econstor Make Your Publications Visible.

Transcription

1 econstor Make Your Publications Visible. A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Hartung, Joachim; Knapp, Guido Working Paper Confidence intervals for the between group variance in the unbalanced one-way random effects model of analysis of variance Technical Report, SFB 475: Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 2000,04 Provided in Cooperation with: Collaborative Research Center 'Reduction of Complexity in Multivariate Data Structures' (SFB 475), University of Dortmund Suggested Citation: Hartung, Joachim; Knapp, Guido (2000) : Confidence intervals for the between group variance in the unbalanced one-way random effects model of analysis of variance, Technical Report, SFB 475: Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 2000,04, SFB 475, Universität Dortmund, Dortmund This Version is available at: Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.

2 Condence Intervals for the Between Group Variance in the Unbalanced OneWay Random Eects Model of Analysis of Variance Joachim Hartung and Guido Knapp Department of Statistics, University of Dortmund, D44221 Dortmund Abstract A condence interval for the between group variance is proposed which is deduced from Wald's exact condence interval for the ratio of the two variance components in the oneway random eects model and the exact condence interval for the error variance resp. an unbiased estimator of the error variance. In a simulation study the condence coecients for these two intervals are compared with the condence coecients of two other commonly used condence intervals. There, the condence interval derived here yields condence coecients which are always greater than the prescribed level. Key Words: variance components, unbalanced oneway nested model, Wald's condence interval To appear in: Journal of Statistical Computation and Simulation 1

3 1 Introduction In the present paper we consider condence intervals for the between group variance in the unbalanced oneway random eects model of analysis of variance (ANOVA). In the case of a balanced design one method for constructing a condence interval for the between group variance was independently proposed by Tukey (1951) and Williams (1962). The Tukey Williamsmethod is based on two quadratic forms in normal variables which are exactly distributed as multiples of 2 distributed random variables and the expectations of these quadratic forms are parametric functions of the between group variance. Thus, for these two parametric functions of the between group variance exact (1 ; )condence intervals can be calculated and by solving the intersection of these two condence intervals, a condence interval of the between group variance is given which has a condence coecient at least as great as 1 ; 2 due to Bonferroni's inequality. The results of simulation studies conducted by Boardman (1974) indicated that the condence coecient of the Tukey Wiliamsinterval is near 1 ; (cf. also Graybill (1976, p. 620)) and Wang (1990) showed that the condence coecient of this interval is at least 1 ;. Following the TukeyWilliams approach Thomas and Hultquist (1978) proposed a condence interval for the between group variance in the unbalanced case where the distributions of the two involved quadratic forms in normal variables can only be approximated by multiples of a 2 distributed random variable, but again the expected values of the quadratic forms are parametric functions of the between group variance. The approximation to a 2 distribution is, however, not satisfactory if the ratio of between and within group variances is less than 0.25 and the design is rather unbalanced. To overcome this problem Burdick, Maqsood and Graybill (1986) considered a conservative condence interval for the ratio of between and within group variance, which was used in Burdick and Eickman (1986) to construct a condence interval for the between group variance based on the ideas of the TukeyWilliams method. In Burdick and Eickman (1986) a comparison of the condence coecients of the ThomasHultquistinterval and the BurdickEickman interval are given by simulation studies. The results of the simulations studies indicated that the condence coecient is near 1 ; in most cases. If the approximation to a 2 2

4 distribution in the ThomasHultquist approach is not so good, the resulting condence interval can be very liberal, while in these situations the BurdickEickmanintervall can be very conservative. Now we propose a condence interval for the between group variance in the unbalanced design which is constructed from an exact condence interval for the ratio of between and within group variance derived from Wald (1940), cf. also Searle, Casella, and McCulloch (1992, p. 78), Burdick and Graybill (1992, p. 186 f.), and an exact condence interval of the error variance resp. an estimator of the error variance. The structure of the paper is as follows: In section 2 a description of the unbalanced model and the properties of the mean sum of squares are presented. In section 3 the dierent approaches for constructing a condence interval for the between group variance are described in detail. The section 4 explains the conducted simulation studies concerning the condence coecients of the three and contains the results of the simulation study. Finally, some conclusions are given. Note that all mean sum of squares considered in the following are assumed to be positive, which is given with probability one. 2 The Model We consider the unbalanced case of the oneway random eects model of ANOVA, i. e. y ij = + a i + e ij i =1 ::: r j =1 ::: n i > 1 (1) where y ij denotes the observable variable, the xed, but unknown grand mean, a i the unobservable random eect with mean 0 and variance 2 a, and e ij mean 0 and variance 2 e. the error term with We assume that the random variables a 1 ::: a r e 11 :::e rn r are normally distributed and mutually stochastically independent. Furthermore, let n = P r n i denote the number of the total observations. 3

5 In model (1) it holds that the mean sum of squares between the groups, i. e. MS1 = 1 r ; 1 rx with y i: = P n i j=1 y ij=n i and y :: = P r and where 2 freedom. n i (y i: ; y :: ) 2 (2) P E(MS1) =k 2 + a 2 e k = 1 r ; 1 n2 r ; n (r ; 1)MS1 k 2 a + 2 e 2 r;1 if 2 a P ni j=1 y ij=n, has the expected value n2 i (3) =0 (4) denotes a central chisquare distributed random variable with degrees of The mean sum of squares within the groups, i. e. MS2 = 1 n ; r has expected value n rx X i j=1 (y ij ; y i: ) 2 (5) E(MS2) = 2 e (6) and (n ; r)ms2= 2 e 2 n;r : (7) According to (7), a (1 ; )condence interval for 2 e is given by " # (n CI( 2 ): ; r)ms2 (n ; r)ms2 e 2 (8) 2 n;r 1;=2 n;r =2 where 2 denotes the quantile of a 2 distribution with degrees of freedom. Due to (4) the approximation of the distribution of MS1 byamultiple of a 2 distribution is only satisfactory if the between group variance 2 a is close to 0. Thus, the transfer of the condence interval for 2 a in the balanced case independently proposed by Tukey (1951) and Williams (1962) to the unbalanced case is not possible. In the next section we will therefore consider three dierent approaches of constructing a condence interval for 2 a in the unbalanced case. 4

6 3 Condence intervals on 2 a 3.1 The ThomasHultquist condence interval Instead of MS1 from (2) Thomas and Hultquist (1978) considered the sample variance of the group means given by MS3 = 1 r ; 1 rx y i: ; 1 r rx They showed that it holds approximately (r ; 1)MS3 2 a + 2 e=~n y i:! 2 : (9) appr: 2 r;1 (10) where ~n denotes the harmonic mean of the sample sizes of the r groups. Furthermore, Thomas and Hultquist proved that MS2 and MS3 are stochastically independent, so that 2 e 2 a + 2 e=~n MS3 MS2 where F 1 2 appr: F r;1 n;r (11) denotes a F distributed random variable with 1 and 2 degrees of freedom. From (10) and (11) (1;)condence intervals for 2 a +2 e=~n and 2 a= 2 e can be constructed and adopting the ideas of constructing a condence interval by Tukey and Williams to the present situation, Thomas and Hultquist proposed the following condence interval for 2 a: CI TH ( 2 a ): " (r ; 1) 2 r;1 1;=2 MS3 ; MS2 ~n F r;1 n;r 1;=2 (r ; 1) 2 r;1 =2 MS3 ; MS2 ~n F r;1 n;r =2 # : (12) Due to Bonferroni's inequality the condence coecient of (12) is at least (1 ; 2), but one may hope that the actual condence coecient is near(1 ; ). However, in Thomas and Hultquist (1978) it is reported that the 2 approximation in (10) is not good for extremely unbalanced designs where the ratio = 2 a= 2 e is less than Thus, in such situations the condence interval (12) can be a liberal one, i. e. the condence coecient substantially lies below (1 ; ). 5

7 3.2 The BurdickEickman condence interval Burdick, Maqsood and Graybill (1986) suggested a condence interval for the ratio = 2 a= 2 e which overcomes the problem with small ratios in the ThomasHultquist procedure and has a condence coecient of at least 1 ;. This interval is given by CI() : MS3 MS2 1 F r;1 n;r 1;=2 ; 1 n min MS3 MS2 1 F r;1 n;r 1;=2 ; 1 n max with n min = minfn 1 ::: n r g and n max =maxfn 1 ::: n r g. The dierence between (13) and the condence interval for = 2 a= 2 e (13) in the Thomas Hultquist procedure is that due to (11) Thomas and Hultquist subtract 1=~n in both bounds instead of 1=n min and 1=n max, respectively, in (13). Using (13) and the condence interval for 2 a + 2 e=~n due to (10), Burdick and Eickman (1986) investigated the condence interval for 2 a constructed by thetukey-williams method. This interval is given by CI BE ( 2 a ): " ~nl 1+~nL (r ; 1)MS3 2 r;1 1;=2 ~nu 1+~nU # (r ; 1)MS3 (14) 2 r;1 1;=2 with and L = max U = max 0 MS3 MS2 1 F r;1 n;r 1;=2 ; 1 n min 0 MS3 MS2 1 F r;1 n;r =2 ; 1 n max 3.3 Condence interval based on Wald's condence interval for Instead of approximative condence intervals for as in the ThomasHultquist and Burdick-Eickman approach we consider the exact condence interval for given in Wald (1940) to construct a condence interval for 2 a. 6

8 Following Wald (1940) we observe that Var(y i: )= 2 a + 2 e=n i = 2 e=w i (15) with w i = n i =(1 + n i ), i =1 ::: r. Now, Wald considered the sum of squares rx P r (r ; 1)MS4 = w i y w 2 iy i: i: ; P r w (16) i and proved that (r ; 1)MS4= 2 e 2 r;1 : Furthermore, MS4 and MS2 are stochastically independent so that F w () = MS4 MS2 F r;1 n;r : (17) According to (17), an exact condence interval for the ratio can be constructed. Wald showed that (r ; 1)MS4 is a strictly monotonously decreasing function in, and so the bounds of the exact condence interval are given as the solutions of the following two equations: lower bound: F w () =F r;1 n;r 1;=2 (18) upper bound: F w () =F r;1 n;r =2 Since F w () is a strictly monotonously decreasing function in the solution of (18), if it exists, is unique. But due to the fact that is nonnegative, (r ; 1)MS4 is bounded at =0, namely it holds (r ; 1)MS4 rx n i y i: ; Thus, a nonnegative solution of (18) may not exist. P r n 2 P iy i: r n : (19) n=1 i If such a solution of one of the equations in (18) does not exist, the corresponding bound in the condence interval is set equal to zero. Note that the existence of a nonnegative solution in (18) only depends on the chosen. 7

9 Let us denote by L and U the solutions of the equations in (18), so we propose, using the condence bounds from (8) for e, 2 the following condence interval for 2 a (n ; r)ms2 (n CI( 2 ): ; r)ms2 a 2 L n;r 1; 2 U (20) n;r which has a condence coecient of at least (1 ; 2) according to Bonferroni's inequality. But due to the fact that the condence coecient of[ 2 e L 2 e U ] is exactly 1 ;, the resulting condence interval (20) may be very conservative, i. e. the condence coecient is larger than (1 ; ). So, we also consider a condence interval for 2 a with the estimator MS2 for 2 e instead of the bounds of the condence interval for 2 e, i. e. fci( 2 a ):[MS2 L MS2 U ] : (21) 4 Simulation studies In simulation studies we compare the condence coecients of the four dierent condence intervals (12), (14), (20), and (21) for 2 a in the unbalanced oneway random eects model. The simulations are conducted using SAS 6.12 under Windows NT. The means of the r groups, y i:, i = 1 ::: r, are independently generated using the SAS function RANNOR and the sum of squares within the groups, (n ; r) MS2, is generated independently from y i:, i =1 ::: r, using the SAS function RANGAM. During all simulations the error variance 2 e is set equal to one, and for the variance between the groups, a, 2 we consider the values 0 0:01 0:05 0:1 0:25 0:5 0: and10. The dierent unbalanced designs, which we examined, are given in table 1, whereby eight of these patterns were also analysed by Burdick and Eickman (1986). For r =3and r =6we extend the analysis by considering some patterns which are not so extremely unbalanced as in Burdick and Eickman. We consider two-sided condence intervals with = 0:1 and = 0:05, respectively. All estimated condence coecients are based on 10,000 replications for every set of parameters. Based on normal approximation it means that a 95%condence interval for an estimated condence coecient ^p =0:95 is given by [0:9456 0:9541] and for ^p =0:9 by [0:8940 0:9057]. 8

10 Table 1: Patterns of unbalanced designs used in simulations Pattern r n i 1 3 5, 10, , 20, , 10, , 1, , 2, , 10, 15, 5, 10, , 20, 30, 10, 20, , 10, 15, 20, 25, , 1, 1, 1, 1, , 2, 2, 2, 2, , 1, 4, 5, 6, 6, 8, 8, 10, , 2, 4, 5, 6, 6, 8, 8, 10, , 3, 4, 5, 6, 6, 8, 8, 10, 10 These patterns were also considered in Burdick and Eickman (1986) For solving the system of equations (18), we use the bisection method and choose as the precision of the solutions 10 ;14. As starting values of the bisection method we use the ones proposed by Wald (1940), who showed that for the solutions of (18), say ~ = L or U, it holds with = =2 or =1; =2 MS3 1 ; 1 ~ MS3 MS2 F r;1 n;r n 1 ; 1 : (22) min MS2 F r;1 n;r n max Note that the lower bound for L and the upper bound for U coincide with the condence interval for considered by Burdick, Maqsood and Graybill (1986), cf. (13). In table 2 the results of the simulation study are explicitly shown for all values of 2 a with =0:1 in pattern 5, where CI TH denotes the ThomasHultquistinterval from (12), CI BE the BurdickEickmaninterval from (14), CI the interval from (20) and f CI from (21). We choose this pattern as an example to illustrate the characteristics we found in all simulation studies. If 2 =0all condence intervals are rather conservative. For small a 2 a > 0 the estimated 9

11 Table 2: Estimated condence coecients for the four dierent condence intervals in pattern 5 with =0:1 Pattern 2 a CI TH CI BE CI CI f condence coecients of the ThomasHultquist interval lie below 1 ;, but the dierence to 1 ; in this pattern is not so severe. If 2 a becomes larger, the estimated condence coecient lies near 1 ;. The BurdickEickmaninterval possesses for small 2 a > 0 a high condence coecient, i. e. in these situations the interval is very conservative. The estimated condence coecient of this interval declines, if 2 a becomes larger, and lies near 1 ; for large a. 2 The condence interval CI, where the bounds of the (1 ; ) condence interval for 2 e are taken as estimates of the error variance, has a condence coecient near1 ; for small 2 a > 0, and the estimated condence coecient increases if 2 a becomes larger. Thus, for large 2 a this condence interval may be rather conservative. For the condence interval CI f with the mean squared error as the estimator of the error variance we get estimated condence coecients which lie near 1 ; for all 2 a > 0. If an increase or a decline of the estimated condence coecients is found, the estimated 10

12 condence coecients attain a certain level near 2 =1,sothat for a 2 a > 1 there is little variation between the estimated condence coecients. Thus, for simplifying the presentation of all simulations we present in table 3 and 4 ranges of the estimated condence coecients for 0 < 2 a 1 and 2 a > 1 separately for all patterns with = 0:1, and = 0:05, respectively, where in the case 0 < 2 a 1 we omit the values for 2 = 0, a because as stated above the condence intervals are rather conservative in this situation. The most extreme result in table 3 and 4, respectively, isgiven in pattern 11. There, the ThomasHultquistinterval is very liberal for 0 < 2 a 1, whereas the BurdickEickmanninterval produces estimated condence coecients greater than 0.99 in these situations. Generally speaking, the results just described for = 0:1 in pattern 5 are reected in a similar way in all conducted simulations as well for =0:1 as for =0:05. 5 Conclusions In our simulation studies we conrm the results of Burdick andeickman (1986) that the ThomasHultquistinterval may be very liberal for small a, 2 i. e. the condence coecient considerably lies below 1 ;. In these situation the BurdickEickmaninterval has a condence coecient which is always larger than 1 ;, but the interval can be very conservative. If 2 a becomes larger, both intervals are very similar. The condence interval CI deduced from Wald's condence interval for the ratio with the bounds of the condence interval of the error variance as estimates for the error variance has always a condence coecient at least as great as 1 ;, but this interval can be very conservative for large a. 2 A good compromise for the whole range of 2 a is the condence interval CI f from (21), which has a condence coecient at least as great as 1 ; for small a, 2 and for growing 2 a the condence interval only becomes moderately conservative. Acknowledgement The support of the Deutsche Forschungsgemeinschaft (SFB 475, "Reduction of complexity in multivariate data structures") is gratefully acknowledged. 11

13 References Boardman, T. J. (1974). Condence intervals for variance components A comparative Monte Carlo study. Biometrics 30, Burdick, R. K. Eickman, J. (1986). Condence intervals on the among group variance component in the unbalanced onefold nested design. J. Statist. Comput. Simul. 26, Burdick, R. K. Graybill, F. A. (1992). Condence Intervals on Variance Components. Dekker, New York. Burdick, R. K. Maqsood, F. Graybill, F. A. (1986). Condence intervals on the intraclass correlation in the unbalanced oneway classication. Commun. Statist.Theory Meth. 15, Graybill, F. A. (1976). Theory and Application of the Linear Model. Duxbury Press, North Scituate. Searle, S. R. Casella, G. McCulloch, C. E. (1992). Variance components. Wiley, New York. Thomas, J. D. Hultquist, R. A. (1978). Interval estimation for the unbalanced case of the oneway random eects model. Ann. Statist. 6, Tukey, J. W. (1951). Components in regression. Biometrics 7, Wald, A. (1940). A note on the analysis of variance with unequal class frequencies. Ann. Math. Statist. 11, Wang, C. M. (1990). On the lower bound of condence coecients for a condence interval on variance components. Biometrics 46, Williams, J. S. (1962). A condence interval for variance components. Biometrika 49,

14 Table 3: Ranges of estimated condence coecients for the four dierent condence intervals (12), (14), (20), and (21) with =0:1 Pattern 2 a CI TH CI BE CI CI f 1 <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= >

15 Table 4: Ranges of estimated condence coecients for the four dierent condence intervals (12), (14), (20), and (21) with =0:05 Pattern 2 a CI TH CI BE CI CI f 1 <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= > <= >