Analysis of a Composite Endpoint with Missing Data in Components - - PowerPoint PPT Presentation

1

Analysis of a Composite Endpoint with Missing Data in Components

Hui Quan, Daowen Zhang, Ji Zhang & Laure Devlamynck

Sanofi-Aventis February 2007 Rutgers Biostatistics Day

2

Outline

• The use of composite endpoint
• The case of two components
• The general case
• Simulation Results
• Real data example
• Discussion

3

Composite Endpoint

Composite endpoints are often used in clinical trials

• GI outcomes study: perforation, ulcer and GI bleeding
• Major CV trials: MI, stroke and all-cause mortality
• VTE trials: CPMP PtC--DVT, PE and all-cause deaths

Classify a patient as having an event if the patient has events in any components

4

Composite Endpoint (2)

Advantages:

• To increase the overall event rate
• To reduce the size of the trial and achieve desired power
• To shorten the duration and get timely answers
• To avoid multiplicity issue

Conventional approach for analyzing composite endpoint:

• To directly analyze the composite endpoint
• A valid approach when no missing data in any components

As any study endpoints, components could have missing data.

5

Composite Endpoint (3)

A trial to compare two treatments on prevention of venous

thromboembolism in patients with knee surgery Primary endpoint– a composite endpoint

– DVT through venograph at Week 1 (30% missing) – Symptomatic DVT and/or PE (pulmonary embolism) – VTE related death

Two naïve approaches:

• To exclude patients with missing data in components
• To assume patients with missing data in a component to have no

event in the component

Both provide inconsistent estimates.

6

Composite Endpoint (4)

The new approach will:

• Use all available data – consistent with ITT principle
• Derive rates for all potential study outcomes
• Then combine these rates to obtain the rate for the

composite endpoint

• Use a likelihood-based approach
• Provide consistent estimate under the MAR assumption

for the components

7

Two Components

Two components: X and Y (=1 for event, =0 otherwise) When there are no missing data Composite endpoint: V (V=1 if either X=1 or Y=1 V=0 if X=0 and Y=0). Focus: Pr(V=1)=1- 1 X 1 Y

00

π

01

π

10

π

11

π

00

π

8

Two Components (2)

When there are missing data Scenarios 6 and 8: partial missing data Scenario 9: complete missing data

. . . 9 . . 8 1 1 . 7 . . 6 1 . 1 5 4 1 1 3 1 1 2 1 1 1 1 V Y X Scenario

9

Two Components (3) Naïve 1: directly analyze V after excluding missing data (Scenarios 6, 8 and 9) – Is valid only when missing data on V are MAR and MLE is used. – However, even when missing data on all components are MCAR, missing data on V may still not be MAR. – Since some observed X=0 and Y=0 are ignored in analysis, the derived rate for the composite endpoint will not be consistent with the true rate. Naïve 2: assume V=0 for Scenarios 6 and 8 => under estimates the true rate.

10

A new Approach for two Components Let (Xi,Yi), , i=1, …, n be complete data from n patients. Then ~ Multinomial (1; ) The Probability mass of the complete data ] , [ k Y j X I u

i i jk i

= = = ) , , , (

11 10 01 00 i i i i i

u u u u u =

11 10 01 00

, , , π π π π ∏ =

11 11 10 10 01 01 00 00

) , (

i u i u i u i u

u P π π π π π

11

A new Approach for two Components (2) In the case of missing data, define Likelihood

(Completely observed data) (X observed and Y missing) (X missing and Y observed)

] , [ k Y j X I n

i i jk

= = ∑ =

11 11 10 10 01 01 00 00

) (

n n n n

L π π π π π =

.] , [ . = = ∑ =

i i j

Y j X I n ] ., [ . k Y X I n

i i k

= = ∑ =

. 1 11 10 . 01 00

) ( ) (

n n

π π π π + + ×

1 . 11 01 . 10 00

) ( ) (

n n

π π π π + + ×

12

A new Approach for two Components (3)

• Direct iterative approach (Newton-Raphson) for MLE
• No closed form solution in general
• Fisher information for asymptotic variance of 1-

Alternatively, EM algorithm for MLE is pretty straightforward

2

ˆ σ

00

ˆ π

13

A new Approach for two Components (4)

14

Two Components – one with no missing data Let X be component with no missing data and Y be the

• ther component.

Pr(X=1 or Y=1)=Pr(X=1)+Pr(X=0 Y=1) =Pr(X=1)+Pr(Y=1|X=0)Pr(X=0) =n be the # of patients, = the # of (X=1) = the # of (X=0, Y .), = the # of (X=0, Y=1)

The MLE for the overall event rate for the composite endpoint:

1

n

1

m

2

n

2

m

≠

1 1 1 2 2 1 1 1 2 1

) ˆ 1 ( ˆ ˆ ˆ n m n n m n m p p p r − + = − + =

15

General Case

For the case of s components

• Calculate the probability uh of the hth outcomes based on π.
• Count then number nh of patients with the outcome
• Use (uh)nh as a factor in the likelihood

When s=4, there will be 24-1=15 independent π’s and 34-1=80 different factors in the likelihood --- too difficult to handle. Components with non-missing data can be combined to reduce the number of components. More details are given for the case

• f 3 components.

16

Three Components

Three binary components : X, Y and Z

• =1 for event and =0 for no event.
• Due to missing data, three outcomes for each

component

• A total of 27=3x3x3 possible outcomes (3k for k comps)

Composite endpoint V=1 if X=1 or Y=1 or Z=1,

V=0 if X=0 and Y=0 and Z=0, V=. otherwise.

As for the case of two components, there will be a problem if directly analyze V.

17

Three Components (2)

Define Similarly, the number of patients with each outcome. Likelihood for :

) , , Pr( l Z k Y j X

jkl

= = = = π

1

) , Pr( .) , , Pr( .

jk jk jk

k Y j X Z k Y j X π π π + = = = = = = = =

l j l j l j

l Z j X l Z Y j X

1

) , Pr( ) ., , Pr( . π π π + = = = = = = = =

kl kl kl

l Z k Y l Z k Y X

1

) , Pr( ) , ., Pr( . π π π + = = = = = = = = ) Pr( .. j X

j

= = π ) Pr( .. l Z

l

= = π ) Pr( . . k Y

k

= = π

∏ ∏ ∏ ∏ =

l n l k n k jk n jk jkl n jkl

G

.. .. . . . . . . ...

) ( π π π π π

) ,..., , , (

111 010 001 000

π π π π π =

18

Three Components (3) In generally, there is no closed form solution for MLE. Newton-Raphson or EM algorithm can be used to obtain and the corresponding asymptotic variance. When missing data pattern is monotonic A closed form solution for MLE exits.

000

ˆ 1 ˆ π − = r

19

Between-Treatment Comparison

20

Simulation –Two components Assume: The true overall event rate for composite endpoint: In addition, assume a MCAR for X and MAR for Y, (independent with j & k) (independent with k) (independent with k) ) 1 Pr( = = X px ) | 1 Pr( = = = X Y py ) 1 (

X Y X

p p p r − + = ) , | . Pr( k Y j X X pmx = = = = ) , | . Pr( k Y X Y pmY = = = = ) , 1 | . Pr(

1

k Y X Y pmY = = = = .) ., Pr( = = = Y X

21

Simulation Results for 2 components 1 9.79 9.80 11.34 8.31 2 9.79 9.77 13.08 7.60 3 14.50 14.49 16.72 12.36 4 14.50 14.60 19.32 11.44 5 23.50 23.61 26.76 19.53 6 23.50 23.31 28.14 18.89 7 28.00 28.04 31.01 22.60 8 28.00 27.97 33.01 22.36 Same r different missing data rates Case

2

ˆ

na

r

1

ˆ

na

r

mle

r ˆ r

22

Simulation: Between-Treatment Comparison 1 1.00 1.01 1.00 0.92 2 1.00 1.01 0.84 1.03 3 0.42 0.42 0.41 0.38 4 0.42 0.42 0.36 0.42 5 1.19 1.19 1.15 1.11 6 1.19 1.20 1.03 1.22 Same different missing data rates Case

2

ˆ

na

λ

1

ˆ

na

λ

mle

λ ˆ λ λ

23

Example

A study for comparing Fondaparinux and Enoxaparin on prevention of thromboembolic complications in patients with total knee replacement (Bauer et al NEJM 2001).

• Treatment Duration: 5 – 9 postoperative days
• Venograph: Day 5 to Day 11
• X: fatal or non-fatal PE – non-missing
• Y: DVT through venograph – 30% missing
• Primary endpoint: composite endpoint of X and Y.

24

Example (2)

25

Example (3)

) /( ) ( ˆ

. 11 . 1 10 01 1

n n n n n n r

na

− + + + = n n n n n r

na

/ ) ( ˆ

11 . 1 10 01 2

+ + + = n n n n n n n n n n n r

MLE

/ ) )( /( / ) ( ˆ

. 01 00 01 00 01 11 . 1 10

+ + + + + + =

2 1

ˆ ˆ ˆ

na MLE na

r r r ≥ ≥

26

Individual Components

After the demonstration of treatment effect on the composite endpoint, there may be the need to assess treatment effects on individual components. For the case of two components:

• for X component
• for Y component
• Joint model for all components not individual

model for individual components even under MAR

• No need for multiplicity

11 10

π π +

11 01

π π +

27

Individual Components (2)

For the case of three components: {X, Y, Z} {X, Y} and {X, Z} {X}

⇓ ⇓

28

Discussion

• Theoretically and based on simulation, the

conventional approaches of directly analyzing the composite endpoint provide inconsistent estimates.

• The new method provides consistent and

more efficient estimate under MAR for components.

• The new method uses all available data –

consistent with the ITT principle.

• Incorporation of covariates will be in future

research.

29

References

Bauer, K. A. et al. (2001). Fondaparinux compared with enoxaparin for the prevention of venous thromboembolism after elective major knee surgery, The New England Journal of Medicine, 345, 1305-1310. CPMP (2000). Points to consider on clinical investigation of medicinal products for prophylaxis of intra-and post-operative venous thromboemblic risk, http://www.emea.eu.int/pdfs/human/ewp/070798en.pdf. Kessler, K.M. Correspondence and DeMets, D. L. and Califf, R. M. Reponse (2003).Combining composite endpoints: counterintuitive or a mathematical impossible? Circulation,107: e70. Little, R.J. (1995). Modeling the drop-out mechanism in longitudinal studies, J. Am. Statistical Assoc. 90, 1112-1121. Quan, H., Zhang, D., Zhang, J. and Devlamynck, L. (2006). Analysis of a Composite Endpoint with Missing Data in Components. Technical Report, # 2. Rubin, D.B. (1976). Inference and missing data (with discussion), Biometrika, 63, 581-592. Sankoh, A.J., D’Agostino, R. B. and Huque, M. F. (2003). Efficacy endpoint selection and multiplicity adjustment methods in clinical trials with inherent multiple endpoint issues. Statistics in Medicine, 22: 3133-3150.