TABLE FOR UPPER PERCENTAGE POINTS OF THE LARGEST ROOT OF A DETERMINANTAL EQUATION WITH FIVE ROOTS. By William W. Chen

Transcription

1 TABLE FOR UPPER PERCENTAGE POINTS OF THE LARGEST ROOT OF A DETERMINANTAL EQUATION WITH FIVE ROOTS By Willia W. Chen The distribution of the non-null characteristic roots of a atri derived fro saple observations taken fro ultivariate noral populations is fundaental iportance in ultivariate analysis. The Fisher-Girshick-Hsu-Roy distribution(1939), which has interested statisticians ore than si decades, is revisited. Instead of using Pillai s K.C.S. ethod by neglecting higher order ters of the cuulative distribution function(c.d.f.) of the largest root to approiate the percentage points, we siply keep the whole c.d.f. and apply its natural nondecreasing property to calculate the eact probabilities. At the duplicated percentage points we found our coputed percentage points consistent to the eisting tables. However our tabulations have greatly etended the eisting tables. Key words : Characteristic roots, etended tables, Fisher-Girshick-Hsu-Roy Distribution, Percentage points. Willia W. Chen, willia.w.chen@irs.gov Editor: Ahed S. Ejaz, Ahed@ath.uregina.ca

2 1. INTRODUCTION We are concerned here with the distribution of the largest characteristic roots in ultivariate analysis when there are five roots. Fisher-Girshick-Hsu-Roy(1939) discusses this in detail and presents the eact joint probability density function in general. This well-known distribution depends on the nuber of characteristic roots and two paraeters and n. They are defined differently for various situations as described by Pillai (1955). The upper percentage points of the distribution are coonly used in three different types of hypothesis testing in ultivariate analysis, naely, i. test of equality of the variance-covariance atrices of two p-variate noral population, ii. test of equality of the p- diensional ean vectors for k p-variate noral populations and iii. test of independence between a p-set and a q-set of variates in a (p+q)-variate noral population. When the null hypotheses to be tested are true, all the three types of test proposed above have been shown to depend only on the characteristic roots of atrices using observed saples. We could state the proble in the following anner. Using a rando saple fro the ultivariate noral population we could copute the characteristic roots fro a usual su of product atrices of this saple. We then copare the largest characteristic root of the atrices with the percentage points that we have tabulated in this paper to deterine whether or not to reject the null hypothesis at a certain probability confidence. For this reason the percentage points of the largest characteristic roots of the distribution has deeply attracted the attention of atheatical statistician ore than si decades. There already has any published tables that either focus on upper percentage points tabulations or chart the various sizes of roots. Pillai is the ost well known contributor in this area. He gave the general rules of finding the C.D.F. of the largest root and tabulated upper percentage points of 95% and 99% for various sizes of roots. Other contributors including Nanda (1948, 1951), Foster (1957, 1958), Rees (1957), and Heck (196) will be discussed in ore detail in section 2. In section 3 we give the joint distribution of s non-null characteristic roots of a atri in general for

3 and the C.D.F. of the fifth largest characteristic root. In section 4 we discuss in detail the algorith we used to create the tables of this paper. In section 5 we copare Pillai s ethod with ours and also the advantage we have in our approach. 2. CUMULATIVE FUNCTION AND HISTORICAL WORK The joint distribution of s non-null characteristic roots of a atri in ultivariate distribution was given by Fisher-Girshick-Hsu-Roy(1939). This distribution can be epressed in the for of (3.1). In this study we interested in the distribution of the largest characteristic root with the five roots. Even through we know the for of the joint density function it ay not easy to write out its C.D.F. of the largest characteristic root. To solve this proble, there are two ethods that could lead us to find the C.D.F. ore easily. Pillai (1965) suggested that the C.D.F. of the largest characteristic root could be presented in deterinantal for of incoplete beta functions. To overcoe the difficulty of nuerical integration of each of the s! ultiple integrals when the deterinant is epanded, he has suggested an alternative reduction forula. This forula gives eact epressions for the C.D.F. of the largest root in ters of incoplete beta functions or functions of incoplete beta functions for various values of s. Later, Pillai(1956a) epanded C.D.F. by neglecting higher order ters and tabulated the 95% and 99% percentage points. An alternative ethod suggested by Nanda (1948) yield the sae results. He started with the Vanderonde deterinant and epanded the deterinant in inors of a row then repeated applied integration by part to find the C.D.F. of largest characteristics root. In this paper we slightly odified Nanda s notation and presented the case with five roots in equation (3.2). Following this C.D.F. and the algorith described in section 4 we could tabulate the upper percentage points. At this oent it is useful to review soe of the published tables and sees soe reasons to etend the tables. Pillai (1956a, 1959) published tables that focus only

4 on two percentage points i.e. 95% and 99% for s =2,6, = (1)4, and n varying fro 5 to 1. Foster and Rees (1957), have tabulated the upper percentage points 8%, 85%, 9%, 95% and 99%,of the largest root for s=2, =-.5, (1)9, n=1(1)19(5)49,59, 79. Foster(1957, 1958) has further etended these tables for values of s=3 and 4. Heck(196) has given soe charts of upper 95%, 97.5% and 99% points for s=2(1)5, =-.5, (1)1 and n greater than 4. Recently Rencher (reference [18],[19] and [2]) included the percentage point.95 in all of the three references. In reference [18],page 518, he selected = (1), 5,7,1,15 and N = 5(5),3(1),6(2),8,12,24. Sae selection occurs in reference [2], page 575. In Reference [19] he choses = -.5,(1),1,15 while N = 1(2),5(5),3(1), 5,1, 5,1 in Table B.5 page 436. Rencher s ecellent written tetbook [18] gives a well known nuerical eaple that deonstrates how to use these table values. In this eaple, four variables have selected to test its significant difference : trunk girth at 4 years 1, etension growth at 4 years (), trunk girth at 15 years (1), and weight of tree above ground at 15 years (lb1). (For the detailed coputational procedure, see reference [18], page 189.) Another ecellent application can be found in reference [19], a study that copared four groups of rabbits infected with huan tuberculosis. Here, the four groups are unvaccinated control, infected during etabolic depression, infected during heightened etabolic activity, and infected during noral activity. (The data and detailed coputational procedure can be found in page 157, eaple ) These eaples deonstrate how the research and applications in this area of study are still active. At the early age without a odern coputer, it is an understandable difficulty task to copute the whole C.D.F.(3.2) at each percentage points. This is not only tedious but also worthless. Therefore deleting higher order ters and keeping a few lower order ters to approiate the roots is a good and reasonable ethod to solve the proble. But this approach will involve intolerable error at the lower percentage points such as 8%,82.5%,85%,87.5%,9% or 92.5%. These percentage points usually are

5 ignored not because of lack of use but because of difficulty of coputation. Traditional ethods treat issing values by interpolation. However without say 85% or 9% percentage points it is difficult to interpolate 87.5%. In recent years, the coputer has gradually atured in eory, speed, and fleibility in usage. It has greatly changed the ethod we study the statistics. In this study we use one of the ost basic properties of C.D.F. and revisit this ost iportant distribution. We attepted to include as any percentage points as we needed in one coputer run. The included upper percentage points are.8,.825,.85,.875,.89,.9,.91(.5),.995. Different authors have selected different and n paraeter values. We selected these two paraeters in such a way that all eisted table values will include. For the paraeter =(1)15 and the paraeter n=1(1)2(2)3(5)8(1) 15,2(1)1. Our table will give us the eact accuracy percentage points and probabilities and avoid the interpolation proble. 3. THE DISTRIBUTION FUNCTION OF FIFTH CHARACTERISTICS ROOT The joint distribution of s non-null characteristic roots of a atri in ultivariate analysis given by Fisher-Girshick-Shu-Roy(1939) can be written as f ( 1 where,... s ) C(s,,n) s i 1 i (1 ( s s/2 2 2n s i 2 ( ) i 1 C(s,,n) 2. s 2 i 1 2n i 1 i ( ) ( ) ( ) i i ) n i j i j ) ( 1... s 1), (3.1) and the paraeter and n are defined differently for various situations as

6 described by Pillai (1955,pp 118). Following the Nanda (1948), repeated integration by part ethod, the cuulative distribution function of the largest characteristic root for S=5 is given below: Pr( 5 C(5,, n) [ n 5 2 t - IO(, 3, n 1) { [ n 3 ( 2 t dt 2I(,2 5,2n 1) * V(,, n)} n 4 {-IO(, 3, n 1) * seq_21(,, n) - 2I(,2 3,2n 1) * V(, 1, n) 2I(,2 4,2n 1) * V(, 2 t ) 2 4 2) * seq _ 21(,, n)] - 2I(, n 1 2n 1 (1 ) dt dt n 1 * seq _ 4321(,, n) * seq _ 321(, 2 (1 ), n) n 1 ( * t * seq _ 321(,, n)] 2 t 1, n) n 4 dt 2 t 3,2n 1) * V(, 2 6 n 2n ) * seq_321(,, n)} 2 2 2n 1 2, n) dt 2n 1 dt (3.2) All the sybols used in above equation are defined as follows. V(,, n) IO(, I(,,n) 1 ( n t 3,n 1) t [- 2) seq _ 21(,, n) 1 3 n n (1 ) dt dt (1 ) n 1 t n 1 n dt 2 t 2 1 2n 1 dt] (3.3) (3.4) seq _ 321(,, n) 1 { n 3 2 t n 1 (1 ) dt t n 1 1 *seq _ 21(,, n) n dt 2 t 2 3 2n 1 dt t n dt} (3.5)

7 seq _ 4321(,, n) 1 { n 4 2 t n 3 (1 ) 2n 1 dt n 1 [ seq _ 321(, 2 (1 ) ( 2) * seq _ 21(,, n)] 2 t n 1 2 5, n) * t 2 t 2n n dt dt 2 t 2n dt * seq _ 21(,, n)} * seq _ 21(, 2n 1 dt 1, n) (3.6) C(5,, n) 5 5 / 2 2 2n 7 i ( ) i i 1 2n i 1 i ( ) ( ) ( ) i 1 (2 2n 8) (2 2n 5) (2 2n 11) 192 * (2 2) (2n 2) (2 4) (2n 4) ( 3) (n 3) (3.7) 4. THE ALGORITHM In this section we describe in ore detail how we copute the percentage points. In this study no new theory created, instead we apply the fundaental nondecreasing function property of the C.D.F. i.e. if 1 2 then f(1) f ( 2 ). Applying this useful and siple property helps us to find all the needed percentage points. To create the tables, we first set up a calling progra that includes the given starting values,, and n. Then we build up a subroutine and any subprogras. In the subroutine we code equation (3.2) and call a set of subfunctions. Each subfunction carries a function value that represents a value in the equations (3.3), (3.4), (3.5), (3.6) and (3.7). Finally, the coputed values fro subfunctions are substituted into equation (3.2) and produce a probability used to tabulate our table. Let us start with a standard procedure used in coputer algoriths to see how we generate one percentage point. First choose one set of and n values, say = 1 and n =2, and a very sall value, say or.1*1 4 to ensure that there are no

8 issing percentage points we interested are larger than this value. Using these selected values, substitute into equation (3.2) to copute the probability cuulate to this selected value. If the coputed probability say equals then write out this coputed probability,, n, and values in a specified file, say f95.dat. Then loop the pointer back and add a very sall aount on, say 4.1*1, and again copute the probability. If this tie the coputed probability is then write out this coputed probability,, n, and values in a different specified file, say f96.dat. Since we know that the cuulative function is always nondecreasing and continuous it ensures us that any probability ranged fro to 1 will have chance to be reached at least once for soe selected values. It is possible for several specific values round to the sae probability. This eans we could increase either or n by a selected value and reset to or a sall value again to repeat the process of adding a sall aount to to copute the corresponding probability. This process should continue until we fill all by n tables. Our eperience shows that for a chosen fied and n, as increases by the above stated increent the coputed probabilities also increases with ultiple values rounding to the desired probability. The following siple rule has been adopted to select a triplet, and n for a desired probability. Let s say the desired probability y is p and estiate to reach this probability y is i.e Pr( 5 ) p. We need to find a pair say, ' and '' such that ' Pr( 5 ) p Pr( 5 ' '' We then can conclude by onotonicity that, in the interval (, ), is the desired estiated ordinate and report in the attached table. In the attached table we have rounded our results to four decial accurate places. '' )

9 5. SOME CONCLUSION REMARKS Pillai s approiation ethod by neglecting higher order ters has soe liitations. Pillai(1954) has studied these liitations in ore detail for the case s =2,3 and 4. The error of the approiation of the upper percentage points of the distribution is defined as the difference between the approiate and eact probabilities. Pillai s coparative study can obtain the following conclusions. 1. There is greater agreeent between the probabilities for the approiate and eact cases in upper 99% points than in the 95%. 2. The difference between the approiate and eact probabilities in the upper 95% points occur in the fifth decial place, that on rounding gives a difference of only one in the fourth decial place. 3. If we fied the paraeter as constant, the error of approiation increases slowly as the other paraeter increased; such increase occurs only in the sith decial place or at ost is unity in the fifth decial place when rounding. Pillai (1959) also concluded that the approiate forula is only appropriate for percentage points 95% or higher. It ight be adequate for those percentage points slightly below the 95%. In application it is very clear that lower percentage points are needed. Using the algorith suggested in section 4, we can copute any percentage points accurately. Since we used the whole distribution function not a truncated distribution. The table included in this paper is only a sall portion of the table generated fro coputer. For the interested reader could write to author for ore detailed tabulations. References [1] Fisher, R.A.(1939) The sapling distribution of soe statistics obtained fro non-linear equations, Ann. Eugenics, Vol. 9, pp [2] Foster, F.G. and Rees, D.H.(1957) Upper percentage points of the generalized beta distribution. I, Bioetrika, Vol. 44, pp [3] Foster, F.G.(1957) Upper percentage points of the generalized beta distribution. II, Bioetrika, Vol. 44, pp

10 [4] Foster, F.G.(1958) Upper percentage points of the generalized beta distribution. III, Bioetrika, Vol. 45, pp [5] Girshick, M.A.(1939) on the sapling theory of the roots of deterinantal equations, Ann. Math. Stat., Vol. 1, pp [6] Heck, D.L.(196) Charts of soe upper percentage points of the distribution of the largest characteristic root, Ann. Math. Statist., 31, pp [7] Hsu, P.L.(1939) On the distribution of roots of certain deterinantal equations, Ann. Eugenics, Vol. 9, pp [8] Nanda, D.N.(1948) Distribution of a root of a deterinantal equation, Ann. Math. Stat., Vol. 19, pp [9] Nanda, D.N.(1951) Probability distribution tables of the largest root of a deterinantal equation with two roots, J. Indian Soc. Of Agricultural Stat., Vol. 3,, pp [1] Pillai, K.C.S.(1955) Soe new test criteria in ultivariate analysis. Ann. Math. Statist. 26, pp [11] Pillai, K.C.S.(1956a) On the distribution of the largest or sallest root of a atri in ultivariate analysis, Bioetrika, Vol. 43, pp [12] Pillai, K.C.S.(1956b) Soe results useful in ultivariate analysis. Ann. Math. Statist. Vol. 27, pp [13] Pillai, K.C.S.(1957) Concise Tables for Statisticians, The Statistical Center, University of the Philippines, Manila. [14] Pillai, K.C.S. and Bantegui C.G.(1959) On the distribution of the largest of Si roots of a atri in ultivariate analysis, Bioetrika, Vol. 46, pp [15] Pillai, K.C.S. (1965) On the distribution of the largest characteristic root of a atri in ultivariate analysis. Bioetrika 52, [16] Roy, S.N.(1939) p-statistics or soe generalizations in analysis of variance appropriate to ultivariate probles, Sankhya, Vol. 4, pp [17] Roy, S.N.(1945) The individual sapling distribution of the aiu, the iniu and any interediate of the p-statistics on the null hypothesis. Sankhya, Vol. 7, pp [18] Rencher A.C. Methods of ultivariate analysis, 1995 by John Wiley & Sons, Inc. [19] Rencher A.C. Multivariate statistical inference and application by John Wiley & Sons, Inc. [2] Rencher A.C. Methods of ultivariate analysis, 22 by John Wiley & Sons, Inc.

11 Upper percentage points of.8 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

12 Upper percentage points of.8 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

13 Upper percentage points of.85 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

14 Upper percentage points of.85 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

15 Upper percentage points of.9 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

16 Upper percentage points of.9 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

17 Upper percentage points of.915 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

18 Upper percentage points of.915 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

19 Upper percentage points of.925 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

20 Upper percentage points of.925 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

21 Upper percentage points of.95 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

22 Upper percentage points of.95 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

23 Upper percentage points of.975 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

24 Upper percentage points of.975 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n

25 Upper percentage points of.985 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =,when s=5 n

26 Upper percentage points of.985 of theta(p,,n),the largest eigenvalue of B-theta(W+B) =, when s=5 n