Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals

We find the percentage points of the likelihood ratio test of the null hypothesis that a sample of n observations is from a normal distribution with unknown mean and variance against the alternative that the sample is from a mixture of two distinct normal distributions, each with unknown mean and unknown (but equal) variance. The mixing proportion pi is also unknown under the alternative hypothesis. For 2,500 samples of sizes n = 15, 20, 25, 40, 50, 70, 75, 80, 100, 150, 250, 500, and 1,000, we calculated the likelihood ratio statistic, and from these values estimated the percentage points of the null distributions. Our algorithm for the calculation of the maximum likelihood estimates of the unknown parameters included precautions against convergence of the maximization algorithm to a local rather than global maximum. Investigations for convergence to an asymptotic distribution indicated that convergence was very slow and that stability was not apparent for samples as large as 1,000. Comparisons of the percentage points to the commonly assumed chi-squared distribution with 2 degrees of freedom indicated that this assumption is too liberal; i.e., one's P-value is greater than that indicated by chi 2(2). We conclude then that one would need what is usually an unfeasibly large sample size (n greater than 1,000) for the use of large-sample approximations to be justified.     

Cassava (Manihot esculenta) leaf and tuber concentrate in diets for broiler chickens

Experiment 1 was conducted to determine the nutritive quality of cassava tuber leaf concentrate prepared either by mixing cassava tubers and leaves together before grinding or grinding them separately before mixing. Three mixing proportions of 50:50, 60:40 and 80:20 were used. Samples were subjected to either sundrying or oven drying. Physical examination of samples, proximate analysis and cyanide content of each of the samples were determined. In experiment 2 the sample of highest nutritive value was selected for inclusion in four diets at levels of 0%, 10%, 20% and 30% respectively. A total of 120 day-old broilers were divided into four groups and each group further into three subgroups. The four experimental diets were fed to each of the groups for a period of 9 weeks. Growth parameters, carcass characteristics and haematological parameters were also determined. Results showed that in experiment 1 the grinding together of cassava tubers and leaves in the proportion of 50:50 before sun-drying gave the best texture, highest CP content with low HCN content. Inclusion of 10% cassava concentrate gave good performance in terms of growth and feed conversion, with no detrimental effects on haematological parameters and carcass characteristics.     

The prevalence of negative studies with inadequate statistical power: an analysis of the plastic surgery literature.

Studies published in the medical literature often neglect to consider the statistical power needed to detect a meaningful difference between study groups. Small sample sizes tend to produce negative results because of low statistical power. Studies that cannot make conclusive statements about their hypotheses can waste resources, deter further research, and impede advances in clinical treatment. The current study reviewed three of the most frequently read plastic surgery journals from 1976 to 1996 to determine the prevalence of inadequately (<80 percent) powered clinical trials and experimental studies that found no difference (negative studies) in the response variable of interest between comparison groups. The statistical power of 54 negative studies using continuous response variables was calculated to detect a difference of 1 SD (+/-1 SD) in means between the comparative groups. The power of another 57 negative studies with dichotomous response (yes/no) variables was calculated to detect a relative change in proportions of 25 percent and 50 percent from the experimental to the control group. It was found that 85 percent of the studies with continuous response variables had inadequate power to detect the desired mean difference of +/-1 SD. In studies with dichotomous response variables, 98 percent had inadequate power to detect a desired 25 percent relative change in proportions, and 74 percent had inadequate power to detect a desired 50 percent relative change in proportions. These results indicate that many of the studies in the plastic surgery literature lack adequate power to detect a moderate-to-large difference between groups. The lack of power makes the interpretation of the studies with negative findings inconclusive. Proper study design dictates that investigators consider a priori the difference between groups that is of clinical interest, and the sample size per group that is needed to provide adequate statistical power to detect the desired difference.     

Uncertainty in source partitioning using stable isotopes

Stable isotope analyses are often used to quantify the contribution of multiple sources to a mixture, such as proportions of food sources in an animal's diet, or C₃ and C₄ plant inputs to soil organic carbon. Linear mixing models can be used to partition two sources with a single isotopic signature (e.g., '¹³C) or three sources with a second isotopic signature (e.g., '¹⁵N). Although variability of source and mixture signatures is often reported, confidence interval calculations for source proportions typically use only the mixture variability. We provide examples showing that omission of source variability can lead to underestimation of the variability of source proportion estimates. For both two- and three-source mixing models, we present formulas for calculating variances, standard errors (SE), and confidence intervals for source proportion estimates that account for the observed variability in the isotopic signatures for the sources as well as the mixture. We then performed sensitivity analyses to assess the relative importance of: (1) the isotopic signature difference between the sources, (2) isotopic signature standard deviations (SD) in the source and mixture populations, (3) sample size, (4) analytical SD, and (5) the evenness of the source proportions, for determining the variability (SE) of source proportion estimates. The proportion SEs varied inversely with the signature difference between sources, so doubling the source difference from 2‰ to 4‰ reduced the SEs by half. Source and mixture signature SDs had a substantial linear effect on source proportion SEs. However, the population variability of the sources and the mixture are fixed and the sampling error component can be changed only by increasing sample size. Source proportion SEs varied inversely with the square root of sample size, so an increase from 1 to 4 samples per population cut the SE in half. Analytical SD had little effect over the range examined since it was generally substantially smaller than the population SDs. Proportion SEs were minimized when sources were evenly divided, but increased only slightly as the proportions varied. The variance formulas provided will enable quantification of the precision of source proportion estimates. Graphs are provided to allow rapid assessment of possible combinations of source differences and source and mixture population SDs that will allow source proportion estimates with desired precision. In addition, an Excel spreadsheet to perform the calculations for the source proportions and their variances, SEs, and 95% confidence intervals for the two-source and three-source mixing models can be accessed at http://www.epa.gov/wed/pages/models.htm.     

Peak power, ground reaction forces, and velocity during the squat exercise performed at different loads

This study examined the changes in peak power, ground reaction force and velocity with different loads during the performance of the parallel squat movement. Twelve experienced male lifters (26.83 +/- 4.67 years of age) performed the standard parallel squat, using loads equal to 20, 30, 40, 50, 60, 70, 80, and 90% of 1 repetition maximum (1RM). Each subject performed all parallel squats with as much explosiveness as possible using his own technique. Peak power (PP), peak ground reaction force (PGRF), peak barbell velocity (PV), force at the time of PP (FPP), and velocity at the time of PP (VPP) were determined from force, velocity, and power curves calculated using barbell velocity and ground reaction force data. No significant differences were detected among loads for PP; however, the greatest PP values were associated with loads of 40 and 50% of 1RM. Higher loads produced greater PGRF and FPP values than lower loads (p < 0.05) in all cases except between loads equal to 60-50, 50-40, and 40-30% of 1RM for PGRF, and between loads equal to 70-60 and 60-50% of 1RM for FPP. Higher loads produced lower PV and VPP values than lower loads (p < 0.05) in all cases except between the 20-30, 70-80, and 80-90% of 1RM conditions. These results may be helpful in determining loads when prescribing need-specific training protocols targeting different areas of the load-velocity continuum.     

Applications of the Wei-Lachin Multivariate One-Sided Test for Multiple Outcomes on Possibly Different Scales

Many studies aim to assess whether a therapy has a beneficial effect on multiple outcomes simultaneously relative to a control. Often the joint null hypothesis of no difference for the set of outcomes is tested using separate tests with a correction for multiple tests, or using a multivariate T ²-like MANOVA or global test. However, a more powerful test in this case is a multivariate one-sided or one-directional test directed at detecting a simultaneous beneficial treatment effect on each outcome, though not necessarily of the same magnitude. The Wei-Lachin test is a simple 1 df test obtained from a simple sum of the component statistics that was originally described in the context of a multivariate rank analysis. Under mild conditions this test provides a maximin efficient test of the null hypothesis of no difference between treatment groups for all outcomes versus the alternative hypothesis that the experimental treatment is better than control for some or all of the component outcomes, and not worse for any. Herein applications are described to a simultaneous test for multiple differences in means, proportions or life-times, and combinations thereof, all on potentially different scales. The evaluation of sample size and power for such analyses is also described. For a test of means of two outcomes with a common unit variance and correlation 0.5, the sample size needed to provide 90% power for two separate one-sided tests at the 0.025 level is 64% greater than that needed for the single Wei-Lachin multivariate one-directional test at the 0.05 level. Thus, a Wei-Lachin test with these operating characteristics is 39% more efficient than two separate tests. Likewise, compared to a T ²-like omnibus test on 2 df, the Wei-Lachin test is 32% more efficient. An example is provided in which the Wei-Lachin test of multiple components has superior power to a test of a composite outcome.     

Removal of Acid Hematin-Type Pigments from Sections; Efficacy of Alcoholic Picric Acid Solution

Paraffin sections from tissues fixed in nonneutralized 10% formalin or Carnoy's fluid were treated with saturated solutions of picric acid in each of the following solvents: water, 80%, 90%, 95% and absolute ethanol, and with half-saturated solutions in 95% ethanol for 5 min. Tests with the supernatant fluid 30-60 min after mixing the solid and the solvent showed that adequate concentration had not yet occurred; hence more time is needed, with 15 hr recommended. Only saturated solutions of picric acid in absolute and 95% ethanol removed all acid hematin-type pigment. The effectiveness of saturated solutions decreased with decreasing alcohol concentration.     

Comparison of SNPs and microsatellites for fine-scale application of genetic stock identification of Chinook salmon in the Columbia River Basin

Genetic stock identification (GSI) is an important tool in fisheries management. Microsatellites (μSATs) have been the dominant genetic marker for GSI; however, increasing availability and numerous advantages of single-nucleotide polymorphism (SNP) markers make them an appealing alternative. We tested performance of 13 μSAT vs. 92 SNP loci in a fine-scale application of GSI, using a new baseline for Chinook salmon consisting of 49 collections (n = 4014) distributed across the Columbia River Basin. In GSI, baseline genotypes for both marker sets were used independently to analyse a real fishery mixture (n = 2731) representing the total run of Chinook salmon passing Bonneville Dam in the Columbia River. Marker sets were evaluated using three criteria: (i) ability to differentiate reporting groups, (ii) proportion of correct assignment in mixture simulation tests and baseline leave-one-out analyses and (iii) individual assignment and confidence intervals around estimated stock proportions of a real fishery mixture. The μSATs outperformed the SNPs in resolving fine-scale relationships, but all 105 markers combined provided greatest power for GSI. SNPs were ranked by relative information content based on both an iterative procedure that optimized correct assignment to the baseline and ranking by minor allele frequency. For both methods, we identified a subset of the top 50 ranked loci, which were similar in assignment accuracy, and both reached maximum available power of the total 92 SNP loci (correct assignment = 73%). Our estimates indicate that between 100 and 200 highly informative SNP loci are required to meet management standards (correct assignment > 90%) for resolving stocks in finer-scale GSI applications.     

Staining and Mounting Helminths

The following procedure is recommended: Fix ces-todes and trematodes (while held flat between glass slides) 0.5-2.0 hr. in the following mixture: formalin, 15; acetic acid (gl.), 5; glycerol, 10; 95% ethyl alcohol, 24; distilled H₂O, 46; all proportions by volume. After freeing them from the slides, wash thoroughly in running water and stain immediately thereafter. Stock staining solution: ferric ammonium alum (violet cryst.), 2 g.; distilled H₂O (cold) 100 ml.; after solution, add 2 ml. concentrated H₂SO₄, bring to a boil; add 1 g. coelestin blue B (Nat. Aniline), boil 3-5 min.; cool and add 10 ml. absolute methyl alcohol and 10 ml. glycerol. Dilute 1 vol. with 3 vol. distilled H20 for use. Stain 5-30 min., depending on size of specimens. Wash with 2 changes 0.5 hr. each of distilled H₂O, then 50% isopropyl alcohol 12-16 hr., 50% isopropyl alcohol 2 hr., followed by graded isopropyl alcohol for dehydration. Ether: ethyl alcohol (equal parts), 1 hr., is followed by embedding in celloidin in a sheet just thick enough to cover the specimens. Trim embedded specimens and dehydrate with isopropyl alcohol, 80%, 90% and absolute. Clear in beechwood creosote. Mount in balsam with cover glasses that overlap the edges of the celloidin 1-2 mm. While drying at 37°C, refill edges of mount with fresh balsam as needed. When dry, remove excess balsam and ring the edges with ordinary gloss enamel paint.     

Using lod scores to detect sex differences in male-female recombination fractions

Human recombination fraction (RF) can differ between males and females, but investigators do not always know which disease genes are located in genomic areas of large RF sex differences. Knowledge of RF sex differences contributes to our understanding of basic biology and can increase the power of a linkage study, improve gene localization, and provide clues to possible imprinting. One way to detect these differences is to use lod scores. In this study we focused on detecting RF sex differences and answered the following questions, in both phase-known and phase-unknown matings: (1) How large a sample size is needed to detect a RF sex difference? (2) What are "optimal" proportions of paternally vs. maternally informative matings? (3) Does ascertaining nonoptimal proportions of paternally or maternally informative matings lead to ascertainment bias? Our results were as follows: (1) We calculated expected lod scores (ELODs) under two different conditions: "unconstrained," allowing sex-specific RF parameters (theta(female), theta(male)); and "constrained," requiring theta(female) = theta(male). We then examined the DeltaELOD (identical with difference between maximized constrained and unconstrained ELODs) and calculated minimum sample sizes required to achieve statistically significant DeltaELODs. For large RF sex differences, samples as small as 10 to 20 fully informative matings can achieve statistical significance. We give general sample size guidelines for detecting RF differences in informative phase-known and phase-unknown matings. (2) We defined p as the proportion of paternally informative matings in the dataset; and the optimal proportion p(circ) as that value of p that maximizes DeltaELOD. We determined that, surprisingly, p(circ) does not necessarily equal (1/2), although it does fall between approximately 0.4 and 0.6 in most situations. (3) We showed that if p in a sample deviates from its optimal value, no bias is introduced (asymptotically) to the maximum likelihood estimates of theta(female) and theta(male), even though ELOD is reduced (see point 2). This fact is important because often investigators cannot control the proportions of paternally and maternally informative families. In conclusion, it is possible to reliably detect sex differences in recombination fraction.     

Vitrification of mouse oocytes and embryos at various stages of development in an ethylene glycol-based solution by a simple method

Mouse oocytes and embryos at various developmental stages were exposed directly to an ethylene glycol-based vitrification solution (EFS) for 2 or 5 minutes at 20 degrees C. They were then vitrified at -196 degrees C and were warmed rapidly. At the germinal vesicle stage, the proportion of morphologically normal oocytes was 36 to 39% if they had cumulus cells, whereas in cumulus-removed immature oocytes and in ovulated oocytes it was only 2 to 4%. This low survival was attributed to the harmful action of ethylene glycol. After fertilization, on the other hand, the post-warming survival rate of 1-cell zygotes, as assessed by cleavage to the 2-cell stage, increased markedly (62%). As the developmental stage proceeded, higher proportions of vitrified embryos developed to expanded blastocysts; the rates increased up to 77 and 80% in 2-cell and 4-cell embryos, respectively. For embryos at the 8-cell, morula and early blastocyst stages, the proportion of embryos developed after vitrification (90 to 95%) was not significantly different from that of the untreated embryos (95 to 100%) when the period of exposure to EFS solution was 2 minutes. As the blastocoel began to enlarge, however, survival began to decrease again, with rates of 79 and 57% in blastocysts and expanded blastocysts, respectively. After the cryopreserved 2-cell, 4-cell and 8-cell embryos as well as morulae and blastocysts were transferred to recipients, 43 to 57% of the recipients became pregnant, and 48 to 60% of these various stage embryos developed into live young.     

THE POWER OF THE " A" -" NOT A" METHOD

The " A" - " Not A" method is a rating method with two categories. It is often treated as a discrimination method. Unlike forced choice procedures, the Thurstonian model for this method involves a choice criterion. In statistical tests, it is treated as a comparison of two proportions. In this paper, the power for hypothesis tests involving the monadic and replicated monadic " A" - " Not A" method is discussed. The power functions and the sample sizes needed for 80% power are given based on Thurstone's δ. Designs with equal and unequal allocations for A and A (Not A) samples are considered. The power of the method is also compared with that of four forced choice methods under the assumption that the perceptual variance is identical among methods. The comparison shows that, in general, the power for the five methods ranks from high to low: the 3-AFC, 2-AFC, " A" - " Not A", triangular and duo-trio. The comparison also shows that, based on the same number of panelists and/or the same sample size for the A and A⁻ samples for the methods, if the panelists are not too discrepant and the choice criterion in the " A" - " Not A" method is not too strict or too lax, the power of the " A" - " Not A" method is very close to that of the 2-AFC method.     

An Improved Hematoxylin-Eosin Stain for Sections of Marrow Units

The procedure recommended is: Fix “marrow units” (small functional structures of bone marrow) in 10% formol-saline solution for 1-2 hours and dehydrate in 80% alcohol, 95% alcohol and acetone 30 minutes each. Place in fresh 50° and 53°C. paraffin for 30 minutes each. Embed in fresh 53°C. paraffin. Serially section at 5μ thickness and mount with Schleicher's floating solution. Allow to dry for 1 hour in an oven and deparaffinize by passing through xylene I and II, absolute alcohol I and II, and 95% alcohol. Rinse in fresh distilled water and place in dilute Harris' hematoxylin (stock solution 50 ml., distilled water 200 ml.) for 2 to 3 minutes. Rinse well in distilled water and check staining under the microscope. Dip in acid-alcohol 5 times (1 dip to equal about 1 second). Rinse well in weak (0.02%) ammonia water and distilled water. Dip in 2% aqueous phosphotungstic acid about 3 to 5 times (equal to 3-5 seconds). Rinse in fresh distilled water and place in weak ammonia water for 1 minute. Rinse in fresh distilled water I and II. Place in 80% alcohol for 5 minutes and check under the microscope for “blueness” and nuclear differentiation. Place in dilute alcoholic eosin (0.5% alcohol-eosin stock solution 10 parts and 95% alcohol 90 parts) for 1 to 2 minutes. Rinse in 80% alcohol and place for 1 minute in 95% alcohol. Check under the microscope for staining quality. Place in absolute alcohol for 1 minute, alcohol-xylene (equal parts), 10 dips, and xylene I and II. Mount. This hematoxylin-eosin staining schedule brings out minute structural detail of bone marrow tissue heretofore not demonstrable.     

Capacity of large-scale, long-term biodiversity monitoring programmes to detect trends in species prevalence

There is a critical need for monitoring programmes to assess change or trends in species status to inform conservation. A key aspect in developing such programmes is evaluating their statistical power—the ability to detect a real change. Here we examine the capacity of a broad-scale biodiversity monitoring programme in Alberta, Canada to measure changes in species prevalence. Using observed variation in detectability and prevalence for 252 species monitored at 85 sites, we simulated 3% annual declines and evaluated sample size (6 different sizes) and length of monitoring (5 different durations) necessary to detect change with a 90% certainty (power) at an α of 0.1. Our results suggest that after four monitoring cycles (e.g., 20 years for a 5-year cycle) a power of 90% can be expected for 99% of species when monitoring 1,625 sites, 65% of species for 300 sites, 27% of species for 75 sites, and 8% of species for 25 sites. We found that 66% detectability and 50% prevalence were needed to ensure that 3% annual change is detected at 50 sites over a 20-year period. Our results demonstrate that broad-scale monitoring programmes cannot effectively detect trends in all species at all spatial scales. The time period and spatial scale necessary to detect a real change at a specified level needs to be provided to stakeholders to ensure the short-term success of biodiversity monitoring programmes and to ensure that the most robust indicators of biodiversity are selected.     

Staining and Mounting Helminths

The following procedure is recommended: Fix ces-todes and trematodes (while held flat between glass slides) 0.5–2.0 hr. in the following mixture: formalin, 15; acetic acid (gl.), 5; glycerol, 10; 95% ethyl alcohol, 24; distilled H₂O, 46; all proportions by volume. After freeing them from the slides, wash thoroughly in running water and stain immediately thereafter. Stock staining solution: ferric ammonium alum (violet cryst.), 2 g.; distilled H₂O (cold) 100 ml.; after solution, add 2 ml. concentrated H₂SO₄, bring to a boil; add 1 g. coelestin blue B (Nat. Aniline), boil 3–5 min.; cool and add 10 ml. absolute methyl alcohol and 10 ml. glycerol. Dilute 1 vol. with 3 vol. distilled H20 for use. Stain 5–30 min., depending on size of specimens. Wash with 2 changes 0.5 hr. each of distilled H₂O, then 50% isopropyl alcohol 12–16 hr., 50% isopropyl alcohol 2 hr., followed by graded isopropyl alcohol for dehydration. Ether: ethyl alcohol (equal parts), 1 hr., is followed by embedding in celloidin in a sheet just thick enough to cover the specimens. Trim embedded specimens and dehydrate with isopropyl alcohol, 80%, 90% and absolute. Clear in beechwood creosote. Mount in balsam with cover glasses that overlap the edges of the celloidin 1–2 mm. While drying at 37°C, refill edges of mount with fresh balsam as needed. When dry, remove excess balsam and ring the edges with ordinary gloss enamel paint.     

Topographic complexity and the power to detect structural and functional changes in temperate reef fish assemblages: The need for habitat-independent sample sizes

Functional approaches have shown promising results to detect degradation in marine fish assemblages. However, background variability significantly affects the amplitude of change that is detectable by a monitoring plan, and failing to detect such changes can have devastating consequences and carry aggravated recovery costs due to unnoticed degradation. The present study aimed to understand the relationship between topographic complexity in temperate reefs and the power to detect variations in fish-based metrics. Underwater visual census of fish assemblages was performed using strip transects and a Monte Carlo simulation approach was used to generate a large number of replicates and simulate three alternative hypotheses representing different magnitudes of change. Statistical power to detect differences between null and alternative hypotheses was estimated through 10,000 Mann–Whitney tests for numbers of replicates ranging from 2 to 15. Power tended to vary with topographic complexity particularly with small and medium changes in metric values and when using small sample sizes. While power increased with complexity for most metrics, some showed decreasing trends. With a large effect, 5–15 transects were needed, depending on the metric, to stabilise power above 0.80 independently of habitat features. A power of 0.95, however, could not be achieved for most metrics in all sites, even when using 15 transects. The observation that the power to detect degradation and recovery in temperate reef fish seems to vary with habitat features means that a monitoring programme that is correctly planned for a particular area may not be directly applicable in a nearby reef. Adding to the need to maximise power in monitoring, this study highlights the need to take into account habitat variability in these calculations and estimate habitat-independent sample sizes that are appropriate for the scale and location of interest.     

Mixture-model classification in DNA content analysis.

DNA abundance provides important information about cell physiology and proliferation activity. In a typical in vitro cellular assay, the distribution of the DNA content within a sample is comprised of cell debris, G0/G1-, S-, and G2/M-phase cells. In some circumstances, there may be a collection of cells that contain more than two copies of DNA. The primary focus of DNA content analysis is to deconvolute the overlapping mixtures of the cellular components, and subsequently to investigate whether a given treatment has perturbed the mixing proportions of the sample components. We propose a restricted mixture model that is parameterized to incorporate the available biological information. A likelihood ratio (LR) test is developed to test for changes in the mixing proportions between two cell populations. The proposed mixture model is applied to both simulated and real experimental data. The model fitting is compared with unrestricted models; the statistical inference on proportion change is compared between the proposed LR test and the Kolmogorov-Smirnov test, which is frequently used to test for differences in DNA content distribution. The proposed mixture model outperforms the existing approaches in the estimation of the mixing proportions and gives biologically interpretable results; the proposed LR test demonstrates improved sensitivity and specificity for detecting changes in the mixing proportions.     

Growth of ornamental plants in two composts prepared from agroindustrial wastes

Two composts prepared from agroindustrial wastes were assayed as substrates: C1 from brewing waste (yeast and malt) plus lemon tree prunings; and C2 from the solid fraction of olive mill wastewater plus olive leaves. Sixteen substrates were prepared by combining each compost with Sphagnum peat or a commercial substrate (CS) in different proportions. The nutrients (N and K) provided by the composts, which acted as slow-release fertilisers, influenced especially the development of calendula, although the physical and physico-chemical properties such as total pore space and electrical conductivity (EC) were also relevant. On the other hand, in the salt-sensitive calceolaria hybrid, EC and chloride concentration were the main factors influencing growth. Adequate substrates for the development of calendula can be prepared by mixing C1 at up to 75% with peat or at up to 50% with CS, and C2 at up to 50% with peat or CS. For calceolaria, the substrate should have a lower proportion of compost, C1 at up to 50% and C2 at up to 25%, both mixed with peat or CS. Therefore, composts of agroindustrial origin such as these can be used as an alternative to peat and CSs for growing ornamental plants. provided the mixture contains at least 25% peat or CS.     

Homogeneity Test for Correlated Binary Data

In ophthalmologic studies, measurements obtained from both eyes of an individual are often highly correlated. Ignoring the correlation could lead to incorrect inferences. An asymptotic method was proposed by Tang and others (2008) for testing equality of proportions between two groups under Rosner''s model. In this article, we investigate three testing procedures for general g ≥ 2 groups. Our simulation results show the score testing procedure usually produces satisfactory type I error control and has reasonable power. The three test procedures get closer when sample size becomes larger. Examples from ophthalmologic studies are used to illustrate our proposed methods.