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1 This is a author produced versio of a paper published i Commuicatios i Statistics - Theory ad Methods. This paper has bee peer-reviewed ad is proof-corrected, but does ot iclude the joural pagiatio. Citatio for the published paper: Forkma, J. (29) Estimator ad Tests for Commo Coefficiets of Variatio i Normal Distributios. Commuicatios i Statistics - Theory ad Methods. Volume: 38 Number: 2, pp Access to the published versio may require joural subscriptio. Published with permissio from: Taylor & Fracis Epsilo Ope Archive

2 Estimator ad Tests for Commo Coefficiets of Variatio i Normal Distributios Johaes Forkma Departmet of Eergy ad Techology Swedish Uiversity of Agricultural Scieces Box 732, SE-75 7 Uppsala, Swede Johaes.Forkma@vpe.slu.se Abstract Iferece for the coefficiet of variatio i ormal distributios is cosidered. A explicit estimator of a coefficiet of variatio that is shared by several populatios with ormal distributios is proposed. Methods for makig cofidece itervals ad statistical tests, based o McKay s approximatio for the coefficiet of variatio, are provided. Exact expressios for the first two momets of McKay s approximatio are give. A approximate F-test for equality of a coefficiet of variatio that is shared by several ormal distributios ad a coefficiet of variatio that is shared by several other ormal distributios is itroduced. Key words: Coefficiet of Variatio, Cofidece iterval, Hypothesis test, McKay s approximatio, Normal distributio, Statistical iferece 1 Itroductio The coefficiet of variatio c i a sigle sample with observatios y 1, y 2,..., y is defied as c = s/m, where m ad s are m = 1 y j ad s = 1 (y j m) 1 2, (1) j=1 respectively. I this paper we cosider idepedetly ormally distributed observatios with expected value µ >, variace σ 2 ad populatio coefficiet of variatio γ = σ/µ. We discuss three problems: (i) Estimatio of a coefficiet of variatio γ that is shared by k populatios (ii) Cofidece iterval ad test for a coefficiet of variatio γ that is shared by k populatios j=1

3 (iii) Test for equality of a coefficiet of variatio γ 1 that is shared by k 1 populatios ad a coefficiet of variatio γ 2 that is shared by k 2 populatios Give k estimates c 1,..., c k of a commo coefficiet of variatio γ a method is eeded for poolig the estimates ito oe estimate. Explicitly we wat to kow if the sigle estimates shall be weighted by the umber of observatios i, by the degrees of freedom i 1 or by some other fuctio of the sample size. Zeigler (1973) compared several estimators of a commo coefficiet of variatio, but cosidered oly the case of equally large sample sizes, ad did ot discuss hypothesis tests ad cofidece itervals. Tia (25) studied the problem of makig iferece about a commo γ based o k samples ad suggested a repeated samplig method for calculatig a geeralized probability value as defied by Tsui ad Weerahadi (1989). A drawback with resamplig methods is that they do ot give the same result wheever applied. There could be a eed for a method based o explicit expressios. Verrill ad Johso (27) proposed a likelihood ratio based cofidece iterval for a commo coefficiet of variatio. However, the likelihood ratio test is kow to be too liberal for small sample sizes (Doorbos ad Dijkstra, 1983; Fug ad Tsag, 1998; Nairy ad Rao, 23). This is cofirmed i Sectio 4.2 of the preset paper. The likelihood ratio test is computatioally icoveiet whe there are may populatios. Verrill ad Johso (27) provided, for the likelihood ratio test, a web-based program that simulates critical values for small samples. We suggest that cofidece itervals ad tests for a commo coefficiet of variatio γ be based o the trasformatio for the coefficiet of variatio developed by McKay (1932). This trasformatio gives a approximately χ 2 distributed radom variable whe γ < 1/3, as cofirmed by Fieller (1932), Pearso (1932), Iglewicz ad Myers (197) ad Umphrey (1983). Forkma ad Verrill (28) showed that McKay s χ 2 approximatio is asymptotically ormal with mea 1 ad variace slightly smaller tha 2( 1). Vagel (1996) showed, by Taylor series expasio, that the error i McKay s approximatio is small whe the populatio coefficiet of variatio is small. We derive exact expressios for the first two momets of McKay s approximatio. A test is itroduced, based o McKay s trasformatio, for the equality of a coefficiet of variatio that is shared by k 1 populatios ad a coefficiet of variatio that is shared by k 2 populatios. May tests have bee proposed for the special case k 1 = k 2 = 1: the likelihood ratio test (Lohrdig, 1975; 2

4 Beett, 1977; Doorbos ad Dijkstra, 1983), the Wald test ad the score test (Gupta ad Ma, 1996), Beett s test (Beett, 1976; Shafer ad Sulliva, 1986) ad Miller s test (Miller, 1991a; Feltz ad Miller, 1996; Miller ad Feltz, 1997). The three problems listed above are cosidered i Sectios 2 4, respectively. I Sectio 4.2 we make a small Mote Carlo study of the performace of the ew test for equality of coefficiets of variatio compared with the likelihood ratio test, Beett s test ad Miller s test. 2 Estimatio of a Commo Coefficiet of Variatio Cosider samples from k ormally distributed populatios with a commo populatio coefficiet of variatio γ, ad defie the sample coefficiets of variatio as i Defiitio 1. Defiitio 1. Let y ij = µ i + e ij, where e ij are idepedetly distributed N(, σi 2), i = 1, 2,..., k ad j = 1, 2,..., i, with positive expected values µ i ad a positive commo populatio coefficiet of variatio γ = σ i /µ i, i = 1, 2,..., k. The coefficiet of variatio c i of sample i, i = 1, 2,..., k, is defied as c i = s i /m i, where m i ad s i are m i = 1 i y ij ad s i = 1 i (y ij m i ) i i 1 2, (2) respectively. j=1 We shall, throughout this paper, assume that the expected values µ i, i = 1, 2,..., k, are positive, which implies γ >. Sice we are focused o applicatios with positive ad approximately ormally distributed observatios, we also assume that the probability of egative observatios is small. We make this assumptio by requirig µ i 3σ i >, i = 1, 2,..., k. This implies γ < 1/3. The joit probability desity fuctio of the observatios {y ij } ca be writte k ( ( 1 i (2π) i/2 exp µ i γ 2 y ij 1 2µ 2 i γ2 i=1 j=1 i j=1 j=1 yij 2 )) i 2γ 2 i log µ i γ. (3) 3

5 Thus, by the factorizatio theorem, the 2k dimesioal statistic S = { i i y ij, j=1 is sufficiet for η = {1/(µ i γ 2 ), 1/(µ 2 i γ2 )} k i=1, ad sice there is a oe-to-oe correspodece betwee η ad β = (γ, µ 1, µ 2,..., µ k ), S is also sufficiet for β. By writig (3) as ( exp 1 2γ 2 k i (y ij µ i ) 2 i=1 j=1 µ 2 i j=1 y 2 ij } k i=1 k i log µ i γ i=1 k i=1 ) i 2 log(2π), we see that k i (y ij µ i ) 2 i=1 j=1 is complete ad sufficiet for γ 2, whe µ i, i = 1, 2,..., k, are kow. Whe µ i, i = 1, 2,..., k, are ukow, µ i ca be estimated by m i. Thus, with otatio accordig to Defiitio 1, cosider µ 2 i U = 1 k i=1 ( i 1) k i (y ij m i ) 2 i=1 j=1 m 2 i = k i=1 ( i 1) c 2 i k i=1 ( i 1), (4) as a estimator of γ 2. Theorem 1. Let γ be a commo populatio coefficiet of variatio, as defied i Defiitio 1. Let v = k i=1 ( i 1), ad let U v = U as defied by (4). Assume that ( i 1)/v λ i > as v. The v (Uv γ 2 ) d N(, 2γ 4 + 4γ 6 ), as v. Proof. With otatios accordig to Defiitio 1 i (m i µ i, s 2 i σ 2 i ) d N(, V), i = 1, 2,..., k, where ( σ 2 V = i 2σi 4 ). 4

6 Followig Serflig (198, p. 124) i (c 2 i γ 2 ) d N(, g Vg), i = 1, 2,..., k, where, evaluated at {m i, s 2 i } = {µ i, σ 2 i }, ( ) ( c g 2 = i, c2 i 2σ 2 m i s 2 = i i µ 3, 1 ) i µ 2. i Thus i (c 2 i γ 2 ) d N(, 2γ 4 + 4γ 6 ), i = 1, 2,..., k, ad, because k i=1 λ i = 1, v (U γ 2 ) = k i 1 i=1 v i 1 (c 2 i γ 2 ) d N(, 2γ 4 + 4γ 6 ). We ow cosider T = f(u) = U, (5) with U from (4), as a estimator of γ. Accordig to Theorem 2 the estimator (5) is asymptotically ormally distributed with mea γ ad variace (γ 2 /2 + γ 4 )/ k i=1 ( i 1). Theorem 2. Let γ be a commo populatio coefficiet of variatio, as defied by Defiitio 1, ad let v = k i=1 ( i 1). Let T v = U v, where U v = U as defied by (4). The v (Tv γ) d N(, γ 2 /2 + γ 4 ), as v. (6) Proof. By Theorem 1, v U v d N(γ 2, 2γ 4 + 4γ 6 ) as v. The, by applicatio of Theorem 3.1A i Serflig (198), v Tv = v f(u v ) d N(f(γ 2 ), (f (γ 2 )) 2 (2γ 4 + 4γ 6 )), as v, ad (6) follows sice (f (γ 2 )) 2 (2γ 4 + 4γ 6 ) = γ 2 /2 + γ 4. The expected values of the coefficiets of variatio c i are ot defied, because the desities of m i, i = 1, 2,..., k, are positive i eighborhoods of zero. As a cosequece the expected value of the estimator T, as defied by (5), does ot exist. Nevertheless, whe the expected values µ i are sufficietly 5

7 large, say larger tha 3 σ i, the probability of averages m i close to zero are egligible i most applicatios, ad we expect c i, i = 1, 2,..., k, to be close to γ. Accordig to Theorem 2 the estimator T is asymptotically ubiased. We shall ow derive a bias correctio for the case of small sample sizes. By a secod order series expasio of T, as a fuctio of {m i, s 2 i }, i = 1, 2,..., k, E(T ) γ k i=1 ( 2 T (s 2 i )2 V ar(s2 i ) + 2 T m 2 i ) V ar(m i ), (7) where the partial secod order derivatives should be evaluated at {m i, s 2 i }k i=1 = {µ i, σ 2 i }k i=1. By (7), sice V ar(s2 i ) = 2σ4 i /( i 1) ad V ar(m i ) = σ 2 i / i, E(T ) γ = γ k ( (i 1) 2 i=1 4v 2 γ 3 µ 4 i k ( (i 1) γ i=1 = γ γ 4v + γ3 2v k i=1 2v 2 i 1 i 2σi 4 i 1 + ( ( i 1) γ vµ 2 3 ) ) i 1 σ 2 i i v i + γ3 v ( i 1 3 )) i 1 i v ( 3 ) i 1, v where v = k i=1 ( i 1). Thus we expect T to be close to γ(1 1/(4v)) whe γ is small, ad a bias adjusted estimator of a commo populatio coefficiet of variatio γ is give by ˆγ = ( k i=1 ( i 1) ) 1 k i=1 ( i 1) c 2 i k i=1 ( i 1), (8) with otatios as i Defiitio 1. The result of a simulatio study is preseted i Table 1. Three samples of 1, 2 ad 3 observatios from ormal distributios with expected values 1, 1, ad 1,, respectively, ad with a commo populatio coefficiet of variatio γ, were geerated 2, times i MATLAB 6.5 (The Mathworks Ic., Natick, MA, USA). I total 18 combiatios of γ, 1, 2 ad 3 were ivestigated. For each combiatio, the meas of the uadjusted estimator (5), the bias adjusted estimator (8), ad their stadard deviatio were calculated. The study idicates that the bias adjustmet works well for coefficiets of variatio smaller tha 2%. I may chemical applicatios the 6

8 coefficiet of variatio is smaller tha 2%, ad the bias adjusted estimator (8) is the recommeded. For example, i immuoassays the workig rage is ofte defied as the rage of cocetratios for which the coefficiet of variatio is smaller tha 2% (Carroll, 23). Table 1: Averages ad stadard deviatios of simulated estimators T ad ˆγ of a commo coefficiet of variatio γ γ Mea(T ) Mea(ˆγ) SD Cofidece Iterval ad Test for a Commo Coefficiet of Variatio I this sectio we cosider the problems of makig a cofidece iterval for a coefficiet of variatio γ that is shared by k populatios ad testig the statistical hypothesis that γ equals some specific value γ. Cofidece itervals ad tests could be based o the estimator ˆγ from (8). However, the percetiles of the distributio of ˆγ are ot easily obtaied. The uadjusted 7

9 estimator (5) is the square root of a weighted average of squared sample coefficiets of variatio, ad the reciprocal of each sample coefficiet of variatio is ocetral t-distributed (Owe, 1968). We shall build the cofidece iterval ad test o McKay s approximatio, as defied by Defiitio 2. Defiitio 2. McKay s approximatio for the coefficiet of variatio of sample i, as defied by Defiitio 1, is give by K i = (1 + 1γ ) ( i 1)c 2 i c 2 i (. (9) i 1)/ i The expected value of McKay s approximatio (9) exists, ad the distributio of (9) is well approximated by a cetral χ 2 distributio with i 1 degrees of freedom, provided that γ < 1/3. McKay s approximatio is useful for approximately ormally distributed measuremets of variables that oly take positive values. As oted i Sectio 2, if oly positive values ca be obtaied ad the distributio is approximately ormal, the expected value µ caot be smaller tha 3 stadard deviatios σ. The requiremet γ = σ/µ < 1/3, eeded for McKay s approximatio to be approximately χ 2 distributed, is the fulfilled. The distributio of u i = c 2 i 1 + c 2 i ( i 1)/ i, i = 1, 2,..., k, (1) is cosequetly well approximated by a distributio with expected value θ = γ2 1 + γ 2 = γ2 γ 4 + γ 6... (11) ad variace iversely proportioal to i 1. Because the distributio of McKay s approximatio K i is, approximately, cetral χ 2 distributed with i 1 degrees of freedom, k i=1 K i is, approximately, cetral χ 2 distributed with k i=1 ( i 1) degrees of freedom. Thus, k i=1 K i = k i=1 ( i 1)u i /θ ca be used as a approximate pivotal quatity for costructig a cofidece iterval for θ as defied by (11). This approximate 1(1 α)% cofidece iterval for θ ca be writte [ k i=1 ( i 1)u i χ 2 1 α/2, k i=1 ( i 1)u i χ 2 α/2 ], 8

10 where χ 2 α deotes the 1α:th percetile of a cetral χ 2 distributio with k i=1 ( i 1) degrees of freedom, ad u i is defied by (1). The correspodig approximate 1(1 α)% cofidece iterval for γ is [ k i=1 ( ] i 1)u i χ 2 1 α/2 k k i=1 (, i=1 ( i 1)u i i 1)u i χ 2 α/2 k i=1 (. (12) i 1)u i Cosider the statistical ull hypothesis H : γ = γ. This hypothesis is equivalet to the hypothesis H : θ = θ, where θ = γ 2/(1 + γ2 ). Thus we ca use k i=1 ( i 1)u i (13) θ as a approximately cetral χ 2 distributed, with k i=1 ( i 1) degrees of freedom, test statistic of the hypothesis H : γ = γ. The proposed cofidece iterval (12) ad test (13) rely o the adequacy of McKay s approximatio. Sice we are iterested i the adequacy, we ed this sectio with a ivestigatio of the first two momets of the approximatio, as fuctios of the populatio coefficiet of variatio γ ad the sample size. Theorem 4. Let γ be the populatio coefficiet of variatio as defied by Defiitio 1, ad let K = K 1 be McKay s approximatio for the coefficiet of variatio i a sample with = 1 observatios, as defied by Defiitio 2. The first ad secod momets of McKay s approximatio are E(K) = ( 1) h(γ, ) 2θ E(K 2 ) = (2 1)((1 + γ 2 ) h(γ, ) 2γ 2 ) 4θ 2, where θ = γ 2 /(1 + γ 2 ) ad, for t =, 1, 2...,, h(γ, ) = 9

11 γ 2 (1 exp( 1/γ 2 )), = 2 t ( ) γ ( 1) r 2 r+1 Γ(t + 3/2) t + 3/2 Γ(t + 3/2 r) r= +( 1) t+1 ( γ 2 t + 3/2 ) t+3/2 2 Γ(t + 3/2) π d( (t + 3/2)/γ), = 3 + 2t t ( ) γ ( 1) r 2 r+1 Γ(t + 2) t + 2 Γ(t + 2 r) r= ( ) γ +( 1) t+1 2 t+2 Γ(t + 2) (1 exp( (t + 2)/γ 2 )), = 4 + 2t, t + 2 with d(x) = exp( x 2 ) x exp(z 2 ) dz. Proof. Forkma ad Verrill (28) showed that Kθ/ is type II ocetral beta distributed with parameters ( 1)/2, 1/2 ad /γ 2. Cosequetly X = 1 Kθ/ is type I ocetral beta distributed with parameters 1/2, ( 1)/2 ad /γ 2. A type I ocetral beta distributio with parameters a, b ad δ is cetral Beta(a+V, b), where V is Poisso(δ/2). Marchad (1997) provided expressios for the first ad secod momets of the type I ocetral beta distributio with parameters a, b ad δ based o g(a+b, δ/2) = E(1/(a+b+ V )). We let h(γ, ) = g(/2, /(2γ 2 )). The, accordig to Marchad (1997), E(X 2 ) = 1 (2 1) γ 2 2 E(X) = 1 ( 1)h(γ, ) 2, (14) + ( 1)( 3 + ( + 1) γ2 ) h(γ, ), (15) 4 ad the theorem follows sice E(K) = (1 E(X))/θ ad E(K 2 ) = 2 (1 2E(X) + E(X 2 ))/θ 2. The fuctio d, required for odd sample sizes i Theorem 4, is Dawsos s itegral, which has bee tabulated by Abramowitz ad Stegu (1972). Because exp( (t + 2)/γ 2 ), t =, 1, 2..., Theorem 4 makes it easy to calculate approximate first ad secod momets, especially for eve sample 1

12 sizes. For example, E(K) 1 (1 + γ 2 ), = 2 ) 3 (1 + γ2 2 γ4, = 4 2 ) 5 (1 + γ γ γ6, = 6, 9 ad 3 (1 + 2γ 2 + γ 4 ), = 2 E(K 2 15 (1 + γ 2 γ 4 γ 6 ), = 4 ) ) 35 (1 + 8γ γ4 + 6γ 6 + 8γ8, = 6. 3 Notice that whe γ is small, = 2, 4, 6, E(K) approximately equals 1 ad E(K 2 ) approximately equals ( 1)( + 1), which is the exact first ad secod momets, respectively, of a χ 2 distributed radom variable with 1 degrees of freedom. 4 Test for Equality of Two Commo Coefficiets of Variatio We ow itroduce a statistical test for the hypothesis that a coefficiet of variatio γ 1 that is shared by k 1 populatios equals a coefficiet of variatio γ 2 that is shared by k 2 populatios. Defiitio 3 makes clear the settig ad what we mea by coefficiets of variatio i this case. Defiitio 3. Let y rij = µ ri + e rij, where e rij are idepedetly distributed N(, σri 2 ), r = 1, 2, i = 1, 2,..., k r ad j = 1, 2,..., ri, with positive expected values µ ri ad positive populatio coefficiets of variatio γ r = σ ri /µ ri. The coefficiet of variatio c ri, r = 1, 2, i = 1, 2,..., k r, is defied as c ri = s ri /m ri, where m ri ad s ri are m ri = 1 ri y rij ad s ri = 1 ri (y rij m ri ) ri ri 1 2, (16) j=1 j=1 11

13 respectively. 4.1 A Test for Equality of Coefficiets of Variatio Cosider the hypothesis H : γ 1 = γ 2. Let ad u ri = c 2 ri 1 + c 2 ri ( ri 1)/ ri, r = 1, 2; i = 1, 2,..., k r, γ2 r θ r = 1 + γr 2, r = 1, 2, with otatio accordig to Defiitio 3. Because k r i=1 ( ri 1)u ri /θ r is approximately cetral χ 2 distributed with k r i=1 ( ri 1) degrees of freedom, ad θ 1 = θ 2 whe the hypothesis is true, F = k1 i=1 ( 1i 1)u 1i / k 1 i=1 ( 1i 1) k2 i=1 ( 2i 1)u 2i / k 2 i=1 ( 2i 1) (17) is approximately F distributed with k 1 i=1 ( 1i 1) ad k 2 i=1 ( 2i 1) degrees of freedom. Thus F ca be used as a approximately F distributed test statistic for the hypothesis of equal coefficiets of variatio. Whe k 1 = k 2 = 1, the test statistic F, as defied by (17), simplifies to F = c2 1 /(1 + c2 1 ( 1 1)/ 1 ) c 2 2 /(1 + c2 2 ( 2 1)/ 2 ), (18) where c r = c r1 ad r = r1, r = 1, 2, as defied by Defiitio 3. Accordig to Theorem 6 the distributio of the logarithm of F, as defied by (18), equals the distributio of the logarithm of a F distributed radom variable plus some error variables that are i probability of small orders. Let O p deote order i probability, defied as i Azzalii (1996). Theorem 6. Let γ = γ 1 = γ 2 as defied by Defiitio 3, with k 1 = k 2 = 1. Let X be a F distributed radom variable with 1 1 ad 2 1 degrees of freedom, let Z be a stadardized ormal radom variable, ad let U 1 ad U 2 be χ 2 distributed radom variables with 1 1 ad 2 1 degrees of freedom, respectively. Let X, Z, U 1 ad U 2 be idepedet. The log F = d 1 log X + 2Zγ + 1 ( U1 + U ) 2 γ 2 + R( 1, 2, γ) (19)

14 where F is defied by (18) ad R( 1, 2, γ) = O p (max{ 1 1 γ2, 1 2 γ2, γ 4 }). Proof. Write log F as log F = log c 2 1 ( 1 + ( 1 1)c ) 1 ( log c ( 2 1)c 2 ) 1 2. (2) 2 Let W r = U r /( r 1), r = 1, 2, ad let Z 1 ad Z 2 be idepedet stadardized ormal radom variables. The distributios of the averages m r1 ad the stadard deviatios s r1, r = 1, 2, as defied by Defiitio 3, equals the distributios of µ r1 +Z r σ r1 / r ad µ r1 γ W r, respectively. Thus c 2 r equals W r γ 2 /(1 + Z r γ/ ) 2 i distributio. The distributio of the first term i (2) cosequetly equals the distributio of ( log W 1 γ 2 log 1 + 2Z 1γ + Z2 1 γ2 + ( 1 1)W 1 γ 2 ) = log W 1 γ 2 + 2Z 1γ + Z2 1 γ2 + ( 1 1)W 1 γ ( 2Z1 γ + Z2 1 γ2 + ( 1 1)W 1 γ 2 ) = log W 1 γ 2 + 2Z 1γ + ( 1 1)W 1 γ 2 ( ( )) γ 2 + O p max, γ 4. (21) The correspodig calculatios ca be made also for the secod term i (2), ad (19) follows. 4.2 Simulatio Study I this sectio we ivestigate, by Mote Carlo techique, the sigificace levels ad powers of the itroduced approximate F-test (18), for the hypothesis H : γ 1 = γ 2 whe k 1 = k 2 = 1. We also study the likelihood ratio test, Beett s test as modified by Shafer ad Sulliva (1986), ad Miller s test. These tests are, for quick referece, give i the Appedix. I each simulatio two samples with 1 ad 2 observatios, respectively, were radomly geerated 2, times i MATLAB 6.5. The observatios were ormally distributed with expected values 1 ad 1,, ad with coefficiets of variatio γ 1 ad γ 2, respectively. The tests were performed with sigificace level 5% agaist the alterative hypothesis of uequal coefficiets of variatio, that is the tests were two-sided. Eight cases were studied. The type I errors of the tests were ivestigated 13

15 i Cases 1 6, ad the powers of the tests were ivestigated i Cases 7 ad 8. I Cases 1 3, the commo coefficiet of variatio γ was 5%, 15% ad 3%, respectively. The sample sizes were equal, i.e. 1 = 2 =, ad varied varied from 2 to 2. I Cases 4 6, γ was 5%, 15% ad 3%, respectively, but i these cases 1 = 4 ad 2 varied from 2 to 2. I Case 7 oe coefficiet of variatio was 5%, ad the other 15%, ad i Case 8 oe coefficiet of variatio was 15%, ad the other 3%. I Case 7 ad 8, the two sample sizes were equal ad varied from 2 to 2. Thus 19 simulatios were made per case. The results of the simulatio study are preseted i Figures 1 8, with oe figure per ivestigated case. The likelihood ratio test showed too large frequecy of type I error (Figures 1 6), especially for sample sizes smaller tha 1. Whe 2 = 2 the frequecy of rejected hypotheses was larger tha 2% with the likelihood ratio test. Beett s test was also too liberal, though ot as liberal as the likelihood ratio test (Figures 1 6). Miller s test performed better with regard to type I error, except whe oly 2 observatios were sampled per distributio (Figures 1 3). The approximate F-test, itroduced i this paper, was the oly test that produced almost correct frequecy of rejected hypotheses i all cases (Figures 1 6). The likelihood ratio test ad Beett s test, which were too liberal, showed better power for small sample sizes tha Miller s test ad the approximate F-test (Figures 7 ad 8). 5 Discussio I applicatios with costat, or almost costat, coefficiets of variatio it is ofte appropriate to assume that the data follows logormal distributios. Oe should therefore cosider workig o the log scale (Cole, 2). After log trasformatio of the data the usual tests for equality of variaces, such as Bartlett s test, could be applied. However, it is ot always appropriate to assume that the data is logormally distributed. I immuoassays, for example, ormally distributed errors i the volumes of samples could result i ormally distributed measuremets of cocetratio with costat coefficiet of variatio. I this paper we have discussed iferece for the coefficiet of variatio whe there are reasos to believe that the data is ormally, but ot logormally, distributed. 14

16 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 1: Case 1. = 1 = 2. of type I error whe γ 1 = γ 2 = 5% ad 15

17 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 2: Case 2. = 1 = 2. of type I error whe γ 1 = γ 2 = 15% ad 16

18 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 3: Case 3. = 1 = 2. of type I error whe γ 1 = γ 2 = 3% ad 17

19 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 4: Case 4. of type I error whe γ 1 = γ 2 = 5% ad 1 = 4. 18

20 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 5: Case 5. 1 = 4. of type I error whe γ 1 = γ 2 = 15% ad 19

21 Approximate F test Likelihood ratio test Beett s test Miller s test Figure 6: Case 6. 1 = 4. of type I error whe γ 1 = γ 2 = 3% ad 2

22 1 Approximate F test 1 Likelihood ratio test Beett s test 1 Miller s test Figure 7: Case 7. Power whe γ 1 = 5%, γ 2 = 15% ad = 1 = 2. 21

23 1 Approximate F test 1 Likelihood ratio test Beett s test 1 Miller s test Figure 8: Case 8. Power whe γ 1 = 15%, γ 2 = 3% ad = 1 = 2. 22

24 Sice it is ofte ecessary to relate the stadard deviatio to the level of the measuremets, the coefficiet of variatio is a widely used measure of dispersio. Coefficiets of variatio are ofte calculated o samples from several idepedet populatios, ad questios about how to compare them aturally arise. There is a eed for a explicit estimator of a commo coefficiet of variatio. Such a estimator has bee give i this paper. For makig cofidece itervals ad tests we have cosidered McKay s approximatio, which is valid oly whe the populatio coefficiet of variatio γ is smaller tha 1/3. Coefficiets of variatio are usually calculated o positive data, such as measuremets of cocetratio, weight or legth. Give that the positive measuremets are approximately ormally distributed γ is smaller tha 1/3, because otherwise the expected value is smaller tha 3 stadard deviatios, ad the probability of egative observatios is ot egligible. Thus, McKay s approximatio is applicable for positive variables, provided oly that the distributios are approximately ormally distributed. Normality could be checked, e.g. by the Shapiro-Wilk test. Over the years may tests have bee proposed for equality of coefficiets of variatio. I this paper, a additioal test has bee itroduced: the approximate F-test. Ulike may other tests the ew test ca be applied ot oly whe there are oe estimate per populatio coefficiet of variatio, but also whe there are several. The small simulatio study reported i this paper idicated good performace of the approximate F-test, especially with regard to type I error. It would, however, be iterestig to see results from a larger simulatio study, icludig more cases, several tests ad a ivestigatio of robustess. As already poited out, the methods proposed i this paper are iteded for ormally distributed data. Miller (1991b) suggested a oparametric test for equality of coefficiets of variatio. This oparametric test was recommeded for oormal distributios by Fug ad Tsag (1998). The F max -test (Hartley, 195) is a atural extesio of the approximate F-test to more tha two commo coefficiets of variatio (G.E. Miller, persoal commuicatio). For two coefficiets of variatio, the F max -test ad the approximate F-test are idetical for the two-sided hypothesis. The F max - test requires equal degrees of freedom. Tables for the F max distributio were give by Nelso (1987). 23

25 Ackowledgmets I thak the referees for useful commets. The Cetre of Biostochastics, Swedish Uiversity of Agricultural Scieces, ad Pharmacia Diagostics AB, Uppsala, Swede, supported the research. Appedix. The Likelihood ratio test, Beett s ad Miller s Tests Let m r = m r1 ad c r = c r1 deote the average ad the coefficiet of variatio, respectively, i the r:th sample, r = 1, 2, as defied by Defiitio 3. The likelihood ratio test statistic ca be writte 1 ( γ µ 1 ) 2 2 log λ = 1 log ( 1 1) c 2 1 m2 1 2 ( γ µ 2 ) log ( 2 1) c 2 2 m2 2, (22) where λ is the likelihood ratio ad γ, µ 1 ad µ 2 are the maximum likelihood estimates of γ, µ 1 ad µ 2, respectively. These are, accordig to Gerig ad Se (198), 1 m 1 µ 2 µ 1 =, µ 2 = q ( ) µ 2 2 m 2 2p + q 2 4p 2 r p, γ = 1 µ 2 ( 2 1) c 2 2 m m 2 2 m 2 µ 2, (23) where p = ( )c , q = (2 2 c )m 2 ad r = (( 2 2 (c ) 2 1 (c ))m2 2 )/( ). Asymptotically (22) is χ 2 distributed with 1 degree of freedom. Beett s test statistic, as modified by Shafer ad Sulliva (1986), ca be writte ( ( ) log ( ( 1 1)c 2 1 ( 1 1)c 2 )) c 2 1 ( + 1 1)/ c 2 1 ( 1 1)/ 1 ( ( 1 1)c 2 ) 1 ( 1 1) log ( 1 1)(1 + c 2 1 ( 1 1)/ 1 ( ) ( 2 1) log, ( 2 1)c 2 2 ( 2 1)(1 + c 2 2 ( 2 1)/ 2 ) ad is approximately χ 2 distributed with 1 degree of freedom. 24

26 Miller s test statistic (Miller, 1991a),which is approximately N(,1), is ( c 2 (c 1 c 2 ) 2( 1 1) + c c 2 ) 1/2 2( 2 1) + c4, 2 1 where c = (( 1 1)c 1 + ( 2 1)c 2 )/( ). Refereces Abramowitz, M. ad Stegu, I. A. (1972). Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables. New York: Dover. Azzalii, A. (1996). Statistical Iferece. Lodo: Chapma ad Hall. Beett, B. M. (1976). O a approximate test for homogeeity of coefficiets of variatio. I W. J. Ziegler (Ed.), Cotributios to Applied Statistics dedicated to A. Lider. Experietia Supplemetum 22: Basel: Birkhäuser. Beett, B. M. (1977). LR tests for homogeeity of coefficiets of variatio i repeated samples. Sakhya, Series B, 39:4 45. Carroll, R. J. (23). Variaces are ot always uisace parameters. Biometrics 59: Cole, T. J. (2). Sympercets: symmetric percetage differeces o the 1 log e scale simplify the presetatio of log trasformed data. Statistics i Medicie 19: Doorbos R. ad Dijkstra J. B. (1983). A multi sample test for the equality of coefficiets of variatio i ormal populatios. Commuicatios i Statistics Simulatio ad Computatio 12: Feltz, C. J. ad Miller G. E (1996). A asymptotic test for the equality of coefficiets of variatio from k populatios. Statistics i Medicie 15: Fieller, E. C. (1932). A umerical test of the adequacy of A. T. McKay s approximatio. Joural of the Royal Statistical Society 95: Forkma, J. ad Verrill, S. (28). The distributio of McKay s approximatio for the coefficiet of variatio. Statistics & Letters 78:

27 Fug W. K. ad Tsag T. S. (1998). A simulatio study comparig tests for the equality of coefficiets of variatio. Statistics i Medicie 17: Gerig T. M. ad Se A. R. (198). MLE i two ormal samples with equal but ukow populatio coefficiets of variatio. Joural of the America Statistical Associatio 75: Gupta R. C. ad Ma S. (1996). Testig the equality of coefficiets of variatio i k ormal populatios. Commuicatios i Statistics Theory ad Methods 25: Hartley H. O. (195). The maximum F-ratio as a short cut test for heterogeeity of variace. Biometrika 37: Iglewicz B. ad Myers R. H. (197). Compariso of approximatios to the percetage poits of the sample coefficiet of variatio. Techometrics 12: Lohrdig R. K. (1975). A two sample test of equality of coefficiets of variatio or relative errors. Joural of Statistical Computatio ad Simulatio 4: Marchad E. (1997). O momets of beta mixtures, the ocetral beta distributio, ad the coefficiet of determiatio. Joural of Statistical Computatio ad Simulatio 59: McKay, A. T. (1932). Distributio of the coefficiet of variatio ad the exteded t distributio. Joural of the Royal Statistical Society 95: Miller G. E. (1991a). Asymptotic test statistics for coefficiets of variatio. Commuicatios i Statistics Theory ad Methods 2: Miller G. E. (1991b). Use of the squared raks test to test for the equality of the coefficiets of variatio. Commuicatios i Statistics Simulatio ad Computatio 2: Miller G. E. ad Feltz C. J. (1997). Asymptotic iferece for coefficiets of variatio. Commuicatios i Statistics Theory ad Methods 26: Nairy, K. S. ad Rao, K. A. (23). Tests of coefficiets of variatio of ormal populatio. Commuicatios i Statistics Simulatio ad Computatio 32:

28 Nelso L. S. (1987). Upper 1%, 5% ad 1% poits of the maximum F- ratio. Joural of Quality Techology 19: Owe D. B. (1968). A survey of properties ad applicatios of the ocetral t-distributio. Techometrics 1: Pearso E. S. (1932). Compariso of A.T. McKay s approximatio with experimetal samplig results. Joural of the Royal Statistical Society 95: Serflig, R. J. (198) Approximatio Theorems of Mathematical Statistics. New York: Wiley. Shafer N. J. ad Sulliva J. A. (1986). A simulatio study of a test for the equality of the coefficiets of variatio. Commuicatios i Statistics Simulatio ad Computatio 15: Tia L. (25). Ifereces o the commo coefficiet of variatio. Statistics i Medicie 24: Tsui K. W. ad Weerahadi S. (1989). Geeralized P-values i sigificace testig of hypotheses i the presece of uisace parameters. Joural of America Statistical Associatio 84: Umphrey G. J. (1983). A commet o McKay s approximatio for the coefficiet of variatio. Commuicatios i Statistics Simulatio ad Computatio 12: Vagel M. G. (1996). Cofidece itervals for a ormal coefficiet of variatio. The America Statisticia 15: Verrill S. ad Johso R. A. (27). Cofidece bouds ad hypothesis tests for ormal distributio coefficiets of variatio. Commuicatios i Statistics Theory ad Methods 36: Zeigler R. K. (1973). Estimators of coefficiets of variatio usig k samples. Techometrics 15: