ST3242

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Cox proportional hazards model
ST3242: Introduction to Survival Analysis
Alex Cook
September 2008
ST3242 : Cox proportional hazards model 1/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Unique failure times
The partial likelihood for β is
ℓp(β, x) =
m
i=1
φi
j∈R(ti ) φj
ST3242 : Cox proportional hazards model 2/49
δi
.

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Repeated times
Breslow’s approximation
ℓp(β, x) =
I
i=1
j∈D(t (i)) φj
j∈R(t (i)) φj
|D(t(i))| .
ST3242 : Cox proportional hazards model 3/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Repeated times
Efron’s approximation
ℓp(β, x) =
I
i=1
j∈D(t (i)) φj
|D(t (i))|
k=1 j∈R(t (i)) φj − k−1
|D(t (i))|
j∈D(t (i)) φj
.
ST3242 : Cox proportional hazards model 4/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Repeated times
Exact partial likelihood
ℓp(β, x) =
I
i=1
j∈D(t (i)) φj
q∈Qi Φq
where Qi is the set of all |D(t(i))|-tuples that could be
selected from R(t(i)) and Φq is the product of φj for all
members j of |D(t(i))|-tuple q.
ST3242 : Cox proportional hazards model 5/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Repeated times
Example
Suppose individuals labelled 1–5 are at risk at time t(i), i.e. in
R(t(i)), and that of these, individuals 1–3 fail at time t(i)
Breslow:
φ1φ2φ3
,
(φ1 + φ2 + φ3 + φ4 + φ5) 3
Efron:
φ1φ2φ3
(φ1 + φ2 + φ3 + φ4 + φ5)( 2
3 φ1 + 2
3 φ2 + 2
3 φ3 + φ4 + φ5)( 1
3 φ1 + 1
3 φ2 +
Exact:
φ1φ2φ3
.
φ1φ2φ3 + φ1φ2φ4 + φ1φ2φ5 + . . . + φ3φ4φ5
ST3242 : Cox proportional hazards model 6/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
How to maximise these?
In practice
Luckily, R does it for you.
But exact method may take a long time.
Theory behind R’s method
Problem of maximising a (log) (partial) likelihood
common in statistics
For applied problems, often searching over 10+ of
dimensions
“Grid search” okay for 1D or 2D
Deterministic or stochastic methods—see ST4231
Computer Intensive Statistical Methods
ST3242 : Cox proportional hazards model 7/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
How to maximise these?
In practice
Luckily, R does it for you.
But exact method may take a long time.
Theory behind R’s method
Problem of maximising a (log) (partial) likelihood
common in statistics
For applied problems, often searching over 10+ of
dimensions
“Grid search” okay for 1D or 2D
Deterministic or stochastic methods—see ST4231
Computer Intensive Statistical Methods
ST3242 : Cox proportional hazards model 7/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
How to maximise these?
In practice
Luckily, R does it for you.
But exact method may take a long time.
Theory behind R’s method
Problem of maximising a (log) (partial) likelihood
common in statistics
For applied problems, often searching over 10+ of
dimensions
“Grid search” okay for 1D or 2D
Deterministic or stochastic methods—see ST4231
Computer Intensive Statistical Methods
ST3242 : Cox proportional hazards model 7/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
How to maximise these?
In practice
Luckily, R does it for you.
But exact method may take a long time.
Theory behind R’s method
Problem of maximising a (log) (partial) likelihood
common in statistics
For applied problems, often searching over 10+ of
dimensions
“Grid search” okay for 1D or 2D
Deterministic or stochastic methods—see ST4231
Computer Intensive Statistical Methods
ST3242 : Cox proportional hazards model 7/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
How to maximise these?
In practice
Luckily, R does it for you.
But exact method may take a long time.
Theory behind R’s method
Problem of maximising a (log) (partial) likelihood
common in statistics
For applied problems, often searching over 10+ of
dimensions
“Grid search” okay for 1D or 2D
Deterministic or stochastic methods—see ST4231
Computer Intensive Statistical Methods
ST3242 : Cox proportional hazards model 7/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Naïve grid search
ST3242 : Cox proportional hazards model 8/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
This is a deterministic, iterative procedure:
deterministic because no random search involved
at each stage the estimate gets better
hopefully
ST3242 : Cox proportional hazards model 9/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
This is a deterministic, iterative procedure:
deterministic because no random search involved
at each stage the estimate gets better
hopefully
ST3242 : Cox proportional hazards model 9/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Where
U(θ) = ( ∂l(θ)
)
∂θ1
T
2 ∂ l(θ)
I (θ) = −
∂θr∂θs
ST3242 : Cox proportional hazards model 10/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Note
θ (0) is arbitrary: if far
from ˆ θ we might not
converge
Can write ℓp instead of
ℓ and β instead of θ
ST3242 : Cox proportional hazards model 11/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Newton–Raphson
Assume parameters are θ, dim(θ) = p and log-likelihood is
ℓ(θ)
1 k = 1
2 choose θ (k)
3 Solve
I (θ (k) )(θ (k+1) −θ (k) ) = U(θ (k) )
for θ (k+1) .
4 Increment k by one.
5 Go back to step 3 and repeat
until convergence.
Example
i t (i) x (i)
1 6 31.4
2 98 21.5
3 189 27.1
4 374 22.7
5 1002 35.7
6 1205 30.7
7 2065 26.5
8 2201 28.3
9 2421 27.9
h(t, xi) = h0(t)e βxi
ST3242 : Cox proportional hazards model 12/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
lp(β) = β
d
dβ lp(β) =
9
x(i) −
i=1
9
x(i) −
i=1
d 2
dβ 2 lp(β) = −
where A =
⎛
9
i=1
⎝
j∈R(t (i))
⎛
9
log ⎝
i=1
9
i=1
j∈R(t (i))
βxj
j∈R(t xje
(i))
j∈R(t (i)) eβxj
A
j∈R(t (i)) eβxj
2
x 2 j e βxj
⎞ ⎛
⎠ ⎝
j∈R(t (i))
e βxj
e βxj
⎞
⎠
⎞ ⎛
⎠ − ⎝
j∈R(t (i))
xje βxj
ST3242 : Cox proportional hazards model 13/49
⎞
⎠

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
If we write
U(β) = d
dβ lp(β)
I (β) = d2
lp(β)
dβ2 then Newton–Raphson update step is just
β (k+1) = β (k) + U(β (k) )/I (β (k) ).
ST3242 : Cox proportional hazards model 14/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
Try β (0) = 0 first
U(0) = −2.51 and I (0) = 77.13, so our new guess at the
MLE is
β (1) = 0 − 2.51/77.13 = −0.0326.
β (2) = −0.0326 − 0.069/72.83 = −0.0335.
β (3) = −0.0335 − 0.000061/72.70 = −0.0335 = β (2)
MLE
Here it worked, as our first guess was quite close to the MLE.
But. . .
ST3242 : Cox proportional hazards model 15/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
Try β (0) = 0 first
U(0) = −2.51 and I (0) = 77.13, so our new guess at the
MLE is
β (1) = 0 − 2.51/77.13 = −0.0326.
β (2) = −0.0326 − 0.069/72.83 = −0.0335.
β (3) = −0.0335 − 0.000061/72.70 = −0.0335 = β (2)
MLE
Here it worked, as our first guess was quite close to the MLE.
But. . .
ST3242 : Cox proportional hazards model 15/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
Try β (0) = 0 first
U(0) = −2.51 and I (0) = 77.13, so our new guess at the
MLE is
β (1) = 0 − 2.51/77.13 = −0.0326.
β (2) = −0.0326 − 0.069/72.83 = −0.0335.
β (3) = −0.0335 − 0.000061/72.70 = −0.0335 = β (2)
MLE
Here it worked, as our first guess was quite close to the MLE.
But. . .
ST3242 : Cox proportional hazards model 15/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
Try β (0) = 0 first
U(0) = −2.51 and I (0) = 77.13, so our new guess at the
MLE is
β (1) = 0 − 2.51/77.13 = −0.0326.
β (2) = −0.0326 − 0.069/72.83 = −0.0335.
β (3) = −0.0335 − 0.000061/72.70 = −0.0335 = β (2)
MLE
Here it worked, as our first guess was quite close to the MLE.
But. . .
ST3242 : Cox proportional hazards model 15/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Score and information
Try β (0) = 0 first
U(0) = −2.51 and I (0) = 77.13, so our new guess at the
MLE is
β (1) = 0 − 2.51/77.13 = −0.0326.
β (2) = −0.0326 − 0.069/72.83 = −0.0335.
β (3) = −0.0335 − 0.000061/72.70 = −0.0335 = β (2)
MLE
Here it worked, as our first guess was quite close to the MLE.
But. . .
ST3242 : Cox proportional hazards model 15/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 16/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 17/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 18/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 19/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 20/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Non-convergence
ββ
−1.5 −0.5 0.5 1.5
2 4 6 8 10
iteration
ST3242 : Cox proportional hazards model 21/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Replace
by
A solution
for ξ < 1
I (θ (k) )(θ (k+1) − θ (k) ) = U(θ (k) )
I (θ (k) )(θ (k+1) − θ (k) ) = ξU(θ (k) )
ST3242 : Cox proportional hazards model 22/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
A solution
ββ
−1.5 −0.5 0.5 1.5
0 20 40 60 80 100
iteration
ST3242 : Cox proportional hazards model 23/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Easy peasy
coxph
coxph(Surv(times,deltas) covariates)
ST3242 : Cox proportional hazards model 24/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Easy peasy
coxph
coxph(Surv(times,deltas) covariates)
ST3242 : Cox proportional hazards model 24/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Optional arguments
data = aml
subset = 1:100
init = 10
init = c(0.01, 0.01)
method = ’efron’
method = ’breslow’
method = ’exact’
ST3242 : Cox proportional hazards model 25/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Example
Worcester heart attack study
start time = heart attack
end time = death (or censoring)
covariate: body mass index
ST3242 : Cox proportional hazards model 26/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Example
> library(survival)
>
>
cols=c(’id’,’age’,’sex’,’c4’,’c5’,’c6’,’bmi’,’history’,
’c9’,’c10’,’c11’, ’c12’,’c13’,’c14’,’c15’,’c16’,
’c17’,’c18’,’c19’,’c20’,’t’,’delta’)
attach(read.table(’http://www.umass.edu/statdata/statdata
/data/whas500.dat’ ,col.names=cols))
> phm.bmi = coxph(Surv(t,delta) bmi)
> summary(phm.bmi)
ST3242 : Cox proportional hazards model 27/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Example
ST3242 : Cox proportional hazards model 28/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Example—using Breslow
ST3242 : Cox proportional hazards model 29/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Standard error
R provides us with the standard error of β. Note:
asymptotically.
ˆβ ∼ N(β, E{I (β)} −1 )
Actually. . .
R provides an estimated standard error, replacing the expected
information E{I (β)} by the observed information I ( ˆ β).
ST3242 : Cox proportional hazards model 30/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Standard error
R provides us with the standard error of β. Note:
asymptotically.
ˆβ ∼ N(β, E{I (β)} −1 )
Actually. . .
R provides an estimated standard error, replacing the expected
information E{I (β)} by the observed information I ( ˆ β).
ST3242 : Cox proportional hazards model 30/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Confidence interval
CI
1 − α confidence interval for β is
Example
ˆβ
1
± z1−α/2
I ( ˆ β)
i.e. ˆ β ± z1−α/2se( ˆ ˆ β).
−0.0985 ± 1.96 × 0.0148 = (−0.13, −0.07) for effect of BMI
ST3242 : Cox proportional hazards model 31/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Confidence interval
CI
1 − α confidence interval for β is
Example
ˆβ
1
± z1−α/2
I ( ˆ β)
i.e. ˆ β ± z1−α/2se( ˆ ˆ β).
−0.0985 ± 1.96 × 0.0148 = (−0.13, −0.07) for effect of BMI
ST3242 : Cox proportional hazards model 31/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Confidence interval
ST3242 : Cox proportional hazards model 32/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Confidence interval
This CI can be transformed to a confidence interval for the
hazard ratio.
ST3242 : Cox proportional hazards model 33/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Confidence interval
ST3242 : Cox proportional hazards model 34/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
All test:
Three tests
Wald test
score test
likelihood ratio test
H0 : β = 0 versus H1 : β = 0
for h(t, xi) = h0(t, α) exp(βxi)
ST3242 : Cox proportional hazards model 35/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Wald test
If H0 is true, z 2 ∼ χ 2 1.
z 2 = ˆ β 2
ˆV( ˆ β) = ˆβ 2 I ( ˆβ)
ST3242 : Cox proportional hazards model 36/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Wald test
If H0 is true, z 2 ∼ χ 2 1.
z 2 = ˆ β 2
ˆV( ˆ β) = ˆβ 2 I ( ˆβ)
ST3242 : Cox proportional hazards model 36/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Score test
If H0 is true, z ⋆2 ∼ χ 2 1.
z ⋆2 = U(0) 2 I ( ˆ β) −1
ST3242 : Cox proportional hazards model 37/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Score test
If H0 is true, z ⋆2 ∼ χ 2 1.
z ⋆2 = U(0) 2 I ( ˆ β) −1
ST3242 : Cox proportional hazards model 37/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Likelihood ratio test
If H0 is true, G ∼ χ 2 1.
G = 2[ℓp( ˆ β) − ℓp(0)]
But
The tests give different p-values.
ST3242 : Cox proportional hazards model 38/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Likelihood ratio test
If H0 is true, G ∼ χ 2 1.
G = 2[ℓp( ˆ β) − ℓp(0)]
But
The tests give different p-values.
ST3242 : Cox proportional hazards model 38/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
H0 true
Likelihood ratio test
If H0 is true, G ∼ χ 2 1.
G = 2[ℓp( ˆ β) − ℓp(0)]
But
The tests give different p-values.
ST3242 : Cox proportional hazards model 38/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
β has p dimensions
H0 : first q dimensions are equal to specified β ⋆ j
last p − q dimensions are free parameters
H1 : βj = β ⋆ j
for at least one j ≤ q
ST3242 : Cox proportional hazards model 39/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
β has p dimensions
H0 : first q dimensions are equal to specified β ⋆ j
last p − q dimensions are free parameters
H1 : βj = β ⋆ j
for at least one j ≤ q
ST3242 : Cox proportional hazards model 39/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
β has p dimensions
H0 : first q dimensions are equal to specified β ⋆ j
last p − q dimensions are free parameters
H1 : βj = β ⋆ j
for at least one j ≤ q
ST3242 : Cox proportional hazards model 39/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
β has p dimensions
H0 : first q dimensions are equal to specified β ⋆ j
last p − q dimensions are free parameters
H1 : βj = β ⋆ j
for at least one j ≤ q
ST3242 : Cox proportional hazards model 39/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
β has p dimensions
H0 : first q dimensions are equal to specified β ⋆ j
last p − q dimensions are free parameters
H1 : βj = β ⋆ j
for at least one j ≤ q
ST3242 : Cox proportional hazards model 39/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
More general test
Perform the test by fitting two nested models
General model
Lets all p parameters be estimated.
ℓp( ˆ β)
Particular model
Fixes q parameters at hypothesised values.
Lets other p − q parameters be estimated.
ℓp(β ⋆
1:q, ˆ β q+1:p)
Test statistic
If H0 is true, G = 2[ℓp( ˆ β) − ℓp(β ⋆
1:q, ˆ β q+1:p)] ∼ χ 2 q
ST3242 : Cox proportional hazards model 40/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Worcester heart attacks
x b
i is body mass index
x a
i is age at heart attack
x s
i is sex (0 male, 1 female)
x h
i is history of heart problems (0 no history, 1 history)
First fit terms individually
ST3242 : Cox proportional hazards model 41/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Worcester heart attacks
x b
i is body mass index
x a
i is age at heart attack
x s
i is sex (0 male, 1 female)
x h
i is history of heart problems (0 no history, 1 history)
First fit terms individually
ST3242 : Cox proportional hazards model 41/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Commands
ST3242 : Cox proportional hazards model 42/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
BMI
ST3242 : Cox proportional hazards model 43/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Age
ST3242 : Cox proportional hazards model 44/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
Sex
ST3242 : Cox proportional hazards model 45/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
History
ST3242 : Cox proportional hazards model 46/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
BMI, age, sex
Male mean age: 67; female mean age: 75.
ST3242 : Cox proportional hazards model 47/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
BMI, age
ST3242 : Cox proportional hazards model 48/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
BMI, age
ST3242 : Cox proportional hazards model 48/49

Reminder Maximising the partial likelihood Estimation in R Hypothesis testing Example
BMI, age
ST3242 : Cox proportional hazards model 49/49