some new methods for latent outline variable models and ...davidian/newgreenberg.pdf · moments of...
TRANSCRIPT
Some New Methods for LatentVariable Models and Survival
Analysis
Marie Davidian
Department of Statistics
North Carolina State University
http://www.stat.ncsu.edu/∼davidian
(Joint work with X. Huang, L. Stefanski, K. Doehler, L. Tang, M. Zhang)
Greenberg Lecture IV: Latent Variable/Survival 1
Outline
1. Introduction
2. Latent-model theoretical robustness
3. Empirically checking latent-model robustness
4. “Smooth” inference for survival functions with arbitrarily censored
data
5. “Smooth” semiparametric regression analysis with arbitrarily
censored data
Greenberg Lecture IV: Latent Variable/Survival 2
1. Introduction
Two mainstays of biostatistical methodology and practice:
• “Latent-variable models ” – e.g., measurement error models, models
with random effects
• Survival analysis
Two mini-talks: Research by my PhD students
• Tools for checking whether inference in latent variable models is
robust to assumptions on the latent variable distribution – with
Xianzheng Huang and Len Stefanski
• Methods for survival analysis based on mild smoothness assumptions
– with Kirsten Doehler , Lihua Tang , and Min Zhang
Greenberg Lecture IV: Latent Variable/Survival 3
Latent-Model Robustness inStructural Measurement Error
Models
Xianzheng Huang, Len Stefanski, and Marie Davidian
Department of Statistics
North Carolina State University
Greenberg Lecture IV: Latent Variable/Survival 4
2. Latent-model theoretical robustness
Particular latent variable model: Structural measurement error model
Y = observed response
X = true predictor (q × 1), with true density f∗X(x)
W = observed predictor (q × 1)
Usual assumptions: Take q = 1 for simplicity
• Conditional density of Y |X is fY |X(y|x; θ), true value θ∗
• W = X + U , U ∼ N (0, σ2U ), σ2
U known
⇒ conditional density of W |X is fW |X(w|x;σ2U ) (normal )
• fY,W |X(y, w|x; θ) = fY |X(y|x; θ)fW |X(w|x;σ2U ) (surrogacy )
• Interested in inference on θ
Observed data: (Yj ,Wj), j = 1, . . . , n, iid
Greenberg Lecture IV: Latent Variable/Survival 5
2. Latent-model theoretical robustness
X is a latent variable: Assumptions on X?
• One approach to inference on θ: Make a parametric assumption
about the true density of X (i.e., the latent variable model )
• Assumed parametric latent variable model: f(a)X (x; τ (a)), depending
on a parameter vector τ (a)
Likelihood inference: Estimate θ, τ (a) by θ, τ (a) maximizing
L(θ, τ (a)) =
n∏
j=1
fY,W (Yj ,Wj ; θ, τ(a))
=
n∏
j=1
∫fY |X(Yj |x; θ)fW |X(Wj |x;σ
2U )f
(a)X (x; τ (a)) dx
• If f(a)X (x; τ (a)) is correctly specified ⇒ θ is consistent and
asymptotically efficient
Greenberg Lecture IV: Latent Variable/Survival 6
2. Latent-model theoretical robustness
What if f(a)X (x; τ (a)) is incorrectly specified?
• θ can be inconsistent (and hence asymptotically biased )
Our definition of “latent-model robustness:” The estimator θ and
more generally the model are said to be robust if this doesn’t happen !
• I.e., Latent-model robustness means lack of asymptotic bias
• ⇒ The estimator under a correct model is trivially robust
• Asymptotic bias is only possible if both f(a)X (x; τ (a)) is misspecified
and σ2U > 0
• So we are interested in whether there is an “interaction ” between
these factors ⇒ nonrobustness
Greenberg Lecture IV: Latent Variable/Survival 7
2. Latent-model theoretical robustness
Definition: Full latent-model robustness
• Score for assumed model
ψ(y, w, θ, τ (a)) = ∂/∂(θ, τ (a)){ log fY,W (y, w; θ, τ (a)) }
• θ(σU ), τ (a)(σU ) satisfy
E[ψ{Y,W, θ(σU ), τ (a)(σU )} ] = 0 (wrt to the true dist’n)
• Under conditions, θp
−→ θ(σU )
• In general , if f(a)X (x; τ (a)) is incorrect and σU > 0, θ(σU ) 6= θ∗
• The MLE for θ under f(a)X (x; τ (a)) is robust if
θ(σU ) ≡ θ∗ σU ≥ 0
Greenberg Lecture IV: Latent Variable/Survival 8
2. Latent-model theoretical robustness
Remarks: As we noted already
• If f(a)X (x; τ (a)) is correctly specified , then this condition will hold
• . . . but it can also hold when f(a)X (x; τ (a)) is incorrectly specified !
• E.g., if f(a)X (x; τ (a)) is incorrect but is sufficiently flexible to capture
moments of the true model on which θ(σU ) depends
Greenberg Lecture IV: Latent Variable/Survival 9
2. Latent-model theoretical robustness
Full model robustness: Only verifiable in simple models; not very
practically useful
A little easier: First-order latent-model robustness
θ(σU ) = θ∗ + σ2Uθ
′′(0) + o(σ2U )
• Can get by implicit differentiation of E{ψ(·)} as in Stefanski (1985,
Biometrika)
• Implies a necessary , first-order condition for robustness is θ′′(0) = 0
• Example where this holds (and can be shown analytically )
Y |X ∼ N (β0 + β1X,σ2e), f
(a)X (x; τ (a)) = τ
(a)2 h(τ
(a)1 + τ
(a)2 x),
h(·) an arbitrary density (see Huang et al. (2006, Biometrika)
Greenberg Lecture IV: Latent Variable/Survival 10
2. Latent-model theoretical robustness
Realistically: First-order robustness is still too hard to be practically
useful for fancier models arising in real applications
• Need an accessible way to assess robustness to the choice of the
model f(a)X (x; τ (a)) that can be used in data analysis
Idea: Exploit these concepts of theoretical robustness
• If θ is robust , then a plot of
θ(σU ) vs. σU
should be flat ! If not robust, θ(σU ) will change with σU
• Construct an empirical plot in this spirit based on data by exploiting
the simulation step of simulation-extrapolation (SIMEX ). . .
Greenberg Lecture IV: Latent Variable/Survival 11
3. Empirically checking robustness
Remeasured data: Add additional increments of measurement error
• Actual observed data (Y,W ), var(W |X) = σ2U
• “Remeasured data ” {Y,W (λ)}, var{W (λ)|X} = (1 + λ)σ2U
W (λ) = W + λ1/2σUZ, Z ∼ N (0, 1), λ > 0
• Key : If the assumed model∫fY |X(y|x; θ)fW |X(w|x;σ2
U )f(a)X (x; τ (a)) dx
is correct for (Y,W ), then∫fY |X(y|x; θ)fW |X{w|x; (1 + λ)σ2
U}f(a)X (x; τ (a)) dx
is correct for {Y,W (λ)}
Greenberg Lecture IV: Latent Variable/Survival 12
3. Empirically checking robustness
Result: If the assumed model f(a)X (x; τ (a)) is correct or yields robust
inferences, an estimator based on remeasured data should be
approximately unbiased regardless of the size of λ
• Thus, estimators based on remeasured data for different λ should
show no dependence on λ
• ⇒ Inspect such estimators for a range of λ in a plot
• Write θ(λ) for an estimator based on λ-remeasured data
Greenberg Lecture IV: Latent Variable/Survival 13
3. Empirically checking robustness
Observed data: (Yj ,Wj), j = 1, . . . , n, λ = 0
• MLE θ(0) (estimates θ when measurement error variance = σ2U )
Remeasurement method: For each λ on a grid of λ ∈ [0, λmax],
1 ≤ λmax ≤ 3
• Construct B remeasured data sets, where the bth remeasured data
set is {Yj ,Wb,j(λ)}, j = 1, . . . , n
Wb,j(λ) = Wj + λ1/2σUZb,j , Zb,jiid∼ N (0, 1), j = 1, . . . , n
b = 1, . . . , B (B = 50 or 100 suffices)
• For each b, compute MLE θb(λ) using {Yj ,Wb,j(λ)}, j = 1, . . . , n
• Compute θB(λ) = B−1∑B
b=1 θb(λ)
(estimates θ when measurement error variance = (1 + λ)σ2U )
Greenberg Lecture IV: Latent Variable/Survival 14
3. Empirically checking robustness
Proposed plot for checking robustness: Plot θB(λ) vs. λ
• If f(a)X (x; τ (a)) is correct or robust the plot should be
approximately flat across the range λ ∈ [0, λmax]
• If f(a)X (x; τ (a)) is nonrobust the plot will exhibit change with λ
• In fact : Can apply the remeasurement method to any estimation
technique for measurement error models (not just likelihood
estimators)
Greenberg Lecture IV: Latent Variable/Survival 15
3. Empirically checking robustness
Example: Y binary, P (Y = 1|X = x) = Φ(β0 + β1x), θ = (β0, β1)T
• True density of X is a bimodal mixture of two normals
• Three estimators for θ: Take f(a)X (x; τ (a)) to be
– a normal density (n)
– the flexible SNP density (s)
– a normal mixture density (m), which is the correct specification
Plots: For β1 (β0 plots similar)
• Theoretical , β(·)1 (σU ) − β
(m)1 (σU ) vs. σU
• Remeasurement method , β(·)1,B(λ) − β
(m)1,B (λ) vs. λ,
B = 100, σU = 0.4
Greenberg Lecture IV: Latent Variable/Survival 16
3. Empirically checking robustness
Theoretical plot:
0.0 0.2 0.4 0.6 0.8 1.0
-0.3
-0.2
-0.1
0.0
PSfrag replacements β(·)
1(σ
U)−
β(m
)1
(σU
)
σU
λ
Solid = β(n)1 (σU ), Dashed = β
(s)1 (σU )
Greenberg Lecture IV: Latent Variable/Survival 17
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
3. Empirically checking robustness
Empirical plot: λ = 0 corresponds to σU = 0.4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-0.1
2-0
.10
-0.0
8-0
.06
-0.0
4-0
.02
0.0
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
b β(·)
1,B
(λ)−
b β(m
)1
,B
(λ)
Solid = β(n)1 (λ), Dashed = β
(s)1 (λ)
Greenberg Lecture IV: Latent Variable/Survival 18
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
3. Empirically checking robustness
Test statistic: In addition to visual assessment
t(λ∗) =θB(0) − θB(λ∗)
SE{θB(0) − θB(λ∗)}, λ∗ > 0
• Choose λ∗ in accordance with B; we have used λ∗ = 1 or 3 with
little difference
• “Large ” | t(λ∗) | indicates lack of robustness
• Reasonable operating characteristics in simulations
Greenberg Lecture IV: Latent Variable/Survival 19
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
3. Empirically checking robustness
Summary:
• The remeasurement method for empirically checking robustness can
be applied to any measurement error model, e.g., multiplicative
error, additional error-free covariates, etc.
• Currently : Extension to more complicated joint models for
longitudinal data and a primary/survival endpoint
Example/details: Huang, X., Stefanski, L., and Davidian, M. (2006)
Latent-model robustness in structural measurement error models.
Biometrika 93, 53-64.
Greenberg Lecture IV: Latent Variable/Survival 20
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
“Smooth” Inference for ArbitrarilyCensored Time-to-Event Data
Kirsten Doehler, Min Zhang, Lihua Tang and Marie Davidian
Department of Statistics
North Carolina State UniversityPSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
bβ(·)1,B
(λ) − bβ(m)1,B
(λ)
Greenberg Lecture IV: Latent Variable/Survival 21
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Survival analysis: Tradition
• Parametric models too restrictive ⇒
• Nonparametric or semiparametric models and methods
• Advantage : Minimal assumptions ⇔ robustness
• Disadvantage : Includes implausible models as possibilities,
computational/inferential difficulties
Perspective: Impose mild “smoothness ” assumptions
• Disadvantage : More restrictive (but not too much )
• Advantage : Computational ease , unified handling of arbitrary
censoring , possible efficiency gains
Greenberg Lecture IV: Latent Variable/Survival 22
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Assume: Time-to-event random variable T , values in (0,∞)
• Survival function S(t) = P (T > t), 0 < t <∞
• Density f(t), f ∈ H, where H is a class of “smooth ” densities
• Objective : Estimate S(t) under these assumptions
Class H: Gallant and Nychka (1987)
• “Sufficiently differentiable ”
• No “unusual ” behavior, e.g., oscillations, jumps, other weirdness
• May be multimodal , skewed , fat- or thin-tailed
• q-dimensional; we consider q = 1 for now
Greenberg Lecture IV: Latent Variable/Survival 23
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Representation of h ∈ H:
h(z) = P 2∞(z)ψ(z) + lower bound
• Infinite Hermite series + lower bound governing tails
• P∞(·) infinite-dimensional polynomial
• ψ(·) is a density with moment generating function; the “base
density ”
• Almost always in published applications : ψ(·) is the standard
normal density ϕ(·) (but doesn’t have to be. . . )
• “SemiNonParametric ” (SNP )
Greenberg Lecture IV: Latent Variable/Survival 24
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Practical use: Truncate
hK(z) = P 2K(z)ψ(z)
• “Standardized version ”
• E.g., K = 2, PK(z) = a0 + a1z + a2z2
• Flexible : K = 1, 2 often suffices to approximate almost any shape
• Approximation has same support as the base density
•∫hK(z) dz = 1 ensured “automatically ” by a reparameterization of
polynomial coefficients via a spherical transformation (Zhang and
Davidian, 2001) ⇒ in terms of K angles φ
• K selected via information criteria. . .
• Approximate any f ∈ H by shifting/scaling of Z with this density
Greenberg Lecture IV: Latent Variable/Survival 25
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Representation of survival density: Consider two base density
representations:
• log(T ) = µ+ σZ, Z has density h ∈ H
Approximate h by hK(z;φ) with ψ(z) = ϕ(z) = (2π)−1/2e−z2/2,
the standard normal density
• T = µZσ, Z has density h ∈ H
Approximate h by hK(z;φ) with ψ(z) = E(z) = e−z, the standard
exponential density
• Alternatively, on the log scale with extreme value base density
• In either case ⇒ approximate f(t) by fK(t; θ), θ = (µ, σ, φ).
• Evidence : Virtually any plausible survival density can be
approximated with small K and one of these base densities
Greenberg Lecture IV: Latent Variable/Survival 26
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Survival function approximation: SK(t; θ) =∫ ∞
tfK(u; θ) du
• E.g., Normal base density
SK(t; θ) =
∫ ∞
(log t−µ)/σ
P 2K(z)ϕ(z) dz
• Linear combination of easy integrals I(k, r) =
∫ ∞
r
zkϕ(z) dz,
I(0, r) = 1 − Φ(r), I(1, r) = ϕ(r),
I(k, r) = rk−1ϕ(r) + (k − 1)I(k − 2, r), k > 2
• Similar recursion for exponential base representation
Result: Straightforward approximation to S(t)
• Trivial computation
Greenberg Lecture IV: Latent Variable/Survival 27
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Straightforward likelihood: Any censoring/truncation pattern
• Right-censored data: Observe iid (Vi,∆i), i = 1, . . . , n,
Vi = min(Ti, Ci), Ti⊥⊥Ci, ∆i = I(Ti ≤ Ci)
• Likelihood for θ based on observed data for fixed K and base
`K(θ) =
n∑
i=1
[∆i log{fK(Vi; θ)} + (1 − ∆i) log{SK(Vi; θ)}
]
• Interval-censored data: Ti known to lie in [Li, Ri]
`K(θ) =
n∑
i=1
[log{SK(Li; θ) − SK(R+
i ; θ)}]
• Etc.
Greenberg Lecture IV: Latent Variable/Survival 28
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Choosing K-base: Standard information criteria
• Fit K = 0, 1, . . . ,Kmax for each base density, Kmax = 3 generally
suffices; i.e., estimate θ
• Choose K-base optimizing a given information criterion, e.g., AIC,
BIC, HQ = −2`K(θ) + 2dim(θ) log log n
• Starting values over a grid to ensure global maximum
Greenberg Lecture IV: Latent Variable/Survival 29
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Details/remarks:
• For chosen K-base , standard errors for SK(t; θ) via delta method
treating as standard parametric problem work well
• Computation : standard optimization routines (e.g., SAS IML
nlpqn), very fast
• Test statistic for comparing two groups: integrated weighted
difference
T =
∫ t00w(u){S1,K1
(u; θ1) − S2,K2(u; θ2)} du
SE[∫ t0
0w(u){S1,K1
(u; θ1) − S2,K2(u; θ2)} du
]
compare to standard normal critical values
Greenberg Lecture IV: Latent Variable/Survival 30
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Representative Monte Carlo simulations: S(t) is Weibull, n = 200,
1000 data sets, estimation at S(t0) = 0.9, 0.8, . . . , 0.1
30% Right cens 75% Interval cens
25% Right cens
S(t0) Rel eff KM Cov prob Rel eff NPML Cov prob
0.1 1.21 0.93 2.97 0.91
0.2 1.39 0.94 3.49 0.92
0.3 1.41 0.95 2.84 0.93
0.4 1.42 0.95 2.75 0.94
0.5 1.47 0.95 2.57 0.93
0.6 1.43 0.95 2.35 0.95
0.7 1.40 0.95 2.21 0.95
0.8 1.45 0.94 2.15 0.93
0.9 1.87 0.94 – 0.94
• SNP bias < 1.5%
Greenberg Lecture IV: Latent Variable/Survival 31
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Right censored Interval-censored
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
time
prob
abili
ty
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
bβ(·)1,B
(λ) − bβ(m)1,B
(λ)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
time
prob
abili
ty
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
bβ(·)1,B
(λ) − bβ(m)1,B
(λ)
Greenberg Lecture IV: Latent Variable/Survival 32
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
ACTG 175: Time to AIDS or death. ZDV monotherapy (n1 = 617,
68% right censored) vs. combination therapy (n2 = 1847, 80% right
censored)
0 200 400 600 800 1000 1200
0.5
0.6
0.7
0.8
0.9
1.0
ACTG 175 Data
time (days)
Sur
viva
l Pro
babi
lity
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
bβ(·)1,B
(λ) − bβ(m)1,B
(λ)
T 2 = 39.9, p-value < 0.0001 (logrank test = 47.2, p-value < 0.0001)
Greenberg Lecture IV: Latent Variable/Survival 33
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
4. “Smooth” inference for survival functions
Breast Cosmesis data (Finkelstein and Wolfe, 1985): Time to
cosmetic deterioration. Radiation (n1 = 46, 25 RC, 21 IC) vs.
Radiation+chemo (n2 = 48, 13 RC, 35 IC)
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
0.8
1.0
Cosmetic Deterioration Data
time (months)
Sur
viva
l Pro
babi
lity
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λ
bβ(·)1,B
(λ) − bβ(m)1,B
(λ)
T 2 = 7.84, p-value = 0.005 (FW test = 6.83, p-value < 0.01)
Greenberg Lecture IV: Latent Variable/Survival 34
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
5. “Smooth” semiparametric regression
Regression analysis: Consider right-censoring
• Ti, Ci as before, Vi = min(Ti, Ci), ∆i = I(Ti ≤ Ci)
• Observed data : (Vi,∆i, Xi), Xi (p× 1) vector of covariates
• Usual assumption : Ti⊥⊥Ci|Xi
• Interested in a model that describes the association between Ti and
Xi
Greenberg Lecture IV: Latent Variable/Survival 35
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
5. “Smooth” semiparametric regression
Popular models: With meaningful interpretation
• Accelerated failure time (AFT) model
log Ti = XTi β + ei, ei has density f0(t)
⇒ Represent f0(t) by SNP
• Proportional hazards model (PH) – unspecified baseline survival
function S0(t)
S(t|Xi) = S0(t)exp(XT
i β)
⇒ Represent density f0(t) of S0(t) by SNP
• Proportional odds model (PI) – unspecified baseline log odds a0(t)
−logit{S(t|Xi)} = a0(t)+XTi β, a0(t) = −logit[S0(t)/{1−S0(t)}]
⇒ Represent density f0(t) of S0(t) by SNP
Greenberg Lecture IV: Latent Variable/Survival 36
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
5. “Smooth” semiparametric regression
Remarks:
• Arbitrary censoring straightforward
• All models in a common framework ⇒ model selection via
information criteria
• Standard errors , confidence intervals , etc. straightforward
• Easy computation
Greenberg Lecture IV: Latent Variable/Survival 37
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)
5. “Smooth” semiparametric regression
Extensions:
• “Heteroscedastic ” AFT model
• Subject-specific AFT model for clustered data
log Tij = XTijβ + bi + eij , bi ∼ N (0, σ2
b ), eijiid∼ f0(t)
• Bivariate survival data: T1, T2 have “smooth ” density
f(t1, t2) ⇒ represent by bivariate (q = 2) SNP
• Joint longitudinal-survival models
• Etc.
Greenberg Lecture IV: Latent Variable/Survival 38
PSfrag replacements
β(·)1 (σU ) − β
(m)1 (σU )
σU
λbβ(·)1,B
(λ) − bβ(m)1,B
(λ)