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Medical and Pharmaceutical Statistics Research Unit. Seminar, Bordeaux School of Public Health 8 June 2011 Combining endpoints in clinical trials to increase power John Whitehead. Medical and Pharmaceutical Statistics Research Unit Department of Mathematics and Statistics

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slide1

Medical and Pharmaceutical

Statistics Research Unit

Seminar, Bordeaux School of Public Health

8 June 2011

Combining endpoints in clinical trials to increase power

John Whitehead

Medical and Pharmaceutical Statistics Research Unit Department of Mathematics and Statistics

Tel: +44 1524 592350 Fylde College

Fax: +44 1524 592681 Lancaster University

E-mail: j.whitehead@lancaster.ac.uk Lancaster LA1 4YF, UK

slide2
1. Ordinal endpoints in stroke studies
  • Treatments for acute stroke are administered for a few days following diagnosis
  • The primary endpoint is the functional status of the patient, 90 days after the stroke
  • Several scoring systems exist, including the Barthel index, the modified Rankin score and the NIH stroke scale
  • All are ordinal scales from full recovery to vegetative state, to which death before 90 days can be added
slide3
Analysis of an ordinal response

R1 = best response (full recovery)

Rk = worst response (death before day 90)

slide4
Let

Ch = c1 +…+ ch

Ch = the number of controls with response Rhor better

Let

Ch = ch +…+ ck

Ch = the number of controls with response Rhor worse

Similarly define Eh, Eh, Th and Th

slide5
Let

QCh = P(a control has response Rh or better)

QEh = P(an experimental has response Rh or better)

(thenQCk = QEk = 1)

Put

qh is the log-odds ratio for response Rh or better, E:C

h = 1,…, k – 1

slide6
The proportional odds assumptionis

1 = 2 = … = k–1 = 

The common value, , is a measure of the advantage of the experimental treatment

> 0 experimental better

 = 0 no difference

< 0 control better

slide7
Under PO, the most efficient test of treatment advantage

-greatest power for any given sample size

is based on the test statistics

and

For large samples and small , approximately Z ~ N(V, V)

Z is the score statistic and V is Fisher’s information

slide8
To test for treatment difference, refer Z2/V to

 This is the Mann-Whitney test

 Also known as the Wilcoxon test

Under the null hypothesis of no treatment effect, PO is true with q = 0

Thus the hypothesis test and the p-value are valid without assumptions

Estimates of and confidence intervals for q do rely on assumptions, as does adjustment for prognostic factors

slide9
How should investigators choose which scale to use?

An alternative to choosing is to combine more than one stroke scale in the analysis

Tilley et al. (1996) combined four scales in the trial of rTPA as a treatment in acute stroke conducted by the National Institute of Neurological Disorders and Stroke

-the trial was positive and the approach caught on

If the treatment has a beneficial effect on all scales, then combining them will increase the power to demonstrate the advantage of the treatment

slide10
2. Example: The ICTUS trial in stroke
  • Currently ongoing in 60 centres in Europe
  • Patients who have suffered acute stroke
  • Randomised between citicoline and placebo
  • Assessed at 90 days on Barthel index, modified Rankin

score and NIH stroke scale

  • Prognostic factors

- baseline NIHSS

- time from stroke to treatment ( or > 12 hours)

- age ( or > 70 years)

- site of stoke (right or left side)

- use of rTPA (yes or no)

slide11
The approach used by Tilley et al.

Combine the three analyses using GEE

(based on an independence covariance structure: IEE)

That is, analyse as if the three scores were independent, but

adjust the standard error of the treatment effect estimate using

the sandwich estimator

  • complicated to understand
  • no associated sample size formula
  • failed in test data set of 1000 patients with binary responses and adjustment for 60 centres
slide12
An alternative general approach

The log-odds ratio q and the test statistics Z and V, for the analysis of the ith response will be denoted by qi, Zi and Vi

i = 1 is Barthel index

i = 2 is modified Rankin score

i = 3 is NIH stroke score

W will test H0: q1 = q2 = q3 = 0 (no effect of treatment on any of the scales) using

Z = Z1 + Z2 + Z3

slide13
For each scale,

Zi ~ N(qiVi, Vi)

if Vi is large and qi is small

If q1 = q2 = q3 = q, then approximately

where V = V1 + V2 + V3, C = 2(C12 + C23 + C31) and

Cij = cov(Zi, Zj)

slide14
It follows that, if

then

as required for a c2 test and for sample size calculation

What we need to use this is an expression for

Cij = cov(Zi, Zj)

slide16
The binary case, no covariates

- ith of several responses

- assuming that each patient provides all responses

slide17
Covariance between Zi and Zj

For two such statistics, we have

where ti1 is the number of patients succeeding on the ith scale,

tj1 the number succeeding on the jth scale and t(ij),1 the number succeeding on both scales (Pocock, Geller and Tsiatis, 1987)

slide18
The ordinal case, no covariates

- ith of several responses

with Cih = ci1 +…+ cih and Cih = cih +…+ cik

slide19
Covariance between Zi and Zj

For two such statistics, we have

where dfv = -1, 0 or 1 if f <, =, > v respectively,

Kfg = tfi tgj/n2, Hfg = t(ij),(fg)/n - Kfg,

t(ij),(fg) is the count of patients who have both response Rf,i on

the ith scale and response Rg,j on the jth scale

slide20
Adjustment for covariates

The approach can be extended to allow for prognostic factors via stratification and/or linear modelling of covariates

Stratification: sum Z and V statistics over strata, and assume that the treatment effect is constant over strata

Covariate adjustment: use proportional hazards regression, plus binary logistic regression to model the simultaneous occurrence of particular responses on different scales (such as complete recovery on Barthel index and partial recovery on the modified Rankin)

slide21
3. Sample size calculation for the combined test

For power of 90% to detect a log-odds ratio of qR as significant

at level 0.05 (two-sided), we need

for a test based on a single response, and

for a test based on the combined approach

slide22
For a single binary (success/fail) response, with an overall success probability of p,

For three binary responses, each having an overall success probability of p, and with the probability of success on any two responses being g

slide23
Suppose that g = p2 (independence), then

-that is one third of the sample size using only one response

For g = p (responses coincide), then

-that is the same as the sample size using only one response

Otherwise, combining the responses reduces sample size by up to one third, depending on the correlation between the responses

slide24
Now suppose that p = 0.2 and that g = 0.1 (correlation = 0.75)

then for one response

and for three responses

58% of the sample size using a single response

slide25
If the success rate on control is 18%, and the trial is to be powered to detect an improvement to 22%, then the log-odds ratio is

so that, for one response

n = 4200

and for three responses

n = 2450

slide26
ICTUS trial

Fixed sample size using only

Barthel: 2590

modified Rankin: 3584

NIH stroke scale: 5494

Combined test: 2421

This is for dichotomised responses, based on the previous data available

ICTUS is using a sequential design

slide27
Ordinal scales

For sample size calculation for combining several ordinal responses, probabilities of every pair of responses on every pair of responses must be anticipated

- Databases from previous trials can be used

- A mid-trial sample size review can be used

slide28
Evaluation of the combined approach

The first of a series of interim analyses of the ICTUS trial takes place when data from 1000 patients are available

A dataset from four previous studies comparing citicoline with placebo is available (Davalos et al., 2002) comprising 1,372 patients

First, one dataset of 1,000 was extracted and analysed using the combined test and the GEE approach

Then 10,000 datasets of size 200, 500 or 1,000 were randomly selected, the treatment code was removed and randomly reassigned

- in some runs an artificial treatment effect of known

magnitude was introduced

slide30

Results from 10,000-fold simulations of the

combined score test and the GEE approach

slide31
Conclusions

Use of the combined approach can reduce sample size, provided that the treatment effect is apparent on all responses being combined

The score approach used here matches the GEE approach, and is more reliable in small samples

The approach can combine quantitative responses and survival responses, it can also be used to combine different types of response

slide32
References

Bolland, K., Whitehead, J., Cobo, E. and Secades, J. J. (2009). Evaluation of a sequential global test of improved recovery following stroke as applied to the ICTUS trial of citicoline. Pharmaceutical Statistics8, 136-149.

Dávalos A, Castillo J, Álvarez-Sabin J, Secades JJ, Mercadal J, López S, Cobo E, Warach S, Sherman D, Clark WM, Lozano R. (2002). Oral citicoline in acute ischemic stroke. Stroke33, 2850-2857.

Dávalos A. (2007). Protocol 06PRT/3005: ICTUS study: International Citicoline Trial on acUte Stroke (NCT00331890) Oral citicoline in acute ischemic stroke. Lancet Protocol Reviews.

Pocock, S.J., Geller, N. L. and Tsiatis, A. A. (1987). The analysis of multiple endpoints in clinical trials. Biometrics43, 487-498.

Tilley, P. C., Marler, J., Geller, N. L., Lu, M., Legler, J., Brott, T., Lyden, P. and Grotta, J. for the National Institute of Neurological Disorders and Stroke (NINDS) rt-PA Stroke Trial Study Group. (1996). Use of a global test for multiple outcomes in stroke trials with application to the National Institute of Neurological Disorders and t-PA Stroke Trial. Stroke27, 2136-2142.

Whitehead, J., Branson, M. and Todd, S. (2010). A combined score test for binary and ordinal endpoints from clinical trials. Statistics in Medicine 29, 521-532.