# Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

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1 Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA , USA I this article, we poit out some iterestig relatios betwee the exact test ad the score test for a biomial proportio p. Based o the properties of the tests, we propose some approximate as well as exact methods of computig sample sizes required for the tests to attai a specified power. Sample sizes required for the tests are tabulated for various values of p to attai a power of 0.80 at level We also propose approximate ad exact methods of computig sample sizes eeded to costruct cofidece itervals with a give precisio. Usig the proposed exact methods, sample sizes required to costruct 95% cofidece itervals with various precisios are tabulated for p =.05(.05).5 ad precisios. The approximate methods for computig sample sizes for score cofidece itervals are very satisfactory ad the results coicide with those of the exact methods for may cases. Key words: Clopper Pearso iterval; Oe-sided limits; coverage probability; Expected legth; Sizes; Wilso iterval Address correspodece to K. Krishamoorthy, Departmet of Mathematics, Uiversity of Louisiaa at Lafayette, Lafayette, LA 70504, USA; Fax: ;

4 4 where the sample proportio ˆp = X/. This test rejects the ull hypothesis i (2) whe Z(k,, p 0 ) z α, where z α deotes the upper αth quatile of the stadard ormal distributio. For testig two-sided alterative hypothesis, that is, whe H 0 : p = p 0 vs. H a : p p 0, (4) the score test rejects the ull hypothesis i (4) whe Z(k,, p 0 ) z α/ The exact test The exact tests are based o the exact p-value that ca be computed usig the biomial pmf. I particular, the p-value for testig (2) is give by P (X k, p 0 ) = p i i 0(1 p 0 ) i. (5) i=k This exact test that rejects H 0 i (2) whe the p-value i (5) is less tha or equal to α. The p-value for a two-sided alterative hypothesis, that is, for testig (4), is give by 2 mi{p (X k, p 0 ), P (X k, p 0 )}. These exact tests are level α tests i the sese that the Type I error rates ever exceed the omial level α for all (, p) cofiguratios. 2.3 Size ad power properties of the tests The exact power of the score test for testig oe-sided hypotheses i (2) is give by ( ) P (Z(X,, p 0 ) z α, p) = P X [p 0 + z α p0 (1 p 0 )] +, p, (6) where X biomial(, p) ad [x] + deotes the smallest iteger greater tha or equal to x. Note that, whe p = p 0, the above expressio gives the size (Type I error rate) of the score test. The power fuctio for testig (4) is give by P ( X [p 0 + z α/2 p0 (1 p 0 )] +, p ) + P ( X [p 0 z α/2 p0 (1 p 0 )], p (7) where [x] deotes the largest iteger less tha or equal to x. For a give p p 0, power β ad level α, the sample size required for a two-tail score test is the smallest value of for which the above power i (2) is at least β, ad the Type I error rate is at most α. The power of the exact test for testig (2) is give by p k (1 p) k I (P (X k, p 0 ) α), (8) k k=0 ),

5 5 where I(.) is the idicator fuctio. We ote that the expressio i (8) with p = p 0 gives the size of the exact test. A power expressio for a two-tail test ca be obtaied usig (8) with the idicator fuctio replaced by I(2 mi{p (X k, p 0 ), P (X k, p 0 )} α). The exact sizes ad powers of the score test ad the exact test are computed usig (6) ad (8) respectively. The sizes ad powers of both tests are plotted as a fuctio of i Figure 1a for testig H 0 : p.3 vs. H a : p >.3; powers are computed at p =.35. It is clear from these graphs that the sizes of the score test are always greater tha or equal to those of the exact test. I particular, we ote from Figure 1a that wheever the size of the score test is below the omial level it coicides with that of the exact test. The fluctuatio i powers reflects the size behaviors for all. Specifically, wheever the size of the score test is above the omial level.05, it offers more power tha the exact test; otherwise the powers of the tests are the same. These fidigs idicate that the sample sizes eeded for both tests to attai a give power are the same if the score test is required cotrol the Type I error rate withi the omial level. We also computed the sizes ad powers of the tests as a fuctio of for other values of p. As the plots exhibited similar patters as those i Figure 1a, they are ot preseted here. I Figure 1b, we plotted the sizes ad powers of the tests as a fuctio of for a two-tail test. We first observe that the size behaviors are differet from those for the right-tail test give i Figure 1a. We also see from Figure 1b that the sizes of the score test are always larger tha that of the exact test; however, there are may sample sizes for which the Type I error rates of the score test are below the omial level ad greater tha those of the exact test. For these sample sizes, the score test ot oly cotrols the Type I error rates, but also provides large power tha the exact test.

6 exact score Figure 1a. Sizes ad powers of the tests for testig H 0 : p.3 vs. H a : p >.3 at α =.05; powers are computed at p = exact score Figure 1b. Sizes ad powers of the tests for testig H 0 : p =.3 vs. H a : p.3 at α =.05; powers are computed at p =.35

7 7 2.4 Sample size calculatio for hypothesis tests For a give p, p 0 ad α, the sample size required for the score test (for oe-sided hypothesis) to attai a power of β is the smallest value of for which the power i (6) is at least β. That is, the smallest sample size for which ( ) P X [p 0 + z α p0 (1 p 0 )] +, p β. (9) Notice that the power fuctios of both tests are oscillatig with respect to the sample size (see Figure 1a); the power does icrease for a large icrease i sample size, but it may decrease for small chages i sample size. Therefore, to compute a accurate sample size, a forward search method, startig from a small value of, ca be used to fid the smallest value of that satisfies (9). Ay other search method may produce a sample size that is uecessarily larger tha the oe required to attai a specified power. Usig (7), the sample size for a two-tail score test ca be computed similarly. A approximatio to the sample size ca be obtaied usig the ormal approximatio to the biomial(, p) distributio. Specifically, a approximatio to the power i (9) ca be expressed as ( ) p0 + z α p0 q 0 p 1 Φ, (10) pq where q = 1 p, q 0 = 1 p 0 ad Φ is the stadard ormal distributio fuctio. Equatig the above power to β ad solvig for, we get (z β pq zα p0 q 0 ) 2. (11) (p p 0 ) 2 It should be oted that z c, for 0 < c < 1, is the upper cth quatile of the stadard ormal distributio. If the hypotheses are two-sided, the it is ot feasible to obtai a simple approximatio to the sample size usig the approach give for oe-sided hypotheses. A ituitive approximatio is (11) with z α replaced by z α/2. That is, (z β pq zα/2 p0 q 0 ) 2. (12) (p p 0 ) 2 The sample size for the oe-sided exact test is the smallest value of for which the power i (8) is at least β. That is, the least value of for which k=0 p k (1 p) k I (P (X k, p 0 ) α) β. (13) k The sample size for a two-tail test ca be expressed similarly.

9 9 3. Cofidece Itervals We shall ow preset the score itervals, exact itervals ad expressios for computig their expected legths. 3.1 Score itervals ad expected legths The score cofidece iterval is obtaied by ivertig the score test statistic. Specifically, solvig the iequality ˆp p z α/2 p(1 p)/ for p, we get the score iterval as ˆp + z2 α/ z2 α/2 ± z α/2 ˆp(1 ˆp) + zα/2 2 /(4). (14) 1 + z2 α/2 Oe-sided limits ca be obtaied by replacig z α/2 by z α. Oe half of the expected legth of the score iterval i (14) is give by x=0 p x (1 p) x x (x/)(1 x/) + zα/2 2 /(4). (15) 1 + z2 α/2 z α/2 A approximatio to the above expressio ca be obtaied usig the followig lemma. Lemma 3.1 Let f(ˆp) be a real valued fuctio. The, Ef(ˆp) = f(p) + O( 1 ). (16) Proof. Usig a Taylor series expasio aroud p, we have f(ˆp) = f(p) + (ˆp p)f (ˆp) ˆp=p + Now, takig expectatio o both sides, we get (16). (ˆp p)2 f (ˆp) ˆp=p ! Applyig the above lemma, we see that (15) is approximately equal to p(1 p) + zα/2 2 /(4). (17) 1 + z2 α/2 z α/2

10 10 A approximatio to the sample size that is required to costruct a cofidece iterval with a give precisio d ca be calculated as follows. Settig (17) equal to d, ad solvig the resultig equatio for, we get [ zα/2 2 (pq 2d 2 ) + ] (pq 2d 2 ) 2 d 2 (4d 2 1), (18) 2d 2 where q = 1 p. Similarly, a approximate sample size to fid a 1 α upper limit withi a precisio of d from the true p ca be computed as the solutio of the equatio ( zα ) pq + z 2 α /(4) p = d. (19) 1 + z2 α Solvig (19) for, we get [ zα 2 pq + d 2d(p + d) + ] (pq + d 2d(p + d)) 2 d 2 ((2(p + d) 1) 2 1). (20) 2d 2 These above approximatios are very satisfactory as will be show later i Sectio Exact cofidece itervals ad expected legths The edpoits of the Clopper-Pearso (1934) cofidece iterval for a biomial proportio p ca be obtaied as solutios of the followig two equatios. I particular, for a give sample size ad a observed umber of successes k, the lower limit p L for p is the solutio of the equatio i=k p i i L(1 p L ) i = α 2, ad the upper limit p U is the solutio of the equatio k i=0 p i i U(1 p U ) i = α 2. Usig a relatio betwee the biomial ad beta distributios, the edpoits ca be expressed as p L = B 1 (α/2; k, k + 1) ad p U = B 1 (1 α/2; k + 1, k), where B 1 (c; a, b) deotes the cth quatile of a beta distributio with the shape parameters a ad b. The iterval (p L, p U ) is a exact 1 α cofidece iterval for p, i the sese that the coverage probability is always greater tha or equal the specified cofidece level

12 = 40 exact score p = 80 exact score p Figure 2a Figure 2b. Expected legths of exact ad score cofidece itervals as a fuctio of p p =.25 exact score p =.5 exact score Expected legths of exact ad score cofidece itervals as a fuctio of For a give precisio d, value of p ad cofidece level 1 α, the exact sample size required to compute the score iterval is the smallest value of for which (15) is less tha or equal to d ad the coverage probability is at least 1 α. We first computed the sample size s usig the approximatio i (18), ad the usig the s as a iitial value i (15) we carried out forward/backward search to fid the exact sample size required to costruct the score iterval with a specified precisio ad coverage probability at least 1 α. The sample sizes for the exact itervals are computed similarly usig s as a iitial value i (22), ad a forward search method. The sample sizes for 95% cofidece itervals are give i Table 2. We also computed the approximate sample sizes for the score itervals usig (18), ad compared them with the exact oes based o (15). Our compariso idicted that the approximate sample sizes are idetical to those based o the exact approach i may cases. I geeral, the approximatio is very satisfactory. The cases where the approximate sample sizes are ot equal to the exact oes are idicated i Table 2. For

14 14 expressio i (6) because it is easier to compute tha the power of the exact test i (8). However, for two-sided hypotheses, the score test ca be recommeded for applicatios as it requires smaller samples i may situatios. The score method is also preferable to the exact method for costructig cofidece itervals as the former requires cosiderably smaller samples tha the latter to attai the same power. We hope that our fidigs are useful for researcher ad practitioers to choose betwee the score ad exact methods for applicatios. Refereces Agresti A, Coull, B. A. (1998). Approximate is better tha exact for iterval estimatio of biomial proportio. The America Statisticia 52: Blyth C. R., Hutchiso D. W. (1960). Tables of Neyma-shortest cofidece itervals for the biomial parameter. Biometrika 47: Blyth C. R., Still H. A. (1983). Biomial cofidece itervals. J. Amer. Statist. Assoc. 78: Brow L. D., Cai T., Das Gupta A. (2001). Iterval estimatio for a biomial proportio (with discussio). Statistical Sciece 16: Cai T. (2005). Oe-sided cofidece itervals i discrete distributios. J. Statist. Pla. Ifer. 131: Cai Y., Krishamoorthy, K. (2005). A simple improved iferetial method for some discrete distributios. Computatioal Statistics ad Data Aalysis 48: Casella G. (2001). Commet o Brow, Cai, ad DasGupta A. Statistical Sciece 16: Clopper C.J., Pearso E. S. (1934) The use of cofidece or fiducial limits illustrated i the case of biomial. Biometrika 26: Geyer C. J. Meede G. D. (2005). Fuzzy ad radomized cofidece itervals ad p-values. Statistical Sciece 20: Wilso E. B. (1927) Probable iferece, the law of successios ad statistical iferece. J. Amer. Statist. Assoc. 22:

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