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AMERICAN STATISTICAL ASSOCIATION 

[

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Jcfc 

1986 

PROCEEDINGS 

of the 

n|J?.$ R'o V ? r 

BIOPHARMACEUTICAL SECTION 

Papers presented at the Annual Meeting of the 

American Statistical Association, Chicago, Illinois, 

August 15-18, 1986, under the sponsorship of the 

Biopharmaceutical Section 

BUDfdP 

Qaot^ 

AMERICA N STATISTICA L ASSOCIATIO N 

8 0 6 15t h Street , N.W . Washington , D.C . 2000 5 
Analysis of (Events/Time at Risk) Ratios From Two-Period Crossover Studies 

Jill Bissette, Greg Carr, Gary G. Koch, Kirkwood F. Adams, David S. Sheps 

Department of Biostatistics, University of North Carolina, Chapel Hill 

P.3 

Abstract 

For some two-period crossover studies, there is 

interest in a time to event response variable with 

potential censoring for the time at risk. For 

example, a gastrointestinal study might be concerned with time to relief of heartburn for 

patients who consumed two symptom-provoking meals 

on two different days with two different treatments. In this paper, the usage of weighted 

least squares methods to analyze (events/time at 

risk) ratios is discussed. Two examples are provided for illustrative purposes. 

1. Introduction 

The two-period crossover design is often used to 

compare treatments when it is desirable to have 

subjects serve as their own controls. Subjects 

are randomly assigned to one of two sequence 

groups for two successive treatment periods. If 

the study seeks to compare an active treatment 

with a placebo, then one sequence group receives 

the active treatment in the first period and the 

placebo in the second period. The remaining sequence group receives the placebo in the first 

period and the active treatment in the second period. 

Treatment, period, and (treatment x period) interaction (i.e. carryover effects) are the sources 

of variation of interest. If there are no carryover effects, the interpretation of treatment comparisons is straightforward. If carryover effects 

do exist, then the interpretation of findings is 

somewhat more difficult as the treatment and period 

effects are no longer additive. [Grizzle, (1965), 

Brown (1980), Koch, Gitomer, and Skalland (1983)]. 

In crossover studies, baseline measurements are 

sometimes taken before the treatment sequence is 

administered. The baseline measurements are helpful in strengthening the tests of treatment and 

carryover effects. When a baseline measurement is 

taken for each subject before the treatment sequence is administered, the study may be said to 

have a two-period crossover design with baseline. 

Standard parametric methods may be used to analyze the data from a crossover design when assumptions involving normal distributions apply [Grizzle 

(1965)]. If the sample size is small and normality does not seem reasonable, then nonparametric 

methods may be applied to the ranks of certain • 

sums and differences of the responses of each subject [Koch (1972)]. In the case of time to event 

data with possible right hand censoring, other 

methods are needed in order to account for the 

variation of time at risk both within subjects and 

across subjects. This paper describes one way of 

addressing this question. It involves the description of time to event data with summary measures 

for the number of events per unit time at risk and 

the analysis of their variation through weighted 

least squares methods. 

2. Weighted Least Souares Methods for Model Fitting 

For time to event data, a useful summary measure 

is the ratio: 

^ _ number of people experiencing the event 

total time at risk 

This ratio is often called an incidence density 

rate since it gives the number of events per unit 

time. When the time to event data have an exponential distribution, it is the maximum likelihood 

estimator for the hazard (scale) parameter. Also, 

the rate X can often be useful in more general 

situations. For crossover studies with moderate 

sample sizes, this rate can be determined for each 

period for each sequence group together with the 

corresponding covariance matrix. 

Define: 

1 if the j-th subject in the i-th 

treatment sequence experiences the 

yijk ~ ' event during the k-th period 

[o otherwise 

time at risk for the j-th subject 1nthe 

i-th treatment sequence during the 

k-th period 

xijk 

ni 

i = 1,2 = s 

j = 1,2 

k = 0,1,2 = d (0 for baseline) 

For multi-center studies or other situations where 

subjects are from two or more strata, the i subscript is generalized to hi where the h=l,2,...,q 

indexes the strata. 

For d=2 periods and baseline, construct the 

multivariate response vector 

?ij

 = ^ijO' Aij0' 'ijl' A i j T J'ij2' 

the mean vectors n. 

j=l 

x ij2] 

h 

and covariance matrices 

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

j=l 

For the composite vector "g = ?•.' • ? 2 ' . 

estimated covariance matrix Vg has block diagonal 

structure with the {V^.} as the diagonal blocks. 

-21 

When there is only one stratum, the matrix VW is 

given by 

the 

-i 0 

% 

For two period studies, each of the {Vg^} is (4x4); 

and for two period studies with baseline, each is (6x6). 

Once the vector of means, g^ has been_ created, 

it is of interest to form the ratios (y-j/x,) for 

each (period x sequence). These computations can 

be expressed asj_ _ _ _ 

?i

 = ^iO^iO ' y i l / x i T yi2 / x i 2 ] " ? = [ M] 

if the study contains a baseline measurement and 

only one stratum. The ratios R provide a way to 

deal with the time to event data during the exposure periods when right censoring is a possible 

outcome. Also, it can be desirable to base analyses on the logarithms of the ratio of the means 7, 

and X-j so as to enhance the applicability of normal 

i 

104 
P.4 

3 

- unit 

ponenx)d 

Also, 

'al 

'ate 

3Ch 

the 

:h 

; the 

.vhere 

ub-

..,q 

the 

mai 

s. 

is 

x4); 

6x6). 

ed, 

for 

can 

nd 

to 

DOilyrma! 

.: in the 

•e w f 

approximations for the resulting distributions. 

Sample sizes of 10 or more in each sequence group 

can reasonably support approximate normality of 

the logarithms of the ratios R and thereby provide 

a suitable framework for statistical tests. 

A vector of functions, f, such as the ratios B 

can be created from J by using matrix operations 

involving linear, logarithmic and exponential 

operations. This construction can be implemented 

using the program MISCAT [Stanish, Koch and Landis 

(1978)]. The logarithms F of (events/time at risk) 

ratios are determined as the functions f=A Ujg (g). 

Their covariance matrix is consistently estimated 

through VF.^ftCPg)~*Vg(Dg)-xA^ where A=[1.-1](B) l2Su 

with u=d for studies without baseline and u=d+l 

for those with baseline. The symbol ® denotes 

Kronecker product multiplication in which the 

matrix on the left multiplies every element ofthe 

matrix on the right; Vg is the estimated covariance 

matrix for g and (Dg~)~~is a diagonal matrix with 

g on the diagonal. -

~ A linear model to describe the variation among 

the functions F can be expressed as £A(F ) = Xg 

where X is the~(su x t) model specification matrix 

with full rank t <. su and g is the (t x 1) vector 

of unknown parameters; also, E^O denotes expected 

values for large samples. The~ weighted least 

squares estimates, b, for g are given by 

b = (Op^PrVp'F . for which E&(b) = § and Varfi(b) 

is consistently estimated by Vb = (X'V^X)" 1. 

Goodness of fi t can be assessed by the Wald Statisti c 

Qw = (E - XbJ'VpMf - Xb) = F'W'(WyFW')_,W F where 

W' is an orthocomplement to X (i.e. ,~WX=0, Rank(W)=? 

(su-t); the statisti c Qy is distributed approximately as chi-square with (su-t) = (su - Rank X) 

degrees of freedom under HQ. For models which ~ 

adequately describe the variation in the data, 

linear hypotheses of the form HQ: Cg=Q where C 

is a (c x t) matrix of ful l rank c may be tested 

via the statistic , Qc = b/Cl£VbC')_1£fe;

 tn e

 statisti c Qr is approximately distributed as chisquare with c degrees of freedom under HQ. 

3. Examples 

Two examples of crossover studies are presented. 

The first example is from a two-period crossover 

design with baseline. Subjects with documented 

ischemia were exposed to 100 parts per million 

carbon monoxide on a randomized exposure day. 

Time to onset of angina was the event of interest. 

Carryover and period effects were not present but 

treatment effects were significant. 

The second example is from a two-period crossover study which was conducted at two different 

treatment centers. Study protocol was identical 

at each center. Subjects prone to heartburn ate 

symptom-provoking meals with the meals corresponding to the two periods. The event of interest was 

whether the subject would experience relief from 

heartburn within 20 minutes of the treatment 

administered. 

3.1 Example 1: Cardiology Study with Baseline 

Thirty subjects with documented ischemia participated in a 3 day crossover study. The first 

day was a training day (baseline) to familiarize 

the subjects with the study equipment and exercise 

protocol. The next 2 days were exposure days. 

Subjects were randomly assigned, upon entry into 

the study, to either the AC or the CA treatment 

sequence where A indicates exposure to chamber air 

and C indicates exposure to chamber air mixed with 

100 parts per million (ppm) carbon monoxide. 

The event of interest in all periods was whether 

or not the subject would experience chest pain 

(angina). Patients performed supine exercise 

until angina or several other defined events which 

terminated the exercise period. 

For all subjects the time to angina (TTANG), an 

indicator for the occurrence of angina (ANG), and 

the duration of exercise (DUR) were recorded. The 

event indicator variable, y, was identical to the 

variable ANG. The time at risk variable, x, took 

the value of TTANG for those experiencing angina 

and DUR for those who did not. 

- The analysis was_based on^ the construction of 

f=A log (g) where g = [g^.g^ ] ' and fl= [1 -1] (x)I6. 

The consistent estimator, Yf, for Var F 1s given as 

Vp = MlU^VBg)"^ ' where Y- is~the estimated 

covariance matrix for g" and Dg- is a diagonal matrix 

with g on the diagonaK 

The~preliminary model fit to the data included 

parameters for period 1 and period 2 effects, treatment effects and carryover effects. Subjects 1n the 

different treatment sequences were assumed to be 

equivalent at baseline. Model parameterizations 

for example 1 are presented in Table 1. 

The goodness of fit for the preliminary model 1s 

the same as a test of subjects within treatment 

sequences being equivalent at baseline (Qy = 0.02 

df=l). The nonsignificance of this test (p=0.90) 

indicates a good fit to the data and supports baseline equivalence of the subjects in the two treatment sequence groups. 

Tests of carryover effects (Qc = 0.78,df=l) and 

for period 1 and period 2 effects (QQ = 1.99, df=2) 

were both nonsignificant (p=0.38 and p=0.37, respectively). Treatment effects for this model approached 

significance (Qc = 3.36,.df=l, p=0.07). 

A final model which eliminated carryover and 

period effects was then fit. The model provided a 

good fit to the data (Qw = 2.35, df=4, p=0.67);and 

a test of treatment effects (Qr = 9.40, df=l) was 

highly significant with p < 0.01. Tests of the parameters for the respective models are presented in Table 2. 

In conclusion, subjects experienced angina earlier 

on days with 100 ppm carbon monoxide exposure than on 

chamber air exposure days. There were no period 

effects and no carryover effects (i.e. no (exposure 

x period) interaction). Estimates and their standard 

errors for the respective models are presented in Table 3. 

3.2 Example 2: Gastrointestinal Study 

Two centers participated in a two-period crossover 

study which dealt with the relief from heartburn 

after a symptom-provoking meal. The meals corresponded to the two periods. Of the thirty patients 

who participated at each center, fifteen received 

the active/placebo (AP) treatment sequence while 

the other fifteen received the placebo/active (PA) 

treatment sequence. Study protocol was the same at 

each period and at each center. After the meal was 

ingested, subjects were given their appropriate 

treatment. Time to relief from heartburn was noted 

if relief actually did occur. The two periods were one 

week apart and the study was doubl e bl i nd at each center. 

If the subject experienced relief from heartburn 

in less than 20 minutes from the time of treatment, 

105 
P.5 

the event indicator variable, y, took the value of 

s-i one. Otherwise, a value of zero was recorded for 

V that variable.. The time at risk with heartburn or 

time to relief from heartburn was the second variable constructed. This variable, x, took the value 

of the actual number of minutes before relief occurred for those experiencing relief in less than 

20 minutes. For those who did not experience relief in the allocated time, a value of 20 was 

recorded. The data for this example are given in 

Koch, Gitomer, and Ska11 and [1983J. 

In order to enhance the applicability of normal 

approximations to the distributions of the logarithms of (events/time at risk) ratios, the analysis was directed at the across site averages 

W - hi lQ9 [fii(g)] where § represents the compound 

vector ofsample means for the two sequence groups 

at the two_ sites and has the structure 

S

 = Lln''9i2''92 r922' ]

'

 w lt h 

?hi

 = EhlT XhiT h.l' xh12]

'

 fo r

 h»1.2and 1-1.2; 

also A. = [0.5(I8), 0.5(I 8 )]; and A,= [l,-1] ®I 4 . 

The variance of £ is consistently estimated with, 

h " *2(5al

,"1-W(5al,",fi2' ' where % is the 

estimated covariance matrix for q~, and D , is a 

2' ~al 

diagonal matrix with a, = log[A,(g")] on tRe diagonal. 

Since no apriori model was specified, the initial 

model fit, F = Xg, was the cell mean model with 

X=X4- The ce11~mean model allows no reduction in 

the dimension of F and so there is no lack of fit 

by definition. Model parameterizations for this 

example are presented in Table 4. 

An initial test of the hypothesis 

H«: p1=u_=y-=y. of no variation among the four 

(period x treatment) logarithms of (events/time at 

risk) ratios (Qr = 14.09, df=3) was significant 

(p < 0.01). Additional testing found that average 

treatment effects (Qc = 10.87, df=l) and carryover 

effects (Qc = 6.92, df=l) were both significant 

(p < 0.01) while period effects (Qc = 2.47, df=l) 

were not. A test of equality of active treatments 

across periods was nonsignificant as well (Qc = 

0.71, df=l, p=0.40). 

A reduced model without period effects and with 

separate effects for placebo during the two periods 

was then fit. The goodness of fit for the reduced 

model (Qw = 0.71, df=l, p=0.40) was satisfactory. 

The parameter estimates from the reduced model help 

clarify the nature of the treatment effects and 

their interaction with period. Subjects who received the active treatment in the second period 

had more favorable experiences than those who received placebo in the second period or either 

treatment in the first period. Tests of model parameters are summarized for this example in Table 5. 

A test of whether the parameter for the comparison of treatments in the first period (Qc = 1.22, 

df=l, p=0.27) was nonsignificant, and this enabled 

a final model to be fit. The final model contained 

only an overall constant and an increment for 

placebo in the second period. Goodness of fit for 

this model (Qu = 1.93, df=2, p=0.38) was reasonable 

even though the parameter space was reduced in 

dimension. Estimates and their standard errors for 

the respective models are presented for this example 

in Table 6. 

Acknowledgements 

This research was partially supported by the U.S. 

Bureau of the Census through Joint Statistical 

Agreement JSA84-5, EPA Co-operative Agreement 

CR 807392, and Health Effects Institute Contract 

Number 85-9. 

References 

Brown, B.W. [1980]. The crossover experiment for 

clinical trials. Biometrics 36, 69-79. 

Grizzle, J.E. [1965]. The two-period change-over 

design and its use in clinical trials. Biometrics 

21, 467-480. 

Koch, G.G. [1972]. The use of non-parametric 

methods in the statistical analysis of the twoperiod change-over design. Biometrics 28. 577-584. 

Koch, G.G., Gitomer, S.L., Skalland, L., Stokes, M.E. 

[1983]. Some non-parametric and categorical data 

analysis for a change-over design study and discussion of apparant carry-over effects. Statistics 

in Medicine 2, 397-412. 

Stanish, W.M., Koch, G.G., Landis, J.R. [1978]. A 

computer program for multivariate ratio analysis 

(MISCAT). Computer Programs in Biomedicine8, 197-207. 

.* 

Table 1. Model Parameterizations for Example 1 

Models 

TAC 

TCA 

Preliminary 

1 2 

\1 + Tt 

Final 

1 

y + Up + T 

W + Tt, + T U + TTj + X 

TAC \i v \i + . 

TCA y y + T y 

106 f

tJ 
P.6 

with 

eriods 

riluced 

tory. 

el help 

and 

reriod 

o rer 

pa ramie 5. 

npari1.22, 

labled 

tained 

it for 

.onable 

n 

)rs for 

.'xample 

eU.S. 

1 

ct 

for 

over 

atrics 

two7-584. 

M.E. 

ata 

cus3tics 

1. A 

iysis 

197-207. 

Table 2. Statistical Tests for Example 1 

Model Test Qc df p-value 

H0: A 0 0.78 1 0.378 

Preliminary HQ: i^ = ir2 = 0 1.99 2 0.369 

HQ: T-'O 3.36 1 0.067 

Final T = 0 9.40 1 0.002 

Table 3. Estimates of Logarithms of (Events/Time at Risk) Ratios for Example 1 

Models 

Per. 

0 

1 

2 

0 

1 

2 

Treat. 

T 

A 

P 

T 

P 

A 

Identity 

Estimate 

-6.31 

-6.39 

-6.20 

-6.37 

-5.98 

-6.31 

Standard 

Error 

0.27 

0.29 

0.26 

0.32 

0.24 

0.32 

Prelimi 

Estimate 

-6.33 

-6.40 

-6.22 

-6.33 

-5.96 

-6.29 

nary 

Standard 

Error 

0.21 

0.26 

0.21 

0.21 . 

0.18 

0.26 

Final 

Estimate 

-6.23 

-6.23 

-6.03 

-6.23 

-6.03 

-5.23 

Standard 

Error 

0.18 

0.18 

0.17 

0.18 

0.17 

0.18 

Table 4. Model Parameterizations for Example 2 

Models 

Identity 

1 2 

Reduced 

1 

Final 

1 

AP 

PA 

v

. 

y3 

^2 

H 

AP 

PA 

y 

y + T., 

y + T 2 

y 

AP 

PA 

y 

y 

y + T 

y 

107 
W^iT^w:. 

Table 5. Statistical Tests for Example 2 

P.7 

Model Test Contrast Q_ df p-value 

H 0: yl= u2= y3=VJ4 1 0 0-1 

0 10- 1 

0 0 1-1 

14.09 3 <0.01 

Identity 

Reduced 

Final 

H

0

: 

H

0

: 

H

o

: 

H

o

: 

H

o

: 

H0: 

HQ: 

HQ: 

Average Period 

Average Trmt. 

Trmt. x 

V

.

=v4 

T2 = 0 

v° 

T

l

 =T 2 

T = 0 

(1 -1 1 -1) 

(1 -1 -1 1) 

Period (1 1-1-1 ) 

( 1 0 0 -1) 

(0 1 0) 

(0 0 1) 

(0 1 -1) 

(0 1) 

2.47 

10.87 

6.92 

0.71 

13.26 

1.22 

6.46 

12.15 

0.12 

<0.01 

0.01 

0.40 

<0.01 

0.27 

0.01 

1 <0.01 

Summ: 

Ii 

tion: 

care: 

been 

randi 

the i 

of tl 

dencp 

Haen: 

test 

with 

grourate 

cons 

3% t 

0.05 

be n 

tumo 

tile 

to t 

dist 

will 

Stat 

and 

Table 6. Estimates of Logarithms of (Events/Time at Risk) Ratios for Example 2 

The' 

for 

in e 

Models 

Per. 

1 

2 

1 

2 

Treat. 

A 

P 

P 

A 

Identity 

Estimate 

-3.30 

-4.73 

-3.55 

-3.04 

Standard 

Error 

0.24 

0.42 

0.24. 

0.19 

Reduced 

Estimate 

-3.14 

-4.71 

-3.49 

-3.14 

Standard 

Error 

0.15 

0.41 

0.23 

0.15 

Final 

Standard 

Estimate Error 

-3.26 0.10 

-4.73 0.41 

-3.26 0.10 

-3.26 0.10 

Numb 

Tun 

108