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## AMERICA N STATISTICA L ASSOCIATIO N 8 0 6 15t h Street , N.W . Washington , D.C . 2000 5Analysi s o f (Events/Tim e a t Risk ) Ratio s Fro m Two-Perio dCrossove r Studie s Jil l Bissette , Gre g Carr , Gar y G . Koch , Kirkwoo dF . Adams , Davi dS . Shep s Departmen t o f Biostatistics , Universit y o f Nort hCarolina , Chape l Hil l P. 3

Abstrac t Fo r some two-perio d crossove r studies , ther e i s interes t ina time t o even t respons e variabl e wit h potentia l censorin g fo r th e time a t risk . Fo r example , a gastrointestina l study migh t b e con cerne dwit htime t o relie f o f heartbur nfo r patient s wh o consume d tw o symptom-provokin g meal s ontwo differen t day s wit htwo differen t treat ments . I nthi s paper , th e usag e o f weighte d leas t square s method s t o analyz e (events/time a t risk ) ratio s i s discussed . Two example s ar e pro vide dfo r illustrativ e purposes . 1 . Introductio n Th e two-perio d crossove r desig ni s ofte nuse dt o compar e treatment s whe ni t i s desirabl e t o hav e subject s serv e a s thei r ow ncontrols . Subject s ar e randoml y

assigne d t o on e o f two sequenc e group s fo r two successiv e treatmen t periods . I f th e study seek s t o compare a nactiv e treatmen t wit ha placebo , the non e sequenc e grou p receive s th e activ e treatmen t inth e firs t perio d an dth e placebo i

nth e secon d period . Th e remainin g se quenc e grou preceive s th e placeb o i nth e firs t perio dan dth e activ e treatmen t i nth e secon d period . Treatment , period , and (treatmen t xperiod ) in teractio n(i.e . carryove r effects ) ar e th e source s o f variatio no f interest . I f ther e ar e n o carry ove r effects , th e interpretatio no f treatmen t com parison s i s straightforward . I f carryove r effect s do exist , the nth e interpretatio no f finding s i s somewha t mor e difficul t a s th e treatmen t an dperio d effect s ar e n o longe r additive . [Grizzle , (1965) , Brow n(1980) , Koch , Gitomer , an d Skallan d (1983)] . Incrossove r studies , baselin e measurement s ar e sometime s take nbefor e th e treatmen t sequenc e i s administered . Th e baselin e measurement s ar e help fu l i nstrengthenin g th e test s o f treatmen t an d carryove r effects . Whe na baselin e measuremen t i s take nfo r eac hsubjec t befor e th e treatmen t se quenc e i s administered , th e study ma y be sai dt o hav e a two-perio d crossove r desig nwit hbaseline . Standar dparametri c method s ma y b e use d t o ana lyz e th e dat a fro

m a crossove r desig nwhe nassump tion s involvin g norma l distribution s appl y [Grizzl e (1965)] . I f th e sampl e siz e i s smal l and normali ty doe s no t seem

reasonable , the nnonparametri c method s ma y

b e applie d t o th e rank s o f certai n• sum s an ddifference s o f th e response s o f eac hsub jec t [Koc h(1972)] . Inth e cas e o f time t o even t dat a wit hpossibl e righ t hand censoring , othe r method s ar e neede d inorde r t o accoun t fo r th e variatio no f time a t ris k bot hwithi nsubject s an d acros s subjects . Thi s pape r describe s on e wa y o f addressin g thi s question . I t involve s th e descrip tio no f time t o even t dat a wit hsummar y measure s fo r th e numbe r o f event s pe r uni t time a t ris k an d th e analysi s o f thei r variatio nthroug hweighte d leas t square s methods . 2 . Weighte dLeas t Souare s Method s fo r Mode l Fitting Fo r time t o even t data , a usefu l summar y measur e i s th e ratio : ^

_numbe r o f peopl e experiencin g th e even t

tota l time a t ris k Thi s rati o i s ofte ncalle d anincidenc e densit y rat e sinc e i t give s th e numbe r o f event s pe r uni t time . Whe nth e time t o even t dat a hav e anexponen tia l distribution , i t i s th e maximu m likelihoo d estimato r fo r th e hazar d(scale ) parameter . Also , th e rat e Xca nofte nbe usefu l inmor e genera l situations . Fo r crossove r studie s wit hmoderat e sampl e sizes , thi s rat e ca nbe determine dfo r eac h perio dfo r eac hsequenc e grou ptogethe r wit hth e correspondin g covarianc e matrix . Define : 1 i f th e j-t hsubjec t i nth e i-t h

treatmen t sequenc e experience s th e y ij k ~ ' even t durin g th e k-t hperio d

[ o otherwis e tim e a t ris k fo r th e j-t hsubjec t 1nth e i-t htreatmen t sequenc e durin g th e k-t hperio d x ij k

ni i = 1, 2 = s j = 1, 2 k = 0,1, 2 = d( 0 fo r baseline ) Fo r multi-cente r studie s o r othe r situation s wher e subject s ar e fro

m two o r mor e strata , th e i sub scrip t i s generalize dt o h i wher e th e h=l,2,..., q indexe s th e strata . Fo r d= 2 period s andbaseline , construc t th e multivariat e respons e vecto r ?i j = ^ijO ' A ij0 ' 'ijl ' A i j T J 'ij2 ' th e mea nvector s n . j= l x ij2 ] 

## h

an dcovarianc e matrice s



### v - = (i/n . r T (g-.-g-Hg.-g- ) .

j= l Fo r th e composit e vecto r " g = ?•. ' • ? 2 ' . estimate dcovarianc e matri xV g ha s bloc k diagona l structur e wit hth e {V^. } a s th e diagona l blocks . -2 1 Whe nther e i s onl y

on e stratum , th e matri xV W i s give nb y th e



## -i 0 % Fo r two perio dstudies , eac ho f th e {Vg^ } i s (4x4) ; an dfo r two perio dstudie s wit hbaseline , eac hi s (6x6) . Onc e th e vecto r o f means , g ^ ha s been

_created , i t i s o f interes t t o form th e ratio s (y-j/x, ) fo r eac h(perio dxsequence) . Thes e computation s can be expresse dasj

____ ? i = ^iO^iO ' y i l / x i T yi2 / x i 2 ] " ? = [ M ] i f th e study

contain s a baselin e measuremen t an d onl y on e stratum . Th e ratio s Rprovid e a wa y t o dea l wit hth e time t o even t dat a durin g th e expo sur e period s whe nrigh t censorin g i s a possibl e outcome . Also , i t ca nbe desirabl e t o bas e analy se s o nth e logarithm s o f th e rati o o f th e mean s 7 , an dX- j s o a s t o enhanc e th e applicabilit y o f norma l i

104P. 4 

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n d t o DO



### ily

rma ! . : inth e

• e w 

## f approximation s fo r th e resultin g distributions .

Sampl e size s o f 1 0 o r mor e i n eac hsequenc e grou p ca nreasonabl y

suppor t approximat e normalit y o f th e logarithm s o f th e ratio s R andthereby provid e a suitabl e framewor k fo r statistica l tests . A vecto r o f functions , f , suc ha s the ratio s B ca nb e create d from

J b y usin g matri xoperation s involvin g linear , logarithmi c an dexponentia l operations . Thi s constructio nca nb e implemente d usin g th e program

MISCA T [Stanish , Koc han dLandi s (1978)] . Th e logarithm s F o f (events/time a t risk ) ratio s ar e determine d a s th e function s f= A Uj g (g) . Thei r covarianc e matri xi s consistentl y estimate d throug hVF.^ftCPg)~*Vg(Dg)- x A ^ wher e A=[1.-1](B ) l2 S u wit hu= d fo r studie s withou t baselin e andu=d+ l fo r thos e wit hbaseline . Th e symbo l ®denote s Kronecke r produc t multiplicatio ni n whic hth e matri xo n th e lef t multiplie s ever y elemen t ofth e matri xo n th e right ; V

g i s th e estimate dcovarianc e matri xfo r g and (Dg~)~~i s a diagona l matri xwit h g o n th e diagonal . -

~ A linea r mode l t o describ e th e variatio namon g th e function s F ca nb e expresse d a s £A(F ) = X g wher e X i s the~(s u xt ) mode l specificatio nmatri x wit hful l rank t < . s u and g i s th e ( t x1 ) vecto r o f unknow

nparameters ; also , E^ O denote s expecte d value s fo r larg e samples . The ~ weighte dleas t square s estimates , b , fo r g ar e give nb y b= (Op^PrVp'F . fo r whic hE & (b ) = § an d Var fi (b ) i s consistentl yestimate d b yV b = (X'V^X)" 1

. Goodnes s o f fi t ca n b

e assesse d b

yth e Wal d Statisti c Q w= ( E - XbJ'VpM f - Xb) = F'W'(Wy F W') _, W F wher e W ' i s a n orthocomplemen t t o X (i.e

. ,~WX=0 , Rank(W)= ? (su-t) ; th e statisti c Q yi s distribute d approxi

matel yas chi-squar e wit h (su-t ) = (s u - Ran k X ) degree s o f freedo m unde rHQ

. Fo rmodel s whic h ~ adequatel ydescrib e th e variatio n i n th e data , linea rhypothese s o f th e for m HQ : Cg= Q where C i s a ( c xt ) matri xo f ful l ran k c ma yb e teste d vi a th e statistic , Q c= b/Cl£VbC') _1 £fe ; tn e sta
tisti c Q ri s approximatel ydistribute d a s chi

squar e wit h c degree s o f freedo m unde rHQ

. 3 . Example s Two example s o f crossove r studie s ar e presented . Th e firs t exampl e i s from a two-perio dcrossove r desig nwit hbaseline . Subject s wit hdocumente d ischemi a wer e expose d t o 10 0 part s pe r millio n carbo nmonoxid e o n a randomize d exposur e day . Time t o onse t o f angin a wa s th e even t o f interest . Carryove r an

d perio d effect s wer e no t presen t bu t treatmen t effect s wer e significant . Th e secon d exampl e i s from a two-perio d cross ove r study

whic hwa s conducte d a t two differen t treatmen t centers . Study protoco l wa s identica l a t eac hcenter . Subject s pron e t o heartbur nat e symptom-provokin g meal s wit hth e meal s correspond in g t o th e two periods . Th e even t o f interes t wa s whethe r th e subjec t woul d experienc e relie f fro m heartbur nwithi n2 0 minute s o f the treatmen t administered . 3. 1 Exampl e 1 : Cardiolog y Study wit hBaselin e Thirt y subject s wit hdocumente dischemi a parti cipate d i n a 3 da y

crossove r study . Th e firs t day wa s a trainin g da y

(baseline ) t o familiariz e th e subject s wit hth e study equipmen t an

dexercis e protocol . Th e nex t 2 day s wer e exposur e days . Subject s wer e randoml y assigned , upo nentr y

int o th e study , t o eithe r th e A C o r the C A treatmen t sequenc e wher e A

indicate s exposur e t o chambe r ai r an dC indicate s exposur e t o chambe r ai r mixe d wit h 10 0 part s pe r millio n(ppm ) carbo nmonoxide . Th e even t o f interes t i n al l period s wa s whethe r o r no t th e subjec t woul dexperienc e ches t pai n (angina) . Patient s performe dsupin e exercis e unti l angin a o r severa l othe r define d event s whic h terminate dth e exercis e period . Fo r al l subject s th e time t o angin a (TTANG) , a n indicato r fo r th e occurrenc e o f angin a (ANG) , an d th e duratio no f exercis e (DUR ) wer e recorded . Th e even t indicato r variable , y , wa s identica l t o th e variabl e ANG . Th e tim e a t ris k variable , x , too k th e valu e o f TTANGfo r thos e experiencin g angin a an dDU Rfo r thos e wh o di dnot .

- Th e analysi s was_base don ^ th e constructio no f f= A log (g ) wher e g = [g^.g ^ ] ' an dfl= [ 1 -1 ] (x)I 6 . Th e consisten t estimator , Yf , fo r Va r F 1 s give na s V p= MlU^VBg)"^ ' wher e Y - is~th e estimate d covarianc e matri xfo r g " andDg - i s a diagona l matri x wit hg o n th e diagona K The~preliminar y

mode l fi t t o th e dat a include d parameter s fo r perio d1 andperio d 2 effects , treat men t effect s an

dcarryove r effects . Subject s 1 nth e differen t treatmen t sequence s wer e assume dt o b e equivalen t a t baseline . Mode l parameterization s fo r exampl e 1 ar e presente di n Tabl e 1 . Th e goodnes s o f fi t fo r th e preliminar y mode l 1 s th e same a s a tes t o f subject s withi ntreatmen t sequence s bein g equivalen t a t baselin e (Q y = 0.0 2 df=l) . Th e nonsignificanc e o f thi s tes t (p=0.90 ) indicate s a goo dfi t t o th e dat a an d support s base lin e equivalenc e o f th e subject s i n th e two treat men t sequenc e groups . Test s o f carryove r effect s (Q c = 0.78,df=l ) and fo r perio d1 an

dperio d2 effect s (Q Q = 1.99 , df=2 ) wer e bot hnonsignifican t (p=0.3 8 an d p=0.37 , respec tively) . Treatmen t effect s fo r thi s mode l approache d significanc e (Q c = 3.36,.df=l , p=0.07) . A fina l mode l whic heliminate dcarryove r and perio deffect s wa s the nfit . Th e mode l provide d a goo dfi t t o th e dat a (Q w = 2.35 , df=4 , p=0.67);an d a tes t o f treatmen t effect s (Qr = 9.40 , df=l ) wa s highl y significan t wit hp < 0.01 . Test s o f th e param eter s fo r th e respectiv e model s ar e presente d i nTabl e 2 . I nconclusion , subject s experience d angin a earlie r o nday s wit h10 0 pp

m carbo nmonoxid e exposur e tha non chambe r ai r exposur e days . Ther e wer e n o perio d effect s an

dn o carryove r effect s (i.e . n o (exposur e xperiod ) interaction) . Estimate s an d thei r standar d error s fo r th e respectiv e model s ar e presente dinTabl e 3 . 3. 2 Exampl e 2 : Gastrointestina l Study Two center s participate di n a two-perio d crossove r study whic hdeal t wit hth e relie f fro m heartbur n after a symptom-provokin g meal . Th e meal s corre sponde dt o th e two periods . O f th e thirt y patient s wh o participate da t eac hcenter , fiftee nreceive d th e active/placeb o (AP ) treatmen t sequenc e whil e th e othe r fiftee nreceive dth e placebo/activ e (PA ) treatmen t sequence . Study protoco l wa s th e same a t eac hperio dan

da t eac hcenter . Afte r th e mea l wa s ingested , subject s wer e give nthei r appropriat e treatment . Time t o relie f fro m heartbur nwa s note d i f relie f actuall y

di doccur . Th e two period s wer e on e wee k apar t an

dth e stud

y wa s doub l e b

l i n d a t eac hcenter . I f th e subjec t experience drelie f fro m heartbur n inles s tha n2 0 minute s fro m th e time o f treatment ,



### 10 5P. 5



### th e even t indicato r variable , y , too k th e valu e o f s-i one . Otherwise , a valu e o f zer o wa s recorde dfo r V tha t variable. . Th e time a t ris k wit hheartbur no r time t o relie f fro

m heartbur nwa s th e secondvaria bl e constructed . Thi s variable , x , too k th e valu e o f th e actua l numbe r o f minute s befor e relie f oc curre dfo r thos e experiencin g relie f i nles s tha n 2 0 minutes . Fo r thos e wh o di dno t experienc e re lie f inth e allocate dtime , a valu e o f 2 0 wa s recorded . Th e dat a fo r thi s exampl e ar e give ni n Koch , Gitomer , andSka1 1 and[1983J . Inorde r t o enhanc e th e applicabilit y o f norma l approximation s t o th e distribution s o f th e loga rithm s o f (events/time a t risk ) ratios , th e analy si s wa s directe da t th e acros s sit e average s W - hi lQ 9 [fii(g) ] wher e § represent s th e compoun d vecto r ofsampl e mean s fo r the tw o sequenc e group s a t th e two

_site s an dha s th e structur e S = Lln''9i2''9 2 r922' ] ' w l t h ?h i = Ehl T X hi T h.l' x h12 ] ' fo rh»1.2an d 1-1.2 ; als o A. = [0.5(I 8 ) , 0.5(I 8 )] ; and A, = [l,-1 ] ®I 4 . Th e varianc e o f £ i s consistentl yestimate d with , 

## h " *2 ( 

## 5 a 

## l , 

## " 1 

## -W (5 

## al , 

## " , 

## fi2 ' ' wher e 

## % i s th e 

### estimate dcovarianc e matri xfo r q~ , andD , i s a 2' ~a l diagona l matri xwit ha , = log[A,(g") ] ontR e diagonal . Sinc e n o aprior i mode l wa s specified , th e initia l mode l fit , F = Xg , wa s th e cel l mea nmode l wit h X=X4 - Th e ce11~mea nmode l allow s no reductio ni n th e dimensio no f F an

ds o ther e i s n o lac k o f fi t b y definition . Mode l parameterization s fo r thi s exampl e ar e presente di nTabl e 4 . A ninitia l tes t o f th e hypothesi s H« : p 1 

### =u_=y-=y . o f no variatio namon g th e fou r (perio dxtreatment ) logarithm s o f (events/time a t risk ) ratio s (Qr = 14.09 , df=3 ) wa s significan t ( p< 0.01) . Additiona l testin g foundtha t averag e treatmen t effect s (Q c = 10.87 , df=l ) an dcarryove r effect s (Q c = 6.92 , df=l ) wer e bot hsignifican t ( p< 0.01 ) whil e perio deffect s (Q c = 2.47 , df=l ) wer e not . A

tes t o f equalit y

o f active treatment s acros s period s wa s nonsignifican t a s wel l (Q c = 0.71 , df=l , p=0.40) . A reduce dmode l withou t perio deffect s andwit h separat e effect s fo r placeb o durin g th e tw o period s wa s the nfit . Th e goodnes s o f fi t fo r th e reduce d mode l (Q w = 0.71 , df=l , p=0.40 ) wa s satisfactory . Th e paramete r estimate s fro m th e reduce dmode l hel p clarif y th e natur e o f th e treatmen t effect s and thei r interactio nwit hperiod . Subject s wh o re ceive dth e active treatmen t inth e secondperio d hadmor e favorabl e experience s tha nthos e wh o re ceive dplaceb o i nth e secondperio do r eithe r treatmen t i nth e firs t period . Test s o f mode l param eter s ar e summarize dfo r thi s exampl e i nTabl e 5 . A tes t o f whethe r th e paramete r fo r th e compari so no f treatment s i nth e firs t perio d(Q c = 1.22 , df=l , p=0.27 ) wa s nonsignificant , andthi s enable d a fina l mode l t o be fit . Th e fina l mode l containe d onl y anoveral l constan t andanincremen t fo r placeb o i nth e secondperiod . Goodnes s o f fi t fo r thi s mode l (Qu= 1.93 , df=2 , p=0.38 ) wa s reasonabl e eve nthoughth e paramete r spac e wa s reduce din dimension . Estimate s an

dthei r standar derror s fo r th e respectiv e model s ar e presente dfo r thi s exampl e i nTabl e 6 . Acknowledgement s Thi s researc hwa s partiall y supporte db

y th e U.S . Burea uo f th e Censu s throug hJoin t Statistica l Agreemen t JSA84-5 , EP A

Co-operativ e Agreement C R807392 , an

dHealt hEffect s Institut e Contrac t Numbe r 85-9 . Reference s Brown , B.W . [1980] . Th e crossove r experimen t fo r clinica l trials . Biometric s 36 , 69-79 . Grizzle , J.E . [1965] . Th e two-perio dchange-ove r desig nandit s us e i nclinica l trials . Biometric s 21 , 467-480 . Koch , G.G . [1972] . Th e us e o f non-parametri c method s i nth e statistica l analysi s o f th e two perio dchange-ove r design . Biometric s 28 . 577-584 . Koch , G.G. , Gitomer , S.L. , Skalland , L. , Stokes , M.E . [1983] . Some non-parametri c andcategorica l dat a analysi s fo r a change-ove r desig nstud y an ddiscus sio no f apparan t carry-ove r effects . Statistic s i nMedicin e 2 , 397-412 . Stanish , W.M. , Koch , G.G. , Landis , J.R . [1978] . A compute r progra

m fo r multivariat e rati o analysi s (MISCAT) . Compute r Program s i nBiomedicine8 , 197-207 .

. *



### Tabl e 1 . Mode l Parameterization s fo r Exampl e 1

Model s

TA C TCA Preliminar y

1 2
\1 + Tt Fina l 1 y+ U p + T
W + Tt , + T U + TT j + X TA

C \i v \i +
. TC Ayy+ Ty

10 6 

### f 

# t JP. 6

wit h eriod s riluce d tory

. e l hel p an d re rio d o re r p a ram i e 5

. npari

1.22 , lable d taine d i t fo r .onabl e n )r s fo r .'xampl e

eU.S

. 1


### c t

fo r
ove r atric s

two
7-584. M.E

. at a cus 3tic s 1
. A iysi s 197-207

. Tabl e 2

. Statistica l Test s fo rExampl e 1
Mode l Tes t Q cd f p-valu e

H 0 : A 0 0.7 8 1 0.37 8 Preliminar yH Q : i ^

= ir 2 = 0 1.9 9 2 0.36 9 H Q : T-' O 3.3 6 1 0.06 7 Fina l T= 0 9.4 0 1 0.00 2

Tabl e 3

. Estimate s o f Logarithm s o f (Events/Tim e a t Risk ) Ratio s fo rExampl e 1

Model s

Per . 0 1 2 0 1 2 Treat .

T A P T P A Identit y
Estimat e -6.3 1 -6.3 9 -6.2 0 -6.3 7 -5.9 8 -6.3 1 Standar d

Erro r 0.2 7 0.2 9 0.2 6 0.3 2 0.2 4 0.3 2 Prelimi

Estimat e -6.3 3 -6.4 0 -6.2 2 -6.3 3 -5.9 6 -6.2 9 nary

Standar d Erro r 0.2 1 0.2 6 0.2 1 0.2 1 . 0.1 8 0.2 6 Fina l

Estimat e -6.2 3 -6.2 3 -6.0 3 -6.2 3 -6.0 3 -5.2 3 Standar d

Erro r 0.1 8 0.1 8 0.1 7 0.1 8 0.1 7 0.1 8

Tabl e 4 . Mode l Parameterization s fo r Exampl e 2

Model s
Identit y 1 2 Reduce d



### 1 Fina l 1 A P P A v

.
y 3 ^ 2 

## H A P P A y
y + T. , y

+ T 2 y A

P P A y y y + T y



### 10 7W^iT^w:.

Tabl e 5 . Statistica l Test s fo r Exampl e 2 P. 7

Mode l Tes t Contras t Q

_d f p-valu e

H 0 : y l = u 2 = y 3 =VJ 4 1 0 0- 1

0 10- 1 0 0 1- 1 14.0 9 3 <0.0 1

Identit y

Reduced

Fina l H 0 :

H 0 : H 

## o : H

## o :
H

## o : H 0 : H Q : H Q : Averag e Perio d Averag e Trmt

. Trmt

. x

V

. =v 4
T 2 = 0 

# v °

T l =T 2 T = 0 ( 1 - 1 1 -1 )

( 1 - 1 - 1 1 ) Perio d ( 1 1-1-1 ) ( 1 0 0 -1 ) ( 0 1 0 ) ( 0 0 1 ) ( 0 1 -1 ) ( 0 1 ) 2.4 7

10.8 7 6.9 2 0.71
13.2 6 1.2 2 6.4 6
12.1 5 0.1 2

<0.0 1 0.0 1 0.4 0
<0.0 1 0.2 7 0.0 1
1 <0.0 1 Summ : I i tion : care : been rand i th e i o f t l denc p Haen : tes t wit h grou rat e con s 3 % t 0.0 5 be n tum o til e t o t dis t wil l Sta t an d

Tabl e 6 . Estimate s o f Logarithm s o f (Events/Tim e a t Risk ) Ratio s fo r Exampl e 2 The ' fo r in e

Model s

Per

. 1 2 1 2 Treat

.

A P P A Identit y

Estimat e -3.3 0 -4.7 3 -3.5 5 -3.0 4 Standar d

Erro r 0.2 4 0.4 2 0.24

. 0.1 9 Reduced

Estimat e -3.1 4 -4.7 1 -3.4 9 -3.1 4 Standar d

Erro r 0.1 5 0.41 0.2 3 0.1 5 Fina l Standar d

Estimat e Erro r -3.2 6 0.1 0 -4.7 3 0.41 -3.2 6 0.1 0 -3.2 6 0.1 0 Num b Tu n

10 8