Loading...

PPT – Compliance and Causal Analysis Lecture 2 Randomisationbased methods 1: likelihood estimation PowerPoint presentation | free to view - id: 1b2271-YWI3N

The Adobe Flash plugin is needed to view this content

Compliance and Causal Analysis Lecture 2

Randomisation-based methods 1 likelihood

estimation

- Ian White
- MRC Biostatistics Unit, Cambridge, UK
- ian.white_at_mrc-bsu.cam.ac.uk

Causal analysis in randomised trials

- A randomised controlled trial is used to evaluate

an intervention - No confounding because randomised group R is

independent of all predictors of outcome - But we get departures from randomised

intervention actual treatment D differs from

allocated treatment R - Want to infer causal effect of treatment D

Departures from randomised intervention

- Sometimes called non-compliance
- I prefer departures
- avoids implicit value judgements
- more precise includes both
- non-adherence (randomised to X, clinician

prescribes X, patient does Y) - changes in prescribed treatment (randomised to

X, clinician prescribes Y, patient does Y)

Types of departure from randomised intervention

- Switches to other trial treatment
- Changes to non-trial treatment
- Includes changes to nothing in a comparative

trial - Departures may be
- all-or-nothing (either always get A or always get

B) - or quantitative (e.g. dose changes)
- or time-dependent (e.g. emergency operation)

? FOCUS ON SWITCHES

Motivating example MASS trial

- Abdominal aortic aneurysms are often fatal if

they rupture - May be repaired if detected before rupture
- Reliably detected by ultrasound screening
- MASS trial (Lancet, 2002) 67,800 men were

randomised to invitation to screening or control - Main outcome was aneurysm-related mortality

MASS trial Aneurysm-related mortality

Intention-to-treat analysis

- Intention-to-treat analysis compares groups as

randomised, ignoring any departures - respects the randomisation
- avoids selection bias
- Now the standard analysis and rightly so
- Answers an important pragmatic question e.g. what

is the public health impact of prescribing X? - Disadvantage this may be the wrong question!

Disadvantage of ITT

- Doctor doctor, how much will taking this tablet

reduce my risk of heart disease? - I dont know, but prescribing it reduces

disease risk by 10 - on average
- thats on average over whether you take it or not

Example MASS trial (ctd)

- Intention-to-treat (ITT) analysis invitation

to screening reduced aneurysm-related death - hazard ratio 0.58 (95 CI, 0.42 to 0.78), P0.002
- 20 of invited group didnt attend for screening

(all-or-nothing non-compliance) - ITT measures the average benefit of screening in

invitees - What is the benefit of screening in attenders?

Plan for this lecture

- Basic idea for all-or-nothing compliance
- Binary outcome estimation
- Normal outcome
- Simple vs. ML estimation
- Negative weights method
- Back door method

1. Basic idea for all-or-nothing compliance

Notation

- Randomise to E (experimental) or S (standard)

treatment - S could be nothing / placebo
- Potential treatments are E and S assume everyone

gets either all E or all S (all-or-nothing

compliance) - Ri 1/0 indicates randomisation of ith

individual to E/S - Di 1/0 indicates receipt of E/S

Theoretical framework

- Ri 1/0 indicates randomisation of ith

individual to E/S - Di 1/0 indicates actual receipt of E/S
- Di(0) 1/0 indicates receipt of E/S if

randomised to S observed in S arm,

counterfactual in E arm - Di(1) 1/0 indicates receipt of E/S if

randomised to E observed in E arm,

counterfactual in S arm - So actual Di Di(Ri)
- The pair Di(0), Di(1) define an individuals

compliance type.

Compliance types

- Always-takers (A) Di(0) Di(1) 1 always take

E regardless of randomisation - Compliers (C) Di(0) 0, Di(1) 1 take whatever

they were allocated to - Never-takers (N) Di(0) Di(1) 0 never take E

regardless of randomisation - Defiers (D) Di(0) 1, Di(1) 0 take the

opposite of what they were allocated to

Using compliance types

- Note that compliance-type is a characteristic

that is inherent to the individual before

randomisation - unlike actual compliance, which is affected by

randomisation - This means we can meaningfully adjust/stratify by

compliance-type, but not by actual compliance - But unfortunately compliance-type is incompletely

observed requires careful statistical methods - Compliance types are an example of Principal

strata (Frangakis Rubin, 2002)

What do we observe about compliance-types?

What do we observe about compliance-types?

Simplification if S arm cant get E (no

always-takers and no defiers)

Defining true treatment effect

- Could be defined in various ways
- Use potential outcomes Y(r,d) outcome if

randomised to r and received d - Y(r,d) may depend on r
- e.g. never-takers of a counselling intervention

might do worse in the E arm (where they refused

the counselling) than in the S arm (where they

werent offered the counselling) - but in many settings it wont (exclusion

restriction)

Defining true treatment effect

- Average effect of treatment
- EY(1,1)-Y(0,0)
- Average effect of treatment among the treated
- EY(1,1)-Y(0,0) D(1)1
- For an always-taker, we will observe
- Y(0,1) if we randomise to S
- Y(1,1) if we randomise to E
- but we cannot observe Y(0,0)
- Is there a measure of causal effect that involves

potentially observable outcomes?

Complier average causal effect (CACE)

- ITT effect EY(1,D(1)) Y(0,D(0))
- Define average causal effect of treatment

assignment in compliance-type t as EY(1,D(1))

Y(0,D(0)) Tt - Complier Average Causal Effect (CACE)
- EY(1,1) Y(0,0) TC
- also Local Average Treatment Effect (LATE)

(Angrist et al, 1996) - Never-taker average causal effect (NACE?)
- EY(1,0) Y(0,0) TN
- Always-taker average causal effect (AACE?)
- EY(1,1) Y(0,1) TA
- CACE measures treatment efficacy but only

involves potentially observable outcomes (Imbens

Rubin, 1997)

CACE (2)

- Ignore defiers (mainly for simplicity)
- Can write ITT wC CACE wN NACE wA AACE
- where wC P(compliance-type C) etc.
- Its often reasonable to assume NACEAACE0, so

that ITT wC CACE - Give a simple estimate of the CACE

CACE (3)

- For binary outcomes, we can define the CACE on

different scales. - Risk difference scale
- CACE EY(1,1) TC EY(0,0) TC
- Risk ratio scale
- CACE EY(1,1) TC / EY(0,0) TC
- Odds ratio scale
- CACE OY(1,1) TC / OY(0,0) TC
- where OXEX/(1-EX)

2. Estimation

Vitamin A trial (Sommer and Zeger, 1991)

- Vitamin A vs. control in Indonesian children
- randomise villages (24000 children)
- outcome is mortality
- about 20 of villages didnt get their vitamin A

supply - this analysis ignores clustering by village

ITT analysis

ITT odds ratio ? 46/77 0.60

Observed compliance (Vitamin A arm)

Inferred compliance (Control arm)

CACE odds ratio ? 12/43 0.28

Estimation method of subtraction

nNE

if nE nS

Formally

Hence write down and maximise (log-)likelihood. No

te 1-w, w previous wC, wN

MLE

In this simple case, pCE is estimated as

(dCE/nCE) pCS is estimated as (dS dNE

nS/nE) / (nS nNE nS/nE) provided both

terms 0 (Ill assume this is true)

MLE

- Use estimates of pCE and pCS to estimate CACE
- as pCE-pCS on RD scale
- as pCE/pCS on RR scale
- as pCE/(1-pCE) / pCS /(1-pCS) on OR scale
- On RD scale only, can show that MLEs obey CACE

ITT / (1-w)

CACE compared with other quantities

Vitamin A vs. control summary

- ITT 0.38 vs. 0.64, RR0.60
- CACE 0.12 vs. 0.45, RR0.28
- Per-protocol 0.12 vs. 0.64, RR0.19
- As-treated 0.12 vs. 0.77, RR0.16
- On-treatment and as-treated are too extreme

because of strong selection effect 1.41

(untreated in treatment arm) vs. 0.64 (untreated

in control arm)

Comparison of assumptions

- Per-protocol and as-treated analyses assume

random non-compliance - no association between compliance-type and

outcome, once treatment effect is taken into

account - CACE analysis assumes exclusion restriction
- randomisation doesnt affect mean outcome for

never-takers and always-takers - no assumption of comparability of different

compliance-types - usually much more plausible

Extensions to CACE model (1)

- Above we had all the S arm getting S
- Easy to allow for S arm possibly getting E

(Cuzick et al, 1997) - CACE is again estimable under
- 2 exclusion restrictions NACEAACE0
- Either no defiers or same causal effect in

defiers as in compliers (DACE-CACE)

Extensions to CACE model (2)

- Introduce covariates
- Covariates that predict Y improve precision (as

in ITT analysis) - Covariates that predict D
- also improve precision (unlike in ITT analysis)

(Jo, 2002) - enable estimation of NACE etc. as well as CACE

(Hirano et al, 2000)

Extensions to CACE model (3)

- Define g difference in outcome between

never-takers and compliers after allowing for

their differences in actual treatment - selection effect
- difference in counterfactual outcomes
- As-treated analysis assumes g0 (random

non-compliance) - CACE and ITT analyses make no assumption about g
- Could estimate g from data leads to CACE

analysis - Instead, introduce appropriate prior information

about g (White, 2005)

Model for log odds of death (observed risk)

Use informative prior gN(0,s2) for various

values of s

Vitamin A trial Bayesian CACE analyses

as-treated

CACE

3. Normal outcome

Normal outcome

- Can simply modify the method of subtraction

work with means instead of proportions - Link to instrumental variables (IV) method

(lecture 5)

Model

CACE mCE-mCS

Likelihood

CACE mCE-mCS

ML estimation

- No closed form solution
- EM algorithm is easy compliance type as the

missing data - usual problems in estimating standard errors
- Newton-Raphson also fairly straightforward
- No directly available software in Stata, but

gllamm can be used see lecture 6.

4. Method comparisons

Comparison of CACE estimators

- Weve looked at
- Simple estimation using ITT wC CACE (for

difference of means, not risk ratio or odds

ratio) - Method of subtraction
- Maximum likelihood
- For binary outcomes, they all give the same

answer - For continuous outcomes, ML estimation is

different (potentially more efficient see later)

Simple CACE vs. ITT

- They estimate different parameters
- But they test the same null hypothesis
- Significance levels are equal
- obvious from CACE ITT / (1-w)
- binary case explored in detail by Branson and

Whitehead (2003) significance levels are equal

when likelihood ratio test is performed

Vitamin A trial profile likelihoods

Likelihood ratio test has same value for both

models

.. and quadratic approximation (dotted)

Likelihood ratio test has same value for both

models but Wald test doesnt.

ML estimation of CACE vs. ITT

- For binary outcome, significance levels are equal
- For Normal outcome, significance levels arent

equal - CACE is more efficient whenever theres a

non-zero selection effect or a non-zero treatment

effect - the next slides are thanks to Taeko Becque

Asymptotic relative efficiency of CACE vs. ITT

(approximate)

q1 CACE, q2 selection effect, outcome SD 1

Power of CACE and ITT analyses

Compliance rate 50 Selection

effect-0.5 CACE-0.5 Standard deviation1.5

Power and compliance rate

Sample size 300 Selection effect-0.5 CACE-0.5

Standard deviation1.5

Power including covariate weak predictor of

compliance

Power including covariate strong predictor of

compliance

Summary

5. Negative weights method (Kim and White, 2004)

Negative weights method

- For simplicity take nEnS
- Recall that we subtracted the number of

events/people in the NE cell (never-takers

randomised to E) from the number of events/people

in the S cell (all randomised to S) - Can also achieve this by including them in the S

arm but with a weight -1 - nS/nE in general

Negative weights in Stata

- Unfortunately many Stata commands are too

sensible to allow negative weights - Exceptions are regress, logistic, cox
- Illustration uses MASS data pretending outcome is

binary

- . use mass, clear
- . l
- rand screen event n
- 1 1 0 27104
- 1 1 1 43
- 1 0 0 6670
- 1 0 1 22
- 0 0 0 33848
- 0 0 1 113
- . tab rand screen fwn, sum(event) mean
- Means and Number of Observations of

AAA-related death? - Invited to Screened?
- screening? 0 1 Total
- -------------------------------------------
- 0 .00332735 . .00332735

- . CACE via negative weights
- . tab rand fwn
- Invited to
- screening? Freq. Percent Cum.
- -----------------------------------------------
- 0 33,961 50.09 50.09
- 1 33,839 49.91 100.00
- -----------------------------------------------
- Total 67,800 100.00
- . gen w 1
- . replace w -33961/33839 if randgtscreen
- (2 real changes made)
- . l
- rand screen event n w

- . logistic event rand fwn, coef ITT analysis
- Logistic regression Number

of obs 67800 - LR

chi2(1) 12.97 - Prob gt

chi2 0.0003 - Log likelihood -1229.0537 Pseudo

R2 0.0052 - --------------------------------------------------

--------------------- - event Coef. Std. Err. z Pgtz

95 Conf. Interval - -------------------------------------------------

--------------------- - rand -.5508119 .1558631 -3.53 0.000

-.856298 -.2453258 - _cons -5.702247 .094229 -60.51 0.000

-5.886933 -5.517562 - --------------------------------------------------

--------------------- - . logistic event screen iwnw, coef CACE

analysis - --------------------------------------------------

--------------------- - event Coef. Std. Err. z Pgtz

95 Conf. Interval - -------------------------------------------------

---------------------

Bootstrap standard error

- Now we bundle the previous commands into a file

negwt.ado - To speed things up, I use only 1/30 of the

controls

- . . negwt
- Logistic regression

Number of obs 1210 - LR

chi2(1) 15.43 - Prob

gt chi2 0.0001 - Log likelihood -410.28453

Pseudo R2 0.0185 - --------------------------------------------------

---------------------- - event Coef. Std. Err. z Pgtz

95 Conf. Interval - -------------------------------------------------

---------------------- - screen -.7461635 .1952982 -3.82 0.000

-1.128941 -.363386 - _cons -1.787902 .1141955 -15.66 0.000

-2.011721 -1.564083 - --------------------------------------------------

---------------------- - . bootstrap _bscreen, reps(1000) negwt
- --------------------------------------------------

---------------------- - Var Reps Observed Bias Std. Err.

95 Conf. Interval - -------------------------------------------------

----------------------

Negative weights summary

- Agrees exactly with direct method in this simple

case - Easy to generalise e.g. to situations with

covariates (weights would have to depend on

covariates) - Naïve standard errors are too small bootstrap

needed - Formal rationale is via unbiased estimating

equations (Abadie 2002)

6. Back-door method (Nagelkerke 2000)

Back door method

- Idea would like to regress Y on D, adjusting for

U - Its enough to adjust for E that blocks every

indirect path from D to Y (back door criterion) - Approximate E by the residual from a linear

regression of D on R

Properties

- Nagelkerke et al showed that the method agrees

exactly with the instrumental variables method

for a linear model - For non-linear models it only approximately

agrees with the method of subtraction - Not clear whether standard errors are adequate or

whether bootstrapping is needed - Easy to generalise to more complex settings

Back door method for MASS

- . reg screen rand fwn
- Source SS df MS

Number of obs 67800 - -------------------------------------------

F( 1, 67798) . - Model 10908.799 1 10908.799

Prob gt F 0.0000 - Residual 5368.59021 67798 .079185082

R-squared 0.6702 - -------------------------------------------

Adj R-squared 0.6702 - Total 16277.3892 67799 .240083028

Root MSE .2814 - --------------------------------------------------

---------------------------- - screen Coef. Std. Err. t

Pgtt 95 Conf. Interval - -------------------------------------------------

---------------------------- - rand .80224 .0021614 371.16

0.000 .7980037 .8064764 - _cons 5.55e-17 .001527 0.00

1.000 -.0029929 .0029929 - --------------------------------------------------

---------------------------- - . predict E, residual
- . tab rand E fwn

Back door method results

- . logistic event screen E fwn
- Logistic regression

Number of obs 67800 - LR

chi2(2) 20.04 - Prob

gt chi2 0.0000 - Log likelihood -1225.5162

Pseudo R2 0.0081 - --------------------------------------------------

---------------------- - event Odds Ratio Std. Err. z Pgtz

95 Conf. Interval - -------------------------------------------------

---------------------- - screen .4738008 .094594 -3.74 0.000

.3203717 .7007087 - E 1.015178 .295373 0.05 0.959

.5739573 1.795582 - --------------------------------------------------

---------------------- - Note tiny effect of E no evidence of selection

in these data

Method comparison

- . logistic event rand fwn ITT analysis
- --------------------------------------------------

---------------------------- - event Odds Ratio Std. Err. z

Pgtz 95 Conf. Interval - -------------------------------------------------

---------------------------- - rand .5764816 .0898522 -3.53

0.000 .4247315 .7824496 - --------------------------------------------------

---------------------------- - . logistic event screen fwn As-treated

analysis - --------------------------------------------------

---------------------------- - event Odds Ratio Std. Err. z

Pgtz 95 Conf. Interval - -------------------------------------------------

---------------------------- - screen .476156 .0834625 -4.23

0.000 .3377127 .6713534 - --------------------------------------------------

---------------------------- - . logistic event screen E fwn Approximate

CACE analysis - --------------------------------------------------

---------------------------- - event Odds Ratio Std. Err. z

Pgtz 95 Conf. Interval - -------------------------------------------------

----------------------------

Summary

- Weve explored a variety of methods for

all-or-nothing treatment switches - randomise to E or S everyone gets all E or all S
- In lecture 4, we will extend to much more complex

patterns of switching - e.g. get E just for 3 months, then S
- Another problem is where some participants get no

treatment at all (or a treatment other than E/S) - ITT difference depends on 2 effects (E vs.

nothing, S vs. nothing) - Walter (2006) adapted the compliance-type

approach but assumed equality between some

compliance-types