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Robustness of BIBD against the unavailability of data
JoumaIof
Meal PlannJq and hference 6 (19$2)29-32
29
Nor&-Holhurd P .tblishing Company
ES8
Sub& GHOSH &‘at&~~
OF BIBD AGAINST THE OF DATA ”
Institute, Calcutta, JndiP
Rmxhd 28 Fcbmary~J97~revisedmanuscriptreceived&4July 1981 I Recommendal by W.T. Fed= NMuct: In this paper we study a robustnessproperty of balanced incomplete block designs (BIBD) with parameters u, b, r, k and A against the unavailability of dntp, in the sense that, when any t, a positive integer, observationsare unavailablethe design remains connected w,r.t. treatment. We show that a BIBD (& b, r, k, A) is robust against the unavailability of &y P- 1 observations, We observe that it is robust against the unavailability of all observations in any F- 1 blocks. We also prove that when any k- 1 observationsare unavailablethe design remains connected w.r.t. treatment and block. AMS 1970 subjective ci&s@#&tion:Secondary G2K10, OSBQS. Key WWU?S and phmes: Balanced incomplete block designs; Bipartite graphs; Robust designs;
Unavailabledata.
I. Introduction
The statistical planning of an experiment is always a difficuit task. Many things can go wrong even in a well planned experiment. One sucl~thing is the ‘availability of data’* The unavailability of data may arise not only from the loss of data or the data being missing but also from the deficit of budget during the experiment. The unavailability of even one observation may destroy the whole purpose of experimentation. It is never possible to anticipate beforehand which observations are going to be unavailable. These facts, which are very common in OUTreal life, motivated the auth&, Ghosh (1979), to study tlhe ro3.%Wness of designs against the unavailability of any number of observations and w.r,t. the estimability of parameters. In this paper we study the robustness property in a block design set-up. If a single observation is unavailable in an experiment using a BI D then the resulting design is not a BIBD. Block designs (BD) are normally used in comparative experiments where the main interest is in comparing several treatments. Connectedness of designs therefore plays an importazit role in comparative experiments. b, r, ic, 1) are well known, see agaiwt the loss of one treatment and * Present address: University of California, Riverside, CA92521, WA.
037803758/82/oooO-
;$02.75 0 1982 North-Holland
w.r.t. variance balance was considered in Hedayat and John (1974) and John (1976). The same property was studied by Most (1973) in case of loss of any subset of a spec: Ec set of n treatments, IQ< k< iu. The robustness of connected balanced block designs against the loss of one treatment and w&t, variance balance sidered in Kageyama (1980). The robustness property against the unavailability of data and w.r. t. the estimability of parameters eogsidered in this paper is completely different from the robustness against the loss of treatments and w.r.t, variance balance in the above mentioned works. This paper therefore establishes a further property of BIBD. A consideration of optimality in addition to the estimability of parameters for designs with the unavailable data is made in Ghosh (1981).
2. Robustness of BIRD Consider a BD with u treatments in b blocks. We can associate with it a bipartite graph G in the following manner: The vertices of 43, treatments and blocks, can be partitioned into two subsets T = {1,2, . . . . v} and B = {1,2, . . . . b} consisting of ‘3 treatments and b blocks, respectively; a point in T is joined to a point in B if the corresponding treatment occurs in the corresponding block. It can be seen that a BD is completely connected (i.e. connected w.r.t. treatment and block) !f and only if the corresponding bipartite graph @ is connected. Moreover, a BD is connected w,r.t. treatment if and only if all points in T of G are connected with each other. In this paper we consider a special Lind of block designs namely B’IBD(v, b, r, k, L). A BIBD is always a completely connected design. If some observations in a BIBD are unavailable then the resulting design may not be another BIBD. Furthermore, the resulting BD may or may not even be connected w.r.t. treatment. Definition 1. A BIBD (v, 6, r, k, jl) is said to be robust against the unavailability of ;anv t (a positive integer) observations if the block design obtained by omitting any t observations remains connected w.r.t. treatment. Definition 2. A BIBD (v, b, r, k, A) is said to be strongly robust against the unavaillability of any t observations if the block desi n obtained by omitting any t observations remains completely connected, An observation corresponding to a treatment ii in a block bl is unavailabl the line joining il and bl in G is removed. I. A BiB.D (ti, b, r, k, A) is robust against the ~nav~i&a~~~it~ of any robsmwr’ions.
e the subset of points in
1
which are joined to lt
II= a, t&,1= f&l = r- il, where I&f denotes the cardie any set of r - 1 observations are unavailable, Le. the oved from G. If for the resulting ected. Consider the si = 2,3, the degrees of d 42 be another subset of points in Bj, e for the resulting graph and G. Let IJ=II and lBjtl=l~. We have k@, &O, kI +&=I, +12= - 1, We now prove the existence of a point ia in r which is se there does not exist such a point d i2, is joined to u points in B2 then to u points in Bs. Thus ~11Ihe points T other than iI and i2 which are B2,. It follows th t &kl. Therefo:e, r - R = I2+ I1s k, + 1,~ r- A- I. This is impossible! Thus the existence of the point i3 is proved, Mence the points ii and is are connected. Therefore the resulting design is oannected. This completes the proof. A BIBD is not robust against the unavailabiiity of any r observations because if all r observations corresponding to a parti :ular treatment are unavailable then the treatment will be disconnected from the other treatments. A BIBD is obviously robust against the unavailability of any t (s r - 1) observations. The maximum value of t fcr BIBS) in Definition 1 is therefore r - 1.
Theorem2.A BIBD (u, b, r, k, A) .is robust against the unavaihbility of aN observations in any r- 1 blocks. Proad Suppose all observations in r- i blocks are unavailable. We prove the connectedness of any two points il and i2 in 7”by showing the existence of a point ij in T which is connected to a point in Bz2 and a point in BJZFas in the proof of Theorem 1. This completes the proof. ainst the unavailability of all observatic,ns in any bO +
(u, b, r, k, A) is strongly robust against the unavailability of any
rot& Similar to the proof of the
the minimum number of lines whose removal results in a disesnnected graph) is k.
32
S. Ghtwh / Robumess
ofBIRD
Theorem 3 can also be proved i&sing: the celebrated Mar&s Theorem (sac ia Harary (sn2)).
Bose, R.C. (1973). Lecture notes in statisticaldesign of experiments.Cslemdcl State thbnity, Fort Colins. Ghosh, S. {1979). On robustness of designs against incomplete data, Sankhyu8 Ser. B, pts. 3 and 4, 204-208, Ghosh, S. (1981). information in observation in robust designs. To appearin G’ontm.Stutist. A. Harary, F. (1972). Graph Theory. Addison-Wtsiey,iheading,MA. Hedayat,A. and John, P.W.M. (1974). Resistant and susceptible BIB designs. Ann. St&& 2,148-138, John, P.W.M. (19t6). Robustness of balanw! incomplete block designs.Ann. St&t, 4,96&962, Kageyama, S. (1980). Robustness of connected balanced block designs. Ann. ftrs?. Stutist. Moth. 32, A, 255-261. Kiefer, J.C. (1958). On the nonrandomized optimality and randomized nonoptimality of symmetrical designs. Ann. Math. Stat. 675-699. Mmt, B.M. (1975). Resistance of balanced imcompkte block designs. Ann. Statist. 3, 1149-1162. Roy, J. (1958). On the efficiency factor of block designs. Sankhyu* 19, 181-188. Srivastava, J.N. (1974). Lecture notes in statistical design of experiments. Colorado State University, Colorado State Ilniversity, Fort Collins.
Meal PlannJq and hference 6 (19$2)29-32
29
Nor&-Holhurd P .tblishing Company
ES8
Sub& GHOSH &‘at&~~
OF BIBD AGAINST THE OF DATA ”
Institute, Calcutta, JndiP
Rmxhd 28 Fcbmary~J97~revisedmanuscriptreceived&4July 1981 I Recommendal by W.T. Fed= NMuct: In this paper we study a robustnessproperty of balanced incomplete block designs (BIBD) with parameters u, b, r, k and A against the unavailability of dntp, in the sense that, when any t, a positive integer, observationsare unavailablethe design remains connected w,r.t. treatment. We show that a BIBD (& b, r, k, A) is robust against the unavailability of &y P- 1 observations, We observe that it is robust against the unavailability of all observations in any F- 1 blocks. We also prove that when any k- 1 observationsare unavailablethe design remains connected w.r.t. treatment and block. AMS 1970 subjective ci&s@#&tion:Secondary G2K10, OSBQS. Key WWU?S and phmes: Balanced incomplete block designs; Bipartite graphs; Robust designs;
Unavailabledata.
I. Introduction
The statistical planning of an experiment is always a difficuit task. Many things can go wrong even in a well planned experiment. One sucl~thing is the ‘availability of data’* The unavailability of data may arise not only from the loss of data or the data being missing but also from the deficit of budget during the experiment. The unavailability of even one observation may destroy the whole purpose of experimentation. It is never possible to anticipate beforehand which observations are going to be unavailable. These facts, which are very common in OUTreal life, motivated the auth&, Ghosh (1979), to study tlhe ro3.%Wness of designs against the unavailability of any number of observations and w.r,t. the estimability of parameters. In this paper we study the robustness property in a block design set-up. If a single observation is unavailable in an experiment using a BI D then the resulting design is not a BIBD. Block designs (BD) are normally used in comparative experiments where the main interest is in comparing several treatments. Connectedness of designs therefore plays an importazit role in comparative experiments. b, r, ic, 1) are well known, see agaiwt the loss of one treatment and * Present address: University of California, Riverside, CA92521, WA.
037803758/82/oooO-
;$02.75 0 1982 North-Holland
w.r.t. variance balance was considered in Hedayat and John (1974) and John (1976). The same property was studied by Most (1973) in case of loss of any subset of a spec: Ec set of n treatments, IQ< k< iu. The robustness of connected balanced block designs against the loss of one treatment and w&t, variance balance sidered in Kageyama (1980). The robustness property against the unavailability of data and w.r. t. the estimability of parameters eogsidered in this paper is completely different from the robustness against the loss of treatments and w.r.t, variance balance in the above mentioned works. This paper therefore establishes a further property of BIBD. A consideration of optimality in addition to the estimability of parameters for designs with the unavailable data is made in Ghosh (1981).
2. Robustness of BIRD Consider a BD with u treatments in b blocks. We can associate with it a bipartite graph G in the following manner: The vertices of 43, treatments and blocks, can be partitioned into two subsets T = {1,2, . . . . v} and B = {1,2, . . . . b} consisting of ‘3 treatments and b blocks, respectively; a point in T is joined to a point in B if the corresponding treatment occurs in the corresponding block. It can be seen that a BD is completely connected (i.e. connected w.r.t. treatment and block) !f and only if the corresponding bipartite graph @ is connected. Moreover, a BD is connected w,r.t. treatment if and only if all points in T of G are connected with each other. In this paper we consider a special Lind of block designs namely B’IBD(v, b, r, k, L). A BIBD is always a completely connected design. If some observations in a BIBD are unavailable then the resulting design may not be another BIBD. Furthermore, the resulting BD may or may not even be connected w.r.t. treatment. Definition 1. A BIBD (v, 6, r, k, jl) is said to be robust against the unavailability of ;anv t (a positive integer) observations if the block design obtained by omitting any t observations remains connected w.r.t. treatment. Definition 2. A BIBD (v, b, r, k, A) is said to be strongly robust against the unavaillability of any t observations if the block desi n obtained by omitting any t observations remains completely connected, An observation corresponding to a treatment ii in a block bl is unavailabl the line joining il and bl in G is removed. I. A BiB.D (ti, b, r, k, A) is robust against the ~nav~i&a~~~it~ of any robsmwr’ions.
e the subset of points in
1
which are joined to lt
II= a, t&,1= f&l = r- il, where I&f denotes the cardie any set of r - 1 observations are unavailable, Le. the oved from G. If for the resulting ected. Consider the si = 2,3, the degrees of d 42 be another subset of points in Bj, e for the resulting graph and G. Let IJ=II and lBjtl=l~. We have k@, &O, kI +&=I, +12= - 1, We now prove the existence of a point ia in r which is se there does not exist such a point d i2, is joined to u points in B2 then to u points in Bs. Thus ~11Ihe points T other than iI and i2 which are B2,. It follows th t &kl. Therefo:e, r - R = I2+ I1s k, + 1,~ r- A- I. This is impossible! Thus the existence of the point i3 is proved, Mence the points ii and is are connected. Therefore the resulting design is oannected. This completes the proof. A BIBD is not robust against the unavailabiiity of any r observations because if all r observations corresponding to a parti :ular treatment are unavailable then the treatment will be disconnected from the other treatments. A BIBD is obviously robust against the unavailability of any t (s r - 1) observations. The maximum value of t fcr BIBS) in Definition 1 is therefore r - 1.
Theorem2.A BIBD (u, b, r, k, A) .is robust against the unavaihbility of aN observations in any r- 1 blocks. Proad Suppose all observations in r- i blocks are unavailable. We prove the connectedness of any two points il and i2 in 7”by showing the existence of a point ij in T which is connected to a point in Bz2 and a point in BJZFas in the proof of Theorem 1. This completes the proof. ainst the unavailability of all observatic,ns in any bO +
(u, b, r, k, A) is strongly robust against the unavailability of any
rot& Similar to the proof of the
the minimum number of lines whose removal results in a disesnnected graph) is k.
32
S. Ghtwh / Robumess
ofBIRD
Theorem 3 can also be proved i&sing: the celebrated Mar&s Theorem (sac ia Harary (sn2)).
Bose, R.C. (1973). Lecture notes in statisticaldesign of experiments.Cslemdcl State thbnity, Fort Colins. Ghosh, S. {1979). On robustness of designs against incomplete data, Sankhyu8 Ser. B, pts. 3 and 4, 204-208, Ghosh, S. (1981). information in observation in robust designs. To appearin G’ontm.Stutist. A. Harary, F. (1972). Graph Theory. Addison-Wtsiey,iheading,MA. Hedayat,A. and John, P.W.M. (1974). Resistant and susceptible BIB designs. Ann. St&& 2,148-138, John, P.W.M. (19t6). Robustness of balanw! incomplete block designs.Ann. St&t, 4,96&962, Kageyama, S. (1980). Robustness of connected balanced block designs. Ann. ftrs?. Stutist. Moth. 32, A, 255-261. Kiefer, J.C. (1958). On the nonrandomized optimality and randomized nonoptimality of symmetrical designs. Ann. Math. Stat. 675-699. Mmt, B.M. (1975). Resistance of balanced imcompkte block designs. Ann. Statist. 3, 1149-1162. Roy, J. (1958). On the efficiency factor of block designs. Sankhyu* 19, 181-188. Srivastava, J.N. (1974). Lecture notes in statistical design of experiments. Colorado State University, Colorado State Ilniversity, Fort Collins.