sample size calculation for crossover design

[la volta statistics]

Dear All

does anybody have a formula for the sample size calculation of a two-sample
crossover design comparing between proportions?

Any help will be appreciated.
Thanks, Christian

*******************************
la volta statistics
Christian Schmidhauser, Dr.phil.II
Weinbergstrasse 108
Ch-8006 Zürich
Tel: [phone]
Fax: [phone]
email: mailto:[hidden email]
internet: http://www.lavolta.ch/

[Marta Garcia-Granero]

la volta statistics escribió:
> does anybody have a formula for the sample size calculation of a two-sample
> crossover design comparing between proportions?
>
Hi Christian:

You don't give a lot of details, but (correct me if I'm wrong) I think
this is the general picture:

1) You are studying a binary variable (yes/no)
2) You are going to use a related samples design (half the patients will
test treatment A first and treatment B after, and the other half the
other way).

This means that you are going to use McNemar test for statistical
analysis. The problem with that test (in relation to sample size
determination) is that  concordant pairs (no/no and yes/yes patters) are
discarded for the computation of the p-value. This makes sample size
determination difficult. I had a copy of good paper that covered that
topic, but, unfortunately it's in Spanish and besides, I lent it to
someone who didn't return it (we say in Spain that there are two types
of stupid people: those who lend books and those who return them, I
suppose the same can be said about photocopies of rare papers...). I was
then going to give you the link to a program (in Spanish, but language
can be switched toEnglish after installation) that would compute the
requiered sample size using a complicated formula I had read in the lost
paper, but the program isn't there anymore... (I can't believe that
today isn't Monday, but Tuesday...).

Anyway, I did a bit of Googling and run into these papers with formulas
for McNemar test sample size:

http://findarticles.com/p/articles/mi_qa3899/is_200110/ai_n8956223
http://links.jstor.org/sici?sici=0006-341X(198906)45%3A2%3C629%3ATSASSF%3E2.0.CO%3B2-5

Unfortunately, right now I'm not at the University (where I have access
to a variaty of papers), but at home, and I can't download the papers
for you. Perhaps you (or some kind soul that reads this message) can
download them (in that case, I'd REALLY appreciate a copy of both, thaks
a lot, because I don't inted to go to the university in several days).

If anyone can send me those papers, I promise to write some SPSS syntax
to simplify the task.

Best regards,
Marta

[Patricia Rego]

I've forwarded Marta the papers she requested.

Patricia

___________________________________
Patricia Régo
Centre for Medical Education
School of Medicine
The University of Queensland
Australia
ph: 61-7-3346-4683;  fax: [phone]
[hidden email]

Evaluation Consultant to
Qld Health Skills Development Centre
ph: 3636-6449
[hidden email]


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Marta Garcia-Granero
Sent: Wednesday, 1 August 2007 2:48 AM
To: [hidden email]
Subject: Re: sample size calculation for crossover design

la volta statistics escribió:
> does anybody have a formula for the sample size calculation of a two-sample
> crossover design comparing between proportions?
>
Hi Christian:

You don't give a lot of details, but (correct me if I'm wrong) I think
this is the general picture:

1) You are studying a binary variable (yes/no)
2) You are going to use a related samples design (half the patients will
test treatment A first and treatment B after, and the other half the
other way).

This means that you are going to use McNemar test for statistical
analysis. The problem with that test (in relation to sample size
determination) is that  concordant pairs (no/no and yes/yes patters) are
discarded for the computation of the p-value. This makes sample size
determination difficult. I had a copy of good paper that covered that
topic, but, unfortunately it's in Spanish and besides, I lent it to
someone who didn't return it (we say in Spain that there are two types
of stupid people: those who lend books and those who return them, I
suppose the same can be said about photocopies of rare papers...). I was
then going to give you the link to a program (in Spanish, but language
can be switched toEnglish after installation) that would compute the
requiered sample size using a complicated formula I had read in the lost
paper, but the program isn't there anymore... (I can't believe that
today isn't Monday, but Tuesday...).

Anyway, I did a bit of Googling and run into these papers with formulas
for McNemar test sample size:

http://findarticles.com/p/articles/mi_qa3899/is_200110/ai_n8956223
http://links.jstor.org/sici?sici=0006-341X(198906)45%3A2%3C629%3ATSASSF%3E2.0.CO%3B2-5

Unfortunately, right now I'm not at the University (where I have access
to a variaty of papers), but at home, and I can't download the papers
for you. Perhaps you (or some kind soul that reads this message) can
download them (in that case, I'd REALLY appreciate a copy of both, thaks
a lot, because I don't inted to go to the university in several days).

If anyone can send me those papers, I promise to write some SPSS syntax
to simplify the task.

Best regards,
Marta

[Marta Garcia-Granero]

Hi Patricia

Many thanks, I'm working on some syntax based on Lehr's paper to compute
sample size for McNemar test.

Best regards,
Marta

> I've forwarded Marta the papers she requested.
>
> Patricia
>
>
> -----Original Message-----
> From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Marta Garcia-Granero
> Sent: Wednesday, 1 August 2007 2:48 AM
> To: [hidden email]
> Subject: Re: sample size calculation for crossover design
>
> la volta statistics escribió:
>
>> does anybody have a formula for the sample size calculation of a two-sample
>> crossover design comparing between proportions?
>> Hi Christian:
>>
>> You don't give a lot of details, but (correct me if I'm wrong) I think
>> this is the general picture:
>>
>> 1) You are studying a binary variable (yes/no)
>> 2) You are going to use a related samples design (half the patients will
>> test treatment A first and treatment B after, and the other half the
>> other way).
>>
>> This means that you are going to use McNemar test for statistical
>> analysis. The problem with that test (in relation to sample size
>> determination) is that  concordant pairs (no/no and yes/yes patters) are
>> discarded for the computation of the p-value. This makes sample size
>> determination difficult. I had a copy of good paper that covered that
>> topic, but, unfortunately it's in Spanish and besides, I lent it to
>> someone who didn't return it (we say in Spain that there are two types
>> of stupid people: those who lend books and those who return them, I
>> suppose the same can be said about photocopies of rare papers...). I was
>> then going to give you the link to a program (in Spanish, but language
>> can be switched toEnglish after installation) that would compute the
>> requiered sample size using a complicated formula I had read in the lost
>> paper, but the program isn't there anymore... (I can't believe that
>> today isn't Monday, but Tuesday...).
>>
>> Anyway, I did a bit of Googling and run into these papers with formulas
>> for McNemar test sample size:
>>
>> http://findarticles.com/p/articles/mi_qa3899/is_200110/ai_n8956223
>> http://links.jstor.org/sici?sici=0006-341X(198906)45%3A2%3C629%3ATSASSF%3E2.0.CO%3B2-5
>>
>> Unfortunately, right now I'm not at the University (where I have access
>> to a variety of papers), but at home, and I can't download the papers
>> for you. Perhaps you (or some kind soul that reads this message) can
>> download them (in that case, I'd REALLY appreciate a copy of both, thanks
>> a lot, because I don't intend to go to the university in several days).
>>
>> If anyone can send me those papers, I promise to write some SPSS syntax
>> to simplify the task.
>>
>> Best regards,
>> Marta
>>
>>

[Marta Garcia-Granero]

OK

Quick and dirty solution. It can be easily turned into a cute MACRO.
Also, output can be improved (report input conditions, intermediate
values computed...).

Regards,
Marta

* Table layout:
*               |    Treatment B
*               |-------------------------
* Treatment A   |Negative  Positive Total
* ----------------------------------------
*    Negative   |    a         b     a+b
*    Positive   |    c         d     c+d
* ----------------------------------------
*       Total   |   a+c       b+d     n

* Proportion with outcome in A
*  p1=(c+d)/n

* Proportion with outcome in B
*  p2=(b+d)/n

* p2-p1=(b-c)/n

* Sample size equation (alpha=0.05 and power=0.80):
* n = 16pq(1-r²)/d²

* where:
* p=(p1+p2)/2
* q=1-p
* r=phi correlation coefficient:
* r=(ad-bc)/SQRT[(a+b)(c+d)(a+c)(b+d)]

* also:
* r=(   p1+p2-phi-2p1p2)
    ------------------------  (phi=proportion of discordant pairs)
    2*SQRT[p1(1-p1)p2(1-p2)]

* maximum r is (1-delta)/(1+delta)
* where delta=p2-p1

* minimum r value is 0

* Therefore, in order to compute sample size for McNemar test, it
* is advisable to have a small pilot study used to estimate phi .

* Instead of 16, use:
* 21   if alpha=0.05 & power=0.90
* 26   if alpha=0.05 & power=0.95
* 23.5 if alpha=0.01 & power=0.80
* 30   if alpha=0.01 & power=0.90
* 36   if alpha=0.01 & power=0.95
* see Table 3 of Lehr's paper for other values (one tailed tests)

* BEGINNING OF SYNTAX *.

MATRIX.
* Example data from Lehr's paper (table 2), replace by your own *.
COMPUTE p1 = 0.9 .
COMPUTE p2 = 0.8 .
COMPUTE phi= 0.14 .
* Coefficient for alpha=0.05 & power =0.80 *.
COMPUTE coeff=16.
* Intermediate computations *.
COMPUTE delta=ABS(p2-p1).
COMPUTE p=(p1+p2)/2.
COMPUTE q=1-p.
COMPUTE minr = 0.
COMPUTE maxr = (1-delta)/(1+delta).
COMPUTE r    = (p1+p2-phi-2*p1*p2)/(2*SQRT(p1*(1-p1)*p2*(1-p2))).
* Computing possible values for n *.
COMPUTE minn = TRUNC(coeff*p*q*(1-maxr)/(delta**2)).
COMPUTE maxn = TRUNC(coeff*p*q*(1-minr)/(delta**2)).
COMPUTE realn= TRUNC(coeff*p*q*(1-   r)/(delta**2)).
* Report *.
PRINT {minn,realn,maxn}
 /FORMAT='F8.0'
 /CLABELS='Min','Estim.','Max'
 /TITLE='Estimated sample size required'.
END MATRIX.

* If only delta is given (no p1 or p2), then the only possibility is compute
* max&min n (based on minr=0 and maxr=(1-delta)/(1+delta) and average them.

MATRIX.
* Input data (replace by your own) *.
COMPUTE delta=0.1.
COMPUTE p=0.5.
COMPUTE q=0.5.
* Coefficient for alpha=0.05 & power =0.80 *.
COMPUTE coeff=16.
* Computing possible values for n *.
COMPUTE minr =  0.
COMPUTE maxr = (1-delta)/(1+delta).
COMPUTE minn = TRUNC(coeff*p*q*(1-maxr)/(delta**2)).
COMPUTE maxn = TRUNC(coeff*p*q*(1-minr)/(delta**2)).
COMPUTE Estn= (minn+maxn)/2.
* Report *.
PRINT {minn,estn,maxn}
 /FORMAT='F8.0'
 /CLABELS='Min','Estim.','Max'
 /TITLE='Estimated sample size required'.
END MATRIX.

[la volta statistics]

Hi Marta and Patricia
Many thanks for the syntax. I asked Patricia to send me the papers as well.
Meanwhile a colleague of mine has sent me a PDF file with class notes from
Lehana Thabane about sample size determination in clinical trials. He told
me that he found the notes on the web a while a go, but unfortunately they
are not there any more.

Lehana Thabane mentions the formula for a two sample cross over as follows:


        (z_alpha - z_beta)^2 * sigma^2
ni = -----------------------------------
        2(pi_1 - pi_2) ^2

where  the Null-hypothesis is: pi_1 - pi_2 = 0

If you are interested I can send you the PDF file off-list as attachment.
Unfortunately, there are no references where the formulae came from.

Thanks again
Christian

-----Ursprüngliche Nachricht-----
Von: SPSSX(r) Discussion [mailto:[hidden email]]Im Auftrag von
Marta Garcia-Granero
Gesendet: Mittwoch, 1. August 2007 11:57
An: [hidden email]
Betreff: Re: PAPERS SENT - RE: sample size calculation for crossover
design


OK

Quick and dirty solution. It can be easily turned into a cute MACRO.
Also, output can be improved (report input conditions, intermediate
values computed...).

Regards,
Marta

* Table layout:
*               |    Treatment B
*               |-------------------------
* Treatment A   |Negative  Positive Total
* ----------------------------------------
*    Negative   |    a         b     a+b
*    Positive   |    c         d     c+d
* ----------------------------------------
*       Total   |   a+c       b+d     n

* Proportion with outcome in A
*  p1=(c+d)/n

* Proportion with outcome in B
*  p2=(b+d)/n

* p2-p1=(b-c)/n

* Sample size equation (alpha=0.05 and power=0.80):
* n = 16pq(1-r²)/d²

* where:
* p=(p1+p2)/2
* q=1-p
* r=phi correlation coefficient:
* r=(ad-bc)/SQRT[(a+b)(c+d)(a+c)(b+d)]

* also:
* r=(   p1+p2-phi-2p1p2)
    ------------------------  (phi=proportion of discordant pairs)
    2*SQRT[p1(1-p1)p2(1-p2)]

* maximum r is (1-delta)/(1+delta)
* where delta=p2-p1

* minimum r value is 0

* Therefore, in order to compute sample size for McNemar test, it
* is advisable to have a small pilot study used to estimate phi .

* Instead of 16, use:
* 21   if alpha=0.05 & power=0.90
* 26   if alpha=0.05 & power=0.95
* 23.5 if alpha=0.01 & power=0.80
* 30   if alpha=0.01 & power=0.90
* 36   if alpha=0.01 & power=0.95
* see Table 3 of Lehr's paper for other values (one tailed tests)

* BEGINNING OF SYNTAX *.

MATRIX.
* Example data from Lehr's paper (table 2), replace by your own *.
COMPUTE p1 = 0.9 .
COMPUTE p2 = 0.8 .
COMPUTE phi= 0.14 .
* Coefficient for alpha=0.05 & power =0.80 *.
COMPUTE coeff=16.
* Intermediate computations *.
COMPUTE delta=ABS(p2-p1).
COMPUTE p=(p1+p2)/2.
COMPUTE q=1-p.
COMPUTE minr = 0.
COMPUTE maxr = (1-delta)/(1+delta).
COMPUTE r    = (p1+p2-phi-2*p1*p2)/(2*SQRT(p1*(1-p1)*p2*(1-p2))).
* Computing possible values for n *.
COMPUTE minn = TRUNC(coeff*p*q*(1-maxr)/(delta**2)).
COMPUTE maxn = TRUNC(coeff*p*q*(1-minr)/(delta**2)).
COMPUTE realn= TRUNC(coeff*p*q*(1-   r)/(delta**2)).
* Report *.
PRINT {minn,realn,maxn}
 /FORMAT='F8.0'
 /CLABELS='Min','Estim.','Max'
 /TITLE='Estimated sample size required'.
END MATRIX.

* If only delta is given (no p1 or p2), then the only possibility is compute
* max&min n (based on minr=0 and maxr=(1-delta)/(1+delta) and average them.

MATRIX.
* Input data (replace by your own) *.
COMPUTE delta=0.1.
COMPUTE p=0.5.
COMPUTE q=0.5.
* Coefficient for alpha=0.05 & power =0.80 *.
COMPUTE coeff=16.
* Computing possible values for n *.
COMPUTE minr =  0.
COMPUTE maxr = (1-delta)/(1+delta).
COMPUTE minn = TRUNC(coeff*p*q*(1-maxr)/(delta**2)).
COMPUTE maxn = TRUNC(coeff*p*q*(1-minr)/(delta**2)).
COMPUTE Estn= (minn+maxn)/2.
* Report *.
PRINT {minn,estn,maxn}
 /FORMAT='F8.0'
 /CLABELS='Min','Estim.','Max'
 /TITLE='Estimated sample size required'.
END MATRIX.