On the minimum expected quantity for the validity of the chi-squared test in 2 2 2 tables

A 2 2 2 contingency table can often be analysed in an exact fashion by using Fisher's exact test and in an approximate fashion by using the chi-squared test with Yates' continuity correction, and it is traditionally held that the approximation is valid when the minimum expected quantity E is E S 5. Unfortunately, little research has been carried out into this belief, other than that it is necessary to establish a bound E>E*, that the condition E S 5 may not be the most appropriate (Martín Andrés et al., 1992) and that E* is not a constant, but usually increasing with the growth of the sample size (Martín Andrés & Herranz Tejedor, 1997). In this paper, the authors conduct a theoretical experimental study from which they ascertain that E* value (which is very variable and frequently quite a lot greater than 5) is strongly related to the magnitude of the skewness of the underlying hypergeometric distribution, and that bounding the skewness is equivalent to bounding E (which is the best control procedure). The study enables estimating the expression for the above-mentioned E* (which in turn depends on the number of tails in the test, the alpha error used, the total sample size, and the minimum marginal imbalance) to be estimated. Also the authors show that E* increases generally with the sample size and with the marginal imbalance, although it does reach a maximum. Some general and very conservative validity conditions are E S 35.53 (one-tailed test) and E S 7.45 (two-tailed test) for alpha nominal errors in 1% h f h 10%. The traditional condition E S 5 is only valid when the samples are small and one of the marginals is very balanced; alternatively, the condition E S 5.5 is valid for small samples or a very balanced marginal. Finally, it is proved that the chi-squared test is always valid in tables where both marginals are balanced, and that the maximum skewness permitted is related to the maximum value of the bound E*, to its value for tables with at least one balanced marginal and to the minimum value that those marginals must have (in non-balanced tables) for the chi-squared test to be valid.     

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.     

New critical region tables for Fisher's exact test

The best-known non-asymptotic method for comparing two independent proportions is Fisher's 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 and 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 Fisher's 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% and 1% and values for N (the total size of both samples) of less than or equal to 40. This article shows how to calculate the P value in a specific problem, using the tables as a means of partial checking and as a preliminary step to determining the exact P value.     

The Multisectoral Trilemma of Network Management

This article presents an analysis of different network coordinationstrategies. The article extends network management scholarshipby integrating sector-based differences within a theoreticalframework encompassing extant conceptions of network management.Even as the emergent field of network management scholarshipadvances, current research tends to generalize network managementapproaches based on assumptions that organizations behave similarlywithin a network regardless of whether the organizations aregovernmental, nonprofit, or commercial. Consequently, existingresearch does not fully account for whether sector-based differenceshave implications for network management. This article providesevidence that sector-based differences within a network matterbecause the differences provide strategic opportunities andconstraints for managers involved in coordinating mixed-sectornetworks. This article makes several contributions to networkmanagement scholarship. First, this article provides a frameworkthat reviews and situates current conceptions about networkcoordination within a passive-to-active continuum of managerialapproaches. Sectoral differences are situated and integratedwithin this framework. Second, this article provides an empiricallybased investigation of a quasi-natural experiment that examinessector-based differences in mixed-sector workforce developmentnetworks in Boston. The article's findings suggest that integratingsector-based orientations within a passive-to-active networkmanagerial continuum helps clarify and categorize the strategicoptions and trade-offs that managers may consider in coordinatingmultisectoral networks.     

Erratum to: Child at Risk of Poverty or Social Exclusion: Comparative View Between Spain and Portugal in the European Context

The aim of this study is to investigate the relations between identity statuses and the perceived quality of life in adolescents aged 16–19. The research methods include the Ego Identity Process Questionnaire to assess identity status of a sample covering 233 participants (148 girls, 85 boys), and the Youth Quality of Life Instrument to assess their subjective quality of life. Diffused identity is linked to the lowest level of subjective quality of life, whereas foreclosed identity to the highest. Five patterns of the connection between identity dimensions and perceived quality of life have been distinguished through cluster analysis. The results indicate that different patterns of identity processes in adolescents coexist with different levels of quality of life.     

One-sided asymptotic inferences for a proportion

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 (Wald's classic method applied to the data increased in the values proposed by Borkowf: adding a single imaginary failure or success).     

Optimal correction for continuity in the chi-squared test in 2×2 tables

The most common asymptotic procedure for analyzing a 2 × 2 table (under the conditioning principle) is the ‰ chi-squared test with correction for continuity (c.f.c). According to the way this is applied, up to the present four methods have been obtained: one for one-tailed tests (Yates') and three for two-tailed tests (those of Mantel, Conover and Haber). In this paper two further methods are defined (one for each case), the 6 resulting methods are grouped in families, their individual behaviour studied and the optimal is selected. The conclusions are established on the assumption that the method studied is applied indiscriminately (without being subjected to validity conditions), and taking a basis of 400,000 tables (with the values of sample size n between 20 and 300 and exact P-values between 1% and 10%) and a criterion of evaluation based on the percentage of times in which the approximate P-value differs from the exact (Fisher's exact test) by an excessive amount. The optimal c.f.c. depends on n, on E (the minimum quantity expected) and on the error α to be used, but the rule of selection is not complicated and the new methods proposed are frequently selected. In the paper we also study what occurs when E ≥ 5, as well as whether the chi-squared by factor (n-1).