Assessing bias of multicenter trials with incomplete treatment allocation

Journal of Statistical Planning and Inference 96 (2001) 83–107

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Assessing bias of multicenter trials with incomplete treatment allocation a Division

Stuart A. Ganskya;∗ , Gary G. Kochb;1

of Oral Epidemiology & Dental Public Health, School of Dentistry, University of California, 707 Parnassus Avenue, San Francisco, CA 94143-1361, USA b Department of Biostatistics, School of Public Health, University of North Carolina, Chapel Hill, NC 27599-7400, USA

Abstract Some multicenter randomized controlled trials (e.g. for rare diseases or with slow recruitment) involve many centers with few patients in each. Under within-center randomization, some centers might not assign each treatment to at least one patient; hence, such centers have no within-center treatment e1ect estimates and the center-strati2ed treatment e1ect estimate can be ine3cient, perhaps to an extent with statistical and ethical implications. Recently, combining complete and incomplete centers with a priori weights has been suggested. However, a concern is whether using the incomplete centers increases bias. To study this concern, an approach with randomization models for a 2nite population was used to evaluate bias of the usual complete center estimator, the simple center-ignoring estimator, and the weighted estimator combining complete and incomplete centers. The situation with two treatments and many centers, each with either one or two patients, was evaluated. Various patient accrual mechanisms were considered, including one involving selection bias. The usual complete center estimator and the weighted estimator were unbiased under the overall null hypothesis, even with selection bias. An actual c 2001 Elsevier Science B.V. dermatology clinical trial motivates and illustrates these methods.
All rights reserved. MSC: 62G35; 62K99; 92C50; 92B15 Keywords: Selection bias; Randomization model; Patient accrual; Missing data; Weighted estimator; Randomized clinical trial

1. Introduction Clinical trials sometimes utilize a large number of centers with each having a relatively small number of patients. Usually, these multicenter studies have within-center randomization, which can ease problems such as certain centers withdrawing from the ∗

Corresponding author. Tel.: +1-415-502-8094; fax: +1-415-502-8447. E-mail address: [email protected] (S.A. Gansky). 1 Tel.: +1-919-966-7282.

c 2001 Elsevier Science B.V. All rights reserved. 0378-3758/01/$ - see front matter  PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 3 2 7 - X

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study or adhering to the study protocol di1erently; thus, despite any potential di1erences among centers, patients are nearly equally split amongst treatments both within centers and overall. However, a methodologic dilemma can arise when some centers have no patients assigned to some treatments (i.e. incomplete designs). Although proper analysis should be based on the actual study design, the strati2cation applied during design can be too extensive to be fully incorporated in analyses without losing e3ciency. The main dilemma from sparse strata is fewer patients per stratum with empty cross-classi2ed cells as the extreme. With fully strati2ed analyses, discarding patients in incomplete strata can lead to ine3ciency and to ethical problems from excluding treated patients’ data. An actual randomized controlled trial (RCT) (modi2ed to protect con2dentiality for proprietary reasons) motivated these methods. A multicenter, double-blind, randomized parallel groups trial of two medication regimens for a dermatological disorder was performed in 426 adults. One of two treatment regimens (standard or test) was assigned to each patient within 232 study sites. Centers enrolled one or two patients, so that 16% had incomplete assignment; i.e. both treatment groups were not assigned. The response variable was the natural log of the number of days (2–7, 10, 14, or 21) until rash subsided. This situation with two treatments and many centers, each with either two patients (one per treatment) or one patient, is the same as a matched pairs design with missing data in the centers with only one patient (called singleton or unpaired centers). Such data would usually be missing completely at random (MCAR) (Little and Rubin, 1987) since missingness is only related to the presumably random fact that the center had one patient instead of two; missingness would be non-random only if the center did not enroll the second patient because of some circumstance related to the 2rst patient’s baseline status or response outcome or to the second patient’s baseline status. In each center with two treatments, the 2rst treatment assignment is random, while the second is for the other treatment (i.e., deterministic). For convenience, refer to the pattern of within-center treatment assignment as the accrual “order”. Although this incomplete pairs structure may appear simple, it actually has more estimability problems than scenarios with more than two patients per center, as in an AB=BA crossover trial. Various authors have examined model-based methods for incomplete paired data (e.g. Lachenbruch and Myers, 1983; Ekbohm, 1976; Lin and Stivers, 1974); some have advocated using weighted combinations of the paired and unpaired data for better e3ciency (Bhoj, 1978, 1991; Gansky and Koch, 1996), but some questions about bias remain. Minimal assumption methods which draw conclusions about the particular sample randomized could have more validity in this scenario than the model-based methods. Bouza (1983) addressed the situation of incomplete paired data from a survey sampling perspective with a 2nite population model. Although RCTs are not probability samples or simple random samples from populations of known size but actually convenience samples with random allocation to treatments, many analytic approaches (e.g. random e1ects models and generalized estimating equations) assume patients represent some population of interest through hypothetical arguments. Although Bouza’s

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approach with optimal weight derivation applies for multicenter RCTs with incomplete assignment, it ignores the complete pair assignment order. Bouza subsamples missing data to address nonresponse, which is unethical in RCTs (i.e., patient participation is voluntary). In addition, Wei (1983) developed a class of rank tests for interchangeability of two correlated and possibly incomplete responses in a similar spirit to the above weighted methods combining paired and unpaired data. After ranking pooled responses, two independent components—a paired quantity and an unpaired one—are combined with weights based on the proportion of pairs, but without accounting for pair assignment order. Various studies (e.g. VACURG, 1967; Eisenhauer et al., 1994) had patient baseline values change during recruitment which may bias combining complete and incomplete centers. Thus, clinical trials involving two treatments with one or two patients in each of many centers, can be examined under a 2nite population framework with the indicator function method, as in Corn2eld (1944), to permit a thorough bias investigation. Letting h = 1; 2; : : : ; H index the centers, i = 1; 2 index accrual order of patients in the hth center, and j = A; B index the treatments, consider yhij , the response of the ith patient (i.e., the patient in the ith accrual order) in the hth center if assigned to the jth treatment, as a 2xed constant for each patient. (Note only one j is actually observed per (h; i)). Two indicator functions, one as a recruitment index (Lh ) and the other as a treatment assignment index (Uh ), are used to develop a randomization model:
1 if center h has 2 patients (paired); Lh = 0 if center h has 1 patient (singleton=unpaired);
1 if patient 1 in center h is assigned treatment A; Uh = 0 if patient 1 in center h is assigned treatment B;
1 if center h assigned treatments in order A : B; = 0 if center h assigned treatments in order B : A: Lh and Uh can be independent study design components. Sometimes, the Lh are 2xed by the study design; other times they are consequences. Usually the Lh and Uh are assumed statistically independent. When the Lh are not independent of the Uh , compatibility of unpaired=singleton and paired centers is a general concern; speci2cally selection bias poses a problem. Regardless of the relationship between Lh and Uh , the Uh are evaluated after 2xing the Lh , either via design or conditional expectations. The number of centers (H ) is 2xed. The following indicator sums and 2gure relate to sample sizes:   Lh = np ; (1 − Lh ) = ns ; Lh h h   (1 − Lh )Uh = nSA ; (1 − Lh )(1 − Uh ) = nSB ; 0 1 h h 0 nSB nPB nB   Lh Uh = nPA ; Lh (1 − Uh ) = nPB ; Uh h h 1 nSA nPA nA   Uh = nA ; (1 − Uh ) = nB ; nS np H h

h

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De2ne the following group sums: SP·A ≡ np yQ P·A = nPA yQ P1A + nPB yQ P2A =

 h

{Lh Uh yh1A + Lh (1 − Uh )yh2A }

= SP1A + SP2A ; SP·B ≡ np yQ P·B = nPB yQ P1B + nPA yQ P2B =

 h

{Lh (1 − Uh )yh1B + Lh Uh yh2B }

= SP1B + SP2B ; SS1A ≡ nSA yQ S1A = SS1B ≡ nSB yQ S1B =

 h

 h

(1 − Lh )Uh yh1A ; (1 − Lh )(1 − Uh )yh1B ;

where Sgij is the sum for the gth group (g = P for paired centers; g = S for singleton= unpaired centers) with assignment to the jth treatment in the ith order and ngj is the number of centers in the gth group which assigned treatment j 2rst. Then, the estimated treatment di1erences accounting for allocation order, separately for paired and unpaired centers, are as follows. For paired centers,   Lh (1 − Uh )(yh2A − yh1B ) h (yh1A − yh2B ) h Lh U  + h dQp = 2 h Lh Uh 2 h Lh (1 − Uh ) =

SP1A − SP2B SP2A − SP1B + 2nPA 2nPB

(1)

and for singleton=unpaired centers   h (1 − Lh )Uh yh1A h (1 − Lh )(1 − Uh )yh1B −  dQs =  (1 − L )U h h h h (1 − Lh )(1 − Uh ) = {SS1A =nSA − SS1B =nSB }:

(2)

An overall treatment di1erence estimator adjusting for the accrual order is a weighted sum of ratio estimators with a priori weights w: dQ = wdQp + (1 − w) dQs ;

(3)

where w can be chosen from various options, such as the proportion of complete centers or the proportion of patients in complete centers (Koch and Gansky, 1996; Gansky and Koch, 1996). An alternate, heuristic estimator is the simple di1erence of treatment means dQ∗ = =

H−

+ SP·A S SS1B + SP·B  S1A  − (1 − L )(1 − U ) H − h h h h (1 − Lh )Uh

SS1A + SP·A SS1B + SP·B − ; H − nSB H − nSA

(4)

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which directly uses the sums de2ned earlier, instead of combining separate estimators for the paired and singleton=unpaired center treatment di1erences, but it ignores allocation order. For these estimators, the nonnull scenario is evaluated, along with two null hypothesis scenarios, to assess bias under various conditions via the indicator functions. The nonnull hypothesis HA is that patients assigned to one treatment have a greater preponderance of larger responses than patients assigned to the other treatment. The general null hypothesis H0g : yhiA = yhiB = yhi is the response for the ith patient in the hth center is equivalent for the two possible treatment assignments. Alternatively, relative to an underlying full model E(y) =  +

h

+ i + j + ( )hi + ( )hj + ()ij + ( )hij ;

where  is the overall mean, h are the center e1ects, i are the accrual (period=order) e1ects, j are the treatment e1ects, (••)•• are the pairwise interactions, and (• • •)••• are the three-way interactions, this hypothesis is H0g : A − B + ()iA − ()iB + ( )hA − ( )hB + ( )hiA − ( )hiB = 0: The parameters of the full model are restricted to sum to zero over any complete      index; e.g. h h = 0. The average null hypothesis, H0a : h i yhiA = h i yhiB (or alternatively H0a : A = B ), is that the responses are equivalent for the two possible treatment assignments after aggregating over center and accrual order. 2. Fixed patient accrual and random treatment allocation The 2rst conditions are that the patient accrual indices (Lh ) are 2xed constants by design and that the allocation indices (Uh ) are independent and identically distributed (i.i.d.) according to a Bernoulli distribution with probability one-half, denoted i:i:d: Uh ∼ Bernoulli( 12 ). So, the sum of the allocation indices across centers (the number of patients assigned to treatment A 2rst) is distributed binomially with probability  1 one-half, denoted h Uh ∼ Binomial(H; 2 ). Although np and ns are 2xed since the Lh are 2xed, each particular center has its 2rst patient assigned to treatment A or B at random, making nPA ; nPB ; nSA , and nSB random as well. Although E(Uh ) = 1=2 unconditionally, the following conditional expectations apply: nPA nPB E(Uh | Lh = 1; nPA ; nPB ) = ; E(1 − Uh | Lh = 1; nPA ; nPB ) = ; np np E(Uh | Lh = 0; nSA ; nSB ) =

nSA ; ns

E(1 − Uh | Lh = 0; nSA ; nSB ) =

nSB : ns

These results follow from nPA and nSA each being distributed Binomial(ng ; 12 ), where g = P for paired or g = S for unpaired=singleton. Conditioning on either of them and patient accrual indices (Lh ) induces a hypergeometric distribution for the treatment allocation indicator (Uh ).

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Under the nonnull scenario, HA , that the yhij are distinct for each treatment group, the estimators are evaluated. Expectations of sums, such as  E(SP1A | HA ; nPA ; nPB ) = (nPA =nP ) h Lh yh1A ; when nPA and nPB are 2xed, and  E(SP1A | HA ) = 12 h Lh yh1A , when they are random, are used to evaluate the estimators. (A full table is provided in the Appendix.) The estimators for paired (1) and singleton=unpaired (2) centers have the following expectations. E(dQp | Lh ; HA ) = E nPA ;nPB {E(dQp | Lh ; HA ; nPA ; nPB )}
E(SP1A | Lh ; nPA ; nPB ) − E(SP2B | Lh ; nPA ; nPB ) 1 = E nPA ;nPB 2 nPA  E(SP2A | Lh ; nPA ; nPB ) − E(SP1B | Lh ; nPA ; nPB ) + nPB     1 = Lh yhiA − Lh yhiB 2nP h i h i = yQ P·A − yQ P·B = P·A − P·B = A − B + ( Q )PA − ( Q )PB ;  where ( Q )Pj = (1=np ) h Lh ( )hj . (Details of all other derivations are shown in the Appendix; only results are given in the text.) If there is no center×treatment e1ect or if the center×treatment e1ect “averages out” in the paired centers (as in the parametrization restrictions across all centers), then E(dQp | Lh ; HA ) = A − B , which shows the estimator is unbiased. Alternatively, if one can assume the Lh are random by design or are essentially random phenomena with 2xed np and ns , then E(dQp | HA ; np ; ns ) = A −B : Under the global null hypothesis (H0g ) responses are equivalent for each treatment assignment; thus, the paired estimator is unbiased under the general null without simplifying assumptions. However, under the average null hypothesis (H0a : A = B ), the paired estimator is unbiased only if the center×treatment interaction is null or the Lh are actually random. Table 1 summarizes the assumptions needed for unbiasedness for each estimator and hypothesis. For the singleton=unpaired centers, the expected values are as follows: E(dQs | Lh ; HA ) = A − B + ()1A − ()1B + ( Q )SA − ( Q )SB 

+ ( Q )S1A − ( Q )S1B ;

 where ( Q )Sj = h (1 − Lh )( )hj and ( Q )S1j = h (1 − Lh )( )hij . Assuming no center×treatment and no three-way interactions (or at least that the model restrictions across all centers also hold in the unpaired centers; i.e. ( Q )Sj = 0 and ( Q )S1j = 0), as well as no accrual order×treatment interaction, the singleton estimator is unbiased. Again, an alternative scenario (random Lh ; 2xed np and ns ) can be considered. E(dQs | HA ; np ; ns ) = A − B + ()1A − ()1B ;

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Table 1 Assumptions needed for unbiasedness with 2xed accrual Statistic

Hypothesis

Required assumptions for unbiasednessa

dQp

HA =H0a H0g

Nob center×treatment or Lh essentially random None

dQs

HA =H0a

No interactionsc involving treatment or (no accrual×treatment and Lh essentially random) None

H0g dQ

HA =H0a H0g

dQ∗

HA =H0a

H0g

No interactionsc involving treatment or (no accrual×treatment and Lh essentially random) None (No interactionsc involving treatment and equal allocation in unpaired centers) or (Lh essentially random, no accrual×treatment and no accrual e1ect) or (Lh essentially random, no accrual×treatment and equal allocation in unpaired centers) (No accrual e1ect, no center e1ect and no accrual×center) or equal allocation in unpaired centers

a E(· | H

A ) = A − B ; E(· | H0g ) = 0; E(· | H0a ) = 0 are unbiased. interactions involving center “average out” within those centers as model restrictions do in all centers together. c No accrual×treatment, nob center×treatment and nob 3-way interaction. b Or

which shows unbiased dQs under HA with random Lh and null accrual×treatment effects. Under the general null hypothesis (H0g ), the terms in the previous result are zero, so E(dQs | Lh ; H0g ) = 0; indicating unbiasedness. Under the average null hypothesis (H0a ); A = B , the same assumptions as for HA are needed for unbiasedness. Then, the overall weighted estimator (3) with 2xed w, can be evaluated. Assuming interactions involving centers sum to zero separately for paired and for unpaired= singleton centers, as well as no accrual interaction with treatment (in unpaired=singleton centers), the overall weighted estimator is unbiased. Under the general null hypothesis (H0g ); E(dQ | H0g ) = 0, showing unbiasedness without additional assumptions; under the average null (H0a ) with no accrual×treatment, no center×treatment, and no three-way interaction, E(dQ | H0a ) = 0, showing unbiasedness with the same restrictions as for HA . If nSA and nSB are large, Taylor series linearization (TSL) can be applied, so nSA is approximately the same as nSB and each is approximately ns =2; thus, the heuristic simple di1erence in means estimator (4) has the large sample expectation: np H E(dQ∗ | HA ) ≈ A − B + [()1A − ()1B ] + [()2A − ()2B ] N N np np + [( Q )PA − ( Q )PB ] + [( Q )P2A − ( Q )P2B ]; N N  where N = H + np is the total number of patients and ( Q )Pj = (1=np ) Lh ( )hj . h

So even if nSA = nSB = ns =2, interactions of treatment with accrual, center, and accrual

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by center must be null for the heuristic estimator to be unbiased. Alternatively, assuming random Lh with 2xed np and ns , the heuristic estimator is unbiased if the accrual×treatment interaction is null and either the accrual e1ect is null or treatment allocation is equal (by design or by TSL approximation in large samples). Under the global=general null hypothesis (H0g ), E(dQ∗ | H0g ) = 0; if nSA = nSB or yQ S1 = yQ P· (i:e: Q S − Q P + 1 + ( Q )S1 = 0); where yQ S1 =

1  (1 − Lh )yh1 ; ns h

Q = 1  (1 − Lh ) h ; S ns h

yQ P· =

1  Lh yhi ; 2np h i

Q = 1  Lh P np h

h:

So under the general null, either equal treatment allocation in the unpaired= singleton centers yields unbiasedness or null center, accrual, and accrual×center effects give unbiasedness. Under the average null hypothesis (H0a ); A = B , the same assumptions as for HA are required for unbiasedness: no accrual, no accrual×treatment interaction, and random Lh ; or both no center×treatment and equal unpaired center sample size.

3. Random accrual independent of treatment allocation The next situation involves both random accrual and random treatment allocation with these two components being independent (⊥). This reTects designs which rani:i:d: domly assign the number of patients a particular center will enroll. Suppose Lh ∼  i:i:d: Bernoulli(), Uh ∼ Bernoulli( 12 ), and Lh ⊥Uh ∀h so np = h Lh ∼ Binomial(H; ). As in Section 2, the conditional expectations are calculated; for example, E(Lh Uh | ∼ n ∗ =  nPA =H , with ∼ n ∗ ≡ (nPA ; nPB ; nSA ; nSB ) being distributed multinomially. Conditioning on these group totals induces a multivariate hypergeometric distribution for the indicator functions. Further, the group totals are binomial: nPA ∼ Binomial(H; =2) and nSA ∼ Binomial(H; (1 − )=2). So, the expectations of sums under the nonnull HA for 2xed  and random group totals can be calculated; e.g. E(SP1A | HA ; ∼ n ∗ ) = (nPA =H ) h yh1A  and E(SP1A | HA ) = (=2) h yh1A . Estimators for paired (1) and singleton=unpaired (2) centers have expectations as follows. E(dQp | HA ) = A − B , which is unbiased. Under both the general (H0g ) and average (H0a ) null hypotheses, the paired estimator is unbiased with independent random accrual and allocation. Table 2 summarizes unbiasedness assumptions for estimators and hypotheses for random accrual independent of random allocation. For the unpaired=singleton centers: E(dQs | HA ) = yQ 1A − yQ 1B = A − B + ()1A − ()1B , which is unbiased assuming no accrual×treatment interaction.

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Table 2 Unbiasedness assumptions with random accrual independent of allocation Statistic

Hypothesis

Required assumptions for unbiasednessa

dQp

HA =H0g =H0a

None

dQs

HA =H0a H0g

No accrual×treatment interaction None

dQ

HA =H0a H0g

No accrual×treatment interaction None

dQ∗

HA =H0a

No accrual e1ect or equal group allocationb No accrual×treatment No accrual e1ect or equal group allocationb

H0g a E(· | H bn

PA

A ) = A − B ; E(· | H0g ) = 0; E(· | H0a ) = 0 are unbiased. = nPB and nSA = nSB .

Under the general null hypothesis (H0g ), both the treatment e1ect and accrual× treatment interaction are null (as in Section 2), so the singleton=unpaired estimator is unbiased. Under the second null hypothesis, the average null (H0a ), however, the same assumption as for the nonnull hypothesis is needed for unbiasedness: null accrual×treatment interaction. With 2xed w, the expected value of (3) is E(dQ | HA ) = A − B + (1 − w)[()1A − ()1B ], so null accrual×treatment is needed for the overall weighted estimator to be unbiased. For the general null hypothesis (H0g ), the weighted estimator is unbiased without extra assumptions, while for the average null hypothesis (H0a ), accrual× treatment must be null for unbiasedness. Assuming no accrual and no accrual×treatment e1ects, then   yh1j = yh2j for j = A; B (5) h

h

(or equivalently yQ 1j = yQ 2j or from the underlying model 1 + ()1j = 2 + ()2j ), and the expectation of the heuristic simple di1erence in means estimator (4), ignoring accrual, reduces to E(dQ∗ | HA ) = A − B . If only group totals are equal (i.e. nPA = nPB and nSA = nSB ), no accrual×treatment interaction is still needed for unbiasedness. Under the general null hypothesis (H0g ), E(dQ∗ | H0g ) = 0 if yQ 1· − yQ 2· = 0 or 
nPA (nPA + nSA ) − nPB (nPB + nSB ) = 0; En ∗ ∼ (np + nSA )(np + nSB )  where yQ i· = (1=H ) h yhi =  + i , showing that the heuristic estimator is unbiased under reasonable assumptions: no accrual e1ect (1 = 2 ) or equal group totals (nPA = nPB and nSA = nSB ). Since E(nPA ) = E(nPB ) and E(nSA ) = E(nSB ), a Taylor series linearization yields unbiasedness with large H . Under the average null hypothesis (H0a ), the same assumptions as in the nonnull case are required for unbiasedness: no accrual×

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treatment interaction along with either equal treatment group allocation within paired and within singleton=unpaired centers separately or no accrual e1ect.

4. Accrual dependent on allocation: selection bias Next, the previous section is extended so the accrual and allocation functions are both random but not independent ( ⊥). This permits exploration of selection bias in which patients are allocated to treatments di1erentially. Investigators enrolling patients based on previous patients’ responses or current patients’ underlying conditions can cause imbalances (intentionally or unintentionally) in treatment allocation (Blackwell and Hodges, 1957). A trial enrollment mechanism can introduce bias if not consistently applied to all potential participants in a standardized protocol (Wei and Cowan, 1988). 4.1. General selection bias The scenario in Section 3 can be generalized to allow dependent accrual and treatment allocation indices (i.e. Lh⊥ Uh ); Lh ∼ Bernoulli(Qh ) but they are not i.i.d. (i.e. i:i:d:

each Lh has its own probability parameter Qh ) and Uh ∼ Bernoulli( 12 ), as before. Selection bias can then be modelled separately for each treatment and each center. De2ne the following: Pr{Lh = 1 | Uh = 1} = hA ;

Pr{Lh = 0 | Uh = 1} = 1 − hA ;

Pr{Lh = 1 | Uh = 0} = hB ;

Pr{Lh = 0 | Uh = 0} = 1 − hB :

Qh = (hA + hB )=2

With no selection bias Pr{Uh = 1 | Lh = 1} = 1=2, i.e. equal probability of treatment allocation regardless of the number of patients in the center. More generally, using Bayes Theorem, these conditional probabilities can be rewritten conditioning on the number of patients in each center: hA hA = ; Pr{Uh = 1 | Lh = 1} = hA + hB 2Qh Pr{Uh = 0 | Lh = 1} = Pr{Uh = 1 | Lh = 0} =

hB hB = ; hA + hB 2Qh

1 − hA ; 2 − (hA + hB )

Pr{Uh = 0 | Lh = 0} =

1 − hB : 2 − (hA + hB )

Then under the nonnull hypothesis HA , conditional and unconditional expectations of sums, such as E(SP1A | Lh ; HA ) = apply.

 h

hA Lh yh1A hA + hB

and

E(SP1A | HA ) =

1 hA yh1A ; 2 h

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The group allocation totals have the following expectations: E{nPA } =

1 1 hA ; E{nPB } = hB ; 2 h 2 h

E{nSA } =

1 1 (1 − hB ): (1 − hA ); E{nSB } = 2 h 2 h

But expectations for the paired (1) and singleton=unpaired (2) centers under the nonnull hypothesis cannot be assessed as in the previous two sections because the Lh are not i.i.d. Instead, in large samples, expectations can be approximated with TSL. For the paired estimator E(dQp | HA )



≈ A − B +  +

h hB {(

h hA {(

)h1 − ( )h2 + ( )hA − ( )hB + ( )h1A − ( )h2B }  2 h hA

)h2 − ( )h1 + ( )hA − ( )hB + ( )h2A − ( )h1B }  ; 2 h hB

which is unbiased only with no center×accrual, no center×treatment and no three-way interactions. If hj = h for each j then the probability a center has two patients actually only depends on center characteristics, such as size, reputation, or case mix;   so E(dQp | HA ; hj = h ) ≈ A − B + { h h {( )hA − ( )hB }}={ h h }, which is unbiased if the center×treatment interaction is null. However, if hj = j for all h then the probability a center has two patients only depends on the treatment assigned to the 2rst patient; so the paired estimator is unbiased in large samples. Under the general null hypothesis the paired estimator has approximate unbiased expectation if there is no center×accrual interaction or if either hj = j for all h or hj = h for each j. Under the average null hypothesis, the same assumptions as for the nonnull hypothesis are needed for unbiasedness. Assumptions required for unbiased approximate expected values for all estimators and hypotheses are summarized in Table 3. Next, the estimator for unpaired=singleton centers has the large sample expectation: E(dQs | HA ) ≈ A − B + ()1A − ()1B  (1 − hA ){ h + ( )h1 + ( )hA + ( )h1A }  + h h (1 − hA )  (1 − hB ){ h + ( )h1 + ( )hB + ( )h1B }  − h ; h (1 − hB ) it would be unbiased under assumptions of no center e1ect along with no accrual× treatment, no center×accrual, no center×treatment and no three-way interactions. If hj = h for each j, then the large sample expectation would be unbiased if pairwise

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Table 3 Large sample unbiasedness with random accrual dependent on allocation (general selection bias) Statistic

Hypothesis

Required assumptions for unbiasednessa

dQp

HA =H0a H0g

No interactionsb involving center or hj = j ∀h No center×accrual or hj = h ∀j or hj = j ∀h

dQs

HA =H0a

(No interactions and no center e1ect) or (hj = h ∀j and no interactionsc involving treatment) or (hj = j ∀h and no accrual×treatment) (No center e1ect and no center×accrual) or hj = h ∀j or hj = j ∀h

H0g dQ

HA =H0a H0g

dQ∗

HA =H0a H0g

No interactions and no (unpaired) center e1ect or (hj = h ∀j and no interactionsc involving treatment) or (hj = j ∀h, no accrual, and no accrual×treatment) (No (unpaired) center e1ect and no center×accrual) or hj = h ∀j or hj = j ∀h (No accrual, no center and no interactions) or (hj = h ∀j and no interactionsc involving treatment) or (hj = j ∀h and no accrual×treatment) (No accrual e1ect, no center e1ect and no center×accrual) or hj = h ∀j or (hj = j ∀h and no accrual e1ect)

a E(· | H

A ) ≈ A − B ; E(· | H0g ) ≈ 0; E(· | H0a ) ≈ 0 are asymptotically unbiased. center×accrual, no center×treatment and no 3-way interactions. c No center×treatment, no accrual×treatment and no 3-way interactions. b No

and three-way interactions involving treatment are null. Furthermore, if hj = j for all h then the unpaired=singleton estimator would be unbiased in large samples if the accrual×treatment interaction is actually null. Under the general null hypothesis (H0g ), the singleton=unpaired estimator is unbiased if there is no center e1ect and no center×accrual interaction or if either hj = j for all h or hj = h for each j. The average null hypothesis (H0a ) has the same requirements as the nonnull hypothesis. Then, with 2xed w, the weighted estimator (3) would be unbiased for HA only in the absence of pairwise and three-way interactions as well as center e1ect in the singleton=unpaired centers. If hj = h for each j, then the weighted estimator has large sample unbiasedness if center×treatment is null and in the unpaired=singleton centers the accrual×treatment and the three-way interactions are null. If hj = j for all h, the weighted estimator would be unbiased, in the large sample sense, with null accrual×treatment in the unpaired=singleton centers. Under the general null hypothesis (H0g ), the expected value of the weighted estimator is unbiased with no center×accrual interaction and no center e1ect in the singleton=unpaired centers or if either hj = j for all h or hj = h for all j. With the average null hypothesis, however, the pairwise and three-way interactions again pose a problem with selection bias.

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Additionally, expectations of heuristic estimator (4) can be derived, as shown.    H h (hA − hB ) Q   E(d∗ | HA ) ≈ A − B + (1 − 2 ) (H + h hB )(H + h hA )   ()2A h hB ()2B h hA H ()1A H ()1B     + − + − H + h hB H + h hA H + h hB H + h hA  hB { h + ( )h2 + ( )hA + ( )h2A }  + h H + h hB  hA { h + ( )h2 + ( )hB + ( )h2B }  − h ; H + h hA so the heuristic estimator would only be unbiased under the nonnull hypothesis with no accrual e1ect, no center e1ect, and no pairwise or three-way interactions. If hj = h for each j then the no treatment×accrual, no center×treatment and no three-way interactions assumptions are still needed for unbiasedness. If hj = j for all h then null accrual e1ect and accrual×treatment assumptions support unbiasedness. Under the general null hypothesis (H0g ), the heuristic estimator would be unbiased with no accrual e1ect (i.e. 1 = 2 ), no center e1ect and no center×accrual interaction; alternatively, hj = h for each j or both hj = j for all h and no accrual e1ect are su3cient for unbiasedness. Under the average null hypothesis (H0a ), the same assumptions as the nonnull case apply. 4.2. Selection bias in treatment sequences A selection bias model directed more towards the treatment e1ect allows the selection bias to be di1erent for each treatment sequence (i.e. A : B versus B : A) but assumes it is the same across the centers with the same sequence; this speci2cation is a simpli2cation of the general selection bias model so expectations need not be approximations. So in terms of the previous model: hj = j for all h; thus, the Lh have the same i:i:d:

i:i:d:

Q and Uh ∼ Bernoulli( 12 ), but with Lh⊥ Uh probability parameter, so Lh ∼ Bernoulli() for all h. Then, selection bias can be modelled separately for each treatment allocation sequence. Again, the group allocation totals (nPA ; nPB ; nSA , and nSB ) are random quantities which are jointly distributed multinomially with parameters A =2, B =2, (1−A )=2, and (1 −B )=2, respectively. The appropriate conditional probabilities of Lh and Uh and expectations of sums (Sgij ) can be found by simply dropping the center index h from those in Section 4.1; so, for example, Pr{Uh = 1 | Lh = 1} = A =(A + B ) = A =2Q and  E(SP1A | HA ) = (A =2) h yh1A . Also, the following expectations, conditioning on both number of patients and group totals, apply: E{Uh = 1 | Lh = 1; nPA ; nPB } =

nPA ; nP

E{Uh = 0 | Lh = 1; nPA ; nPB } =

nPB ; nP

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Table 4 Unbiasedness with random accrual dependent on allocation (treatment allocation selection bias) Statistic

Hypothesis

Required assumptions for unbiasednessa

dQp

HA =H0g =H0a

None

dQs

HA =H0a H0g

No accrual×treatment interaction None

dQ

HA =H0a H0g

No accrual×treatment interaction None

dQ∗

HA =H0a H0g

No accrual e1ect and no accrual×treatment interaction No accrual e1ect

a E(· | H

A ) = A

− B ; E(· | H0g ) = 0; E(· | H0a ) = 0 are unbiased.

E{Uh = 1 | Lh = 0; nSA ; nSB } = E{Lh = 1 | np } =

nP ; H

nSA ; ns

E{Uh = 0 | Lh = 0; nSA ; nSB } =

E{Lh = 0 | ns } =

nSB ; ns

ns : H

So, under the nonnull hypothesis HA , expectations of sums conditioning on both number of patients and group totals modify those expectations of sums conditioning only on number of patients by altering the fractional multiplicative factor; thus, in this case, for example E(SP1A | Lh ; HA ) =

A  Lh yh1A A +  B h

and E(SP1A | Lh ; HA ; nPA ; nPB ) =

nPA  Lh yh1A : nP h

Then, expectations of the separate estimators for paired (1) and singleton=unpaired (2) centers under the nonnull hypothesis are as follows.    1  E(dQp | HA ) = yhiA − yhiB = A − B ; 2H h i h i which is unbiased even in the presence of selection bias. Under both the general and average null hypotheses, the paired estimator is unbiased regardless of selection bias. Table 4 tabulates expected values’ assumptions needed for unbiasedness for all estimators and hypotheses. And E(dQs | HA ) = A − B + ()1A − ()1B , which would be unbiased assuming no accrual×treatment interaction. Selection bias could result from a nonnull accrual× treatment interaction (i.e. ()1A = ()1B ). Under the general null hypothesis (H0g ), the unpaired estimator is unbiased. Under the average null hypothesis (H0a ), accrual× treatment interaction could make the singleton estimator biased.

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97

Then, with 2xed w, the expected value for the weighted estimator (3) is E(dQ | HA ) = A − B + (1 − w)[()1A − ()1B ]; showing that the weighted estimator would be unbiased for HA only in the absence of accrual×treatment interaction. Under the general null hypothesis (H0g ), the expected value of the weighted estimator is zero; i.e. the treatment estimator is unbiased. With the average null hypothesis, however, the accrual×treatment interaction again poses a problem with selection bias. Additionally, expectations of the heuristic estimator (4) can be derived, as shown. E(dQ∗ | HA ) =

1 [(1 + A + ()1A ) + B (2 + A + ()2A )] 1 + B −

1 [(1 + B + ()1B ) + A (2 + B + ()2B )] 1 + A

= A − B + (1 − 2 )

 A − B (1 + A )(1 + B )

+ {()1A + B ()2A }=(1 + B ) − {()1B + A ()2B }=(1 + A ); so the heuristic estimator would only be unbiased under the nonnull hypothesis with no accrual e1ect and no accrual×treatment interaction. The same assumptions for unbiasedness as above are required even if it is conditional on the four-fold multinomial counts instead. Under the general null hypothesis (H0g ), E(dQ∗ | H0g ) = (1 + B 2 )=(1 + B ) − (1 + A 2 )=(1 + A ) = (1 − 2 ) = 0;

 A − B ; (1 + A )(1 + B )

if 1 − 2 = 0;

showing that the heuristic estimator would be unbiased with no accrual=order e1ect (or if A = B ; i.e. no selection bias). Under the average null hypothesis (H0a ), as with the nonnull hypothesis, the accrual e1ect and accrual×treatment interaction need to be null for the heuristic estimator to be unbiased. 4.3. Notes on selection bias Actual multicenter trials with possibly incomplete paired data can be assessed for indications of selection bias. Assumptions can be examined, but never completely veri2ed. Association between recruitment indices (Lh ) and treatment assignment indices (Uh ) for the H centers can be evaluated for evidence of selection bias, though lack of association does not necessarily imply lack of selection bias. A Fisher’s exact test can assess the independence of {Uh } and {Lh } in a 2×2 table. (With at least a moderate number of centers, the chi-square approximation can be used.) Comparing nPA versus

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the Normal approximation to a binomial distribution with parameters np and 1=2 (or parameters H and np =2H ) can shed light on the association between center treatment sequence and number of centers. Moreover, the Spearman correlation between the recruitment indices (Lh ) and the responses for the 2rst patient in each center (yh1j ), separately for each treatment group should be near zero and similar for each treatment group; the partial Spearman correlation accounting for treatment also should be near zero. Any nonzero Spearman correlation between response and recruitment would be worrisome and indicative of di1erential recruitment in centers, based upon the response of the 2rst subject. Further, absolute values of the Spearman correlations of the random allocation indices (Uh ) with the 2rst patients’ responses (yh1j ) and with the second patients’ responses (yh2j ) should be similar; otherwise selection bias might be suspected. Finally, methods used to assess carryover e1ect (period×treatment interaction) in crossover trials (Jones and Kenward, 1989, pp. 22–28; Senn, 1993, pp. 44 –54) can be applied. In this situation accrual×treatment, which is analogous to carryover e1ect, can  be examined with the sums of responses within complete centers ( h (yh1A + yh2B )  versus h (yh1B + yh2A )) for each sequence (Uh ). Graphical methods for carryover e1ect (Jones and Kenward, 1989, pp. 43– 45) can be used to examine accrual×treatment interaction as well. With one or two patients per center, however, the power to detect the accrual×treatment interaction is usually not very high, just as with carryover in crossover RCTs. A two-stage procedure (Grizzle, 1965) would inTate type I error just as in crossover trials (Freeman, 1989) and should be avoided. More powerful extensions to situations with more than one or two patients per center can be considered. 4.4. Example Using the example described in the Introduction, the methods suggested in the previous subsection are applied to examine the dermatology RCT for evidence of noncompatible incomplete and complete centers. The two-sided Fisher’s exact test for lack of independence of {Lh } and {Uh } had p = 0:60, which does not show an association between number of patients per center and treatment sequence. The number of complete centers assigning A 2rst tested using the Normal approximation to the binomial distribution with probabilities of 12 and np =2H = 0:42 yielded      1 1 nPA − 2 np  nPA − np = 0:18; = 0:16 and Pr z6  Pr z6 1 √ 2  2H −n  1 2 np np H p 2 respectively, which do not contradict nPA having the binomial distribution. Spearman correlation results, summarized in Table 5, do not provide any evidence of worrisome relationships between indicator functions and responses. Moreover, they do not show a strong relationship between treatment indicator and response, if only the paired data were analyzed separately for each sequence. Additionally, “carryover” (accrual order by treatment interaction) analysis through the Wilcoxon rank sum tests for comparing

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99

Table 5 Spearman correlation (rs ) results for example RCT Comparison {Lh }, {yh1A } {Lh }, {yh1B } {Lh }, {yh1j }| j = A; B | {Uh }; {yh1j }| | {Uh }; {yh2j }|

rs

p-Value

0:07 −0:10 −0:01

0.44 0.28 0.83

0:06 0:04

0.36 0.56

Fig. 1. Order×treatment plot.

within-center response sums for accrual order in the complete centers does not show “carryover” (p = 0:82). The graphical display (Fig. 1) does not suggest a substantial accrual order by treatment interaction. Overall, there was no evidence of bias in this study. 5. Summary and conclusions This paper investigates bias for several estimators of the di1erence between treatments for possibly incomplete pairs of patients in a 2xed number of centers (H ). Although the heuristic di1erence in means estimator is attractive for its simplicity, it pools responses for each treatment group while ignoring accrual order. Under the nonnull (HA : yhiA = yhiB ), general null (H0g : yhiA = yhiB = yhi ), and average null     (H0a : h i yhiA = h i yhiB ) hypotheses, the heuristic estimator is unbiased with reasonable assumptions for 2xed or random recruitment (Lh ); the no accrual×treatment assumption is not needed for the general null hypothesis with random recruitment. With selection bias, however, these assumptions may not hold.

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The weighted estimator, which combines the paired and unpaired=singleton estimators with a priori weights, accounts for accrual order and makes use of more information by not discarding data from unpaired=singleton centers. (Since the paired and singleton=unpaired estimators are components of the weighted estimator, only the weighted estimator will be summarized here.) Under both the average null and nonnull hypotheses, the weighted estimator is unbiased with fewer assumptions than the heuristic estimator with either 2xed or random recruitment; with selection bias, however, even these reasonable assumptions are no longer tenable. (The paired estimator is unbiased under the nonnull and average null hypotheses even with selection bias as considered here; the weighted estimator is biased for these hypotheses because it includes the unpaired=singleton data.) Thus, missing data in the form of incomplete treatment block allocation from trials with a design for paired data should be examined for patterns of missingness reTecting selection bias. Under the general null hypothesis, the weighted estimator is unbiased with either 2xed or random recruitment indices (Lh ), even in the presence of selection bias. Acknowledgements The authors wish to thank Dr. Kant Bangdiwala, Dr. Harry Guess, Dr. Ron Horner, and Dr. Dana Quade for their comments and suggestions on an early version of this manuscript, as well as the reviewers for their helpful recommendations. Any remaining ambiguities or errors are our own. Stuart Gansky was supported by the Veterans A1airs O3ce of Academic A1airs’ Health Services Research Predoctoral Fellowship and the University of North Carolina at Chapel Hill’s Biometric Consulting Laboratory. Appendix Further details of conditional and unconditional expectation derivation results presented in the text are shown below. Expected values of group sums for combinations of center size, patient order and treatment from Section 2 are displayed in Table 6 with both 2xed and random group sample sizes. The derivations of expectations of estimators from Section 2 follow. E(dQp | HA ; np ; ns ) = =

1  (yhiA − yhiB )E(Lh | np ; ns ) 2nP h i 1  (yhiA − yhiB ) = A − B ; 2H h i


E(SS1A | Lh ; nSA ; nSB ) E(SS1B | Lh ; nSA ; nSB ) − nSA nSB    1  = (1 − Lh ) yh1A − (1 − Lh ) yh1B ns h h

E(dQs | Lh ; HA ) = E nSA ;nSB



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101

Table 6 Sum (Sgij )

Fixed nPA ; nPB ; nSA ; nSB E(Sgij | HA ; ngA ; ngB )

Random nPA ; nPB ; nSA ; nSB E(Sgij | HA )

SP1A

nPA nP

Lh yh1A

1 2

SP2A

nPB nP

h

Lh yh2A

1 2

SP1B

nPB nP

h

Lh yh1B

SP2B

nPA nP nSA ns nSB ns

h

SS1A SS1B




where g =





h

Lh yh1A

h

Lh yh2A

2

h

Lh yh1B

Lh yh2B

2

h

Lh yh2B

h

(1 − Lh )yh1A

2

h

(1 − Lh )yh1A

h

(1 − Lh )yh1B

2

h

(1 − Lh )yh1B



h



  



 1

 1

 1  1

S for singleton=unpaired centers ; j = A; B. P for paired centers

= yQ S1A − yQ S1B = S1A − S1B = A − B + ()1A − ()1B + ( Q )SA − ( Q )SB + ( Q )S1A − ( Q )S1B ; E(dQs | HA ; np ; ns ) = =

1 ns 1 H

 E(dQ∗ | HA ) = En ∗




 h




 h

yh1A −

 h

h

 yh1B

 yh1B E(1 − Lh | np ; ns )

= A − B + ()1A − ()1B ;

np + nSA

n ∗ } + E{SP·B | Lh ; HA ; ∼ n ∗ } E{SS1B | Lh ; HA ; ∼



np + nSB

n  SA

h

ns

= En ∗

(1 − Lh )yh1A +

nSB ns

nPA nP

 h

Lh yh1A +

np + nSA

∼

−



E{SS1A | Lh ; HA ; ∼ n ∗ } + E{SP·A | Lh ; HA ; ∼ n ∗ }

∼

−

yh1A E(1 − Lh | np ; ns ) −

 h

(1 − Lh )yh1B +

nPB nP

 h

Lh yh1B +

np + nSB


nPA nP

nPB nP

 h

 h

Lh yh2A

Lh yh2B



  nSA 1 (1 − Lh )yh1A + Lh yhiA ns (np + nSA ) h 2(np + nSA ) h i    nSB 1 − (1 − Lh )yh1B − Lh yhiB ; 2(np + nSB ) h i ns (np + nSB ) h

= E nSA ;nSB

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where ∼ n ∗ ≡ (nPA ; nPB ; nSA ; nSB ), since E(nPA ) = E(nPB ) = np =2. If nSA and nSB are large, by Taylor series, E(dQ∗ | HA ) ≈

1  1  (1 − Lh )yh1A + Lh yhiA N h i N h 1  1  (1 − Lh )yh1B − Lh yhiB N h i N h

−

np H [()1A − ()1B ] + [()2A − ()2B ] N N np np + [( Q )PA − ( Q )PB ] + [( Q )P2A − ( Q )P2B ]; N N  where N = H + np is the total number of patients and ( Q )Pj = (1=np ) h Lh ( )hj .
 np yQ P· + nSA yQ S1 np yQ P· + nSB yQ S1 − E(dQ∗ | H0g ) = E nSA ;nSB np + nSA np + nSB 
np (nSA − nSB ) = (yQ S1 − yQ P· )E nSA ;nSB (np + nSA )(np + nSB ) = A − B +

if nSA = nSB or yQ 1S1 = yQ 1P· (i:e: Q S − Q P + 1 + ( Q )S1 = 0);

=0 where yQ S1 =

1  (1 − Lh )yh1 ; ns h

Q = 1  (1 − Lh ) S ns h

h

yQ P· = and

1  Lh yhi ; 2np h i

Q = 1  Lh P np h

h:

Details about expected values from Section 3 are given below: n ∗ )} = En ∗ {E(dQp | HA ; ∼ n ∗ )} E(dQp | HA ) = En ∗ {E(dQp | HA ; ∼ ∼

=

1 En  2 ∼∗ +



∼

E(SP1A | HA ; ∼ n ∗ ) − E(SP2B | HA ; ∼ n ∗ ) nPA

E(SP2A | HA ; ∼ n ∗ ) − E(SP1B | HA ; ∼ n ∗ ) nPB


= =



 h

1 2H

(yh1A − yh2B ) +



 h

= A − B

i

yhiA −

 h

 h

i

 (yh2A − yh1B )  yhiB

where ∼ n ∗ ≡ (nPA ; nPB ; nSA ; nSB ):

2H

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 E(dQs | HA ) =

En ∗ {E(dQs | HA ; ∼ n ∗ )} = En ∗ ∼ ∼

E(SS1A | HA ; ∼ n ∗ ) nSA

−

n ∗ ) E(SS1B |HA ; ∼

103



nSB

  1  h yh1A − h yh1B = yQ 1A − yQ 1B H = A − B + ()1A − ()1B : =

 E(dQ∗ | HA ) =

n ∗ )} = En ∗ En ∗ {E(dQ∗ | HA ; ∼ ∼ ∼ −

E{SS1A | HA ; ∼ n ∗ } + E{SP·A |HA ; ∼ n ∗ }

n ∗ } E{SS1B | HA ; ∼ n ∗ } + E{SP·B | HA ; ∼



np + nSA

np + nSB

  yh1A + nPA h yh1A + nPB h yh2A = En ∗ ∼ H (np + nSA )     nSB h yh1B + nPB h yh1B + nPA h yh2B ; − H (np + nSB )


nSA



h

so under (6) the expectation reduces to  
(nSA + nPA + nPB ) h i yhiA Q E(d∗ | HA ) = En ∗ ∼ 2H (np + nSA )    (nSB + nPB + nPA ) h i yhiB − 2H (np + nSB )
  1  = yhiA − yhi B = A − B : 2H h i h i  nPA + nSA nPB + nSB yQ 1· − ∼ np + nSA np + nSB    nPB nPA yQ 2· + − np + nSA np + nSB 
nPA (nPA + nSA ) − nPB (nPB + nSB ) ; = (yQ 1· − yQ 2· )En ∗ ∼ (np + nSA )(np + nSB )

E(dQ∗ | H0g ) = En ∗




where yQ i· =

1  yhi =  + i ; H h

if yQ 1· − yQ 2· = 0 
nPA (nPA + nSA ) − nPB (nPB + nSB ) or En ∗ ∼ (np + nSA )(np + nSB )

=0

= 0: Expected values of group sums for combinations of center size, patient order and treatment from Section 4.1 are displayed in Table 7; conditional expectations (on the

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Table 7 E(Sgij | Lh ; HA )

Sum (Sgij )

E(Sgij | HA )



SP1A

hA L y h hA +hB h h1A hB L y h hA +hB h h2A hB L y h hA +hB h h1B hA L y h hA +hB h h2B

1 2

1−hA (1 h 2−(hA +hB )

− Lh )yh1A

1 2

1−hB (1 h 2−(hA +hB )

− Lh )yh1B

2



SP2A

1 2



SP1B

1 2



SP2B



SS1A



SS1B



where g =

S for singleton=unpaired centers P for paired centers

1 2



 y h hA h1A



 y h hB h2A



 y h hB h1B



 y h hA h2B



h

(1 − hA )yh1A

h

(1 − hB )yh1B

 1

; j = A; B:

recruitment index) as well as unconditional ones are shown. E(dQp | HA )
 SP1A − SP2B 1 SP2A − SP1B = E + 2 nPA nPB    1 12 h hA yh1A − 12 h hA yh2B ≈ +  1 2 h hA 2 

− yh2B )





1 h hB yh1B h hB yh2A − 2  1 h hB 2





− yh1B ) h hB (y h2A  2  hA hB h h  hA {( )h1 − ( )h2 + ( )hA − ( )hB + ( )h1A − ( )h2B }  = A − B + h 2 h hA  hB {( )h2 − ( )h1 + ( )hA − ( )hB + ( )h2A − ( )h1B }  + h : 2 h hB =

h hA (y h1A

1 2

2

E(dQp | H0g ) ≈



+

h hA {(

) − ( )h2 }  h1 − 2 h hA



h hB {(

2

if ( )h1 = ( )h2 ; hj = j ∀h; or hj = h ∀j:

E(dQs | HA ) = E




SS1A SS1B − nSA nSB



) − ( )h2 }  h1 = 0; h hB

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  1 h (1 − hA )yh1A h (1 − hB )yh1B 2 ≈ 1 − 1 h (1 − hA ) h (1 − hB ) 2 2   h (1 − hA )yh1A h (1 − hB )yh1B =  −  h (1 − hA ) h (1 − hB )  (1 − hA ){ = A − B + ()1A − ()1B + h

105

1 2

 −

h

(1 − hB ){

h

h

+ ( )h1 + ( )hA + ( )h1A }  h (1 − hA )

+ ( )h1 + ( )hB + ( )h1B }  : h (1 − hB )

E(dQ | HA ) = E(wdQp ) + E[(1 − w)dQs ] = w E(dQp ) + (1 − w)E(dQs ) ≈ A − B + (1 − w)[()1A − ()1B ]  w h hA {( )h1 −( )h2 +( )hA −(  + 2 h hA  w h hB {( )h2 −( )h1 +( )hA −(  + 2 h hB  (1 − w) h (1 − hA ){ h + ( )h1 + (  + h (1 − hA )  (1 − w) h (1 − hB ){ h + ( )h1 + (  − h (1 − hB )

)hB + ( )h1A −( )h2B } )hB + ( )h2A −( )h1B } )hA + ( )h1A } )hB + ( )h1B }

:

 SS1A + SP1A + SP2A SS1B + SP1B + SP2B − np + nSA np + nSB    1 (1 − hA )yh1A + 12 h hA yh1A + 12 h hB yh2A ≈ 2 h 1   1 1 h (1 − hA ) + 2 h hA + 2 h hB 2    1 1 1 hA yh2B h (1 − hB )yh1B + 2 h hB yh1B + 2 2 −    h 1 1 1 (1 −  ) +  +  hB h h hB h hA 2 2 2     yh1B + h hA yh2B h yh1A + h hB yh2A  = − h H + h hB H + h hA    H h (hA − hB )   = A − B + (1 − 2 ) (H + h hB )(H + h hA )   ()2A h hB ()2B h hA H ()1A H ()1B     + − + − H + h hB H + h hA H + h hB H + h hA

E(dQ∗ | HA ) = E




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h hB { h

+



h hA { h

−

+ ( )h2 + ( )hA + ( )h2A }  H + h hB + ( )h2 + ( )hB + ( )h2B }  : H + h hA

Moreover, from Section 4.2, more detailed derivations are as follows:
 E(SP1A | Lh ; nPA ; nPB ) − E(SP2B | Lh ; nPA ; nPB ) 1 Q E(dp | HA ) = ELh | np E nPA ;nPB 2 nPA  E(SP2A | Lh ; nPA ; nPB ) − E(SP1B | Lh ; nPA ; nPB ) + nPB 1  (yh1A − yh2B + yh2A − yh1B )E{Lh | np } 2np h    1  yhiA − yhiB = A − B : = 2H h i h i =






E{SS1A | Lh ; nSA ; nSB } E{SS1B | Lh ; nSA ; nSB } − nSA nSB
  1  1  = (yh1A − yh1B )E{1 − Lh | ns } = yh1A − yh1B ns h H h h

E(dQs | HA ) = ELh | ns



E nSA ;nSB

= A − B + ()1A − ()1B : E(dQ∗ | HA ) = ELh | n ∗ ∼

−





En ∗ ∼

E(SS1A | Lh ; ∼ n ∗ ) + E(SP1A | Lh ; ∼ n ∗ ) + E(SP2A | Lh ; ∼ n ∗ ) np + nSA

n ∗ ) + E(SP1B | Lh ; ∼ n ∗ ) + E(SP2B | Lh ; ∼ n ∗ ) E(SS1B | Lh ; ∼



np + nSB 
nSA  1 = En ∗ |np +nSA (1 − Lh )yh1A ∼ (np + nSA ) H h  nPA  nPB  + Lh yh1A + Lh yh2A H h H h 
nSB  1 −En ∗ |np +nSB (1 − Lh )yh1B ∼ (np + nSB ) H h  nPB  nPA  + Lh yh1B + Lh yh2B H h H h         1 1 = yh1A + B yh2A − yh1B + A yh2B H (1 + B ) h H (1 + A ) h h h

S.A. Gansky, G.G. Koch / Journal of Statistical Planning and Inference 96 (2001) 83–107

=

107

1 [(1 + A + ()1A ) + B (2 + A + ()2A )] 1 + B −

1 [(1 + B + ()1B ) + A (2 + B + ()2B )] 1 + A

= A − B + (1 − 2 )

 A − B (1 + A )(1 + B )

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