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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Cox proportional hazards model<br />
<strong>ST3242</strong>: Introduction to Survival Analysis<br />
Alex Cook<br />
September 2008<br />
<strong>ST3242</strong> : Cox proportional hazards model 1/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
<strong>ST3242</strong> : Cox proportional hazards model 3/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
<strong>ST3242</strong> : Cox proportional hazards model 4/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
<strong>ST3242</strong> : Cox proportional hazards model 5/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
<strong>ST3242</strong> : Cox proportional hazards model 6/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
NUS news<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Plan for today<br />
Go over mock mid-term test<br />
3pm feedback<br />
Lecture: confidence intervals for the Cox phm<br />
<strong>ST3242</strong> : Cox proportional hazards model 8/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Exam advice<br />
Read all questions at start of exam<br />
Read all instructions<br />
Ensure do everything asked and nothing more<br />
Write as neatly as you can<br />
Don’t “do a Ravi”<br />
Avoid rounding too much too soon<br />
<strong>ST3242</strong> : Cox proportional hazards model 9/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Exam advice<br />
INSTRUCTIONS TO CANDIDATES:<br />
1 This test contains EIGHT questions and comprises<br />
FOUR printed pages.<br />
2 Answer ALL questions for a total of 100 marks.<br />
3 This is a closed-book test; only non-programmable<br />
calculators are allowed.<br />
<strong>ST3242</strong> : Cox proportional hazards model 10/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q1: [3]<br />
If the survival function is S(t), what is the hazard function<br />
h(t)?<br />
<strong>ST3242</strong> : Cox proportional hazards model 11/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q2: [6]<br />
If the hazard function is h(t) = θt for a parameter θ > 0,<br />
what are the survival and density functions?<br />
<strong>ST3242</strong> : Cox proportional hazards model 12/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q3: [10]<br />
If survival times in the absence of censoring are distributed<br />
according to a log-logistic distribution with parameters κ and<br />
λ, the hazard and survival functions are<br />
h(t) = λκtκ−1<br />
1 + λtκ S(t) =<br />
1<br />
1 + λtκ respectively. If we have observed data of the form (ti, δi),<br />
where δi = 1 if individual i fails at time ti and δi = 0 if i is<br />
right-censored at ti, for i = 1, . . . , m, what is the<br />
log-likelihood function?<br />
<strong>ST3242</strong> : Cox proportional hazards model 13/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q4: [6]<br />
Consider a study into the length of time people can survive<br />
without purchasing or receiving plastic goods. The study<br />
starts at time Tstart and ends at time Tend. Three nus<br />
students are among the participants.<br />
Student A is recruited to the study at time Tstart. She<br />
buys no plastic until time tA < Tend, when she buys a<br />
bottle of Pocari Sweat.<br />
For each student, say if the datum is uncensored,<br />
right-censored, left-censored or interval-censored.<br />
<strong>ST3242</strong> : Cox proportional hazards model 14/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q4: [6]<br />
Consider a study into the length of time people can survive<br />
without purchasing or receiving plastic goods. The study<br />
starts at time Tstart and ends at time Tend. Three nus<br />
students are among the participants.<br />
Student B is recruited to the study at time Tstart. He<br />
reports to the organisers at time tB < Tend that he has<br />
bought no plastics, but they never hear from him again.<br />
For each student, say if the datum is uncensored,<br />
right-censored, left-censored or interval-censored.<br />
<strong>ST3242</strong> : Cox proportional hazards model 15/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q4: [6]<br />
Consider a study into the length of time people can survive<br />
without purchasing or receiving plastic goods. The study<br />
starts at time Tstart and ends at time Tend. Three nus<br />
students are among the participants.<br />
Student C is recruited to the study at time<br />
tC ∈ (Tstart, Tend). She buys no plastic for the remainder<br />
of the study.<br />
For each student, say if the datum is uncensored,<br />
right-censored, left-censored or interval-censored.<br />
<strong>ST3242</strong> : Cox proportional hazards model 16/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q5: [12]<br />
Use the delta method to approximate the mean and variance<br />
of √ X , where X is a random variable with mean µ and<br />
variance σ 2 . When do you expect these will provide a good<br />
approximation to the true mean and variance of √ X ?<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q6(i): [36]<br />
Calculate and sketch the Kaplan–Meier estimate of S(t) for<br />
the following data, where δi = 1 if individual i died at time ti<br />
and δi = 0 if i was censored at that time.<br />
i ti δi<br />
1 0.6 1<br />
2 0.9 1<br />
3 1.1 0<br />
4 1.5 1<br />
5 2.0 0<br />
6 2.0 0<br />
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Q6(ii): [36]<br />
Recall that<br />
<br />
ˆV{ˆS(t)} = ˆS(t)<br />
2<br />
d(i)<br />
n(i<br />
t (i)≤t<br />
− )(n(i − ) − d(i))<br />
where t(i) is the ith ordered event time, d(i) is the number of<br />
deaths at that time and n(i − ) is the number at risk an instant<br />
before that time.<br />
Construct the “naïve” 95% confidence interval described in<br />
the lecture notes for S(1), the probability of surviving at least<br />
to one time unit. What is the probability the true value of<br />
S(1) lies within your confidence interval?<br />
<strong>ST3242</strong> : Cox proportional hazards model 19/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q7: [12]<br />
The following are data on the times to recurrence of breast<br />
cancer following treatment in German women. These data<br />
have been grouped according to whether the patients have<br />
undergone the menopause (g = 2) or not (g = 1); t(i) is the<br />
time in years until the ith unique recurrence time, ng,(i − ) is the<br />
total number at risk in group g an instant before time t(i),<br />
dg,(i) is the number in group g that had a recurrence at time<br />
t(i), while n(i − ) = n1,(i − ) + n2,(i − ) and d(i) = d1,(i) + d2,(i). The<br />
columns marked êg,(i) and ˆvg,(i) are the expected value and<br />
variance of dg,(i) under the hypothesis that menopause is<br />
independent of recurrence, respectively. Note that for your<br />
convenience, the data have been grouped into years. [To find<br />
out more about these data, see Hosmer et al., 2008.]<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q7: [12]<br />
i t (i) n 1,(i − ) n 2,(i − ) n (i − ) d 1,(i) d 2,(i) d (i) ê 1,(i) ê 2,(i) ˆv 1,(i)<br />
1 1 290 396 686 29 27 56 23.7 32.3 12.6<br />
2 2 245 357 602 44 65 109 44.4 64.6 21.6<br />
3 3 183 275 458 20 39 59 23.6 35.4 12.4<br />
4 4 140 191 331 16 23 39 16.5 22.5 8.4<br />
5 5 98 130 228 4 18 22 9.5 12.5 4.9<br />
6 6 56 65 121 6 5 11 5.1 5.9 2.5<br />
7 7 14 22 36 0 3 3 1.2 1.8 0.7<br />
Test the hypothesis that the menopause is independent of<br />
breast cancer recurrence.<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q8(i): [15]<br />
Given that the Cox proportional hazards model for a single<br />
continuous covariate xi is<br />
h(t, xi) = h0(t, α) exp{βxi}<br />
for individual i, what is the corresponding survival function<br />
incorporating this covariate?<br />
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Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Q8(ii): [15]<br />
What is the hazard ratio comparing an individual with<br />
covariate x1 to one with covariate x2 at time t?<br />
<strong>ST3242</strong> : Cox proportional hazards model 23/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
So far. . .<br />
Saw an equation for asymptotic distribution of ˆβ:<br />
ˆβ ∼ N(β, E{I(β)} −1 )<br />
Can replace by observed information:<br />
ˆβ ∼ N(β, I(ˆβ) −1 )<br />
R gives us standard errors for the individual βs<br />
<strong>ST3242</strong> : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
So far. . .<br />
Saw an equation for asymptotic distribution of ˆβ:<br />
ˆβ ∼ N(β, E{I(β)} −1 )<br />
Can replace by observed information:<br />
ˆβ ∼ N(β, I(ˆβ) −1 )<br />
R gives us standard errors for the individual βs<br />
<strong>ST3242</strong> : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
So far. . .<br />
Saw an equation for asymptotic distribution of ˆβ:<br />
ˆβ ∼ N(β, E{I(β)} −1 )<br />
Can replace by observed information:<br />
ˆβ ∼ N(β, I(ˆβ) −1 )<br />
R gives us standard errors for the individual βs<br />
<strong>ST3242</strong> : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
<strong>ST3242</strong> : Cox proportional hazards model 25/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
We want more!<br />
CIs for β<br />
CIs for hazard ratio between two categories<br />
CIS for hazard ratio between two individuals<br />
<strong>ST3242</strong> : Cox proportional hazards model 26/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β<br />
The asymptotic distribution of β can be estimated using the<br />
observed information<br />
ˆβ ∼ N(β, I( ˆ β) −1 )<br />
If β is vector of length p, I( ˆ β) is p × p matrix<br />
i, j element is<br />
− d2 lp(ˆβ)<br />
dβi dβj<br />
<strong>ST3242</strong> : Cox proportional hazards model 27/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
where<br />
⎛<br />
⎝<br />
Example<br />
⎛<br />
⎝<br />
ˆβ1<br />
ˆβ2<br />
ˆβ3<br />
⎞<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
⎛⎛<br />
⎠ ∼ N ⎝⎝<br />
⎞<br />
⎠ =<br />
⎛<br />
⎜<br />
⎝<br />
β1<br />
β2<br />
β3<br />
⎞<br />
⎛<br />
⎠ , ⎝<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
⎞⎞<br />
⎠⎠<br />
− d2 lp(ˆβ)<br />
dβ1 dβ1 − d2 lp( ˆ β)<br />
dβ1 dβ2 − d2 lp( ˆ β)<br />
dβ1 dβ3<br />
− d2 lp(ˆβ)<br />
dβ2 dβ1 − d2 lp(ˆβ)<br />
dβ2 dβ2 − d2 lp(ˆβ)<br />
dβ2 dβ3<br />
− d2 lp( ˆ β)<br />
dβ3 dβ1 − d2 lp( ˆ β)<br />
dβ3 dβ2 − d2 lp( ˆ β)<br />
dβ3 dβ3<br />
<strong>ST3242</strong> : Cox proportional hazards model 28/44<br />
⎞<br />
⎟<br />
⎠<br />
−1
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
If ⎛<br />
⎝<br />
ˆβ1<br />
ˆβ2<br />
ˆβ3<br />
⎞<br />
⎛⎛<br />
⎠ ∼ N ⎝⎝<br />
β1<br />
β2<br />
β3<br />
⎞<br />
the marginal distribution of ˆβ1 is<br />
R<br />
gives √ σii as standard output!<br />
⎛<br />
⎠ , ⎝<br />
ˆβ1 ∼ N(β1, σ11)<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
⎞⎞<br />
⎠⎠<br />
<strong>ST3242</strong> : Cox proportional hazards model 29/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
If ⎛<br />
⎝<br />
ˆβ1<br />
ˆβ2<br />
ˆβ3<br />
⎞<br />
⎛⎛<br />
⎠ ∼ N ⎝⎝<br />
β1<br />
β2<br />
β3<br />
⎞<br />
the marginal distribution of ˆβ1 is<br />
R<br />
gives √ σii as standard output!<br />
⎛<br />
⎠ , ⎝<br />
ˆβ1 ∼ N(β1, σ11)<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
⎞⎞<br />
⎠⎠<br />
<strong>ST3242</strong> : Cox proportional hazards model 29/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for β<br />
<br />
p β1 − 1.96 √ σ11 < ˆ β1 < β1 + 1.96 √ <br />
σ11 = 0.95<br />
<strong>ST3242</strong> : Cox proportional hazards model 30/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for β<br />
<br />
p ˆβ1 − 1.96 √ σ11 < β1 < ˆ β1 + 1.96 √ <br />
σ11 = 0.95<br />
<strong>ST3242</strong> : Cox proportional hazards model 31/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
A 95% confidence interval for βbmi is<br />
(−0.0985 − 1.96 × 0.0148, −0.0985 + 1.96 × 0.0148)<br />
= (−0.13, −0.07)<br />
<strong>ST3242</strong> : Cox proportional hazards model 32/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for g(β)<br />
If (a, b) is a confidence interval for β,<br />
(e a , e b ) is a confidence interval for e β<br />
(e ax , e bx ) is a confidence interval for e βx<br />
What about a CI for e β1−β2 ?<br />
<strong>ST3242</strong> : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for g(β)<br />
If (a, b) is a confidence interval for β,<br />
(e a , e b ) is a confidence interval for e β<br />
(e ax , e bx ) is a confidence interval for e βx<br />
What about a CI for e β1−β2 ?<br />
<strong>ST3242</strong> : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for g(β)<br />
If (a, b) is a confidence interval for β,<br />
(e a , e b ) is a confidence interval for e β<br />
(e ax , e bx ) is a confidence interval for e βx<br />
What about a CI for e β1−β2 ?<br />
<strong>ST3242</strong> : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Confidence interval for g(β)<br />
If (a, b) is a confidence interval for β,<br />
(e a , e b ) is a confidence interval for e β<br />
(e ax , e bx ) is a confidence interval for e βx<br />
What about a CI for e β1−β2 ?<br />
<strong>ST3242</strong> : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
h(t, xla, xad, xsm, xsq) = h0(t) exp(βlaxla+βadxad+βsmxsm+βsqxsq)<br />
where xc = 1 if the individual has cell type c and xc = 0 if<br />
not. Output relative to one category (adeno) as baseline<br />
<strong>ST3242</strong> : Cox proportional hazards model 34/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
h(t, xla, xad, xsm, xsq) = h0(t) exp(βlaxla+βadxad+βsmxsm+βsqxsq)<br />
where xc = 1 if the individual has cell type c and xc = 0 if<br />
not. Output relative to one category (adeno) as baseline<br />
<strong>ST3242</strong> : Cox proportional hazards model 34/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Multivariate normal distribution<br />
If x ∼ N(µ, Σ) then any subset of the rows of x also has a<br />
Normal distribution with corresponding rows of µ and rows<br />
and columns of Σ.<br />
For example<br />
⎛ ⎞ ⎛⎛<br />
ˆβ1<br />
⎝ ˆβ2<br />
⎠ ∼ N ⎝⎝<br />
ˆβ3<br />
<br />
ˆβ1<br />
⇒ ∼ N<br />
ˆβ3<br />
β1<br />
β2<br />
β3<br />
β1<br />
β3<br />
⎞<br />
⎛<br />
⎠ , ⎝<br />
<br />
,<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
<br />
σ11 σ13<br />
σ31 σ33<br />
<br />
⎞⎞<br />
⎠⎠<br />
<strong>ST3242</strong> : Cox proportional hazards model 35/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Multivariate normal distribution<br />
If x ∼ N(µ, Σ) then any subset of the rows of x also has a<br />
Normal distribution with corresponding rows of µ and rows<br />
and columns of Σ.<br />
For example<br />
⎛ ⎞ ⎛⎛<br />
ˆβ1<br />
⎝ ˆβ2<br />
⎠ ∼ N ⎝⎝<br />
ˆβ3<br />
<br />
ˆβ1<br />
⇒ ∼ N<br />
ˆβ3<br />
β1<br />
β2<br />
β3<br />
β1<br />
β3<br />
⎞<br />
⎛<br />
⎠ , ⎝<br />
<br />
,<br />
σ11 σ12 σ13<br />
σ21 σ22 σ23<br />
σ31 σ32 σ33<br />
<br />
σ11 σ13<br />
σ31 σ33<br />
<br />
⎞⎞<br />
⎠⎠<br />
<strong>ST3242</strong> : Cox proportional hazards model 35/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Q: what is V(X + Y )?<br />
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent<br />
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X , Y )<br />
Q: what is V(X − Y )?<br />
<strong>ST3242</strong> : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Q: what is V(X + Y )?<br />
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent<br />
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X , Y )<br />
Q: what is V(X − Y )?<br />
<strong>ST3242</strong> : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Q: what is V(X + Y )?<br />
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent<br />
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X , Y )<br />
Q: what is V(X − Y )?<br />
<strong>ST3242</strong> : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Q: what is V(X + Y )?<br />
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent<br />
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X , Y )<br />
Q: what is V(X − Y )?<br />
<strong>ST3242</strong> : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V(X − Y )?<br />
A:<br />
V(X − Y ) = V(X ) + V(−Y ) + 2C(X , −Y )<br />
= V(X ) + V(Y ) − 2C(X , Y )<br />
<strong>ST3242</strong> : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V(X − Y )?<br />
A:<br />
V(X − Y ) = V(X ) + V(−Y ) + 2C(X , −Y )<br />
= V(X ) + V(Y ) − 2C(X , Y )<br />
<strong>ST3242</strong> : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V(X − Y )?<br />
A:<br />
V(X − Y ) = V(X ) + V(−Y ) + 2C(X , −Y )<br />
= V(X ) + V(Y ) − 2C(X , Y )<br />
<strong>ST3242</strong> : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V(X − Y )?<br />
A:<br />
V(X − Y ) = V(X ) + V(−Y ) + 2C(X , −Y )<br />
= V(X ) + V(Y ) − 2C(X , Y )<br />
<strong>ST3242</strong> : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V(X − Y )?<br />
A:<br />
V(X − Y ) = V(X ) + V(−Y ) + 2C(X , −Y )<br />
= V(X ) + V(Y ) − 2C(X , Y )<br />
<strong>ST3242</strong> : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Properties of Cov<br />
Properties of the covariance<br />
C(X , Y ) = C(Y , X )<br />
C(X , X ) = V(X )<br />
C(aX + b, cY + d) = acC(X , Y )<br />
Q: what is V( ˆ βi − ˆ βj)?<br />
A:<br />
V( ˆ βi − ˆ βj) = V( ˆ βi) + V(− ˆ βj) + 2C( ˆ βi, − ˆ βj)<br />
= V( ˆ βi) + V( ˆ βj) − 2C( ˆ βi, ˆ βj)<br />
<strong>ST3242</strong> : Cox proportional hazards model 38/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for βi − βj<br />
A 95% CI for βi − βj is<br />
ˆβi − ˆ βj ± 1.96 σii + σjj − 2σij<br />
R does it for us!<br />
The coxph functions silently outputs var, the<br />
variance–covariance matrix<br />
phm.cell=coxph(Surv(t,delta)∼factor(Cell))<br />
phm.cellvar<br />
phm.cellcoefficients<br />
<strong>ST3242</strong> : Cox proportional hazards model 39/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for βi − βj<br />
A 95% CI for βi − βj is<br />
ˆβi − ˆ βj ± 1.96 σii + σjj − 2σij<br />
R does it for us!<br />
The coxph functions silently outputs var, the<br />
variance–covariance matrix<br />
phm.cell=coxph(Surv(t,delta)∼factor(Cell))<br />
phm.cellvar<br />
phm.cellcoefficients<br />
<strong>ST3242</strong> : Cox proportional hazards model 39/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
<strong>ST3242</strong> : Cox proportional hazards model 40/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
95%CI for βsq − βla is<br />
−0.77 − 1.00 ± 1.96 √ 0.0642 + 0.0638 − 2 × 0.0256<br />
i.e. (−0.31, 0.77)<br />
A 95% CI for the hazard ratio comparing squamous to large<br />
cells is<br />
(e −0.31 , e 0.77 ) = (0.73, 2.17)<br />
<strong>ST3242</strong> : Cox proportional hazards model 41/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for β, hazard ratios<br />
We want more!<br />
CIs for β<br />
CIs for hazard ratio between two categories<br />
CIS for hazard ratio between two individuals<br />
<strong>ST3242</strong> : Cox proportional hazards model 42/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
CIs for hazard ratios<br />
Suppose we want a hazard ratio comparing individuals with x a<br />
and x b<br />
squamous cell, performance of 60, aged 70, with no prior<br />
therapy<br />
large cell, performance of 20, aged 50, with no prior<br />
therapy<br />
Same approach<br />
obtain ˆ β and covariance matrix via coxph<br />
evaluate ˆ β T<br />
xa and ˆ β T<br />
xb evaluate V( ˆ β T<br />
x a − ˆ β T<br />
x b )<br />
95% CI is ˆβ T<br />
x a − ˆβ T<br />
x b ±<br />
<br />
V(ˆβ T<br />
x a − ˆβ T<br />
x b )<br />
<strong>ST3242</strong> : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov<br />
Example<br />
x a = (x a 1 , x a 2 ) and x b = (x b 1 , x b 2 ) Variance of ˆ βx a − ˆ βx b is<br />
V( ˆ βx a − ˆ βx b ) = V{ ˆ β1(x a 1 − x b 1 ) + ˆ β2(x a 2 − x b 2 )}<br />
= (x a 1 − x b 1 ) 2 V( ˆ β1) + (x a 2 − x b 2 ) 2 V( ˆ β2)<br />
+2C{ ˆ β1(x a 1 − x b 1 ), ˆ β2(x a 2 − x b 2 )}<br />
= (x a 1 − x b 1 ) 2 V( ˆ β1) + (x a 2 − x b 2 ) 2 V( ˆ β2)<br />
+2(x a 1 − x b 1 )(x a 2 − x b 2 )C{ ˆβ1, ˆβ2}<br />
<strong>ST3242</strong> : Cox proportional hazards model 44/44