We will first discuss the parametric AFT model, which is often written as: where log(t i ) = β Z i + σɛ i ɛ i is a random error term with known distribution; σ is a scale factor. By choosing different distributions for ɛ, we can obtain different parametric regression models: Exponential Weibull Logistic Log-logistic Normal Log-normal These can be fitted using the R function survreg(). 10
Both the Exponential and the Weibull regression models discussed earlier can be written as an AFT model, if we choose the proper distribution for ɛ. For the Exponential Model: log(t i ) = β ez i + ɛ i, where ɛ follows an extreme value distribution, i.e. e ɛ follows a unit exponential distribution. So β = β e (including intercept), σ = 1. For the Weibull Model: log(t i ) = σβ wz i + σɛ i, where ɛ again follows an extreme value distribution, and σ = 1/κ. So β = σβ w (including intercept). 11
AFT model with normal error (Log-normal regression) Here we let ɛ N(0, 1). log(t i ) = β Z i + σɛ i This is a very appealing model, because it is the same as the linear regression model with normal error, where the response is the log of the survival time. Therefore the interpretation of the model is straightforward and familiar. Recall that the distribution of T i is called log-normal. This family of distributions have non-monotone hazards. How would you fit the model? If there are no censored observations, how would you fit the model? 12
log-logistic regression model Here we assume that ɛ has a logistic distribution with density f(ɛ) = e ɛ (1 + e ɛ ) 2. If ɛ has a logistic distribution, so does log(t i ) (with non-zero mean). Then T i has a log-logistic distribution. Log-logistic can also have non-monotone hazards. In addition, as t, the hazard goes to zero. The log-logistic model has a simple survival function (Ex.) S(t Z) = 1 1 + (λt) γ where γ = 1/σ and λ = exp( β Z). 13
After some algebra, it can be shown that log where β = β/σ. S(t Z) 1 S(t Z) = β Z γ log(t) If t is fixed, the above is a logistic regression model (why?). Since S(t) is the probability of surviving to time t, S(t)/{1 S(t)} is the odds of surviving to time t. Furthermore, for individuals i and j, S i (t) 1 S i (t) = c S j (t) ij 1 S j (t) for all t, where c ij = exp{β (Z i Z j )}. Therefore the log-logistic model is also called a proportional odds (PO) model, since c ij does not depend on t. 14
It can also be shown that, as t, the hazard ratio λ i (t)/λ j (t) 1. Therefore the log-logistic regression (PO) model can be used to model attenuating hazard ratios, which provides a useful alternative to the PH model. When does Proportional hazards = AFT? We have seen before that the Weibull regression model (which includes the Exponential regression model as a special case), is both a PH model and an AFT model. It turns out that the Weibull (and Exponential) regression model is the only one for which the accelerated failure time and proportional hazards models coincide. See Chan et al. (2018) for examples with R 2 values. 15
(4) Semiparametric AFT model log(t ) = β Z + ɛ If we leave the distribution of ɛ unspecified, then it leads to the semiparametric AFT model. Inference under this model is much more difficult, since there is not immediately a likelihood for this model. Rank-based estimating equations were proposed in the literature, but numerical solutions to these equations are challenging. Semiparametric AFT model has rarely (if ever) been used in practice. 16
(5) Semiparametric transformation model Replace the log transformation on the survival times to be any unspecified monotone transformation g( ): g(t ) = β Z + ɛ and this leads to the semiparametric transformation model. Here ɛ still comes from a parametric distribution. In fact, when ɛ follows the extreme value distribution, the above is equivalent to the PH model. A useful family of distributions for ɛ is the G ρ family of Harrington and Fleming. Inference under this model is like under the PH model, using the nonparametric MLE (NPMLE). When ɛ is logistically distributed, this is the semiparametric proportional odds model. 17
(6) Additive hazards model This is a class of models that is gaining popularity (Aalen 1980, 1989): λ i (t Z) = λ 0 (t) + β Z(t), where λ 0 (t) is an unspecified baseline hazard. What do you think of the model? A: Indeed one needs to make sure that the hazard is not negative. Inference for β is based on the estimating equation (Lin and Ying, 1994) 0 = = n i=1 n i=1 0 0 Z i (t)dm i (t) Z i (t){dn i (t) Y i (t)λ 0 (t) Y i (t)β Z i (t)dt}. And similar to the Breslow estimate, if β were known, Λ 0 (t) = t 0 λ 0(s)ds can be estimated by t n i=1 ˆΛ 0 (t) = {dn i(u) Y i (u)β Z i (u)du} n i=1 Y. i(u) 0 Plugging this estimate back into the above estimating equation (this is known as profiling out Λ 0 ), we have after some algebra: 18
U(β) = where Z(t) = This gives [ n ˆβ = i=1 0 n i=1 0 n Z l Y l (t)/ l=1 l=1 {Z i Z(t)}{dN i (t) Y i (t)β Z i (t)dt}, n Y l (t). Y i (t){z i Z(t)} ] 1 [ n 2 dt i=1 Notice that the above is no longer just rank based. 0 {Z i Z(t)}dN ] i (t). The cumulative baseline hazard function Λ 0 (t) is then estimated by ˆΛ 0 (t) = t 0 n i=1 dn i(u) n i=1 Y i(u) ˆβ t 0 Z(u)du. Note that ˆΛ 0 (t) can be negative. Lin and Ying (1994) suggested to use a modified ˆΛ 0(t) = max 0 s t ˆΛ0 (s). Martingale theory applies so that ˆβ is asymptotically normal with variance estimated by a sandwich of the form A 1 BA 1. R package timereg fits this model. 19
Part of the reason the additive hazards model became popular (over the PH model) is the following: The PH models, unlike the normal linear regression models, are not nested. This is sometimes called non-collapsible (in causal inference). This is also true for other non-linear models like logistic regression. That is, when adjusting or not adjusting for covariate(s), at most one of the two models might be valid (Lancaster and Nickell 1980, Gail et al. 1984, Struthers and Kalbfleisch 1986, Bretagnolle and Huber-Carol 1988, Anderson and Fleming 1995, Ford et al. 1995). Suppose that we have λ(t z 1, z 2 ) = λ 0 (t) exp(β 1z 1 + β 2z 2 ), where z 1, z 2 are vectors of covariates. This implies that S(t z 1, z 2 ) = exp{ Λ(t z 1, z 2 )} = exp{ Λ 0 (t)e β 1z 1 +β 2z 2 }. Then S(t z 1 ) = S(t z 1, z 2 )dg 2 (z 2 z 1 ), where G 2 is the conditional distribution function of Z 2 given Z 1. This gives λ(t z 1 ) = S(t z 1) /S(t z 1 ) t S(t z1, z 2 ) = dg 2 (z 2 z 1 )/ t S(t z 1, z 2 )dg 2 (z 2 z 1 ) = λ 0 (t)e β 1z 1 e β 2z 2 exp{ Λ 0 (t)e β 1z 1 +β 2z 2 }dg 2 (z 2 z 1 ). exp{ Λ0 (t)e β 1 z 1+β 2 z 2 }dg2 (z 2 z 1 ) 20
It is clear that unless the ratio of the two integrals in the last line above can be written as a function of t multiplied by a function of z 1, the PH assumption will be violated. One such example is when Z 2 is the logarithm of a positive stable random variable (see e.g. Feller 1966, Hougaard 1986), then with or without z 2, the model will always be PH, though the estimated coefficients of Z 1 will have changed. [Read] Example 1 Let Λ 0 (t) = t, β 2 = 1. Denote ξ = exp(β 1Z 1 ). Let Z 1 and Z 2 be independent, and Z 2 = log α, where α has a positive stable distribution. A distribution is called stable if for each n and X 1, X 2,..., X n i.i.d. from this distribution, there exists a constant c n, with D(X 1 + X 2 +... + X n ) = D(c n X 1 ), where D(X) means the distribution of X. It turns out that the only constants possible for c n are n 1/γ, γ (0, 2]. The stable distributions with finite variance are the normal, γ = 2, and the degenerate distributions, γ = 1. The stable distributions on the positive numbers have γ (0, 1] and apart from scale factors have Laplace transform E(e sx ) = exp( s γ ), s 0. Then from the calculation above, S(t z 1, z 2 ) = exp( αξt), S(t z 1 ) = 0 e αξt dg γ (α) = exp( ξ γ t γ ), where G γ ( ) is the distribution function of α. Therefore λ(t z 1 ) = γt γ 1 ξ γ = γt γ 1 e γβ 1z 1, which follows a PH model, but the coefficient of z 1 is now γβ 1 instead of β 1. 21
In general, one is almost certain to end up with non-ph rather than PH models after deleting covariates. Below is one example here from Ford, Norrie and Ahmadi (1995). Ford, Norrie and Ahmadi (1995) also provided examples in which adding covariates makes a PH model into a non-ph one. Example 2 Let Λ 0 (t) = t, β 2 = 1. Denote ξ = exp(β 1Z 1 ). Let Z 1 and Z 2 be independent, and Z 2 be the logarithm of an Exp(1) random variable. Then S(t z 1 ) = = 0 0 exp( te β 1z 1 +z 2 ) exp( e z 2 )e z 2 dz 2 exp{ (tξ + 1)e z 2 }e z 2 dz 2 = e (tξ+1) /(tξ + 1), and this is a non-ph model. The fact that the Cox model is non-collapsible has important implications: 1. If an important covariate is missed, the estimated effects of (other) covariates including treatment is biased towards zero, which can lead to less efficient (i.e. powerful) tests (Lagakos and Schoenfeld, 1984). This is also known for other non-linear models like logistic regression. 2. Another side of the coin of the above is: even in randomized trials, adjusting for important covariates can lead to more efficient tests of the treatment effect. 3. Interpretation becomes difficult when there are unobserved heterogeneity or confounders. 22
On the other hand, the additive hazards model is collapsible. [Ex.] 23