# Hierarchical Missing Data and Multivariate Behrens–Fisher Problem

Authors: Jianqi Yu
Journal: Journal of Mathematics
 (2021)

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## Abstract

This article firstly defines hierarchical data missing pattern, which is a generalization of monotone data missing pattern. Then multivariate Behrens–Fisher problem with hierarchical missing data is considered to illustrate that how ideas in dealing with monotone missing data can be extended to deal with hierarchical missing pattern. A pivotal quantity similar to the HotellingT2 is presented, and the moment matching method is used to derive its approximate distribution which is for testing and interval estimation. The precision of the approximation is illustrated through Monte Carlo data simulation. The results indicate that the approximate method is very satisfactory even for moderately small samples.


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## Body

## 1. Introduction

Inferences with incomplete data have aroused lots of interest among statisticians in the past as well as present. The causes for missing data could be various which will not be discussed in this article. However, to ignore the process that causes missing data, it is usually assumed that the data are missing at random (MAR). For an exposition of such issues, we refer to Little and Rubin [1] or Little [2]. Lu and Copas [3] pointed out that inference from the likelihood method ignoring the missing data mechanism is valid if and only if the missing data mechanism is MAR.There are a few missing patterns considered in the literatures, but the incomplete data with monotone pattern (see display (1) and (2)) not only occur frequently in practice but also it allows the exact calculation of the maximum likelihood estimators (MLEs) and the likelihood ratio statistics and relevant distributions if multivariate normality is assumed. Anderson [4], one of the earliest authors in this area, gave a simple approach to derive the MLEs and present them for a special case of monotone pattern. Krishnamoorthy and Pannala [5, 6] provided an accurate, simple approach to construct a confidence region for a normal mean vector. Hao and Krishnamoorthy [7] developed an inferential procedure on a normal covariance matrix. Yu et al. [8] considered the problem of testing equality of two normal mean vectors with the assumption that the two covariance matrices are equal, while Krishnamoorthy and Yu [9] considered the Behrens–Fisher problem. Yu et al. [10] considered the problem of testing equality of two normal covariance matrices with monotone missing data.Besides, Batsidis [11–13] extends the inferences on monotone missing data to the assumption of elliptically contoured distributions of which the multivariate normal is a special case. For theory and methods of multivariate analysis based on the elliptically contoured distributions, we refer to Fang and Zhang [14].Most of the papers mentioned above use a similar strategy in dealing with the monotone missing data. To illustrate this, consider the data matrices with 2-block monotone pattern as shown below:(1)x1,…,xnxn+1…,xn+my1,…,yn.The strategy is as follows: if we do not have the extradata ony, i.e., we have only the first n samples on x,y, usually we already have a statistics, say Q, out of the complete data. Similarly, if we have only n+m sample on x, we also have a similar statistics, say Q1, for the lower-dimensional problem. We then decompose Q into two parts Q1′+Q2, which correspond to the sample data on x and y, respectively. However, since we have extradata on y, Q1′ should be replaced with Q1. Hence, we get the final statistics for inference Q1+Q2.In this article, we will define a new data missing pattern, the hierarchical data missing pattern, which is a generalization of monotone missing pattern. Moreover, the strategy just mentioned can also be used. To see this, we consider the multivariate Behrens–Fisher problem with hierarchical missing data. The approach that we will employ is based on the one due to Krishnamoorthy and Yu [9] for the monotone missing data.The article is organized as follows: in the following section, we define the hierarchical data missing pattern. In Section3, an approximate method for the multivariate Behrens–Fisher problem with hierarchical missing data is outlined. The accuracy of the approximation is investigated using the Monte Carlo simulation in Section 4. The methods are illustrated using an example in Section 5, and some concluding remarks are given in Section 6.


## 2. Hierarchical Data Missing Pattern

Supposex1i,x2i,…,xki′ is the i-th observation of the random vector x1,x2,…,xk′, i=1,2,…,N1. The monotone pattern of missing data is like following data:(2)x11,…,x1Nk,…,x1N2,…,x1N1,x21,…,x2Nk,…,x2N2,……,xk1,…,xkNk,where xij is a pi×1 vector, N1≥N2≥⋯≥Nk, i=1,…,k. In other words, there are N1 observations available on the first p1 components, N2 observations available on the first p1+p2 components, and so on. Notice that N1≥N2≥⋯≥Nk and p1+⋯+pk=p.We define the hierarchical data missing pattern of as the following pattern:(3)x1,…,xnxn+1,…,xn+mxn+m+1,…,xn+m+lxn+m+1+1,…,xNy1,…,ynyn+1,…,yn+mzn+m+1,…,zn+m+lun+m+1+1,…,uNv1,…,vnwn+1,…,wn+m,where the index sets satisfy following conditions:(1)
The index set of the first row, i.e1,…,n,n+1,…,n+m,n+m+1,…,n+m+1,n+m+1+1…,N, is the union of the index sets of all the other rows.(2)
The index sets of two different rows are either disjoint, or inclusive. It is easy to see that the monotone pattern is a special case of the hierarchical pattern.Now we consider the Behrens–Fisher problem with hierarchical missing data.To formulate the problem, letx follows a p-variate normal distribution with mean vector μ and covariance matrix Σ, and we write this as x∼Npμ,Σ. Meanwhile, let y∼Npβ,Δ, and y is independent of x. It is assumed that Σ and Δ are unknown and arbitrary positive definite matrices. Let us consider the problem of testing:(4)H0:μ=βvs.Ha:μ≠β.Suppose that we have a sample ofN1 observations available on x and a sample of M1 observations available on y. We consider a simple 3-block hierarchical data as shown below (it is easy to extend the ideas and procedures for 3-block data to general case as in (3), but the notation will become very complicated):(5)x11,…,x1N2,x1,N2+1,…,x1N1y11,…,y1M2,y1,M2+1,…,y1M1x21,…,x2N2,y21,…,y2M2,x3,N2+1,…,x3N1y3,M2+1,…,y3M1,where xij is a pi×1 vector, j=1,…,Ni, while yij is a qi×1 vector, j=1,…,Mi, i=1,2,3. In other words, in the x sample, there are N1 observations available on the first p1 components, N2 observations available on the first p1+p2 components, and N1−N2 observations available on the first p1 and the last p3 components. Notice that N1≥N2, M1≥M2, and p1+p2+p3=q1+q2+q3=p.As pointed in Yu et al. [8], we do not need to consider the case of unequal pattern, i.e, pi≠qi, for some i=1,2,3, since any type of unequal patterns data can be rearranged to form an equal monotone pattern. For example, assume that p1>q1,p2>q2,p3<q3, and p1<q1+q2. Let r1=q1,r2=p1−q1,r3=q1+q2−p1,r4=p1+p2−q1−q2,r5=p3, then it is obvious that we have a 5-block equal pattern.Hence, without loss of generality, we assume thatpi=qi,i=1,2,3.


## 3. Inference onμ−β

### 3.1. Preliminaries

Consider the data matrices in (5) and partition the data matrices as follows:(6)X1=x11,…,x1N2,…,x1N1p1×N1,X2=x11,…,x1N2x21,…,x2N2p1+p2×N2,X3=x1,N2+1,…,x1N1x3,N2+1,…,x3N1p1+p3×N3,where N3=N1−N2.Partition the matrixY similarly. That is,(7)Y1=y11,…,y1M2,…,y1M1p1×M1,Y2=y11,…,y1M2y21,…,y2M2p1+p2×M2,Y3=y1,M2+1,…,y1M1y3,M2+1,…,y3M1p1+p3×M3,where M3=M1−M2.Letx¯l and Sl denote, respectively, the sample mean vector and the sum of squares and sum of products matrix based on Xl, l=1,2,3. Similarly, let y¯l and Vl denote, respectively, the sample mean vector and the sums of squares and products matrix based on Yl, l=1,2,3. We partition these means and matrices accordingly as follows:(8)x¯1=x¯11,x¯2=x¯21x¯22,x¯3=x¯31x¯33,S1=S11,1,S2=S21,1S21,2S22,1S22,2,S3=S31,1S31,3S33,1S33,3.The statisticsyl¯ and Vl based on the data matrix Y in (7) are also partitioned like x¯l and Sl:(9)y¯1=y¯11,y¯2=y¯21y¯22,y¯3=y¯31y¯33,V1=V11,1,V2=V21,1V21,2V22,1V22,2,V3=V31,1V31,3V33,1V33,3.Finally, we partition the parameters as follows:(10)μ=μ1μ2μ3,Σ=Σ11Σ12Σ13Σ21Σ22Σ23Σ31Σ32Σ33,β=β1β2β3,Δ=Δ11Δ12Δ13Δ21Δ22Δ23Δ31Δ32Δ33.where μi is pi dimensional, i=1,2,3.Furthermore, defineδ=μ−β so that(11)δ1=μ1−β1,δ2=μ2−β2,δ3=μ3−β3.Letni=Ni−1 and mi=Mi−1, i=1,2,3.The following summary statistics are needed to define the pivotal quantity that we will use for hypothesis testing aboutδ. Let(12)C1=S11,1n1N1+V11,1m1M1,C2=S2n2N2+V2m2M2,C3=S3n3N3+V3m3M3,B2.1=S22,1n2N2+V22,1m2M2S21,1n2N2+V21,1m2M2−1,B3.1=S33,1n3N3+V33,1m3M3S31,1n3N3+V31,1m3M3−1,C2.1=S22,2n2N2+V22,2m2M2−B2.1S21,2n2N2+V21,2m2M2,C3.1=S33,3n3N3+V33,3m3M3−B3.1S31,3n3N3+V31,3m3M3.Furthermore, let(13)δ^1=x¯11−y¯11,δ^2=x¯22−y¯22,δ^3=x¯33−y¯33,δ^=δ^1′,δ^2′,δ^3′′.The pivotal quantity that we propose for hypothesis testing and confidence estimation ofδ which is given by(14)Q=δ^1−δ1′C1−1δ^1−δ1+δ^2−δ2−B2.1x¯21−y¯21−δ1′C2.1−1δ^2−δ2−B2.1x¯21−y¯21−δ1+δ^3−δ3−B3.1x¯31−y¯31−δ1′C3.1−1δ^3−δ3−B3.1x¯31−y¯31−δ1=Q1+Q2+Q3,say.The idea behindQ is as follows: if there are only N2M2 observations on the first p1 components of XY, the appropriate statistic for hypothesis testing and confidence estimation of δ1′,δ2′′=μ1−β1′,μ1−β2′′ can be decomposed as the sum of two parts after some algebra:(15)x¯21−y¯21−δ1′S21,1n2N2+V21,1m2M2−1x¯21−y¯21−δ1+Q2.Since there are additional observations on the firstp1 components, the first part above should be replaced by Q1.Similarly, If there are only the lastN3M3 observations on the first p1 components of XY, the appropriate statistic for hypothesis testing and confidence estimation of δ1′,δ3′′ can be decomposed as the sum of two parts after some algebra:(16)x¯31−y¯31−δ1′S31,1n3N3+V31,1m3M3−1x¯31−y¯31−δ1+Q3.Again, the first part should also be replaced byQ1.


### 3.2. Hypothesis Test and Confidence Region forμ−β

BecauseQ is resembling the Hotelling-T2 statistic, and its distribution is free of any parameters, it is reasonable to approximate its distribution by the distribution of dFp,ν, where d is a positive constant and Fa,b denotes the F random variable with numerator degrees of freedom a and the denominator degrees of freedom b.To find an approximation to the distribution ofQ, we evaluated its first two approximate moments in the Appendix. Then, using the “moment matching” method, the distribution of Q is approximated by dFp,ν, where d is a positive constant, and Fa,b denotes the F random variable with numerator degrees of freedom a and the denominator degrees of freedom b. The unknown constants d and v can be determined so that the first two moments of Q are equal to those of dFp,ν. Using the modified Wishart approximation (see Lemma A.1 in Appendix) and following the lines of Krishnamoorthy and Pannala [6], we evaluated an approximation G1 for EQ and an approximation G2 for EQ2 in Appendix. To express G1 and G2, we need the following terms.LetS˜1=S11,1/n1N1,V˜1=V11,1/m1M1,C1=S˜1+V˜1,and(17)f1=p1+p121/n1trS˜1C1−12+trS˜1C1−12+1/m1trV˜1C1−12+trV˜1C1−12.LetS˜2=S2/n2N2, V˜2=V2/m2M2, C2=S˜2+V˜2, and(18)f2=p1+p2+p1+p221/n2trS˜2C2−12+trS˜2C2−12+1/m2trV˜2C2−12+trV˜2C2−12.LetS˜3=S3/n3N3, V˜3=V3/m3M3, C3=S˜3+V˜3, and(19)f3=p1+p3+p1+p321/n3trS˜3C3−12+trS˜3C3−12+1/m3trV˜3C3−12+trV˜3C3−12.In terms of the above quantities, we have(20)G1=f1p1f1−p1−1+p2f2f2−1f2−p1−p2−1f2−p1−1+p3f3f3−1f3−p1−p3−1f3−p1−1G2=p1p1+2f12f1−p1−1f1−p1−3+2f1p1f1−p1−1p2f2f2−1f2−p1−p2−1f2−p1−1+2f1p1f1−p1−1p3f3f3−1f3−p1−p3−1f3−p1−1+2p2f2f2−1f2−p1−p2−1f2−p1−1p3f3f3−1f3−p1−p3−1f3−p1−1+p2p2+2f22f2−1f2−3f2−p1−p2−1f2−p1−p2−3f2−p1−1f2−p1−3+p3p3+2f32f3−1f3−3f3−p1−p3−1f3−p1−p3−3f3−p1−1f3−p1−3,and an approximation to the distribution of the pivotal quantity Q in (10) is given by Q∼dFp,ν approximately, where(21)ν=4pG2−2p+2G12pG2−p+2G12,d=G1ν−2ν.Thus, for a given levelα and an observed value Q0 of Q, the null hypothesis that δ=μ−β=0 will be rejected whenever the p value(22)PQ0>dFp,ν|H0<α.Furthermore, an approximate1−α confidence set for μ−β is the set of values of δ that satisfy(23)Q≤dFp,ν1−α,where Q is given in (9) and Fp,ν1−α is the 1−αth quantile of the Fp,ν distribution.


## 3.1. Preliminaries

Consider the data matrices in (5) and partition the data matrices as follows:(6)X1=x11,…,x1N2,…,x1N1p1×N1,X2=x11,…,x1N2x21,…,x2N2p1+p2×N2,X3=x1,N2+1,…,x1N1x3,N2+1,…,x3N1p1+p3×N3,where N3=N1−N2.Partition the matrixY similarly. That is,(7)Y1=y11,…,y1M2,…,y1M1p1×M1,Y2=y11,…,y1M2y21,…,y2M2p1+p2×M2,Y3=y1,M2+1,…,y1M1y3,M2+1,…,y3M1p1+p3×M3,where M3=M1−M2.Letx¯l and Sl denote, respectively, the sample mean vector and the sum of squares and sum of products matrix based on Xl, l=1,2,3. Similarly, let y¯l and Vl denote, respectively, the sample mean vector and the sums of squares and products matrix based on Yl, l=1,2,3. We partition these means and matrices accordingly as follows:(8)x¯1=x¯11,x¯2=x¯21x¯22,x¯3=x¯31x¯33,S1=S11,1,S2=S21,1S21,2S22,1S22,2,S3=S31,1S31,3S33,1S33,3.The statisticsyl¯ and Vl based on the data matrix Y in (7) are also partitioned like x¯l and Sl:(9)y¯1=y¯11,y¯2=y¯21y¯22,y¯3=y¯31y¯33,V1=V11,1,V2=V21,1V21,2V22,1V22,2,V3=V31,1V31,3V33,1V33,3.Finally, we partition the parameters as follows:(10)μ=μ1μ2μ3,Σ=Σ11Σ12Σ13Σ21Σ22Σ23Σ31Σ32Σ33,β=β1β2β3,Δ=Δ11Δ12Δ13Δ21Δ22Δ23Δ31Δ32Δ33.where μi is pi dimensional, i=1,2,3.Furthermore, defineδ=μ−β so that(11)δ1=μ1−β1,δ2=μ2−β2,δ3=μ3−β3.Letni=Ni−1 and mi=Mi−1, i=1,2,3.The following summary statistics are needed to define the pivotal quantity that we will use for hypothesis testing aboutδ. Let(12)C1=S11,1n1N1+V11,1m1M1,C2=S2n2N2+V2m2M2,C3=S3n3N3+V3m3M3,B2.1=S22,1n2N2+V22,1m2M2S21,1n2N2+V21,1m2M2−1,B3.1=S33,1n3N3+V33,1m3M3S31,1n3N3+V31,1m3M3−1,C2.1=S22,2n2N2+V22,2m2M2−B2.1S21,2n2N2+V21,2m2M2,C3.1=S33,3n3N3+V33,3m3M3−B3.1S31,3n3N3+V31,3m3M3.Furthermore, let(13)δ^1=x¯11−y¯11,δ^2=x¯22−y¯22,δ^3=x¯33−y¯33,δ^=δ^1′,δ^2′,δ^3′′.The pivotal quantity that we propose for hypothesis testing and confidence estimation ofδ which is given by(14)Q=δ^1−δ1′C1−1δ^1−δ1+δ^2−δ2−B2.1x¯21−y¯21−δ1′C2.1−1δ^2−δ2−B2.1x¯21−y¯21−δ1+δ^3−δ3−B3.1x¯31−y¯31−δ1′C3.1−1δ^3−δ3−B3.1x¯31−y¯31−δ1=Q1+Q2+Q3,say.The idea behindQ is as follows: if there are only N2M2 observations on the first p1 components of XY, the appropriate statistic for hypothesis testing and confidence estimation of δ1′,δ2′′=μ1−β1′,μ1−β2′′ can be decomposed as the sum of two parts after some algebra:(15)x¯21−y¯21−δ1′S21,1n2N2+V21,1m2M2−1x¯21−y¯21−δ1+Q2.Since there are additional observations on the firstp1 components, the first part above should be replaced by Q1.Similarly, If there are only the lastN3M3 observations on the first p1 components of XY, the appropriate statistic for hypothesis testing and confidence estimation of δ1′,δ3′′ can be decomposed as the sum of two parts after some algebra:(16)x¯31−y¯31−δ1′S31,1n3N3+V31,1m3M3−1x¯31−y¯31−δ1+Q3.Again, the first part should also be replaced byQ1.


## 3.2. Hypothesis Test and Confidence Region forμ−β

BecauseQ is resembling the Hotelling-T2 statistic, and its distribution is free of any parameters, it is reasonable to approximate its distribution by the distribution of dFp,ν, where d is a positive constant and Fa,b denotes the F random variable with numerator degrees of freedom a and the denominator degrees of freedom b.To find an approximation to the distribution ofQ, we evaluated its first two approximate moments in the Appendix. Then, using the “moment matching” method, the distribution of Q is approximated by dFp,ν, where d is a positive constant, and Fa,b denotes the F random variable with numerator degrees of freedom a and the denominator degrees of freedom b. The unknown constants d and v can be determined so that the first two moments of Q are equal to those of dFp,ν. Using the modified Wishart approximation (see Lemma A.1 in Appendix) and following the lines of Krishnamoorthy and Pannala [6], we evaluated an approximation G1 for EQ and an approximation G2 for EQ2 in Appendix. To express G1 and G2, we need the following terms.LetS˜1=S11,1/n1N1,V˜1=V11,1/m1M1,C1=S˜1+V˜1,and(17)f1=p1+p121/n1trS˜1C1−12+trS˜1C1−12+1/m1trV˜1C1−12+trV˜1C1−12.LetS˜2=S2/n2N2, V˜2=V2/m2M2, C2=S˜2+V˜2, and(18)f2=p1+p2+p1+p221/n2trS˜2C2−12+trS˜2C2−12+1/m2trV˜2C2−12+trV˜2C2−12.LetS˜3=S3/n3N3, V˜3=V3/m3M3, C3=S˜3+V˜3, and(19)f3=p1+p3+p1+p321/n3trS˜3C3−12+trS˜3C3−12+1/m3trV˜3C3−12+trV˜3C3−12.In terms of the above quantities, we have(20)G1=f1p1f1−p1−1+p2f2f2−1f2−p1−p2−1f2−p1−1+p3f3f3−1f3−p1−p3−1f3−p1−1G2=p1p1+2f12f1−p1−1f1−p1−3+2f1p1f1−p1−1p2f2f2−1f2−p1−p2−1f2−p1−1+2f1p1f1−p1−1p3f3f3−1f3−p1−p3−1f3−p1−1+2p2f2f2−1f2−p1−p2−1f2−p1−1p3f3f3−1f3−p1−p3−1f3−p1−1+p2p2+2f22f2−1f2−3f2−p1−p2−1f2−p1−p2−3f2−p1−1f2−p1−3+p3p3+2f32f3−1f3−3f3−p1−p3−1f3−p1−p3−3f3−p1−1f3−p1−3,and an approximation to the distribution of the pivotal quantity Q in (10) is given by Q∼dFp,ν approximately, where(21)ν=4pG2−2p+2G12pG2−p+2G12,d=G1ν−2ν.Thus, for a given levelα and an observed value Q0 of Q, the null hypothesis that δ=μ−β=0 will be rejected whenever the p value(22)PQ0>dFp,ν|H0<α.Furthermore, an approximate1−α confidence set for μ−β is the set of values of δ that satisfy(23)Q≤dFp,ν1−α,where Q is given in (9) and Fp,ν1−α is the 1−αth quantile of the Fp,ν distribution.


## 4. Accuracy of the Approximations

We have used two approximations, one for approximating the sum of two Wishart matrices with different scale matrices and another for approximating the moments ofQ to derive the distribution of Q. So, to understand the accuracy of the approximation, we estimated the sizes of the test for hypotheses in (4) when the nominal level is 0.05 using the Monte Carlo simulation.To select the parameter configurations for Monte Carlo simulation, we note that the distribution ofQ is location invariant, and so without loss of generality, we can assume that μ=β=0 to estimate the sizes. As pointed out in the study of Krishnamoorthy and Yu [9], we can also take Σ as a diagonal matrix with positive elements and Δ as a correlation matrix.The estimated sizes are presented in Table1 for the case of p1=2,p2=1,p3=1, and a few selected sample sizes. The sample sizes are chosen so that the number of data missing is relatively small in some cases and large in other cases. It is clear from Table 1 that the coverage probabilities are very close to 0.95 for all the cases considered. In the worst situations, the coverage probabilities are around 0.93.Table 1
Monte Carlo estimates of the coverage probabilities of the confidence region;Σ=diagλ1,λ2,λ3, Δ=1ρ12ρ13ρ121ρ23ρ13ρ231p1,p2,p3=1,1,1;α=0.05.
N1,N2,N3,M1,M2,M3λ1,λ2,λ3,ρ12,ρ13,ρ23(13, 6, 14, 7)(14, 6, 20, 12)(18, 10, 20, 10)(25, 13, 15, 8)(1, 2, 3, 0.3, 0.2, −0.4)0.9540.9490.9520.956(8, 2, 6, −0.4, −0.5, 0.3)0.9480.9670.9540.953(0.4, 3, 9, 0.9, 0.3, −0.1)0.9480.9410.9470.962(1, 1, 1, 0.2, 0.2, 0.2)0.9570.9520.9540.953(3, 3, 3, 0.9, 0.9, 0.9)0.9460.9420.9490.951(0.4, 0.3, 0.1, −0.4, −0.2, −0.1)0.9550.9510.9500.954(2, 0.1, 12, 0.1, −0.9, −0.5)0.9520.9540.9490.944(0.5, 0.9, 0.1, 0.9, 0.4, 0.5)0.9520.9440.9500.955(0.6, 0.5, 0.1, −0.7, 0.5, 0.2)0.9540.9500.9520.954(1, 25, 50, −0.5, 0.5, 0.1)0.9440.9390.9460.956(23, 33, 55, 0.1, −0.1, 0.2)0.9430.9360.9540.951(1, 1, 40, −0.4, 0.4, 0.2)0.9540.9490.9480.955


## 5. An Illustrative Example

We shall now illustrate the methods using “Fisher’s Iris Data” which represent measurements of the sepal length and width and pedal length and width in centimeters of fifty plants for each of three types of iris: Iris setosa, Iris versicolor, and Iris virginica. The data sets are posted in many websites, and we downloaded them fromhttp://javeeh.net/sasintro/intro151.html. For illustration purpose, we use the data on virginica x and setosa y. Since the sample size is large enough, we simply assume that the data are following approximately a multivariate normal distribution.We created hierarchical patterns by discarding the last 15 measurements onx3 (pedal length of virginica) and the first 35 measurements on x4 (pedal width of virginica), the last 30 measurements on y3 (pedal length of setosa) and the first 20 measurements on y4 (pedal width of setosa). That is, we have p1=2,p2=1,p3=1, and N1,N2,N3=50,15,35,M1,M2,M3=50,20,30. Let μ′=μ1,μ2,μ3= (average of the sepal length and width, pedal length, and pedal width) of virginica and β′=β1,β2,β3= (average of the sepal length and width, pedal length, and pedal width) of setosa. We want to test(24)H0:μ−β=0vs.H0:μ−β≠0.After careful calculation, we getQ1=536.925,Q2=1299.851,andQ3=1121.327, so Q=2958.103. The required values to compute the critical value are G1=EQ=4.307, G2=EQ2=31.341, d=4.250, and v=150.759. The critical value dFp,ν0.95=11.325.SinceQ is much larger than the critical value, we have sufficient evidence to reject H0 at 95% confidence level.


## 6. Concluding Remarks

In this article, we define hierarchical data missing pattern and point out that the strategy in many papers dealing with monotone missing data can be extended to deal with hierarchical missing data. To illustrate this, the multivariate Behrens–Fisher problem is considered. Based on the procedures due to Krishnamoorthy and Yu [9] dealing with the monotone missing data, we proposed a Hotelling T2 type test for Behrens–Fisher problem. The test is simple to use, and the hierarchical patterns of the two samples are not necessarily the same.As pointed out by two reviewers, this paper is based on multivariate normal population. Like what did in Batsidis [11–13] for monotone missing data, an extension of the results given in this paper for hierarchical missing data from elliptic distribution is an interesting open problem. Moreover, the proposed study can be extended for the neutrosophic statistics as future research. For details of neutrosophic statistics, see Aslam [15, 16] and Kashif et al. [17].



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*Source: 8837044-2021-01-27.xml*