research lessons from survival analysis
TRANSCRIPT
Research Lessons from Survival Analysis
Joshua Naranjo
Statistical Consulting CenterDepartment of Statistics
Western Michigan University
Joshua Naranjo Research Lessons from Survival Analysis
A modeling framework
Common elements
1 Data
2 Model
3 Fit criterion
Data Model↖ ↗Fit Criterion
Joshua Naranjo Research Lessons from Survival Analysis
Student’s t
Ex. The two-sample problem
1 Data: {xi}, {yj}2 Model: X ∼ N(µ1, σ
2), Y ∼ N(µ2, σ2)
3 Fit criterion: Likelihood
Data Model↖ ↗Fit Criterion
Model : X ∼ N(x , s2p), Y ∼ N(y , s2
p)
Test : y−xsp√
1/n1+1/n2> tα
Estimate : y − x(
SE = sp√
1/n1 + 1/n2
)
Joshua Naranjo Research Lessons from Survival Analysis
Student’s t
Ex. The two-sample problem
1 Data: {xi}, {yj}2 Model: X ∼ N(µ1, σ
2), Y ∼ N(µ2, σ2)
3 Fit criterion: Likelihood
Data Model↖ ↗Fit Criterion
Model : X ∼ N(x , s2p), Y ∼ N(y , s2
p)
Test : y−xsp√
1/n1+1/n2> tα
Estimate : y − x(
SE = sp√
1/n1 + 1/n2
)
Joshua Naranjo Research Lessons from Survival Analysis
Welch-t
1 Data: {xi}, {yj}2 Model: X ∼ N(µ1, σ
21), Y ∼ N(µ2, σ
22)
3 Fit criterion: Likelihood
Data Model↖ ↗Fit Criterion
Model: X ∼ N(µ1, s21 ) , Y ∼ N(µ2, s
22 )
Test : y−x√s2
1/n1+s22/n2
> tα
Estimate : y − x
(SE =
√s2
1/n1 + s22/n2
)
Joshua Naranjo Research Lessons from Survival Analysis
Wilcoxon Rank-Sum
1 Data: {xi}, {yj}2 Model: X ∼ F (t), Y ∼ F (t −∆)
3 Fit criterion: Rank dispersion
Data Model↖ ↗Fit Criterion
Model: X ∼ F (t), Y ∼ F (t − ∆), where ∆ = med {yj − xi}
Test :∑
Ri−E0√V0
> zα
Estimate : ∆ = med {yj − xi}
Joshua Naranjo Research Lessons from Survival Analysis
Evaluation
Data Model↖ ↗Fit Criterion
Pros and cons1 Efficiency
Null (size, etc.)Alternative (power under various configurations)
2 Robustness
Broadness of assumptionsSensitivity to violations
3 Simplicity and interpretability
4 Computing complexity
Joshua Naranjo Research Lessons from Survival Analysis
Evaluation
Example: The two-sample problem
Student’s t Welch t Rank Sum1. Efficiency X X X2. Robustness x X X3. Interpret X X x4. Computing X X X
Joshua Naranjo Research Lessons from Survival Analysis
Beyond two-sample
More data structures
The k-sample problem (Y1,Y2,Y3)
Regression (X1,X2,X3,Y )
Correlation (X ,Y )
Categorical data (fx , fy )
Survival data (Tx ,Ty )
Longitudinal data (Σy )
Time series data (Yt)
Mixture data (pY1 + (1− p)Y2)
· · ·
Joshua Naranjo Research Lessons from Survival Analysis
Two-sample vs Two-sample
Age at death (from all causes)
Study 1 (Estrogen) Study 2 (Vitamin E)
Treatment Control Treatment Control(n = 1680) (n = 1722) (n = 1819) (n = 1994)
66 75 64 5568 58 90 8972 79 83 57...
......
...... 51
......
44 63...
60
Joshua Naranjo Research Lessons from Survival Analysis
Two-sample vs Two-sample
Crucial difference
1 Study 1 is randomized2 Study 2 is not randomized
Treatment group has higher educationTreatment group more female
Need to adjust for education and gender⇒((((((hhhhhhTwo-sample Regression
Joshua Naranjo Research Lessons from Survival Analysis
Survival data
The two-sample problem
Response is time-to-event (e.g. survival time)
Patients enter the study serially (e.g. patients are referred tostudy upon diagnosis)
Two groups of patients (e.g. randomized treatment arms)
Joshua Naranjo Research Lessons from Survival Analysis
Survival data
subj days status group1 8 1 22 23 1 13 23 1 24 52 0 25 60 1 1:
74 828 1 175 976 0 2
days=time from randomization to death or censoring
status=
{1 if death0 if censored (e.g. subject 4 enrolled 52 days ago)
group=
{1 if treatment 12 if treatment 2
Joshua Naranjo Research Lessons from Survival Analysis
Survival data
subj days group1 8 22 23 13 23 24 >52 25 60 1:
74 828 175 >976 2
days=time from randomization to death or censoring
status=
{1 if death0 if censored (e.g. subject 4 enrolled 52 days ago)
group=
{1 if treatment 12 if treatment 2
Joshua Naranjo Research Lessons from Survival Analysis
Kaplan-Meier Curve
Q: How do we accommodate censoring? (L1)
subj days group1 8 22 23 13 23 24 >52 25 60 1:
74 828 175 >976 2
The KM estimate of S(t) = P[T > t]:S(0) = 1S(8) = 74/75 = .9867S(23) = (74/75)(72/74) = .9600S(60) = (74/75)(72/74)(70/71) = .9465
Joshua Naranjo Research Lessons from Survival Analysis
Kaplan-Meier Curve
Joshua Naranjo Research Lessons from Survival Analysis
Kaplan-Meier Curve
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
H0 : S1(·) = S2(·) vs H1 : Not
Two early approaches
1 Gehan (1965)Adapt rank tests to accommodate censoring
2 Mantel (1968)Adapt frequency tables to accommodate censoring (L2)
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
1 Adapt rank tests to accommodate censoring
The Wilcoxon Rank-Sum Test (1945)
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
1 Adapt rank tests to accommodate censoring
Idea: Add up the ranks of the Group 1 subjects
subj days group Rank1 8 22 23 1 2.53 23 24 >52 25 60 1 ?:
74 828 175 >976 2
?
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
1 Adapt rank tests to accommodate censoring
Idea: Add up the ranks of the Group 1 subjects
group 1 group 2days rank days rank
23 2.5 8 160 ? 23 2.5
: >52: :: :: :: >976 75
828 74
?
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
The Mann-Whitney version of Wilcoxon Rank-Sum test
W =∑i
∑j
Uij ,
where Uij =
−1 xi < yj
0 xi = yj1 xi > yj
Let
U∗ij =
−1 xi < yj or xi < y∗j
0 xi = yj or x∗i < yj or xi > y∗j1 xi > yj or x∗i > yj
”Gehan (1965) generalization of Wilcoxon”
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
The Mann-Whitney version of Wilcoxon Rank-Sum test
W =∑i
∑j
Uij ,
where Uij =
−1 xi < yj
0 xi = yj1 xi > yj
Let
U∗ij =
−1 xi < yj or xi < y∗j
0 xi = yj or x∗i < yj or xi > y∗j1 xi > yj or x∗i > yj
”Gehan (1965) generalization of Wilcoxon”
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
”Efron (1967) modification of Gehan-Wilcoxon”
subj days status group Rank1 8 1 22 23 1 1 2.53 23 1 24 52 0 25 60 1 1 ?:
74 828 1 175 976 0 2
?
While it is unlikely that 60 > 52∗, it is more likely that 828 > 52∗,but Gehan treats them as equally uninformative (Uij∗ = 0).
U∗ij = 1− 2S(xi )
S(yj)when xi > y∗j
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
2 Adapt frequency tables to accommodate censoring
Recall the χ2 test
Yes NoGrp 1 a b n1
Grp 2 c d n2
m1 m2 n
Then
χ2 =(a− E )2
V
whereE =
n1m1
nand V =
n1n2m1m2
n2(n − 1)
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
”Cochran-Mantel-Haentszel test (1959)”
When data are stratified according to control variables (e.g. raceor gender)
Yes NoGrp 1 a1 b1
Grp 2 c1 d1
Yes Noa2 b2
c2 d2
· · ·Yes NoaK bKcK dK
χ2CMH =
(∑
ai −∑
Ei )2∑
Vi
Q: Why not∑(
ai−EiVi
)2
Be cynical. (L3)
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
“Logrank test” (Mantel, 1966)
Day 8: 1 death, and it came from Group 2
Grp 1 0 40 40Grp 2 1 34 35
⇒ a1 = 0, E1 = 4075
Day 23: 2 deaths, one from each group
Grp 1 1 39 40Grp 2 1 33 34
⇒ a2 = 1, E2 = 2(
4074
)Day 60: 1 death, and it came from Group 1
Grp 1 1 38 39Grp 2 0 32 32
⇒ a3 = 1, E3 = 3971
Z =∑
(aj−Ej )√∑Vj
, sum over distinct event times
Joshua Naranjo Research Lessons from Survival Analysis
Test for equality of groups
Pros and cons? Not obvious.
While the Mantel logrank test is
ZL =
∑(aj − Ej)√∑
Vj
the Gehan-Wilcoxon can be written as
ZW =
∑nj(aj − Ej)√∑
n2j Vj
where nj is the total number of subjects at risk at event time j .
⇒ The Wilcoxon gives more weight to early differences
Joshua Naranjo Research Lessons from Survival Analysis
Regression
Example: Survival times for heart transplant patients
(T) (X1) (X2) (X3)
Subj Survive Trans Prior Age
1 5 0 0 52
2 15 1 0 55
3 37 1 0 41
4 48 0 1 31
:
:
44 541 1 0 46
45 670 1 0 48
Joshua Naranjo Research Lessons from Survival Analysis
Suppose that survival time T has probability density function f (t).The cumulative density function is
F (t) = P[T ≤ t] =
∫ t
0f (u) du
The survivor function is
S(t) = P[T > t] = 1− F (t)
Define the hazard function as
h(t) =f (t)
S(t)
It can be shown that
f (t) = h(t)e−∫ t
0 h(u) du
Joshua Naranjo Research Lessons from Survival Analysis
Some hazard models:
h(t) =
λ , Exponentialλγt , Gompertzλtα , Weibull
Incorporating covariates like transplant, prior surgery, and age?
h(t) =
λ eβ1x1+···+βkxk , Exponentialλγt eβ1x1+···+βkxk , Gompertzλtα eβ1x1+···+βkxk , Weibull
= λ0(t) eβ′x
Proportional hazards: covariates are multiplicatively related to thehazard
Joshua Naranjo Research Lessons from Survival Analysis
Estimate parameters by maximum likelihood
L =n∏
i=1
fi (ti )
wherefi (t) = hi (t)e−
∫ t0 hi (u) du
hi (t) = λ0(t) eβ′xi
and
λ0(t) =
λ , Exponentialλγt , Gompertzλtα , Weibull
Problem: Specify λ0(t)?
Joshua Naranjo Research Lessons from Survival Analysis
Partial Likelihood
Cox (1972) proposed an estimation method for the βs withoutneeding to specify λ0(t). Maximize the partial likelihood
PL =n∏
i=1
Li
where Li is the conditional probability of failure at time ti giventhe number of cases at risk at time ti .
Joshua Naranjo Research Lessons from Survival Analysis
Example: Survival times for heart transplant patients
(T) (X1) (X2) (X3)
Subj Survive Trans Prior Age
1 5 0 0 52
2 15 1 0 55
3 37 1 0 41
4 48 0 1 31
:
:
44 541 1 0 46
45 670 1 0 48
Joshua Naranjo Research Lessons from Survival Analysis
A death occurred 5 days after enrollment. What is the probabilitythat it happens to patient 1 instead of to one of the others at-risk?
L1 =h1(5)
h1(5) + · · ·+ h45(5)
L2 =h2(15)
h2(15) + · · ·+ h45(15)
:
:
L44 =h44(541)
h44(541) + h45(541)
L45 = 1
Similar to
P(Ai |B) =P(Ai ∩ B)
P(A1 ∩ B) + · · ·+ P(An ∩ B)
Joshua Naranjo Research Lessons from Survival Analysis
L1 =h1(5)
h1(5) + · · ·+ h45(5)
=λ0(5) eβ
′x1
λ0(5) eβ′x1 + · · ·+ λ0(5) eβ′x45
=eβ
′x1
eβ′x1 + · · ·+ eβ′x45
L2 =eβ
′x2
eβ′x2 + · · ·+ eβ′x45
The combination of PH assumption and partial likelihood
PL =n∏
i=1
Li
allows estimation of β without specifying the baseline hazard!
Joshua Naranjo Research Lessons from Survival Analysis
Efficiency
2 Efron (1977) “The Efficiency of Cox’s Likelihood Function forCensored Data”.
Joshua Naranjo Research Lessons from Survival Analysis
SAS output
The PHREG Procedure
Dependent variable: Survive
Maximum Likelihood Estimates
Standard Wald Pr> Risk
Var DF Estimate Error Chi-sq Chi-sq Ratio
Trans 1 -1.708140 0.27860 37.590 0.0001 0.181
Prior 1 -0.421402 0.37100 1.290 0.2560 0.656
Age 1 0.058609 0.01505 15.163 0.0001 1.060
Joshua Naranjo Research Lessons from Survival Analysis
The hazard ratio for age is
h(t;Age = x + 1)
h(t;Age = x)=λ0(t)e−1.708x1−0.421x2+0.0586(x3+1)
λ0(t)e−1.708x1−0.421x2+0.0586x3
= e .0586
= 1.0603
so every additional year of age increases hazard of failure by 6%same for all t.
(Implications and consequences of PH)
Joshua Naranjo Research Lessons from Survival Analysis
Example: Survival times for heart transplant patients
Standard Risk
Var DF Estimate Error Ratio
Trans 1 -1.708140 0.27860 0.181
Prior 1 -0.421402 0.37100 0.656
Age 1 0.058609 0.01505 1.060
“Age increases hazard of death by 6%, and getting a transplantreduces hazard of death by 82%.”
While the age effect may be real, the magnitude of transplanteffect is likely false.
Transplant⇒ Late death
No Transplant⇒ Early death
Joshua Naranjo Research Lessons from Survival Analysis
Example: Survival times for heart transplant patients
Standard Risk
Var DF Estimate Error Ratio
Trans 1 -1.708140 0.27860 0.181
Prior 1 -0.421402 0.37100 0.656
Age 1 0.058609 0.01505 1.060
“Age increases hazard of death by 6%, and getting a transplantreduces hazard of death by 82%.”
While the age effect may be real, the magnitude of transplanteffect is likely false.
Transplant ⇒ Late deathNo Transplant ⇐ Early death
Joshua Naranjo Research Lessons from Survival Analysis
PH assumption
Joshua Naranjo Research Lessons from Survival Analysis
PH assumption
Joshua Naranjo Research Lessons from Survival Analysis
Characterizations of Proportional Hazards
Q: How do we detect PH? (L4: Unobservables)
Two sample:
h(t) = h0(t)eβxi =
{h0(t) Group 1 (x = 0)h0(t)ψ Group 2 (x = 1)
S(t) =
exp{∫ t
0 h0(u)du}
Group 1
exp{∫ t
0 h0(u)ψdu}
Group 2
implies
S2(t) = [S1(t)]ψ
Joshua Naranjo Research Lessons from Survival Analysis
Characterizations of Proportional Hazards
S2(t) = [S1(t)]ψ
Early separation
Late separation
Joshua Naranjo Research Lessons from Survival Analysis
Characterizations of Proportional Hazards
S2(t) = [S1(t)]ψ
Joshua Naranjo Research Lessons from Survival Analysis
The logrank test and Cox Regression
Quotes:Assume that xi is a binary variable and the onlycovariate. Then the score test for Cox-regression isthe same as the log-rank if there are no ties.LR test is aymptotically efficient for the two-sampleproblem when there is noninformative censoring.
Joshua Naranjo Research Lessons from Survival Analysis
The logrank test and Cox Regression
Joshua Naranjo Research Lessons from Survival Analysis
The logrank test and Cox Regression
Joshua Naranjo Research Lessons from Survival Analysis
Extensions 1 (Applications)
What about (L5: Extensions)
Tied event times
Repeated measures
Fixed covariatesTime-dependent covariates (e.g. co-meds, hypertension)
Time-dependent coefficients (e.g. treatments that are lesseffective in later stages)
Large p
Joshua Naranjo Research Lessons from Survival Analysis
Extensions 2 (Tools)
Extend to Cox PH regression
R2
Extra SS
Partial plots
Outlier diagnostics
Mallows Cp
AIC
LASSO...
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
“Cancer immunotherapy activates the immune system to combatcancer. This poses the possibility of a delayed onset of clinicaleffect and violation of the PH assumption.” (Z. Xu, FDA)
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
“Cancer immunotherapy activates the immune system to combatcancer. This poses the possibility of a delayed onset of clinicaleffect and violation of the PH assumption.” (Z. Xu, FDA)
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
JSM 2019 (Denver CO)
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
JSM 2019 (Denver CO)
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
The NPH Working Group
Collaboration initiated by FDA in 2016
Methodical evaluation of available methods is needed
Appropriate analysis for various patterns of non-PH
How to optimally test for treatment differences?How to summarize nonconstant treatment effect?
Organizations representedAZ, BMS, Merck, Boehringer Ingelheim, Novartis, Eli Lilly,Abbvie, Genentech, Roche, Bayer, Janssen, Takeda, Amgen,Pfizer, GSK, Celgene, Sanofi, Johnson & Johnson, FDA
Joshua Naranjo Research Lessons from Survival Analysis
Non-Proportional Hazards
Available methods
Weighted logrank (Fleming and Harrington 1987)
Piecewise logrank (Xu et al. 2016)
Combination tests
Karrison 2016 (Max-Combo)Gares et al. 2017Yang and Prentice 2010Logan 2008
Restricted mean survival time (Uno et al. 2014)
Generalized pairwise comparisons Uij
Peron et al. 2018Buyse 2010
Weighted KM test (Pepe and Fleming 1989, 1991)
Joshua Naranjo Research Lessons from Survival Analysis
Weighted logrank tests
Logrank test (Mantel, 1968)
L =
∑(aj − Ej)√∑
Vj
Wilcoxon (Gehan, 1965)
W =
∑nj(aj − Ej)√∑
n2j Vj
.=
∑S(tj)(aj − Ej)√∑
S2(tj)Vj
Weighted Logrank (Fleming and Harrington, 1987)
FH =
∑W (tj)(aj − Ej)√∑
W 2(tj)Vj
where W (t) =[S(t)
]ρ [1− S(t)
]γJoshua Naranjo Research Lessons from Survival Analysis
Weighted logrank tests
FH(ρ, γ) =
Logrank (ρ, γ) = (0, 0)Wilcoxon (1, 0)Late difference (0, 1)
Pros and cons
Easy to understand and implement
With correct choice of weight, can be much more efficientthan either logrank or Wilcoxon
Low efficiency with incorrect choice of weight
Joshua Naranjo Research Lessons from Survival Analysis
Max-Combo
Karrison (2016): Reject H0 : S1 = S2 if
Zm = max(|FH(0, 0)|, |FH(1, 0)|, |FH(0, 1))| > gα
gα is the upper α percentile of the null distribution ofmax(|Z1|, |Z2|, |Z3|)Calculating gα
α = P[Zm > gα]= 1− P[Zm ≤ gα]= 1− P[|Z1| ≤ gα, |Z2| ≤ gα, |Z3| ≤ gα]
(Z1,Z2,Z3) ∼ MN(0,Σ),Gill (1980), Kosorok and Lin (1999), Karrison (2016)
Joshua Naranjo Research Lessons from Survival Analysis
Max-Combo
FH(0, 0) : p = .022FH(1, 0) : p = .062FH(0, 1) : p = .015
Zm : p = .029
Joshua Naranjo Research Lessons from Survival Analysis
Max-Combo
FH(0, 0) : p = .190FH(1, 0) : p = .045FH(0, 1) : p = .938
Zm : p = .082
Joshua Naranjo Research Lessons from Survival Analysis
Lessons:
1 Accommodate (certain data features)
2 Adapt (existing methods)
3 Be cynical
4 Unobservable assumptions (diagnostics)
5 Extensions (broader applications)
6 Try combining methods (balanced properties)
Joshua Naranjo Research Lessons from Survival Analysis
Recent dissertation topics (Statistics, WMU)
”Nonparametric Tests for Ordered Alternatives in aTwo-Stage Nested Design” (Bang Le, 2019)
”A Two-Stage Bayesian Variable Selection for Geo-ReferencedCount Data” (Yuqian Shen, 2019)
”Testing for Association in Stratified 2× 2 ContingencyTables” (Eunice Ampah, 2018)
”Statistical Models for Correlated Data” (Xiaomeng Niu,2017)
”Diagnostics for Choosing Stratified Logrank or WilcoxonTests” (Jhoanne Gatpatan, 2017)
”Subgroup Analysis and Growth Curve Models forLongitudinal Data” (Nichole Andrews, 2017)
Joshua Naranjo Research Lessons from Survival Analysis