I. Herranz Tejedor

Exact unconditional non-classical tests on the difference of two proportions

Abstract The paper analyzes the three possible unconditional exact tests on the difference (d=p2−p1) of two independent binomial proportions: H SG : d⩽δ vs. K SG : d>δ,H PEG : d∉(δ 1 ,δ 2 ) vs. K PEG : d∈[δ 1 ,δ 2 ] and H SDG : d∈[δ 1 ,δ 2 ] vs. KSDG:d∉[δ1,δ2], where H(K) refers to the null (alternative) hypothesis. For this purpose the methodological bases of such tests are established; the properties of convexity and symmetry of the classic statistic z-pooled are studied; and, finally, the article indicates how to reduce computing problems by decreasing the parametric space and sample space involved (which can reduce the computing time up to tenfold). These tests are a generalization of the “classic” case (δ=0 in SG; δ1=δ2=0 in SDG) and the “non-classic” case (δ 0 in SG for establishing substantial superiority; δ2=−δ1>0 in PEG for establishing equivalence). The paper also proves that the present test PEG is more powerful than the one based on the classic TOST (two one-sided tests) method. Finally, the above tests were used to obtain exact confidence intervals.

Is Fisher's exact test very conservative?

Abstract There aer various two-tailed test versions of Fishers exact test for analyzing a 2×2 table. In this paper, the optimal version is selected on the basis of the concept of mean power (arranging in order from the smallest to the largest hypergeometrical probability, and in the case of a tie, arranging in order from the largest to the smallest value of the odds-ratio), and this selection is as valid when considering it as a conditional test as it is when considering it as an unconditional test. The comparison of the power of the version selected (with one and two tails), with that of the more common unconditional tests (Barnard, 1947, and McDonald et al., 1977), shows that the loss of power produced by using Fishers test is very slight in the majority of situations, and this is acceptable in return for the greater ease of computation and a more generic validity (for all types of sample).

Inferences about a linear combination of proportions

Statistical methods for carrying out asymptotic inferences (tests or confidence intervals) relative to one or two independent binomial proportions are very frequent. However, inferences about a linear combination of K independent proportions L = Σβipi (in which the first two are special cases) have had very little attention paid to them (focused exclusively on the classic Wald method). In this article the authors approach the problem from the more efficient viewpoint of the score method, which can be solved using a free programme, which is available from the webpage quoted in the article. In addition the article offers approximate formulas that are easy to calculate, gives a general proof of Agresti’s heuristic method (consisting of adding a certain number of successes and failures to the original results before applying Wald’s method) and, finally, it proves that the score method (which verifies the desirable properties of spatial and parametric convexity) is the best option in comparison with other methods.Statistical methods for carrying out asymptotic inferences (tests or confidence intervals) relative to one or two independent binomial proportions are very frequent. However, inferences about a linear combination of K independent proportions L = Σβipi (in which the first two are special cases) have had very little attention paid to them (focused exclusively on the classic Wald method). In this article the authors approach the problem from the more efficient viewpoint of the score method, which can be solved using a free programme, which is available from the webpage quoted in the article. In addition the article offers approximate formulas that are easy to calculate, gives a general proof of Agresti’s heuristic method (consisting of adding a certain number of successes and failures to the original results before applying Wald’s method) and, finally, it proves that the score method (which verifies the desirable properties of spatial and parametric convexity) is the best option in comparison with other methods.

A numerical comparison of several unconditional exact tests in problems of equivalence based on the difference of proportions

In order to compare two independent proportions (p 1 and p 2) there are several useful tests for the parameter d=p 2−p 1:H SG:d≤δ vs. K SG:d>δ, H SG2:d=δ vs. K SG2:d≠δ (where−1Δ (where Δ≥0) and H PE:|d|≥Δ vs. K PE:|d|0). The exact unconditional test requires an ordering statistic, which is usually the z-pooled statistic, to be defined. The paper gives the definition of 10 new ordering statistics with a similar computational time, and compares the number of points which each introduces into the critical region obtained to error α=5%. The article reaches the conclusion that the most generally powerful statistics are: the z-pooled one with a small continuity correction (c=1/N if n 1≠n 2 or c=2/N if n 1=n 2, where N={n 1+1} {n 2+1} and n i are the sample sizes) and those z-pooled with Yates’ continuity correction (c={n 1+n 2}/{2n 1 n 2} ). In the paper the authors also show that Barnards two classic convexity conditions are redundant, because when one of them is verified the other is also verified. The programs for these tests may be obtained free of charge from the site http://www.ugr.es/local/bioest/software.htm.

Asymptotic two-tailed confidence intervals for the difference of proportions

In order to obtain a two-tailed confidence interval for the difference between two proportions (independent samples), the current literature on the subject has proposed a great number of asymptotic methods. This paper assesses 80 classical asymptotic methods (including the best proposals made in the literature) and concludes that (1) the best solution consists of adding 0.5 to all of the data and inverting the test based on the arcsine transformation; (2) a solution which is a little worse than the previous one (but much easier and even better when both samples are balanced) is a modification of the adjusted Wald method proposed by Agresti and Caffo (usually adding to all of the data and then applying the classical Wald CI); (3) surprisingly, the classical score method is among the worst solutions, since it provides excessively liberal results.

Perfil clínico de una población hipercolesterolémica española y diferencias entre sexos. Estudio LIPYCARE

Introduccion El estudio LIPYCARE es un registro epidemiologico multicentrico espanol disenado para conocer las caracteristicas clinicas y analiticas y evaluar las diferencias entre sexos de una poblacion hipercolesterolemica (colesterol total serico >240 mg/dl y/o colesterol lipoproteina de baja densidad >160 mg/dl, sin tratamiento hipolipemiante) que acude a la consulta de Atencion Primaria. Material y metodos Se incluyeron 3.424 pacientes por muestreo consecutivo, con una edad media de 58,1±10,5 anos; el 52,7 % mujeres. Resultados Las cifras medias de lipidos fueron: colesterol total, 288,4±32,1 mg/dl; colesterol ligado a lipoproteina de baja densidad, 190,2 ± 37,2; colesterol ligado a lipoproteina de alta densidad, 50,4 ± 14,5; trigliceridos, 171,4 ± 65,4 mg/dl. El 67,4 % presentaron al menos otro factor de riesgo asociado; la hipertension arterial fue el factor de riesgo mas prevalente (48 %), seguido de la edad (43,4 % eran varones >55 anos o mujeres >65). Un 28 % de los sujetos presentaba ya enfermedad cardiovascular, siendo la cardiopatia isquemica la mas frecuente (9,3 %). El riesgo coronario medio de la poblacion sin enfermedad cardiovascular segun diferentes tablas de Framingham fue alrededor del 20 %, siendo mayor en los varones (27,6 % frente a 13,5 %). Segun las tablas de Framingham clasicas, el 39,8 % de la poblacion eran de riesgo bajo ( 30%). Las cifras de presion arterial, colesterol total y colesterol ligado a lipoproteina de baja densidad fueron similares en ambos sexos, mientras que el colesterol ligado a lipoproteina de alta densidad fue significativamente superior en las mujeres. Ademas, las mujeres fueron mas anosas, presentaban menos habitos toxicos (tabaquismo, alcohol) y mayor indice de masa corporal. La prevalencia de cardiopatia isquemica fue superior en varones (12,6 % frente a 5,7 %). Discusion Se puede observar que los valores lipidicos en esta poblacion muestra cifras elevadas de colesterol ligado a lipoproteina de alta densidad y valores de colesterol total y de colesterol ligado a lipoproteina de baja densidad no excesivamente altos. Se demuestra la presencia muy frecuente de otros factores de riesgo asociados, principalmente hipertension arterial. La distribucion de factores de riesgo muestra diferencias importantes entre sexos, de forma similar a las observadas en poblacion normolipemica. El riesgo coronario medio de esta poblacion calculado por las tablas de Framingham fue moderado- alto, pero casi la mitad de los sujetos (45,1 %) eran de riesgo alto o muy alto, lo que puede tener importantes implicaciones terapeuticas.

New critical region tables for Fisher's exact test

The best-known non-asymptotic method for comparing two independent proportions is Fishers exact text. The usual critical region (CR) tables for this test contain one or more of the following defects:they distinguish between rows and columns; they distinguish between the alternatives H = p1 p2; they assume that the error for the two-tailed test is twice that of the one-tailed test; they do not use the optimal version of the test; they do not give both CRs for one and two tails at the same time. All this results in the unnecessary duplication of the space required for the tables, the construction of tables of low-powered methods, or the need to manipulate two different tables (one for the one-tailed test, the other for the two-tailed test). This paper presents CR tables which have been obtained from the most powerful version of Fishers exact test and which occupy the minimum space possible. The tables, which are valid for one- or two-tailed tests, have levels of significance of 10%, 5% a...

The Equivalence of Two Proportions Revisited

The classic conditional test for checking that the difference between two independent proportions is not null may not be appropriate in many circumstances. Dunnett & Gent (1977) showed that in clinical trials, in studies of drugs, etc, the aim is to prove the practical equality (equivalence) of both proportions. On other occasions the aim may be the opposite: i.e. to prove that the two proportions are substantially different (biologically significant). Both cases are usually solved by two one-sided tests (TOST test). In this article, this procedure is shown to be conservative and two true two-sided tests for each case are proposed.The classic conditional test for checking that the difference between two independent proportions is not null may not be appropriate in many circumstances. Dunnett & Gent (1977) showed that in clinical trials, in studies of drugs, etc, the aim is to prove the practical equality (equivalence) of both proportions. On other occasions the aim may be the opposite: i.e. to prove that the two proportions are substantially different (biologically significant). Both cases are usually solved by two one-sided tests (TOST test). In this article, this procedure is shown to be conservative and two true two-sided tests for each case are proposed.

Efectividad antihipertensiva y tolerabilidad de lercanidipino en diabéticos, en pacientes con hipertensión sistólica aislada y en ancianos. Análisis de subgrupos del estudio ELYPSE

El estudio ELYPSE, recientemente publicado, es un estudio de farmacovigilancia que analizo la eficacia y tolerabilidad de lercanidipino (10 mg una vez al dia) en 9.059 pacientes (61± 11 anos; 58 % mujeres) con hipertension grado 1 o 2, durante 3 meses de seguimiento. Este estudio demostro que lercanidipino es un farmaco antihipertensivo eficaz y bien tolerado en la practica clinica diaria, confirmando los datos observados previamente en ensayos clinicos. Ante los resultados del ELYPSE nos planteamos realizar un analisis de subgrupos especiales de pacientes, con el fin de determinar si los buenos resultados observados con el farmaco se demostraban tambien en poblaciones de alto riesgo. Analizamos tres subgrupos considerados de mayor riesgo cardiovascular: los diabeticos (n= 1.269), los pacientes con hipertension sistolica aislada (HSA) (n = 1.024) y los ancianos (> 65 anos) (n = 3.533). Comparamos la efectividad y tolerabilidad de lercanidipino en estos grupos con el resto de los pacientes del estudio (diabeticos frente a no diabeticos, HSA frente a no HSA y >65 anos frente a

One-sided asymptotic inferences for a proportion

ABSTRACT Two-sided asymptotic confidence intervals for an unknown proportion p have been the subject of a great deal of literature. Surprisingly, there are very few papers devoted, like this article, to the case of one tail, despite its great importance in practice and the fact that its behavior is usually different from that of the case with two tails. This paper evaluates 47 methods and concludes that (1) the optimal method is the classic Wilson method with a correction for continuity and (2) a simpler option, almost as good as the first, is the new adjusted Wald method (Walds classic method applied to the data increased in the values proposed by Borkowf: adding a single imaginary failure or success).