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A new framework of statistical inferences based on the valid joint sampling distribution of the observed counts in an incomplete contingency table

Statistical Methods in Medical Research 2017, Vol. 26(4) 1712–1736 ! The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0962280215586591 journals.sagepub.com/home/smm

Guo-Liang Tian1 and Hui-Qiong Li2

Abstract Some existing confidence interval methods and hypothesis testing methods in the analysis of a contingency table with incomplete observations in both margins entirely depend on an underlying assumption that the sampling distribution of the observed counts is a product of independent multinomial/binomial distributions for complete and incomplete counts. However, it can be shown that this independency assumption is incorrect and can result in unreliable conclusions because of the under-estimation of the uncertainty. Therefore, the first objective of this paper is to derive the valid joint sampling distribution of the observed counts in a contingency table with incomplete observations in both margins. The second objective is to provide a new framework for analyzing incomplete contingency tables based on the derived joint sampling distribution of the observed counts by developing a Fisher scoring algorithm to calculate maximum likelihood estimates of parameters of interest, the bootstrap confidence interval methods, and the bootstrap testing hypothesis methods. We compare the differences between the valid sampling distribution and the sampling distribution under the independency assumption. Simulation studies showed that average/expected confidence-interval widths of parameters based on the sampling distribution under the independency assumption are shorter than those based on the new sampling distribution, yielding unrealistic results. A real data set is analyzed to illustrate the application of the new sampling distribution for incomplete contingency tables and the analysis results again confirm the conclusions obtained from the simulation studies. Keywords Bootstrap, confidence interval, incomplete contingency tables, sampling distribution, test statistics

1 Introduction In biomedical studies, especially in clinical trials, we often encounter data such as counts in a contingency table with incomplete observations in both margins. Statistical inferences on cell probabilities or a function of cell probabilities have received a lot of attention in recent years. For example, Choi and Stablein1 presented various asymptotic procedures to test the equality of two binomial proportions with partially incomplete paired data, while Tang and Tang2 proposed unconditional exact procedures for testing the equality of two paired binomial proportions with partially incomplete paired data. To date, some existing confidence interval (CI) methods and hypothesis testing methods in this field (Choi and Stablein,1 p. 258; Tang and Tang,2 pp. 73–74; Tang et al.,3 p. 2922; Tang et al.,4 p. 627; Tang et al.,5 p. 227; Miller and Looney,6 p. 3300) are based on an underlying assumption that the sampling distribution of the observed counts is a product of independent multinomial/binomial distributions for complete and incomplete counts. However, it can be shown that this independency assumption is incorrect and can result in unreliable conclusions because of the under-estimation of the uncertainty. Thus, one major objective of this paper is to derive the valid joint sampling distribution of the observed counts in a 1 2

Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, P. R. China Department of Statistics, Yunnan University, Kunming, P. R. China

Corresponding author: Hui-Qiong Li, Department of Statistics, Yunnan University, Kunming 650091, P. R. China. Email: [email protected]

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contingency table with incomplete observations in both margins. In Section 1.1, from the viewpoint of statistical inferences on cell probabilities, we will present three reasons why the (valid) joint sampling distribution of the complete and incomplete counts is needed.

1.1

Why do we need the joint sampling distribution of fN, mx , my g?

Examples 1–2 in Section 2 motivate us to consider an r  c contingency table with incomplete counts in both margins. Let X and Y be two correlated categorical variables with the joint distribution ij ¼ Pr(X ¼ ¼ j) for Pi,r Y P c i ¼ 1, . . . , r and j ¼ 1, . . . , c. The observed counts and the corresponding cell probabilities for n ¼ i¼1 j¼1 nij Pr Pc complete observations and mx þ my ¼ i¼1 mix þ j¼1 myj partially incomplete observations are displayed in Table 1, where nij denotes the number of subjects who go through both variables with X ¼ i (1  i  r) and Y ¼ j (1  j  c), mix is the number of subjects who go through only variable X with X ¼ i, and myj is the number of subjects who go through only variable Y with Y ¼ j. Let Yobs ¼ fN, mx , my g denote the observed counts, where N ¼ ðnij Þ, mx ¼ ðm1x , . . . , mrx Þ> and my ¼ ðmy1 , . . . , myc Þ> , and h ¼ (11, . . . , 1c, . . . , r1, . . . , rc)> be the cell probability vector. Under the mechanism of missing at random (MAR; Rubin7), the likelihood function of h based on the observed data Yobs is given by ! ! ! r Y c r c Y Y Y myj nij mix ij iþ þj , h 2 Trc ð1Þ LðhjYobs Þ ¼ i¼1 j¼1

i¼1

j¼1

P P where iþ ¼ cj¼1 ij , þj ¼ ri¼1 ij , and Trc denotes the rc-dimensional hyperplane defined by (4). We wonder why the observed-data likelihood function of h under MAR is given by (1), what is the valid joint sampling distribution of Yobs ¼ fN, mx , my g, and why we need this valid joint distribution? In Section 3, we will answer the first two questions. To answer the third question, we note that in general the statistical inferences on h involve at least three aspects: (i) to find the maximum likelihood estimates (MLEs) of {ij}; (ii) to find the CIs of {ij}; (iii) to test whether or not there exists an association between X and Y. First, to find the MLEs of h, both the expectation–maximization (EM) and Newton–Raphson algorithms can be applied to the observed-data likelihood specified by (1), which does not involve the joint distribution of Yobs. However, if the Fisher scoring algorithm is utilized, we must calculate expectations such as E(nij), E(mix) and E(myj), which involve the joint distribution of Yobs. Second, there are two potential problems associated with the 100(1 )% asymptotic Wald-type CI of ij with the following form: h i ^ij  z=2  sbeð^ij Þ, ^ij þ z=2  sbeð^ij Þ ð2Þ where z is the th upper quantile of the standard normal distribution. The first is that even though for large sample sizes, the lower bound of (2) may be less than zero when the true value of ij is close to zero; while the upper bound may be beyond 1 when the true value of the ij is near to 1. The second is that for small to moderate sample sizes, the asymptotic CI (2) is not reliable. To overcome the two difficulties resulted from the bound of the unit interval [0, 1] and from the large-sample theory, respectively, we could employ the parametric bootstrap approach, where the essential step is how to generate a bootstrap sample. To generate a parametric bootstrap sample, we must know the sampling distribution of Yobs. Third, if the score test is used to test whether there is an association between X and Y, we must calculate the Fisher information, which again involves the joint distribution of Yobs.

Table 1. Observed counts and cell probabilities for an r  c table with incomplete observations. Y¼1

...

Y¼c

Missing on Y

..X ¼ 1 . X¼r

..n11(11) . nr1(r1)

.. .. . . ...

..n1c(1c) . nrc(rc)

..m1x(1þ) . mrx(rþ)

Missing on X

my1(þ1)

...

myc(þc)

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1.2

Statistical Methods in Medical Research 26(4)

The objectives of this paper

The first aim of this paper is to derive the valid joint sampling distribution of Yobs ¼ fN, mx , my g. The key idea consists of three steps. First we introduce a three-category response random variable R following a categorical distribution. Second, based on the two categorical variables X and Y, we construct a new rc-category random variable Z following another categorical distribution. Finally, we augment the Z by the response variable R to form a new 3  rc contingency table with incomplete observations. Under the assumption of MAR, the resultant joint sampling distribution of Yobs is a special multinomial distribution with equality constraints in all cell probabilities. From this valid joint sampling distribution, we can immediately obtain the likelihood function of h given by (1). The second aim of this paper is to provide a new framework for analyzing incomplete contingency tables based on the derived joint sampling distribution of Yobs by developing a Fisher scoring algorithm to calculate MLEs of parameters of interest, the bootstrap CI methods, and the bootstrap testing hypothesis methods. A new contribution is that we provide quite simple formulae to calculate the observed and Fisher information matrices for incomplete r  c tables, where a direct computation of the Hessian matrix in such an incomplete r  c table is challenging in practice although it seems feasible theoretically. This article is organized as follows. Section 2 presents two motivating examples. In Section 3, we derive the valid sampling distribution of the observed counts in a contingency table with complete and incomplete observations. In Section 4, we develop a Fisher scoring algorithm to calculate MLEs of parameters of interest. Section 5 considers the construction of bootstrap CIs for an arbitrary function of the cell probabilities. In Section 6, we study testing hypothesis problem via the bootstrap method. In Section 7, we compare the differences between the valid sampling distribution and the sampling distribution under the independency assumption. Simulation studies are conducted to investigate the performance of the new sampling distributions and independent sampling distribution under different criteria in section 8. A real data set is analyzed in Section 9 to illustrate the application of the new sampling distribution for incomplete contingency tables. A discussion is given in Section 10.

2 Two motivating examples Example 1. (Neurological complication data). In a neurological study conducted at St Louis Children’s Hospital, 33 young meningitis patients were given neurological tests at the beginning and the end of a standard treatment on neurological complication (Choi and Stablein1). In this study, some observations were made only on the first response and some only on the second response because of refusal by patients, migration, omission due to recording errors or dropping out due to adverse effect caused by treatments. Thus, the resultant observations include two parts (i.e. the complete and incomplete counts) and are displayed in Table 2 in which 0 and 1 respectively denote the absence and presence of any neurological complication. The primary objective of this study is to detect if there is a substantial proportion difference before and after the treatment. Let Yobs,1 ¼ {n1, . . . , n4; n12, n34; n13, n24} be the observed frequencies and h ¼ (1, . . . , 4)> the cell probability vector. We are interested in the estimation of the parameter 23 ¼ Pr(Y ¼ 1)  Pr(X ¼ 1) ¼ 2  3, which is the difference between the proportions of patients with neurological complication after and before the treatment. Under the assumption of missing at random (MAR, Rubin7), the likelihood function of h based on the observed data Yobs,1 is ! 4 Y nj L1 ðhjYobs,1 Þ ¼ j ð1 þ 2 Þn12 ð3 þ 4 Þn34 ð1 þ 3 Þn13 ð2 þ 4 Þn24 , h 2 T4 ð3Þ j¼1

Table 2. Neurological complication data from Choi and Stablein.1 Complication status at the end of the treatment Complication status at the beginning of the treatment

Y ¼ 0 (Absent)

Y ¼ 1 (Present)

Missing on Y

X ¼ 0 (Absent) X ¼ 1 (Present) Missing on X

6 (n1, 1) 8 (n3, 3) 2 (n13, 1 þ 3)

3 (n2, 2) 8 (n4, 4) 0 (n24, 2 þ 4)

2 (n12, 1 þ 2) 4 (n34, 3 þ 4)

Note: ‘X ¼ 0(1)’ means that patient’s complication at the beginning of the treatment is absent (present), ‘Y ¼ 0(1)’ means that patient’s complication at the end of the treatment is absent (present). The observed frequencies and probabilities are in parentheses.

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where ( >

^ ð1 , . . . , n Þ : j  0, j ¼ 1, . . . , n, Tn ¼

n X

) j ¼ 1

ð4Þ

j¼1

denotes the n-dimensional hyperplane. We wonder why the observed-data likelihood function of h under MAR is given by (3), what is the valid joint sampling distribution of Yobs,1. Example 2. (Child’s wheezing data in six cities). To investigate the effects of maternal smoking on respiratory illness in children, Ware et al.8 conducted a wheeze study in six cities of United States including: Watertown, Massachusetts; Kingston-Harriman, Tennessee; the southeast tip of St Louis, Missouri; Steubenville and Mingo Junction, Ohio; Portage, Wisconsin and Topeka, Kansas. Table 3 displays the number of children cross-classified according to maternal smoking status (X ¼ 1, none; X ¼ 2, moderate; X ¼ 3, heavy) and child’s wheezing status (Y ¼ 1, no wheeze; Y ¼ 2, wheeze with cold; Y ¼ 3, wheeze apart from cold). Note that for some children their maternal smoking status is missing whereas for some mothers their children’ wheezing status is missing. The objective is to examine if association exists between maternal smoking and child’s wheezing status. Let Yobs,2 ¼ {n1, . . . , n9; m1, m2, m3; m4, m5, m6} denote the observed frequencies and h ¼ (1, . . . , 9)> the cell probability vector. Under the assumption of MAR, the likelihood function of h based on the observed data Yobs,2 is ! 9 Y nj j ð1 þ 2 þ 3 Þm1 ð4 þ 5 þ 6 Þm2 ð7 þ 8 þ 9 Þm3 L2 ðhjYobs,2 Þ ¼ j¼1

 ð1 þ 4 þ 7 Þm4 ð2 þ 5 þ 8 Þm5 ð3 þ 6 þ 9 Þm6 ,

h 2 T9

ð5Þ

We wonder why the observed-data likelihood function of h under MAR is given by (5), what is the valid joint sampling distribution of Yobs,2.

3 The valid joint distribution of the complete and incomplete counts 3.1 The joint distribution of the observed counts in incomplete 2  2 tables We P assume that in Table 2 there are a total of N (N is pre-specified and non-random) patients containing n ¼ 4j¼1 nj complete counts, n12 þ n34 incomplete counts on X (or missing on Y), and n13 þ n24 incomplete counts on Y (or missing on X). The aim of this subsection is to derive the joint distribution of the observed counts Yobs,1 ¼ {n1, . . . , n4; n12, n34; n13, n24} under MAR. To this end, we first introduce a three-category response random variable R, where R ¼ 1(or 12) if a patient’s complication status (absent or present) at the beginning and at the end of the treatment are recorded; R ¼ 2(or 12) if a patient’s complication status (absent or present) only at the beginning of the treatment is recorded; R ¼ 3(or 12) if a patient’s complication status (absent or present) only at the end of the treatment is recorded. Let /i ¼ Pr(R ¼ i) for i ¼ 1, 2, 3 and  ¼ ð1 , 2 , 3 Þ> . Note that / 2 T3 , then R follows a categorical distribution, denoted by R  Categorical3(/), where / is called the parameter vector associated with the missing-data mechanism. The categorical distribution is a generalization of the Bernoulli distribution and is a special case of the general finite distribution. (Let x ¼ (x1, . . . , xm)> and

Table 3. Child’s wheezing data from Lipsitz and Fitzmaurice.9 Child’s wheezing status Maternal smoking

Y¼1 (No wheeze)

Y¼2 (Wheeze with cold)

Y¼3 (Wheeze apart from cold)

X ¼ 1 (None) X ¼ 2 (Moderate) X ¼ 3 (Heavy) Missing on X

287 (n1, 1) 18 (n4, 4) 91 (n7, 7) 59 (m4)

39 (n2, 2) 6 (n5, 5) 22 (n8, 8) 18 (m5)

38 (n3, 3) 4 (n6, 6) 23 (n9, 9) 26 (m6)

Missing on Y 279 (m1) 27 (m2) 201 (m3)

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Statistical Methods in Medical Research 26(4)

p ¼ ð p1 , . . . , pm Þ> 2 Tm . A discrete random variable X is said to have the general finite distribution, denoted by X  Finitem(x, p), if Pr(X ¼ xk) ¼ pk for k ¼ 1, . . . , m. In particular, when xk ¼ k, the general finite distribution reduces to the categorical distribution, symbolized as X  Categorical(p1, . . . , pm) or X  Categoricalm(p) accordingly.) Next, based on the two binary variables X and Y, we can construct a new four-category random variable Z as follows: 8 1, if ðX, YÞ ¼ ð0, 0Þ > > > < 2, if ðX, YÞ ¼ ð0, 1Þ Z¼ > 3, if ðX, YÞ ¼ ð1, 0Þ > > : 4, if ðX, YÞ ¼ ð1, 1Þ Note that Pr{(X,Y) ¼ (0, 0)} ¼ 1, Pr{(X,Y) ¼ (0, 1)} ¼ 2, Pr{(X,Y) ¼ (1, 0)} ¼ 3, Pr{(X,Y) ¼ (1, 1)} ¼ 4, and h ¼ ð1 , . . . , 4 Þ> 2 T4 , then Z  Categoricalð1 , . . . , 4 Þ, where h is called the model parameter vector. Finally, we augment the Z by the response variable R to form a new 3  4 contingency table with incomplete observations. The joint distribution of R and Z is defined by p ¼ ðij Þ, where ij ¼ PrðR ¼ i, Z ¼ j Þ for i ¼ 1, 2, 3 and j ¼ 1, . . . , 4 Table 4 displays the observed counts, missing counts, the marginal probabilities of R and Z, and the joint distribution of (R,Z). The exact observations in the first row are fnj g4j¼1 corresponding to cell probabilities f1j g4j¼1 , while the exact observed counts in the second (or third) row corresponding to cell probabilities f2j g4j¼1 (or f3j g4j¼1 ) are not available. Naturally, we introduce eight latent counts fn0j g4j¼1 and fn00j g4j¼1 as shown in Table 4. However, the sum of the counts in the fifth cell (i.e. n01 ) and in the sixth cell (i.e. n02 ) is indeed observed and is equal to n12. And the sum of the counts in the seventh cell (i.e. n03 ) and in the eighth cell (i.e. n04 ) is also observed and equals n34. Similarly, n001 þ n003 ¼ n13 and n002 þ n004 ¼ n24 . Hence, only n01 , n03 , n001 , n002 are missing values while n02 ¼ n12  n01 , n04 ¼ n34  n03 , n003 ¼ n14  n001 and n004 ¼ n24  n002 . Thus, the complete data Ycom ¼ fYobs,1 , n01 , n03 , n001 , n002 g ¼ fn1 , . . . , n4 ; n01 , . . . , n04 , n001 , . . . , n004 g follow a multinomial distribution and the joint distribution is given by (Tian et al.,10 p. 196; Chang11) ! ! ! 4 4 4 Y Y Y 0 00 n n n 1jj 2jj 3jj , p 2 T12 f ðYobs,1 , n01 , n03 , n001 , n002 ; pÞ ¼ C1 1  j¼1

j¼1

j¼1

where C1 1

 ¼



N n1 , . . . , n4 , n01 , . . . , n04 , n001 , . . . , n004

Table 4. The observed counts, missing counts, marginal probabilities of R and Z, and joint probabilities of (R, Z). Four-category variable Z Response variable R 1

1

2

3

4

Marginal probability of R

11(n1) 21(n01 )

12(n2) 22(n02 )

13(n3)

14(n4)

1

23(n03 ) 33(n003 )

24(n04 )

2

34(n004 ) 4

3 1

2 31(n001 ) 3 M.P. of Z

1

32(n002 ) 2

3

Observed counts P n ¼ 4j¼1 nj n12 ¼ n01 þ n02 n34 ¼ n03 þ n04 n13 ¼ n001 þ n003 n24 ¼ n002 þ n004 N ¼ n þ n12 þ n34 þ n13 þ n24

Note: Only n01 , n03 , n001 , n002 are missing values while n02 ¼ n12  n01 , n04 ¼ n34  n03 , n003 ¼ n13  n001 and n004 ¼ n24  n002 . M.P. ¼ Marginal probability. R  Categorical(1, 2, 3) and Z  Categorical(1, . . . , 4).

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Since C1 1 ¼



N n1 , . . . , n4 , n12 , n34 , n13 , n24



n12



n01

n34 n03



n13 n001



n24



n002

the joint distribution of the observed data Yobs,1 is obtained by summating the missing value n01 from 0 to n12, n03 from 0 to n34, n001 from 0 to n13, n002 from 0 to n24: f ðYobs,1 ; pÞ ¼

n34 X n13 X n12 X n24 X

f ðYobs,1 , n01 , n03 , n001 , n002 ; pÞ

n01 ¼0 n03 ¼0 n001 ¼0 n002 ¼0

 ¼

 Y 4

N n1 , . . . , n4 , n12 , n34 , n13 , n24 n12

! n 1jj

j¼1

n34

 ð21 þ 22 Þ ð23 þ 24 Þ ð31 þ 33 Þn13 ð32 þ 34 Þn24 ,

p 2 T12

ð6Þ

P where N ¼ 4j¼1 nj þ n12 þ n34 þ n13 þ n24 ¼ n þ n12 þ n34 þ n13 þ n24 . When R and Z are independent, i.e. ij ¼ i j ,

i ¼ 1, 2, 3 and j ¼ 1, . . . , 4

the missing-data mechanism is said to be ignorable or MAR (Little and Rubin12). Under MAR, the joint distribution (6) becomes !" # 4 N Y nj ð1 j Þ f ðYobs,1 ; h, /Þ ¼ n1 , . . . , n4 , n12 , n34 , n13 , n24 j¼1  ½ ð þ 2 Þn12 ½2 ð3 þ 4 Þn34 ½3 ð1 þ 3 Þn13 ½3 ð2 þ 4 Þn24  2 1  N ¼  L1 ðhjYobs,2 Þ  n1 n212 þn34 n313 þn24 n1 , . . . , n4 , n12 , n34 , n13 , n24

ð7Þ ð8Þ

where L1(hjYobs,2) is given by (3), h 2 T4 and / 2 T3 . We have observed several facts: (a) The formula (7) implies Yobs,1  Multinomial8 ðN, pÞ

ð9Þ

where p ¼ ð1 1 , . . . , 1 4 , 2 ð1 þ 2 Þ, 2 ð3 þ 4 Þ, 3 ð1 þ 3 Þ, 3 ð2 þ 4 ÞÞ> with only 5 free parameters. In other words, (9) is a special eight-dimensional multinomial distribution with different equality constraint on each component of p. (b) The joint distribution of Yobs,1 depends on both the model parameter vector h and the mechanism parameter vector / since p is a function of h and /. (c) The formula (8) indicates that h and / can be estimated separately. When R and Z are dependent, the missing-data mechanism is said to be non-ignorable. Under the non-ignorable missing mechanism, the joint distribution is given by (6) with a total of 11 free parameters since p 2 T12 . According to Ma et al.,13 these {ij} are not identifiable in the frequentist framework.

3.2

Sampling distribution of observed counts in incomplete r  c table and its properties

Similarly, we can derive the joint distribution of the observed data Yobs ¼ fN, mx , my g in an incomplete r  c table as shown in Table 1 under MAR and the assumption that r X c r c X X X nij þ mix þ myj ¼ ^ n þ mx þ my ð10Þ N¼ i¼1 j¼1

i¼1

j¼1

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Statistical Methods in Medical Research 26(4)

is fixed. To this end, we need to introduce a three-category response random variable R, where R ¼ 12 if subjects go through both the categorical variables X and Y; R ¼ 12 if subjects only go through the variable X; R ¼ 12 if subjects only go through the variable Y. Let / ¼ (1, 2, 3)> , where 1 ¼ Pr(R ¼ 12), 2 ¼ Pr(R ¼ 12) and 3 ¼ Pr(R ¼ 12). Hence, R  Categorical(/), where / 2 T3 is the parameter vector associated with the missingdata mechanism. Similar to (7) and (8), under MAR, the joint distribution of Yobs is " #" #" # r Y c r c Y Y Y nij mix myj 1 ð1 ij Þ ð2 iþ Þ ð3 þj Þ ð11Þ f ðYobs ; h, Þ ¼ C  i¼1 j¼1 1

¼C

 LðhjYobs Þ 

i¼1

j¼1

x my n1 m 2 3

ð12Þ

where C1 ¼

iþ ¼

Pc

j¼1 ij ,

þj ¼

Pr

i¼1 ij ,





N N, mx , my

 ¼ ^

N



n11 , . . . , nrc , m1x , . . . , mrx , my1 , . . . , myc

LðhjYobs Þ is given by (1), h ¼ ð11 , . . . , rc Þ> 2 Trc , and / 2 T3 . Thus, Yobs  Multinomialrcþrþc ðN, pÞ

ð13Þ

 > where p ¼ 1 h> , 2 1þ , . . . , 2 rþ , 3 þ1 , . . . , 3 þc with rc þ 1 free parameters. 3.2.1 Conditional and marginal distributions The aim of this subsection is to derive the conditional distributions of Njðmx , my Þ, mx jðN, my Þ, my jðN, mx Þ, and marginal distribution of ðmx , my Þ, where the joint distribution of Yobs ¼ fN, mx , my g is specified by (12). Let u ¼ ðu1 , . . . , up Þ> and v ¼ ðv1 , . . . , P vq Þ> be two vectorsP with non-negative integer-valued components. The ‘1 norms of u and v are denoted by kuk1 ¼ pi¼1 ui and ktk1 ¼ qj¼1 vj . It is easy to verify the following two identities:         kt k1 kuk1 þktk1 kuk1 þktk1 ku k1 kuk1 þktk1 ¼ ¼ ð14Þ ku k1 , t u, t u u, ktk1 t which are useful in deriving the conditional and marginal distributions of Yobs. By using (14), we can rewrite (11) as #" #  Y   "Y r Y c r c Y N n nij m m   ð2 iþ Þ ix ð3 þj Þ yj f ðYobs ; h, /Þ ¼ n n, mx , my 1 i¼1 N i¼1 j¼1 ij j¼1 Given (mx,my), then n ¼ N  mx  my is fixed. Thus, the conditional distribution of N given (mx, my) is   Y r Y c n n f ðNjmx , my ; hÞ ¼ ijij N

ð15Þ

i¼1 j¼1

not depending on /, while the marginal distribution of (mx,my) is given by #" #   "Y r c Y N mix myj n ð2 iþ Þ ð3 þj Þ  f ðmx , my ; h, /Þ ¼ n, mx , my 1 i¼1 j¼1 Similarly, the conditional distributions of mxj(N,my) and myj(N,mx) are given by   r mx Y mix with mx being fixed f ðmx jN, my ; hÞ ¼ mx i¼1 iþ   c my Y myj þj with my being fixed f ðmy jN, mx ; hÞ ¼ my j¼1

ð16Þ

ð17Þ

ð18Þ

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respectively. The conditional distributions (15), (17) and (18) do not depend on the missing-data mechanism parameters /.

4 MLEs of parameters via the Fisher scoring algorithm The formula (12) indicates that the model parameters h and the mechanism parameters / can be estimated separately, and the MLEs of / have the following explicit expressions: n ^1 ¼ , N

mx ^2 ¼ , N

my ^3 ¼ N

ð19Þ

where N is defined by (10). Based on the likelihood function L(hjYobs) specified by (1), although Tang et al.3 obtained the MLEs of h via the following EM iteration: ijðtþ1Þ ¼

ðtÞ ðtÞ nij þ mix ijðtÞ =iþ þ myj ijðtÞ =þj , N

i ¼ 1, . . . , r; j ¼ 1, . . . , c

ð20Þ

its convergence could be painfully slow, especially, for large r  c tables or in the setting of calculating the bootstrap CIs of parameters of interest, where the performance of the EM algorithm (20) is needed in ten to one-hundred thousands times. Alternatively, when the log-likelihood function ‘(hjYobs) ¼ log L(hjYobs) is well-behaved, a natural candidate for finding the MLEs is the Newton–Raphson (NR) or Fisher scoring (FS) algorithm because they converge quadratically.12,14 By deleting the last component rc in the model parameter vector h ¼ ð11 , . . . , rc Þ> , we denote the remaining subvector by hrc ¼ ð11 , . . . , 1c , . . . , r1 , . . . , r,c1 Þ> . Since h 2 Trc , we have rc ¼ 1  1> hrc . Let h(0) be an initial ^ the NR and FS algorithms are defined by value and h(t) denote the t-th approximate of h, ðtÞ 1 ðtÞ hðtþ1Þ rc ¼hrc þ Iobs ðhrc Þ

@‘ ðhðtÞ jYobs Þ , @hrc

ðtÞ 1 ðtÞ hðtþ1Þ rc ¼ hrc þ J ðhrc Þ

@‘ ðhðtÞ jYobs Þ , @hrc

ðtþ1Þ rc ¼ 1  1> hðtþ1Þ rc

ð21Þ

ðtþ1Þ rc ¼ 1  1> hðtþ1Þ rc

ð22Þ

respectively, where Iobs(hrc) is the observed information matrix and J(hrc) ¼ E[Iobs(hrc)] is the expectation/ Fisher information matrix. Thus, the key is to calculate Iobs(hrc) and J(hrc). From (1), the log-likelihood function for an incomplete r  c table is given by ! r1 X c c1 r1 X c c1 X X X X nij log ij þ nrj log rj þ nrc log 1  ij  rj ‘ ðhjYobs Þ ¼ i¼1 j¼1

þ

r1 X i¼1

j¼1

mix log iþ þ mrx log 1 

r1 X i¼1

! iþ þ

i¼1 j¼1 c1 X

j¼1

myj log þj þ myc log 1 

j¼1

c1 X

!

ð23Þ

þj

j¼1

It is not difficult to derive the first partial derivatives of ‘(hjYobs) with respect to {ij}. They are given by 8 nij nrc mix mrx myj myc > > þ  þ  , >  > ij rc iþ rþ þj þc > > @‘ ðhjYobs Þ < nrj nrc myj myc ¼  þ  , > @ij rj rc þj þc > > > > n n m m > : ic  rc þ ix  rx , ic rc iþ rþ

i ¼ 1, . . . , r  1 j ¼ 1, . . . , c  1 i ¼ r; j ¼ 1, . . . , c  1

ð24Þ

i ¼ 1, . . . , r  1; j ¼ c

From (24), we can see that the calculation of the second partial derivatives with respect to {ij} are quite tedious. Thus, a direct computation of the observed information matrix for an incomplete r  c table is challenging in practice although it seems feasible theoretically. To address this issue, we consider a more general likelihood function L ðÞ which includes the likelihood function (1) as a special case, while the observed information matrix associated with the L ðÞ is relatively easy to derive.

1720

4.1

Statistical Methods in Medical Research 26(4)

Computation of the observed information matrix for a more general likelihood function

Consider a general incomplete categorical data problem with observed data denoted by Yobs ¼ {a1, . . . , an; b1, . . . , bq}, where all {ai} and {bj} are positive. Assume that the likelihood function of the parameter vector b is of the form ! q n Y Y ai bj i ðd> ð25Þ L ðbja, b, "Þ ¼ j bÞ i¼1

where b ¼ ð1 , . . . , n Þ> 2 Tn , a ¼ ða1 , . . . , an Þ> 4 0, 0 11 12 B 21 22 B " ¼ ðij Þ ¼ B .. B .. nq @ . . n1 n2

j¼1

b ¼ ðb1 , . . . , bq Þ> , dj ¼ ð1j , . . . , nj Þ> 6¼ 0n , 1    1q    2q C C .. C .. C ¼ ðd1 , . . . , dj , . . . , dq Þ . . A    nq

with ij ¼ 0 or 1. Theorem 1 below gives an explicit expression of the observed information matrix associated with the likelihood function (25). Theorem 1. The observed information matrix associated with the likelihood function (25) is   a1 an1 an Iðbn ja, b, "Þ ¼ diag 2 , . . . , 2 þ 2 1n1 1> n1 þ )n1 n 1 n1 where

0 )n1

B B ¼ ð ik Þ ¼ B B @

11

12



1,n1

21

22

2,n1

.. .

.. .

 .. . 

n1,n1

n1,1

n1,2

.. .

ð26Þ

1 C C C C A

is an (n  1)  (n  1) matrix with ik

¼ b> sik ðb, "Þ,

sik ðb, "Þ ¼ ðik,1 , . . . , ik,j , . . . , ik,q Þ> ,

i, k ¼ 1, . . . , n  1 ik,j ¼

ð27Þ

ðij  nj Þðkj  nj Þ , 2 ðd> j bÞ

1jq

ð28Þ

Proof. From (25), the log-likelihood function is

‘ ðbja, b, "Þ ¼

n X i¼1

¼

n1 X i¼1

ai logði Þ þ

q X

bj logðd> j bÞ

j¼1

ai logði Þ þ an log 1 

n1 X

! i þ

i¼1

q X

" bj log

j¼1

n1 X

ij i þ nj 1 

i¼1

The first and second partial derivatives of ‘ ðbja, b, "Þ with respect to fi gn1 i¼1 are given by q @‘ ðbja, b, "Þ ai an X bj ðij  nj Þ ¼  þ , i ¼ 1, . . . , n  1 @i i n j¼1 d> j b @2 ‘ ðbja, b, "Þ ai an X bj ðij  nj Þ2 ¼ 2 2 2 2 i n j¼1 @i ðd> j bÞ q

n1 X i¼1

!# i

:

Tian and Li

1721 ai an   ii , i ¼ 1, . . . , n  1 2i 2n q @2 ‘ ðbja, b, "Þ an X bj ðij  nj Þðkj  nj Þ ¼ 2 2 @i @k n j¼1 ðd> j bÞ an ð27Þ ¼  2  ik , i 6¼ k; i, k ¼ 1, . . . , n  1 n ð27Þ

¼

œ

Hence, the observed information matrix is given by (26).

4.2

Computation of the observed and Fisher information matrices for incomplete r  c tables

The likelihood function (25) is reduced to the likelihood function (1) when n ¼ rc, b ¼ h ¼ ð11 , . . . , 1c , . . . , r1 , . . . , rc Þ> a ¼ ðn11 , . . . , n1c , . . . , nr1 , . . . , nrc Þ> b ¼ ðm1x , . . . , mrx , my1 , . . . , myc Þ> ¼



mx



ð29Þ

my

q¼rþc and 0

1 B B1 B. B. B. B B1 B B B0 B B0 B B. B " ¼ ðij Þ ¼ B .. B rcðrþcÞ B0 B B. B .. B B B0 B B0 B B. B. @. 0 0

Qð1Þ cr

B ð2Þ B Qcr B ¼B . B .. @ QðrÞ cr

Ic

0  0 0  0 . .. . . . .. . 0  0

1 0 .. . 0

1  0

1

1  0 . .. . . . .. .

0 .. .

1  0 . .. . . . .. .

0 .. .

0  1

1

0  1 . .. . . . .. .

0 .. .

0  1

0

1 0  0 C 1  0C .C .. . . C . .. C . C 0  1C C C 0  0C C 1  0C C .. C .. . . C . .C . C 0  1C C .. C .. . . . .C . C C 0  0C C 1  0C C .. C .. . . C . .A . 0  1

1

C Ic C C ¼ ðd1 , . . . , dj , . . . , drþc Þ .. C . C A Ic

ð30Þ

jÞ is a c  r matrix with all elements being zeros but the jth column being 1c, and Ic denotes the c  c where Qðcr identity matrix. Thus, from (26), the observed information matrix Iobs(hrc) for incomplete r  c tables is given by ! n11 n1c nr1 nr,c1 nrc ð31Þ þ 2 1rc1 1> Iobs ðhrc Þ ¼ diag 2 , . . . , 2 , . . . , 2 , . . . , 2 rc1 þ )rc1 rc 11 1c r1 r,c1

where )rc1 ¼ (b> s ik(h, ")), b is given by (29), {sik(h, "): i, k ¼ 1, . . . , rc  1} are defined by (28), and " is given by (30).

1722

Statistical Methods in Medical Research 26(4)

From (13), we have E ðnij Þ ¼ N1 ij , i ¼ 1, . . . , r; j ¼ 1, . . . , c   E ðmx Þ > > E ðb Þ ¼ ¼ ðN2 1þ , . . . , N2 rþ , N3 þ1 , . . . , N3 þc Þ E ðmy Þ where the notation E/() denotes the resultant expectation depending on the parameter vector /. Thus, from (31), the resulting Fisher information matrix   1 > Jðhrc ; /Þ ¼ N1 ½Dðhrc Þ1 þ 1rc1 1> ð32Þ rc1 þ ðE ðb Þsik ðh, "ÞÞ rc is also dependent on /, where Dðhrc Þ ¼ ^ diagð11 , . . . , 1c , . . . , r1 , . . . , r,c1 Þ

ð33Þ

Since the FS algorithm defined by (22) only involves h, we replace / in (32) with their MLEs /^ specified by (19) and obtain the Fisher information matrix on hrc as follows:   > ^ ¼ n ½Dðhrc Þ1 þ 1 1rc1 1> ^ Jðhrc ; /Þ ð34Þ Jðhrc Þ ¼ rc1 þ ðE^ ðb Þsik ðh, "ÞÞ rc where D(hrc) is given by (33), E^ ðb> Þ ¼ ðmx 1þ , . . . , mx rþ , my þ1 , . . . , my þc Þ

ð35Þ

{sik(h, "): i, k ¼ 1, . . . , rc  1} are defined by (28), and " is given by (30).

5 Bootstrap CIs of parameters Let h ¼ h(hrc) be an arbitrary function of the cell probability vector h (e.g. rate difference, odds ratio and so on) in an incomplete contingency table. Constructing CIs of h becomes a standard procedure in medical studies, especially in clinical trials. Let h^ denote the MLEs of h obtained by the FS algorithm (22) or the NR algorithm (21). Although a 100(1  )% asymptotic Wald-type CI for h can be constructed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi

^ ^ ^ ^ h  z=2 VarðhÞ, h þ z=2 VarðhÞ ð36Þ ^ can be obtained based on J1 ðh^ rc Þ or I1 ðh^ rc Þ by the delta method, where h^ ¼ hðh^ rc Þ is the MLE of h and VarðhÞ obs ^ the CI (36) may not be reliable for small to moderate sample sizes. Even worse, I1 obs ðhrc Þ may not exist in some 3 cases such as the rheumatoid arthritis study of Tang et al. Alternatively, the parametric bootstrap method (Efron15; Efron and Tibshirani16) can be used to calculate bootstrap CIs of h for small to moderate sample sizes. Specifically, given the observed data Yobs ¼ fN, mx , my g, the MLEs /^ are given by (19) and the MLEs h^ can be obtained by means of the EM algorithm (20) or the FS algorithm (22) or the NR algorithm (21) accordingly. Based ^ hÞ ^ and (13), we can generate a bootstrap sample on the ð/, ðN ¼ ðn ij Þ, m x ¼ ðm 1x , . . . , m rx Þ> , m y ¼ ðm y1 , . . . , m yc Þ> Þ  Multinomialrcþrþc ðN, p^Þ > Y obs ¼ fN , m x , m y g, we where p^ ¼ ð^1 h^ , ^2 ^1þ , . . . , ^2 ^rþ , ^3 ^þ1 , . . . , ^3 ^þc Þ> . Based on the bootstrap sample first compute the MLEs ð/^ , h^ Þ and then obtain a bootstrap replication h^ ¼ hðh^ rc Þ. Independently repeating g G ^ G the above process G times, we obtain G bootstrap samples fN g , m g x , my gg¼1 and G bootstrap replications fhg gg¼1 . ^ of h^ can be estimated by the sample standard deviation of the G replications; that is The standard error seðhÞ ( ) G h i2 1=2 1 X ^ ^ ^ ^ h  ðh1 þ    þ hG Þ=G sbeðhÞ ¼ ð37Þ G  1 g¼1 g

Tian and Li

1723

If fh^ g gG g¼1 is approximately normally distributed, a 100(1  )% simple bootstrap CI for h is h i ^ h^ þ z=2  sbeðhÞ ^ h^  z=2  sbeðhÞ,

ð38Þ

If fh^ g gG g¼1 is non-normally distributed, a 100(1  )% bootstrap percentile CI for h can be obtained by h i h^ , h^

ð39Þ

BL

BU

where h^ BL and h^ BU are the 100(/2) and 100(1  /2) percentiles of fh^ g gG g¼1 , respectively. Through the use of (39), we can obtain accurate CIs without making the normality assumption. However, (39) does not adjust the CI to account for skewness in the underlying population or other errors that can result in where hˆ is not the sample mean. Thus, we consider the bootstrap percentile-t CI, which can adjust such errors. Based on g G ^ ^ se for g ¼ 1, . . . , G, where sbe the generated G bootstrap sample fN g , m g x , my gg¼1 , we first compute tg ¼ ðhg  hÞ=b g g g g ^ is the estimated standard error of the hg in the gth bootstrap sample ðN , m g e g ¼ ðVarðh^ g ÞÞ1=2 . A x , my Þ, i.e. sb 100(1  )% bootstrap percentile-t CI for h is given by (DiCiccio and Efron17) h i ^ h^  t^=2  sbeðhÞ ^ h^  t^1=2  sbeðhÞ, ð40Þ where t^=2 and t^1=2 are the 100(/2) and 100(1  /2) percentiles of ft g gG g¼1 , respectively.

6 Testing hypothesis via the bootstrap method Let hðhrc Þ ¼ ðh1 ðhrc Þ, . . . , hk ðhrc ÞÞ> and each h‘() be a continuously differentiable function of the cell probability vector hrc with k  rc  1. In this section, we are interested in testing H0 : hðhrc Þ ¼ c against

H1 : hðhrc Þ 6¼ c

ð41Þ

where c is a known k  1 vector. In the follows, we will discuss three test methods, where both the Wald and score tests involve the computation of the Fisher information matrix (see Section 4) and all of the three tests involve the generation of bootstrap samples from a valid joint distribution of the observed counts in an incomplete r  c table (see Section 3.2) in order to calculate the bootstrap p-values for the cases of small to moderate sample sizes. Let Hðhrc Þ ¼ @hðhrc Þ=@h> rc have full row rank k. The delta method suggests that hðh^ rc Þ  hðhrc Þ ! Nk ð0, Hðhrc ÞJ1 ðhrc ÞHðhrc Þ> Þ as

n ! þ1

When H0 is true, the Wald test statistic Tw ¼ ½hðh^ rc Þ  c> ½Hðh^ rc ÞJ1 ðh^ rc ÞHðh^ rc Þ> 1 ½hðh^ rc Þ  c

ð42Þ

asymptotically follows a chi-squared distribution with k degrees of freedom. The Wald test and the likelihood ratio test often give similar conclusions as they are asymptotically equivalent, but they could disagree enough to lead to different conclusions. In addition, the Wald test uses two approximations (i.e. the covariance–variance matrix, and the 2 distribution), whereas the likelihood ratio test uses one approximation (i.e. the 2 distribution). Let h^ and h~ respectively denote the unconstrained MLE and the constrained MLE of h under H0. The Fisher scoring algorithm presented in Section 4 can be used to obtain them. The observed-data log-likelihood function is given by (23), the likelihood ratio statistic for testing H0 versus H1 in (41) is   n o ^ obs Þ  ‘ ðhjY ^ obs Þ Tl ¼ 2 max ‘ ðhjYobs Þ  max ‘ ðhjYobs Þ ¼ 2 ‘ ðhjY hðhÞ¼c ! r X c r c X ^ij X ^iþ X ^þj ¼2 nij log þ mix log þ myj log ð43Þ ~ij ~iþ ~þj i¼1 j¼1

i¼1

j¼1

When n ! þ 1, Tl asymptotically follows the 2 with k degrees of freedom under the H0. However, the likelihood ~ ratio test requires to calculate both h^ and h.

1724

Statistical Methods in Medical Research 26(4)

^ This makes A main advantage of the score test is that it does not require to calculate unconstrained MLE h. ^ testing feasible when the unconstrained MLE h is a boundary point in the parameter space. According to Yi and Wang,18 the score statistic for testing hypotheses in (41) is given by Ts ¼ ½sðh~ rc Þ> J1 ðh~ rc Þsðh~ rc Þ

ð44Þ

where the score vector  > @‘ ðhjYobs Þ @‘ @‘ @‘ @‘ sðhrc Þ ¼ ¼ ,..., ,..., ,..., @hrc @11 @1c @r1 @r,c1 ~ and J–1(h~ rc) is the value of J–1(hrc) evaluated at h ¼ h. ~ The s(h~ rc) is the value of s(hrc) evaluated at h ¼ h, components of the score vector s(hrc) are given by (24) and the Fisher information matrix J(hrc) is given by (34). As n ! þ 1, Ts also follows the 2 distribution with k degrees of freedom. The validity of the above-mentioned three asymptotic test methods depends on the large-sample theory. For small to moderate sample sizes, these asymptotic test procedures may not be reliable. Therefore, a bootstrap method can be employed to calculate the corresponding p-values of these tests. Based on the observed data Yobs ¼ fN, mx , my g, we can obtain the MLEs ^ij and ~ij for i ¼ 1, . . . , r and j ¼ 1, . . . , c by using the Fisher scoring algorithm introduced in Section 4. Thus, we can calculate the observed value t‘ for each test statistic T‘ (‘ ¼ w, l, s). For the generated bootstrap sample Y obs ¼ fN , m x , m y g as in Section 5, we can calculate the bootstrap replication t ‘ for the statistic T‘ (‘ ¼ l,s,w). Independently repeating this process G times, we can obtain G bootstrap replications ft ‘ ð gÞgG g¼1 . Thus, the bootstrap p-value for testing the null hypothesis H0 by the test statistic T‘ can be computed by G 1X Iðt ð gÞ  t‘ Þ, ‘ ¼ w, l, s ð45Þ p‘ ¼ G g¼1 ‘ where I() is the indicator function.

7 Theoretical comparisons In Section 3.2, we have obtained f ðYobs ; h, /Þ specified by (11) or (12), which is the valid sampling distribution of the observed data Yobs ¼ fN, mx , my g in the incomplete r  c table as shown in Table 1 under MAR and the assumption that the total number of observed Pcounts, N ¼ n þ mP is fixed,P where x þ my , P N ¼ ðnij Þ, mx ¼ ðm1x , . . . , mrx Þ> , my ¼ ðmy1 , . . . , myc Þ> , n ¼ ri¼1 cj¼1 nij , mx ¼ ri¼1 mix , and my ¼ cj¼1 myj . On the other hand, under the assumptions that N, mx, my are mutually independent and all n, mx, my are fixed, the joint distribution of Yobs is the product of three multinomial distributions, given by "  # " # " #  r  c r c my Y myj n Y Y nij mx Y mix f1 ðYobs ; hÞ ¼   þj ,   ð46Þ my N i¼1 j¼1 ij mx i¼1 iþ j¼1     ð1Þ n my mx ¼ ð47Þ LðhjYobs Þ my N mx which is free from the mechanism parameter vector /. In this section, we first discuss relationship between the two joint distributions f(Yobs; h, /) and f1(Yobs; h). In fact, we can rewrite (12) as       N my n mx x my f ðYobs ; h, /Þ ¼ LðhjYobs Þ  n1 m 2 3 n, mx , my my N mx ð48Þ ð47Þ

¼ f1 ðYobs ; hÞ  f2 ðn, mx , my ; /Þ

so that the ratio f ðYobs ; h, /Þ ¼ f2 ðn, mx , my ; /Þ f1 ðYobs ; hÞ

ð49Þ

Tian and Li

1725

is a trinomial probability mass function and is free from the model parameter vector h. Next, we compare two Fisher information matrices. For the incomplete r  c table in Table 1, according to Theorem 1, the observed information matrix Iobs(hrc) is given by (31). Under (46), the expectations of nij and b are given by E1 ðnij Þ ¼ nij ,

i ¼ 1, . . . , r; j ¼ 1, . . . , c

E1 ðbÞ ¼ ðmx 1þ , . . . , mx rþ , my þ1 , . . . , my þc Þ> so that the corresponding Fisher information matrix J1(hrc) is   n n n n n ,..., ,..., ,..., þ 1rc1 1> J1 ðhrc Þ ¼ diag rc1 þ E1 ð)rc1 Þ 11 1c r1 r, c1 rc

ð50Þ

with E1 ð)rc1 Þ ¼ ðE1 ð

ik ÞÞ

¼

! rþc X ðij  nj Þðkj  nj ÞE1 ðbj Þ j¼1

2 ðd> j hÞ

where fdj grþc j¼1 are defined by (30). Therefore, the difference of the two Fisher information matrices is   N1  n N1  n N1  n N1  n ð32Þ Jðhrc ; /Þ  J1 ðhrc Þ ¼ diag ,..., ,..., ,..., 11 1c r1 r, c1 N1  n þ 1rc1 1> rc1 þ E ð)rc1 Þ  E1 ð)rc1 Þ rc ^ then we find that the both are identical: However, if we replace / with their MLEs /, ^ ¼ J1 ðhrc Þ Jðhrc Þ ¼ Jðhrc ; /Þ

ð51Þ

8 Simulation studies To further investigate the performance of the proposed new sampling distribution and the independent sampling distribution, we conduct some simulation studies to compare their empirical coverage probabilities (ECP), expected CI widths (ECWs), the type I error rates and powers for various values of parameters. We only consider incomplete 2  2 tables with two margins (see Table 2). The total sample size N is set to be 10, 20, 30, 50, 100, 150, 200, 500. The proportion of the incomplete data is set to be 20%, 30%, 40% and 50%. The parameter vector of the missingdata mechanism (1, 2, 3) is set to be ð0:8, 0:1, 0:1Þ, ð0:7, 0:1, 0:2Þ, ð0:7, 0:2, 0:1Þ, ð0:6, 0:2, 0:2Þ, ð0:5, 0:3, 0:2Þ: To introduce the dependence/correlation between the paired binary outcomes, we assume that bivariate binary observations are coming from a bivariate distribution with the correlation coefficient defined by (Choi and Stablein1)

¼ ½1  ð1 þ 2 Þð1 þ 3 Þ=½ð1 þ 2 Þð3 þ 4 Þð1 þ 3 Þð2 þ 4 Þ1=2

ð52Þ

In our simulations, let ¼ 0.1, 0.3, 0.5. Given (1 þ 2, 1 þ 3, ), from (52), we have 1 ¼ ð1 þ 2 Þð1 þ 3 Þ þ ½ð1 þ 2 Þð1  1  2 Þð1 þ 3 Þð1  1  3 Þ1=2

ð53Þ

In the first simulation, we calculate the MLE and 95% CI of h ¼ 2  3 ¼ Pr(Y ¼ 1)  Pr(X ¼ 1) by computing the ECPs and ECWs. We set h ¼0.1, 0, 0.1 and 1 þ 2 ¼ 0.3, 0.4, 0.5, 0.6, 0.7. Thus, 1 þ 3 ¼ 1 þ 2  h. For

1726

Statistical Methods in Medical Research 26(4)

each configuration (1 þ 2, 1 þ 3, ), we calculate 1 via (53), 2 ¼ (1 þ 2)  1, and 3 ¼ (1 þ 3)  1. For each combination of (N,1, 2, 3), we can compute the ECP and ECW by ECP ¼ ECW ¼

M 1 X Iðh 2 ½hL ðDðmÞ Þ, hU ðDðmÞ ÞÞ M m¼1 M 1 X ½hU ðDðmÞ Þ  hL ðDðmÞ Þ M m¼1

respectively, where I() is the indicator function, hL(D(m)) and hU(D(m)) denote the lower bound and the upper bound of the CI of h based on the mth sample n o ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ , n , n , n , n , n , n , n DðmÞ ¼ nðmÞ 1 2 3 4 12 34 13 24 generated from the following multinomial distribution (see (17)) Multinomial ðN; 1 1 , 1 2 , 1 3 , 1 4 , 2 ð1 þ 2 Þ, 2 ð3 þ 4 Þ, 3 ð1 þ 3 Þ, 3 ð2 þ 4 ÞÞ

We set M ¼ 10,000. In the computation of the bootstrap CIs, G ¼ 5000 bootstrap samples are generated. Based on the generated samples {D(m): m ¼ 1, . . . , 10,000}, we calculated the 95% coverage probabilities, expected widths for the settings under consideration. Results of numerical evaluations of these CIs are presented in Tables 5–7. To calculate the type I error rates, we take h0 ¼ 0.1 and 1 þ 2 ¼ 0.5. Results are reported in Table 8. To calculate the empirical powers, we set h0 ¼ 0.1, h1 ¼ 0.3, 1 þ 2 ¼ 0.5, 0.8 and ¼ 0.5. Results are displayed in Table 9. We summarize the main findings from the simulations as follows. (1) Effect of : Table 5 shows that all mean ECWs decrease as the correlation coefficient increases. Except for PTCI, the mean coverage probability of SCI and PCI increase as the correlation coefficient increases. ECWs based on the sampling distribution under independency assumption are shorter than those based on the new sampling distribution for the same , yielding unreliable CIs. (2) Effect of (1, 2, 3): From Tables 5 and 7, we observe that larger missing proportion would result in wider ECWs for all methods. From Table 9, we can see that power decreases as the missing proportion increases. (3) Effect of 1 þ 2: Table 6 shows that when 1 þ 2 2 [0.3, 0.5], the mean expected widths increase as 1 þ 2 increases. When 1 þ 2 2 [0.6, 0.7], the mean expected widths decrease with 1 þ 2 increasing. As expected, from Table 9, power increases as 1 þ 2 increases. (4) Effect of N: From Table 7, we obtained that ECWs of all methods are shorter as the sample size N increases. Meanwhile, as expected, the power increases with the sample size N increasing according to Table 9. From Table 8, it is clear that the bootstrap tests based on the score statistic for both new and independent sampling distributions are occasionally liberal in small-sample designs (e.g. N  30). They become robust when sample sizes increase (e.g. N  50). (5) Effect of combination: From Table 7, we found that the ECWs for a bootstrap percentile-t CI are identical under both distributions for large sample sizes. ECWs based on the sampling distribution under the independency assumption are shorter than those based on the new sampling distribution for the same parameter. Except for PTCI, the mean coverage probability of SCI and PCI based on the new sampling distribution are larger than those based on the independency assumption sampling distribution. All bootstrap tests based on the Wald statistic, likelihood ratio statistic, and score statistic are similar for both new and independent sampling distributions and can be classified as robust.

9 Neurological complication data analysis In this section, we analyze the neurological complication data set presented in Table 2 to illustrate the proposed methods. The primary objective of this study is to detect if there is a substantial difference in proportion before and

Tian and Li

1727

Table 5. Mean ECP and ECW for various 95% CIs with different values of (1, 2, 3, ). Note: SCI is a simple bootstrap confidence interval defined by equation (38); PCI is a bootstrap percentile confidence interval defined by equation (39); PTCI is a bootstrap percentile-t confidence interval defined by equation (40). New sampling distribution

ECP

ECW

Independent sampling distribution

1

2

3



SCI

PCI

PTCI

SCI

PCI

PTCI

0.8

0.1

0.1

0.7

0.1

0.2

0.7

0.2

0.1

0.6

0.2

0.2

0.5

0.3

0.2

0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50

0.9404 0.9438 0.9502 0.9387 0.9430 0.9502 0.9389 0.9436 0.9504 0.9374 0.9425 0.9501 0.9365 0.9423 0.9514

0.9585 0.9587 0.9599 0.9586 0.9593 0.9615 0.9584 0.9594 0.9612 0.9585 0.9597 0.9617 0.9590 0.9609 0.9647

0.9540 0.9496 0.9451 0.9413 0.9478 0.9442 0.9418 0.9472 0.9453 0.9493 0.9466 0.9453 0.9484 0.9467 0.9573

0.9376 0.9402 0.9450 0.9359 0.9396 0.9455 0.9364 0.9398 0.9459 0.9353 0.9392 0.9456 0.9341 0.9393 0.9474

0.9499 0.9511 0.9522 0.9499 0.9515 0.9539 0.9499 0.9513 0.9540 0.9508 0.9518 0.9547 0.9508 0.9539 0.9571

0.9417 0.9458 0.9410 0.9482 0.9452 0.9505 0.9496 0.9450 0.9410 0.9477 0.9439 0.9413 0.9463 0.9447 0.9541

0.8

0.1

0.1

0.7

0.1

0.2

0.7

0.2

0.1

0.6

0.2

0.2

0.5

0.3

0.2

0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50

0.4053 0.3679 0.3236 0.4158 0.3785 0.3361 0.4157 0.3790 0.3354 0.4278 0.3913 0.3485 0.4386 0.4047 0.3647

0.4044 0.3676 0.3242 0.4153 0.3783 0.3370 0.4147 0.3787 0.3363 0.4263 0.3907 0.3492 0.4369 0.4051 0.3661

0.4376 0.3914 0.3359 0.4506 0.4042 0.3496 0.4502 0.4039 0.3498 0.4633 0.4176 0.3639 0.4748 0.4320 0.3805

0.4043 0.3650 0.3184 0.4157 0.3764 0.3309 0.4151 0.3762 0.3307 0.4270 0.3898 0.3446 0.4384 0.4031 0.3608

0.4027 0.3637 0.3181 0.4137 0.3755 0.3309 0.4132 0.3754 0.3306 0.4258 0.3889 0.3445 0.4368 0.4024 0.3612

0.4361 0.3884 0.3315 0.4479 0.4006 0.3443 0.4481 0.4002 0.3441 0.4604 0.4144 0.3583 0.4721 0.4283 0.3761

after the treatment. Since the MLEs of h do not have closed-form solutions, we can employ the EM algorithm specified by (20). By choosing h(0) ¼ 0.25  1 4 as the initial values, the MLEs of h converged to ^1 ¼ 0.2495, ^2 ¼ 0.1094, ^3 ¼ 0.3422 and ^4 ¼ 0.2989 in 8 iterations for the EM algorithm. Thus, the MLE of the risk difference h ¼ hðh4 Þ ¼ 2  3 is h^ ¼ ^2  ^3 ¼ 0:2328. In the following, we consider CI construction and hypothesis test on h. The 95% asymptotic Wald CI of h is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

^ ^ ^ ^ h  1:96 Varð2  3 Þ, h þ 1:96 Varð^2  ^3 Þ where Varð^2  ^3 Þ ¼ Varð^2 Þ þ Varð^3 Þ  2Covð^2 , ^3 Þ, and Varð^2 Þ, Varð^3 Þ, Covð^2 , ^3 Þ can be obtained via the 1 ^ ^ inverse observed information matrix I1 obs ðh4 Þ or the inverse Fisher information matrix J ðh4 Þ. The derivations 1 ^ 1 ^ of Iobs ðh4 Þ and J ðh4 Þ are given in Appendix A. 1 ^ ^ Based on I1 obs ðh4 Þ and J ðh4 Þ, the 95% asymptotic Wald CIs of h are [0.4662, 0.0007] including zero value 5 and [0.4655,5.9851  10 ] excluding zero value, respectively. Therefore, the two CIs result in totally opposite conclusions. However, this is not surprising since the conclusion of asymptotic Wald CIs depends on the largesample theory, while the sample size in this example is very small. Thus, we apply the bootstrap method to

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Statistical Methods in Medical Research 26(4)

Table 6. Mean ECP and ECW for various 95% CIs with different values of (1, 2, 3, 1þ2). New sampling distribution

ECP

ECW

Independent sampling distribution

1

2

3

1 þ 2

SCI

PCI

PTCI

SCI

PCI

PTCI

0.8

0.1

0.1

0.7

0.1

0.2

0.7

0.2

0.1

0.6

0.2

0.2

0.5

0.3

0.2

0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7

0.9453 0.9419 0.9411 0.9420 0.9457 0.9442 0.9404 0.9388 0.9419 0.9444 0.9457 0.9408 0.9388 0.9410 0.9456 0.9444 0.9395 0.9373 0.9396 0.9438 0.9443 0.9393 0.9372 0.9389 0.9446

0.9599 0.9583 0.9587 0.9585 0.9598 0.9609 0.9584 0.9580 0.9593 0.9604 0.9610 0.9585 0.9575 0.9581 0.9610 0.9610 0.9579 0.9577 0.9585 0.9609 0.9614 0.9600 0.9592 0.9594 0.9626

0.9507 0.9520 0.9529 0.9526 0.9517 0.9593 0.9500 0.9500 0.9509 0.9495 0.9501 0.9499 0.9503 0.9599 0.9506 0.9408 0.9488 0.9484 0.9476 0.9586 0.9590 0.9581 0.9580 0.9565 0.9495

0.9411 0.9392 0.9388 0.9387 0.9410 0.9407 0.9376 0.9369 0.9393 0.9406 0.9412 0.9385 0.9368 0.9381 0.9415 0.9403 0.9374 0.9360 0.9366 0.9397 0.9406 0.9368 0.9359 0.9362 0.9407

0.9518 0.9497 0.9501 0.9508 0.9517 0.9524 0.9503 0.9503 0.9512 0.9519 0.9531 0.9504 0.9501 0.9506 0.9522 0.9526 0.9518 0.9501 0.9510 0.9527 0.9544 0.9521 0.9523 0.9516 0.9545

0.9468 0.9496 0.9509 0.9494 0.9469 0.9453 0.9477 0.9484 0.9477 0.9458 0.9469 0.9478 0.9482 0.9482 0.9466 0.9544 0.9466 0.9468 0.9463 0.9449 0.9459 0.9460 0.9470 0.9453 0.9460

0.8

0.1

0.1

0.7

0.1

0.2

0.7

0.2

0.1

0.6

0.2

0.2

0.5

0.3

0.2

0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7

0.3695 0.3911 0.3984 0.3908 0.3692 0.3796 0.4020 0.4094 0.4022 0.3792 0.3787 0.4019 0.4098 0.4021 0.3794 0.3895 0.4143 0.4219 0.4142 0.3900 0.4019 0.4266 0.4345 0.4261 0.4016

0.3694 0.3906 0.3974 0.3905 0.3690 0.3796 0.4018 0.4090 0.4016 0.3794 0.3785 0.4015 0.4093 0.4016 0.3789 0.3894 0.4132 0.4213 0.4134 0.3897 0.4023 0.4263 0.4336 0.4260 0.4019

0.3923 0.4194 0.4290 0.4197 0.3919 0.4038 0.4335 0.4423 0.4329 0.4032 0.4033 0.4328 0.4419 0.4323 0.4038 0.4152 0.4464 0.4578 0.4469 0.4151 0.4273 0.4602 0.4511 0.4399 0.4073

0.3646 0.3895 0.3982 0.3899 0.3646 0.3756 0.4019 0.4103 0.4015 0.3750 0.3746 0.4012 0.4101 0.4014 0.3747 0.3858 0.4145 0.4243 0.4144 0.3863 0.3981 0.4267 0.4368 0.4265 0.3981

0.3634 0.3881 0.3971 0.3883 0.3633 0.3743 0.4005 0.4088 0.3999 0.3743 0.3736 0.4002 0.4085 0.3999 0.3741 0.3845 0.4131 0.4225 0.4135 0.3849 0.3974 0.4256 0.4348 0.4251 0.3978

0.3886 0.4175 0.4274 0.4175 0.3881 0.3989 0.4301 0.4406 0.4303 0.3995 0.3992 0.4299 0.4404 0.4297 0.3997 0.4108 0.4440 0.4552 0.4438 0.4106 0.4237 0.4580 0.4690 0.4573 0.4231

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Table 7. Mean ECP and ECW for various 95% CIs with different values of (N, 1, 2, 3). New sampling distribution

ECP

ECW

Independent sampling distribution

N

1

2

3

SCI

PCI

PTCI

SCI

PCI

PTCI

10 20 30 50 100 150 200 10 20 30 50 100 150 200 10 20 30 50 100 150 200 10 20 30 50 100 150 200

0.80

0.10

0.10

0.70

0.10

0.20

0.60

0.20

0.20

0.50

0.30

0.20

0.9477 0.9405 0.9410 0.9398 0.9445 0.9460 0.9473 0.9482 0.9400 0.9355 0.9383 0.9442 0.9458 0.9467 0.9500 0.9367 0.9315 0.9355 0.9442 0.9465 0.9465 0.9502 0.9382 0.9307 0.9342 0.9435 0.9448 0.9473

0.9875 0.9665 0.9578 0.9470 0.9452 0.9453 0.9473 0.9902 0.9692 0.9547 0.9463 0.9445 0.9455 0.9452 0.9917 0.9685 0.9527 0.9432 0.9440 0.9458 0.9465 0.9933 0.9713 0.9548 0.9437 0.9452 0.9447 0.9462

0.9460 0.9540 0.9442 0.9435 0.9455 0.9480 0.9487 0.9490 0.9492 0.9493 0.9417 0.9462 0.9477 0.9477 0.9542 0.9422 0.9432 0.9470 0.9463 0.9485 0.9493 0.9525 0.9542 0.9585 0.9432 0.9450 0.9465 0.9492

0.9330 0.9355 0.9407 0.9412 0.9448 0.9460 0.9475 0.9350 0.9353 0.9360 0.9498 0.9438 0.9458 0.9462 0.9382 0.9318 0.9313 0.9358 0.9438 0.9467 0.9468 0.9398 0.9352 0.9297 0.9338 0.9435 0.9442 0.9468

0.9697 0.9525 0.9483 0.9435 0.9447 0.9458 0.9473 0.9728 0.9545 0.9463 0.9425 0.9437 0.9452 0.9453 0.9757 0.9557 0.9443 0.9397 0.9437 0.9458 0.9462 0.9783 0.9597 0.9468 0.9408 0.9432 0.9445 0.9460

0.9492 0.9490 0.9428 0.9422 0.9482 0.9482 0.9493 0.9552 0.9452 0.9460 0.9485 0.9470 0.9478 0.9485 0.9525 0.9502 0.9490 0.9423 0.9465 0.9485 0.9490 0.9507 0.9527 0.9453 0.9495 0.9457 0.9470 0.9488

10 20 30 50 100 150 200 10 20 30 50 100 150 200 10 20 30 50 100 150 200 10 20 30 50 100 150 200

0.80

0.10

0.10

0.70

0.10

0.20

0.60

0.20

0.20

0.50

0.30

0.20

0.5998 0.4877 0.4198 0.3390 0.2460 0.2008 0.1747 0.6085 0.5012 0.4333 0.3523 0.2545 0.2088 0.1820 0.6205 0.5133 0.4477 0.3655 0.2640 0.2177 0.1878 0.6307 0.5270 0.4633 0.3792 0.2753 0.2272 0.1967

0.5993 0.4867 0.4195 0.3390 0.2460 0.2003 0.1747 0.6095 0.5000 0.4328 0.3517 0.2545 0.2088 0.1820 0.6207 0.5120 0.4468 0.3650 0.2638 0.2177 0.1877 0.6322 0.5267 0.4625 0.3792 0.2752 0.2273 0.1967

0.5993 0.4973 0.4310 0.3487 0.2507 0.2047 0.1773 0.6090 0.5145 0.4492 0.3635 0.2607 0.2120 0.1833 0.6212 0.5305 0.4672 0.3798 0.2707 0.2192 0.1907 0.6352 0.5453 0.4820 0.3937 0.2827 0.2288 0.1983

0.5818 0.4860 0.4215 0.3407 0.2460 0.2008 0.1747 0.5937 0.5008 0.4362 0.3528 0.2543 0.2088 0.1817 0.6067 0.5155 0.4512 0.3658 0.2637 0.2175 0.1875 0.6182 0.5290 0.4657 0.3793 0.2748 0.2270 0.1965

0.5787 0.4833 0.4207 0.3400 0.2460 0.2003 0.1747 0.5893 0.4985 0.4352 0.3528 0.2543 0.2088 0.1817 0.6028 0.5142 0.4493 0.3658 0.2637 0.2175 0.1875 0.6155 0.5277 0.4640 0.3793 0.2748 0.2270 0.1963

0.5987 0.4947 0.4282 0.3452 0.2507 0.2045 0.1772 0.6020 0.5100 0.4430 0.3585 0.2512 0.2118 0.1833 0.6173 0.5262 0.4590 0.3723 0.2707 0.2192 0.1907 0.6327 0.5405 0.4742 0.3868 0.2828 0.2288 0.1983

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Table 8. Empirical type I error (%) for testing H0: h ¼ h0 with  ¼ 0.05. New sampling distribution

Independent sampling distribution

N

1

2

3



Tl

Ts

Tw

Tl

Ts

Tw

10 10 10 20 20 20 30 30 30 50 50 50 100 100 100 150 150 150 200 200 200

0.80

0.10

0.10

0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50

5.23 5.40 4.93 5.00 4.40 4.60 5.53 5.47 4.70 4.80 4.63 4.73 5.33 5.07 5.03 4.87 4.37 4.70 4.93 4.60 4.97

5.17 5.13 4.73 4.80 4.53 4.93 5.00 5.40 4.97 4.97 4.60 5.23 5.07 5.23 4.90 4.40 4.17 5.13 5.00 4.60 4.77

5.00 5.23 4.80 4.83 4.40 4.70 4.97 5.17 4.77 5.27 4.87 5.43 5.07 5.13 4.83 4.43 4.13 5.13 4.90 4.57 4.80

5.00 4.63 4.73 4.70 4.77 4.93 5.00 5.10 5.00 5.60 5.50 5.60 4.93 4.23 4.80 4.93 4.60 4.93 4.43 5.20 5.07

5.03 5.03 5.47 5.33 5.27 5.47 5.27 5.03 4.63 5.40 5.47 5.30 5.13 4.73 4.90 4.93 4.70 5.13 5.17 4.80 4.63

4.30 4.83 5.60 5.27 5.13 5.03 5.30 5.00 4.60 5.27 5.47 5.23 5.37 4.93 5.10 4.87 4.83 4.97 5.23 4.67 4.73

10 10 10 20 20 20 30 30 30 50 50 50 100 100 100 150 150 150 200 200 200 10 10 10 20 20 20 30 30 30 50

0.70

0.10

0.20

0.60

0.20

0.20

0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10

4.73 5.07 4.83 5.43 5.53 5.27 5.33 5.70 4.93 5.33 5.30 4.60 5.93 5.07 4.67 5.17 5.00 4.53 5.43 5.43 5.17 5.47 5.43 5.80 5.23 5.00 4.63 5.67 5.17 4.93 4.90

4.33 4.83 4.87 4.80 4.37 4.67 4.67 5.03 4.47 5.40 5.10 4.67 5.07 4.97 4.93 5.30 4.93 4.37 5.37 5.07 5.03 5.67 5.80 5.97 5.03 4.87 4.67 5.43 4.83 5.10 4.97

4.63 5.00 5.23 5.00 5.13 5.17 5.10 5.23 4.70 5.30 4.83 4.57 5.03 4.90 4.87 5.07 4.87 4.47 5.53 5.17 5.03 5.47 5.63 5.60 5.17 4.80 4.40 5.20 4.73 4.73 5.13

5.23 4.97 5.73 5.07 4.77 4.70 4.97 4.47 4.80 5.03 5.07 4.97 5.13 5.13 5.23 4.60 4.67 4.47 5.13 5.30 5.33 5.37 5.40 5.13 5.37 5.23 5.00 5.23 5.00 4.97 5.73

4.77 4.93 5.17 5.10 4.97 4.57 5.10 4.63 5.30 4.93 5.10 4.87 5.07 5.17 5.43 4.70 4.53 4.97 5.47 5.13 5.33 5.23 5.53 5.73 5.53 5.57 5.30 4.73 5.50 5.17 5.13

5.23 5.33 4.70 4.87 4.67 4.43 4.87 4.50 4.83 4.90 4.83 4.83 5.17 5.23 5.47 4.73 4.43 4.90 5.57 5.43 5.30 4.63 4.80 4.97 5.43 4.90 4.77 4.60 5.13 4.97 5.23 (continued)

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Table 8. Continued New sampling distribution N

1

2

3

50 50 100 100 100 150 150 150 200 200 200

Independent sampling distribution



Tl

Ts

Tw

Tl

Ts

Tw

0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50 0.10 0.30 0.50

4.37 4.60 5.57 4.90 4.77 4.53 4.70 4.87 5.37 4.90 5.33

4.57 5.13 4.90 5.13 5.20 4.77 4.83 4.67 5.20 5.40 5.23

4.40 4.97 4.83 4.87 5.13 4.30 4.93 4.73 5.17 5.67 5.27

5.60 5.77 4.63 4.93 4.83 5.13 5.30 4.57 4.93 5.43 5.50

5.53 5.20 4.90 5.17 4.63 5.50 5.53 5.13 5.43 4.87 4.83

5.63 5.43 4.73 4.93 4.77 5.57 5.57 5.40 5.20 5.13 4.87

Table 9. Empirical power for testing H0: h ¼ h0 with  ¼ 0.05 and different values of (N, 1, 2, 3,1 þ 2). New sampling distribution

Independent sampling distribution

N

1

2

3

1 þ  2

Tl

Ts

Tw

Tl

Ts

Tw

10 10 20 20 30 30 50 50 100 100 150 150 200 200

0.80

0.10

0.10

0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

35.50 36.23 38.23 38.77 41.60 41.93 44.37 45.00 45.33 46.80 46.27 48.37 48.27 48.70

35.67 36.47 38.17 39.00 40.97 41.67 44.27 45.07 45.13 46.90 45.97 48.53 48.03 48.70

35.97 36.57 38.23 38.93 41.40 42.10 44.50 45.13 45.47 46.80 45.90 48.53 47.93 48.63

32.57 33.53 40.20 39.67 41.63 40.87 42.63 43.73 45.87 45.93 45.47 47.30 49.10 48.97

33.07 34.37 40.13 39.97 42.23 41.50 42.33 43.63 45.97 45.93 45.57 46.87 48.57 48.87

32.87 33.80 40.47 39.77 41.93 41.03 42.60 43.87 45.83 45.97 45.70 47.13 48.80 48.77

10 10 20 20 30 30 50 50 100 100 150 150 200 200

0.70

0.10

0.20

0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

32.27 33.43 37.43 38.67 40.57 40.97 42.13 43.63 46.30 47.53 47.90 48.43 47.43 48.47

32.00 33.77 37.07 38.80 40.43 40.73 42.00 43.90 46.37 47.50 47.70 48.40 47.33 47.93

32.53 33.77 37.43 38.90 40.57 40.90 42.23 43.67 46.43 47.30 48.00 48.40 47.37 48.00

34.00 34.60 35.67 38.97 39.73 41.00 42.07 42.87 44.83 47.07 45.87 47.70 46.23 48.17

33.43 34.67 35.63 39.07 39.60 40.93 41.40 43.03 45.00 47.43 46.70 47.87 46.60 48.43

33.43 34.27 36.07 39.03 39.73 41.20 41.77 43.07 45.20 47.23 46.60 47.80 46.60 48.43

10 10 20 20

0.60

0.20

0.20

0.5 0.8 0.5 0.8

31.93 34.47 38.67 39.67

31.53 35.30 38.70 39.65

32.07 34.90 38.80 39.57

33.50 34.17 38.30 39.33

33.10 34.33 37.80 39.40

33.40 34.00 38.10 39.47 (continued)

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Statistical Methods in Medical Research 26(4)

Table 9. Continued New sampling distribution N

1

2

30 30 50 50 100 100 150 150 200 200

3

Independent sampling distribution

1 þ  2

Tl

Ts

Tw

Tl

Ts

Tw

0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

40.27 40.87 40.50 42.67 45.87 45.43 48.07 48.57 47.90 48.10

40.23 41.40 40.70 42.57 45.67 45.73 48.30 48.80 48.10 47.87

40.37 40.93 40.53 42.80 46.17 45.33 48.27 48.93 48.27 47.77

40.30 40.63 42.60 44.07 45.40 44.33 47.20 47.30 48.33 46.50

40.23 40.37 42.57 44.27 45.50 44.60 47.60 47.27 48.20 46.93

40.47 40.93 42.70 44.20 45.80 44.43 47.73 47.10 48.30 46.97

Table 10. The 95% bootstrap CIs of 2  3 for the neurological complication data. Distribution

Method

Lower bound

Upper bound

Width

New sampling distribution

Simple bootstrap CI Bootstrap percentile CI Bootstrap percentile-t CI

0.4522 0.4544 0.4690

0.0133 0.0034 0.0092

0.4655 0.4578 0.4782

Independent sampling distribution

Simple bootstrap CI Bootstrap percentile CI Bootstrap percentile-t CI

0.4589 0.4502 0.4639

0.0067 0.0019 0.0016

0.4556 0.4521 0.4655

compute the CIs. Based on the new sampling distribution (see (13)), we can directly generate 10,000 bootstrap samples ðN , m x , m y Þ from Multinomial8 ð33, ð0:1890, 0:0829, 0:2592, 0:2264, 0:0653, 0:1166, 0:0359, 0:0247ÞÞ Based on the independent sampling distribution (see (46)), we also generate 10,000 bootstrap samples: N  Multinomial4 ð25, ð0:2495, 0:1094, 0:3422, 0:2989ÞÞ m x  Multinomial2 ð6, ð0:3589, 0:6411ÞÞ m y  Multinomial2 ð2, ð0:5917, 0:4083ÞÞ The 95% bootstrap CIs for the risk difference are reported in Table 10. Since all of the CIs include the value of 0, we can conclude that the incidence rates of neurological complication before and after the standard treatment are essentially identical. This result is consistent with that reported by Ng et al.19 It is also shown that the CI widths based on the new sampling distribution is wider than those based on the independent sampling distribution. This is not surprising because the independency assumption under-estimates the uncertainty. To examine whether there is a substantial difference in proportion before and after the treatment, we consider testing the null hypothesis H0 : 2  3 ¼ 0 against

H1 : 2  3 6¼ 0

ð54Þ

Under H0, the constrained MLEs ~1 and ~2 of 1 and 2 can be obtained by the Fisher score algorithm (see Appendix B). We obtained ~1 ¼ 0.2457 and ~2 ¼ 0.2203. The test procedures developed in Section 6 can be used to test the above hypothesis. The corresponding p-values are listed in Table 11. These results show that all tests cannot reject the null hypothesis at the significant level  ¼ 5% and claim that the incidence rates of neurological complication before and after the standard treatment are the same.

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Table 11. The p-values for testing H0: 2  3 ¼ 0 for the neurological complication data.

10

Distribution

Method

p-value

New sampling distribution

Wald test Likelihood ratio test Score test

0.6483 0.5053 0.4961

Independent sampling distribution

Wald test Likelihood ratio test Score test

0.6472 0.5077 0.4993

Discussion

In this article, we first derived the valid sampling distribution for observed data in a contingency table with incomplete counts in both margins, see (13). Second, this new joint distribution explains why the observed-data likelihood function of the cell probability vector under MAR is given by (1), see (12). Third, we developed a Fisher scoring algorithm to calculate the MLEs of parameters, where our contribution is to provide two very simple formulae (31) and (34) to compute the observed and Fisher information matrices for incomplete r  c tables. Fourth, we constructed bootstrap CIs for an arbitrary function of cell probabilities, and our simulation studies and real data analysis results showed that average/expected CI widths of parameters based on the sampling distribution under the independency assumption are shorter than those based on the new sampling distribution. Fifth, we theoretically proved that the two Fisher information matrices based on the respective new and independent sampling distributions are identical if the parameter vector / of the missing-data mechanism is replaced with its MLE, see (51). Sixth, we presented bootstrap testing hypothesis methods. Simulations and real data analysis results showed that the corresponding p-values of test statistics (including Wald statistic, likelihood ratio statistic, and score statistic) are approximately equal for both new and independent sampling distributions. Based on the new joint sampling distribution, we can develop new statistical methods for a case–control study with missing observations. Next, we may investigate the multinomial logistic regression models with covariates under the assumption of MAR. Third, the derivation of the valid joint sampling distribution of the observed counts in a more general incomplete categorical data (which include r  c incomplete contingency tables as special cases) is an important topic for our future research. Acknowledgement We thank two anonymous referees for careful reading and helpful suggestions.

Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research of Hui-Qiong LI was fully supported by the National Natural Science Foundation of China (grant number 11201412).

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1 ^ ^ Appendix A: The derivations of I1 obs ðh4 Þ and J ðh4 Þ

From (31), we have 0

I11 B Iobs ðh4 Þ ¼ @ I21

I12 I22

1 I13 C I23 A,

I31

I32

I33

where I11 I12 I13 I22



n1 n4 n12 n34 n13 n24 ¼ 2þ 2þ þ þ þ 1 4 ð1 þ 2 Þ2 ð1  1  2 Þ2 ð1 þ 3 Þ2 ð1  1  3 Þ2

n4 n12 n34 ¼ 2þ þ 4 ð1 þ 2 Þ2 ð1  1  2 Þ2

n4 n13 n24 ¼ 2þ þ 4 ð1 þ 3 Þ2 ð1  1  3 Þ2

n2 n4 n12 n34 ¼ 2þ 2þ þ 2 4 ð1 þ 2 Þ2 ð1  1  2 Þ2 n4 I23 ¼  2 4

n3 n4 n13 n24 I33 ¼  2 þ 2 þ þ 3 4 ð1 þ 3 Þ2 ð1  1  3 Þ2

Tian and Li

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From (19), the MLEs of / are given by n n1 þ n2 þ n3 þ n4 ^1 ¼ ¼ ¼ 0:7576 N n þ mx þ my mx n12 þ n34 ^2 ¼ ¼ ¼ 0:1818 N N my n13 þ n24 ^3 ¼ ¼ ¼ 0:0606 N N Since Eðnj Þ ¼ N1 j ,

j ¼ 1, . . . , 4

Eðn12 Þ ¼ N2 ð1 þ 2 Þ,

Eðn34 Þ ¼ N2 ð1  1  2 Þ

Eðn13 Þ ¼ N3 ð1 þ 3 Þ,

Eðn24 Þ ¼ N3 ð1  1  3 Þ

from (34), we obtain the Fisher information matrix as follows: 0 J11 B ^ Jðh4 Þ ¼ Jðh4 ; /Þ ¼ @ J21 J31

J12 J22 J32

J13

1

C J23 A J33

where Jij ¼ E(Iij), i.e. J11 ¼

n n n12 þ n34 n12 þ n34 n13 þ n24 n13 þ n24 þ þ þ þ þ 1 4 1 þ 2 1  1  2 1 þ 3 1  1  3

J12 ¼

n n12 þ n34 n12 þ n34 þ þ 4 1 þ 2 1  1  2

J13 ¼

n n13 þ n24 n13 þ n24 þ þ 4 1 þ 3 1  1  3

n n n12 þ n34 n12 þ n34 þ þ þ 2 4 1 þ 2 1  1  2 n ¼ 4

J22 ¼ J23

J33 ¼

n n n13 þ n24 n13 þ n24 þ þ þ 3 4 1 þ 3 1  1  3

Appendix B: The derivation of the Fisher information matrix under H0 Under H0 specified by (54), the log-likelihood function is given by ‘H0 ðhÞ ¼ n1 log 1 þ ðn2 þ n3 Þ log 2 þ n4 logð1  1  22 Þ þ ðn12 þ n13 Þ logð1 þ 2 Þ þ ðn34 þ n24 Þ logð1  1  2 Þ The first and second partial derivatives of the ‘H0(h) with respect to 1 and 2 are given by @‘H0 n1 n4 n12 þ n13 n34 þ n24 ¼  þ  @1 1 1  1  22 1 þ 2 1  1  2 @‘H0 n2 þ n3 2n4 n12 þ n13 n34 þ n24 ¼  þ  @2 2 1  1  22 1 þ 2 1  1  2 @2 ‘H0 n1 n4 n12 þ n13 n34 þ n24 ¼ 2   @12 1 ð1  1  22 Þ2 ð1 þ 2 Þ2 ð1  1  2 Þ2 @2 ‘H0 2n4 n12 þ n13 n34 þ n24 ¼   2 2 @1 @2 ð1  1  22 Þ ð1 þ 2 Þ ð1  1  2 Þ2 2 @ ‘H0 n2 þ n3 4n4 n12 þ n13 n34 þ n24 ¼    2 2 2 2 @2 2 ð1  1  22 Þ ð1 þ 2 Þ ð1  1  2 Þ2

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respectively. Then, the Fisher information matrix is given by   J11 J12 J¼ J21 J22 where  2  @ ‘H0 N1 N1 Nð2 þ 3 Þ Nð2 þ 3 Þ J11 ¼ E  þ þ þ ¼ 1 þ 2 1  1  2 1 1  1  22 @12   2 @ ‘H0 2N1 Nð2 þ 3 Þ Nð2 þ 3 Þ J12 ¼ E  þ þ ¼ 1 þ 2 1  1  2 @1 @2 1  1  22  2  @ ‘H0 2N1 4N1 Nð2 þ 3 Þ Nð2 þ 3 Þ J22 ¼ E  þ þ þ ¼ 2 1 þ 2 1  1  2  1    2 @2 2 1 2