of Small Sample Size H. Yassaee, Tehran University of Technology, Iran, and University of North Carolina at Chapel Hill

Transcription

1 On Properties of Estimators of Testing Homogeneity in r x2 Contingency Tables of Small Sample Size H. Yassaee, Tehran University of Technology, Iran, and University of North Carolina at Chapel Hill bstract The problem ~f testing homogeneity in r x 2 contingency tables can be converted to the problem of testing homogeneity of r independent binomial distributions. To estimate the common probability p, of success, we use likelihood function, statistics ~, x~, I(p; p), and x~, the first of which is to be maximized with respect to p and the rest to be minimized with respect to p. In this paper we study the bias and variance of estimators through means of nnalytical and numerical procedures based on approximate rules and computer-obtained results. We compare the value of bias and variance for each method and give a table which presents their order of magnitude. Key words: contingency tables; maximum likelihood estimate; minimum modified x~ estimate; minimum x~ estimate; minimum discrimination information estimate; minimum logit X 2 estimate; bias; mean square error.

2 page 2. Introduction Cochran (952, p.325) has shown that the problem of tesdng homogeneity.' in r X 2contingencytables with cell frequencies xij's, i=l~ 2,..,r; j:i,2,..,c may be converted to the problem stated as follows: Given r independent binomial distributions from which samples of sizes xl ' x 2 "..,X are drawn, respective-.. r. Iy, one is interested in testing the hypothesis that these populations are homogeneous; Le., p.=p, i=l, 2,...,r which is the probability of "success". We de 2 note x. = L x.. by n. and x'i by x. throughout the paper. To estimate p, we.. J J= use the likelihood function, modified \t 2, Pearson Xp 2, information statistic I(p:;), and 0git x L 2, the first of which is to be maximized with respect to p and the rest to be minimized with respect to p. These estimators are asymptotically equivalent. However, they differ for small sample sizes. Yassaee (975, a, b, c, d) has studied properties of estimators and distribution of various statistics in rxc and rxcxt cant ingency tables of small sample sizes. Some other researchers have studied the comparison of estimators and statistics in various models using different approaches. For example, see Harvey (977), Brown and Muentz (976), PraJhan and Sathe (976), Chapman (976), Hommel (978). In this paper we study analytically the bias and mean square error of estimators under consideration, and the criteria, such as biasedness and mean square error are taken into account for comparing estimators. Numerical investigations are restricted to 2x2 contingency tables. 2. Methods of Estimation In this section we give estimators of the parameter p, obtained by the different procedures mentioned in section.

3 page 3 \ The unique maximum likelihood estimate p for p is given by l x. /\ i=l P = ---- N r r N = L n. i= (2.) 0' 2 2 The unique minimum modified M (Neyman X ) estimator for p is given p = M r '" l. W.p. 'i=l J\ W /\ w. = n. r /\ W= l W. i=l (2.2) \ "Xi where p. = l-q. = We use the normal approximation to the binomial distri- n i bution to investigate properties of PM' The minimum Pearson X~ estimate for p is obtained by solving the equation r /\ /\ } 2. {r "2 r l n.(q.-p.) p 2} + 2 I p. n.p - l n.p. = 0 { i=l. i= i= (2.3) The only root which is between 0 and and minimizes X 2 is given by p r r "2 r \ 2 - n.p n.p.][ i l n.q. ] l,,2 + { [ l i=l i=l i=l Pp = r l n. (q. -p.). = 2 Should it happen that the coefficient of p ~ 'H~e~n~c~e' there is only one root between 0 and. }~ is zero, then we take Pp = 2' (2.4) r \ Let n. (q. -p.) = y., Y = l Y. =

4 J?s;ge. 4 Then one deduces the following inequalities r " \ I y.p. i= /'I Y I < Pp < 2 \ if Y > 0 \ if y = 0.!.<p < 2 P r \ /I I y.p.. III = --j\- Y \ if y < 0. These inequalities are helpful to check the computed p values. The minimum p discrimination information estimate p for which information statistic = -- +, q = -p, r n.p r n.q /I I (p: P) I n.p Jl,n Ln.q Q,n i=l x.. n. -x. = is a minimum, is given by p* = a l+a (2.5) where and Q,n a = ( r Ln.) logit \ p., N.III = N = r L n. ' = Since I(p:p) is a convex function of p, it is a minimum at p*, see Kullback (959). where We define logit X 2 = L r /I \ 2 (L.-L.), Lw. j=l /\ \ L. = logit p E. :: log L. :: logit p,. q' \ \ x.(n.-x.) III w. = n.p.q. =. n. \ \

5 page 5 the minimumlogit x 2 estimator PL is given by logit PL = r,~ L w. logit p. i=l r (\ L w. i=l l+ (2.6) r In form'p is the same as p* given in (2.5) L with~. and L~' i=l substituted for n. and N, respectively. 3. On the Bias and Variance of Estimates \ It is well known that p is an unbiased estimate of p for all values of p. In order to compute different estimates of p simultaneously, we replaced 0 by f in the analysis of 2x2 contingency tables. For p = 2' E(p) = 2' exactly. For fixed N, db(p) dp P is unbiased only at p = 2" in the range of p, 0 < P <. is not zero, in general Since [ + dh I i]2 2 I (dp) P = a (p p = l) 2 > == 4N (3.) For r=2 we can write PM given in (2.2) in the form (3.2) which is a function of two independent random variables Xl and x 2 and is a rational function of independent random variables. By the use of Laplace, Mellin, or characteristic function, one can obtain the probability density function and the moments of such a function. But the algebra is somewhat lengthy, and we will not consider the problem as such here. We refer the

6 page 6 reader to Prasad (970, pp ). We approximate the binomial distribution with probability of "success" p by a normal distribution with mean and variance the same as that of the binomial distribution. Numerically, if t is not added to xi when it is zero, orx. is taken as n-- 2 III should have when x.=n., and the normality assumption is valid, one n. on the condition that W. = ~ is used for W. = pq n. X p.q. Let "y. = W. W, y. = W. W where Then Weights y. 's produce an are sufficiently large. w. = independent random variables. n. pq /\/\ PM = I y.p.. I = /\ w. = n. p.q. asymptotically efficient estimate of PM when the ni's /\ "- Under the assumption of normality, y. and p. are Therefore, Let Z. Then ;., W. = w:- I r p. I W. -Zl i=l i r w. I i=l Zi

7 page 7 Now.L; I r { W. '\ i=l Z'"i f.y----_._- [ L Wi J 2 i=l Z. Hence, To obtain this expectation, we applied a theorem on expectation of a function of random variables as stated in Welch (938, pp ). To get more details on what follows, one may write to the author for a long preliminary report. /\2 r "'''- One may estimate op by ~ [+2.I n. Piqi]. M = pplication of theorem mentioned earlier gives EJ.) =. {-2 f p.q. + O( Y..)} w W i=l ni i=l n i =. {-2( Y..) pq + O( f 2 )} W i=l ni i=l n i One can see that this estimate has asymptotically a negative bias which is approximately equal to first estimate with bias of order OC =... 2 term 0 p neglected. M r L,,\ 2 r /'..'\ a p = -W } { +4 L - p. q. M i=l fi i 2 "2 ), then 0 p n. M :l If one wants to obtain an should be estimated by

8 page 8 The exact variance of PM under normal distribution is given by.zl But Zz has an F-distributionwith (nt-i, n 2 -) degrees of freedom, Consequently, we get the expectation 0 2 (P Iz )with respect to the F-distribution. M Finally 2 I o = -- + K PM W \'/here One may refer to (2.4) and expand Pp in Taylor series about values Pi=p to find approximate bias term for Pp or get the variance of pp' Due to the length of the algebra we will not der~ve /'. the bias and variance here, We expand logit Pi = L i in a Taylor series about the true value p, to get 2 3 p.-p. I (P-q)(Pi-P) L. = (Pi-P) (p2 q2) -3 L + + I Y- - pq + + O(n ). ;L pq 2 2 i P q P q

9 p~ge 9 Consequently, according to (2.5) W~ have E(logit p*) - L + 2 l N- (p-q) +.. ri (q-p) n-3pq) 3N n., because /\ 3-2 E(Pi-p) = n i pq (q-p) Neglecting the third term, we have E(logit p*),;, L + ;N (p-q). ccording to this approximation, as p takes small values, i.e., between 0 and.5.e the bias value of logit p* is of negative sign. bias is positive. For p ciently large, the value of bias tends to zero. For p >.50, the value of = 2' logit p* is unbiased. If the n. 's are suffi Following the derivations just presented, we conclude that the biases of MOl and MLG are of different signs. Thus one may see that an effective comparison can not be achieved by using analytical methods. In the next section we further study this issue numerically. 4. Computational Details of 2x2 Contigency Tables Let (n l, n 2 ) be a set of given raw totals. Then all poss~ble tables can be enumerated whenever the set is specified. For each set (n l, n 2 ) there /\ are (n l + I) (n 2 + ) tables to be generated. For each table, P, PM' Pp' p* and P L are computed according to formulae (2.), (2,2), (2,5), (2.6), and (2,7), respect!vely. S~nce some estimators are not admissible for Xi~ 0 or X i = n., following rule is usej: the { I if 0 2" x. = l n.- 2 if x. = n. X. = Xi if x. =,2,.oo,n i - I }

10 page 0 To compare estim~tors under the s~e ~xperimental condit~ons, this rule was applied to all estimator5. To get the mean, variance, standard deviat~on, bias, ratio of absolute value of bias to standard deviation, and exact level of estimators, the following values were used as true values of p. p =.0 (.05),.50 i.e.,.0,.5,...,.50 The mean and variance of each estimator, for each p, was computed by using the definition of mean and variance of a random variable and formula given in section 2. We now study the bias of estimators according to the increasing size of the sample for different true values of p. We computed the bias of estimators for all possible values of p, p =.0(.05),.50. The bias for p >.50 is easily obtained as the negative of bias at l-p. To make the presentation of an overall conclusion for the values of biases self-explanatory, we preferred to give details on what we have observed, rather than drawing diagrams for them. If (nl,n Z ) = (5,5), the bias of p is an increasing function of the true p for p <.50 and it is zero at p =.375 and p =.50. We note that the bias I is due to the fact that 0 or "i cells are replaced by ~ and n i - ~, in the estimation procedure, otherwise p is unbiased. respectively, For MLE,IBI <.034 where < B = b(p) = bias. The bias of PM is a decreasing function of p for p =.5 and < < an increasing function of p for.5 = P =.50. It is zero at p =.50. gain, of the due h'. part b:i,as is to su stltutlng 2 for x. = o and n l - "2 for x.= n.. ;J. Referring to (2,2) and the discussion in section 3, PM would have been unbiased if the populations were normal. The bias of Pp is an increasing function of p for p <.45 and it decreases to zero at p =.50. Referring to formula (2.3), and

11 knowing that,yi l < Y spond~ng IBI of PM' behaves like page we see that 8 of p should be smaller than the correp The maximum 8 of p is equal to,02, The bias of p* p that of PM except that the bias is almost zero at,40, words, p* is an unbiased estimate of p in a small neighborhood ofp =,50, In other 8 of p* is less than IBI of PM' Finally, the bias of PL behaves closely as Pp which is reasonable due to the relationship between X 2 and X L 2 already given.in Yassaee (975). For the case (n,n 2 ) = (0,5), (0,0), (0,5), (IS,lS), (20,20), and (20,25) we conclude that one cannot claim that PL is always less biased than Pp' but other estimators can be arranged in terms of IBI in ascending order P - p*-p M s n l and n 2 increase, the direction of the bias of P becomes the same as -. ~ those of PM and p*, which are always negative and increase to zero. It. is interesting to note that the bias of p* and. that of PL are of different sign for most of the cases except for small values of p, as we have already concluded approximately in sections (3.4) and (3.5) for bias of logit p* and logit PL; 5. Mean Square Error of Estimators (MSE) We briefly report on the MSE of estimators and compare estimators in this regard for various true values of P.5 For p ~,5, the MSEof PM is smaller than for those of others, For < > = P <.30, the MSE of p* is the smallest and for p =.30 the MSE of P L is the samllest.

12 'Page 2 For true values of p, the ascending order of estimators ~n terms of the magn~tude of their MSE is given as follows: -L scending order of MSE < P =.5 PM - - P - P p - PL,r-.5 < P <.30 p* - PM - P - Pp - PL < <,.'\.30 = P =.50 P L - Pp - P - p* - PM P* /\ for p >.50 the order is the same as that of - P shown here. We now present a table which summarizes the ascending order of estimators for the MSE according as (n l,l 2 ) = (5,0), (0,0), (5,0), (5,5). '.

13 page 3 Table of the scending O~der of Estimators P scending order ofmse < /' 5,0 P =.20 PM - p* - p - n 'p - PL P =.25 P - Pp - p*- P L - PM <.25 < P =.50 P L - Pp - P - p* - PM 0,0 < /\ P =.20 P - p* - p - Pp - P '\M L P =.25 P - Pp - P* - PL - PM.25 < P $.50 P - P - P - p* - PM L P '. e < '\ M (5,0) P =.0 P - p* - P - Pp - P L P =.5 p* - PM - P" - Pp - P L P =.20 P - p* - p - p - M P < I'P L.20 < P =.30 Pp - P L - P - P* - \ PM.30 < P <.50 P - P - P - p* - L P PM < (5,5) p =.0 PM - p* - '" P - P p - P L < < \. ]5 = P -.20 p* - P - PM - P p - P /'\ L P =.20 P - Pp - p* - P L - PM \ P =.25 P - p - p - p* P L - PM '" P =.30 P - P - P - p* - PM < P L < \.35 = P =.50 P - Pp - P - p* - PM L

14 ., -.. page 4 REFERENCES 2 Berkson, J. (946), "pproximation of X by probits and logits", JS 4, pp Brown, C. C. and Muentz, L. R. (976), "Reduced mean square error estimation in contingency tables", JS 7, pp Chapman. J.. W. (976), " comparison of the X 2, -2 log R, and multinomial probability criteria for significance tests when expected frequencies are small", JS.. 7, pp Cochran, W. G. (952), "The X 2 test of goodness-of-fit", nn. Math. Stat. 22, pp Harvey,. C. (977), " comparison of prel iminary estimators for robust regressions", JS 72, pp Hommel, G. (978), "Tail probabilities for contingency tables with small expectations", JS 73, pp Kulback, S. (959), Information theory and statistics. John Wiley &Sons, New York (Dover Pub!. Co. (968), Peter Smith (978)). Pradhan, M. and Sathe, Y. S. (976), "nalytical remarks on Cramer's minimum method for finding the better of two binomial populations", JS 7, pp Prasad, R. (970), "Probability distribution of algebraic functions of independent random variables", S.L.M., J. of pp!. Math. 8, pp Welch, B. L. (938), "The significance of the difference between two means when the population variations are unequal", Biometrika 29, pp Yassaee, H. (975a), "On Monte Carlo comparison of estimators in rxc contingency tables of small size", Technical Report, rya-mehr University of Technology. ( 975b), " comparative exact and Monte Carlo study of estimators in multidimensional contingency tables: logit model", Title: Proceedings of multivariate analysis III, North Holland Publ. Co. (975c), "On comparison of various statistics in rxc contingency tables: Test of homogeneity based on small samples", Technical Report, rya-mehr University of Technology. --,-_(975d), "On Monte Carlo comparison of various statistics in rxc contingency tables of small sample size: independence model", Technical Report, rya-mehr University of Tehran, Iran.