The Performance of Goodness of Fit Tests for Logistic Regression with Discrete Covariates

This paper considers the distribution of previously proposed goodness of fit tests when some or all of the covariates are dichotomous variables. The simulations show that of the statistics suggested for testing fit only one appears suitable for use with discrete covariates. This statistic uses conditional maximum likelihood estimates and groups the estimated probabilities into groups of equal size or into groups based on the patterns of the covariates when these are few in number.     

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.     

Unbalanced ranked set sampling for estimating a population proportion

The application of ranked set sampling (RSS) techniques to data from a dichotomous population is currently an active research topic, and it has been shown that balanced RSS leads to improvement in precision over simple random sampling (SRS) for estimation of a population proportion. Balanced RSS, however, is not in general optimal in terms of variance reduction for this setting. The objective of this article is to investigate the application of unbalanced RSS in estimation of a population proportion under perfect ranking, where the probabilities of success for the order statistics are functions of the underlying population proportion. In particular, the Neyman allocation, which assigns sample units for each order statistic proportionally to its standard deviation, is shown to be optimal in the sense that it leads to minimum variance within the class of RSS estimators that are simple averages of the means of the order statistics. We also use a substantial data set, the National Health and Nutrition Examination Survey III (NHANES III) data, to demonstrate the feasibility and benefits of Neyman allocation in RSS for binary variables.     

Testing for familial correlation in age-at-onset

The analysis of family-study data sometimes focuses on whether a dichotomous trait tends to cluster in families. For traits with variable age-at-onset, it may be of interest to investigate whether age-at-onset itself also exhibits familial clustering. A complication in such investigations is that censoring by age-at-ascertainment can induce artifactual familial correlation in the age-at-onset of affected members. A further complication can be that sample inclusion criteria involve the affection status of family members. The purpose here is to present an approach to testing for correlation that is not confounded by censoring by age-at-ascertainment and may be applied with a broad range of inclusion criteria. The approach involves regression statistics in which subjects's covariate terms are chosen to reflect age-at-onset information from the subjects's affected family members. The results of analyses of data from a family-study of panic disorder illustrate the approach.     

Bryophytes for Beginners: The usability of a printed dichotomous key versus a multi-access computer-based key for bryophyte identification

Bryophytes are a rewarding study group in field biology and the UK bryophyte flora has international importance to biodiversity conservation. We designed an identification key to common woodland moss species and compared the usability of two formats, web-based multi-access and printed dichotomous key, with undergraduate students. The rate of correct species identification and identification speed both showed an advantage for the printed dichotomous key. Our findings suggest that, even in the digital age, printed keys remain valuable in biological education and that quality of key design is more important than presentation medium. We discuss the relative advantages of multi-access and dichotomous keys and how to approach bryophyte identification with beginners.     

Notes on Estimation of the General Odds Ratio and the General Risk Difference for Paired‐Sample Data

Under the matched‐pair design, this paper discusses estimation of the general odds ratio OR_G for ordinal exposure in case‐control studies and the general risk difference RD_G for ordinal outcomes in cross‐sectional or cohort studies. To illustrate the practical usefulness of interval estimators of OR_G and RD_G developed here, this paper uses the data from a case‐control study investigating the effect of the number of beverages drunk at “burning hot” temperature on the risk of possessing esophageal cancer, and the data from a cross‐sectional study comparing the grade distributions of unaided distance vision between two eyes. Finally, this paper notes that using the commonly‐used statistics related to odds ratio for dichotomous data by collapsing the ordinal exposure into two categories: the exposure versus the non‐exposure, tends to be less efficient than using the statistics related to OR_G proposed herein.     

Underestimated effect sizes in GWAS: fundamental limitations of single SNP analysis for dichotomous phenotypes

Complex diseases are often highly heritable. However, for many complex traits only a small proportion of the heritability can be explained by observed genetic variants in traditional genome-wide association (GWA) studies. Moreover, for some of those traits few significant SNPs have been identified. Single SNP association methods test for association at a single SNP, ignoring the effect of other SNPs. We show using a simple multi-locus odds model of complex disease that moderate to large effect sizes of causal variants may be estimated as relatively small effect sizes in single SNP association testing. This underestimation effect is most severe for diseases influenced by numerous risk variants. We relate the underestimation effect to the concept of non-collapsibility found in the statistics literature. As described, continuous phenotypes generated with linear genetic models are not affected by this underestimation effect. Since many GWA studies apply single SNP analysis to dichotomous phenotypes, previously reported results potentially underestimate true effect sizes, thereby impeding identification of true effect SNPs. Therefore, when a multi-locus model of disease risk is assumed, a multi SNP analysis may be more appropriate.     

Adaptive Combination of P-Values for Family-Based Association Testing with Sequence Data

Family-based study design will play a key role in identifying rare causal variants, because rare causal variants can be enriched in families with multiple affected subjects. Furthermore, different from population-based studies, family studies are robust to bias induced by population substructure. It is well known that rare causal variants are difficult to detect from single-locus tests. Therefore, burden tests and non-burden tests have been developed, by combining signals of multiple variants in a chromosomal region or a functional unit. This inevitably incorporates some neutral variants into the test statistics, which can dilute the power of statistical methods. To guard against the noise caused by neutral variants, we here propose an ‘adaptive combination of P-values method’ (abbreviated as ‘ADA’). This method combines per-site P-values of variants that are more likely to be causal. Variants with large P-values (which are more likely to be neutral variants) are discarded from the combined statistic. In addition to performing extensive simulation studies, we applied these tests to the Genetic Analysis Workshop 17 data sets, where real sequence data were generated according to the 1000 Genomes Project. Compared with some existing methods, ADA is more robust to the inclusion of neutral variants. This is a merit especially when dichotomous traits are analyzed. However, there are some limitations for ADA. First, it is more computationally intensive. Second, pedigree structures and founders'' sequence data are required for the permutation procedure. Third, unrelated controls cannot be included. We here show that, for family-based studies, the application of ADA is limited to dichotomous trait analyses with full pedigree information.     

Validation of a Method for Quantifying Male Mating Preferences in the Guppy (Poecilia reticulata)

Male and female mating preferences are commonly inferred from association times spent with potential mates in a dichotomous‐choice test. However, this assessment method is rarely validated, particularly so for male mating preferences. Using the Trinidadian guppy (Poecilia reticulata), an important model species in the study of sexual selection, we tested whether a male’s mating preference for either of two stimulus females in a dichotomous‐choice test predicted his mating behaviours directed at the preferred female when he was allowed to swim freely with both females. First, we presented individual males with two females that differed in body length in a dichotomous‐choice apparatus in which the male could only use visual cues to assess the paired females. We quantified male mating preference as the duration of time a focal male spent associating with each female. Immediately following this test, the focal male was allowed to swim freely with both females, and we quantified the time he spent sexually pursuing each female and the number of courtship sigmoid displays and copulation attempts he directed at each female. On average, males did not significantly prefer either of the two stimulus females in either of the two tests; however, the magnitude of male preference for the larger female tended to increase as the size difference between the paired females increased. More importantly, there was a significant positive relationship between male association time in the dichotomous‐choice test and both the time spent sexually pursuing and the number of courtship sigmoid displays directed at the same female initially preferred in the dichotomous‐choice test. Collectively, these results confirm that association time measured in a dichotomous‐choice test is a reliable predictor of male mating preferences in the Trinidadian guppy.     

Type I error and the power of the s-test: old lessons from a new,analytically justified statistical test for phylogenies

We present a new procedure for assessing the statistical significance of the most likely unrooted dichotomous topology inferrable from four DNA sequences. The procedure calculates directly a P-value for the support given to this topology by the informative sites congruent with it, assuming the most likely star topology as the null hypothesis. Informative sites are crucial in the determination of the maximum likelihood dichotomous topology and are therefore an obvious target for a statistical test of phylogenies. Our P-value is the probability of producing through parallel substitutions on the branches of the star topology at least as much support as that given to the maximum likelihood dichotomous topology by the aforementioned informative sites, for any of the three possible dichotomous topologies. The degree of statistical significance is simply the complement of this P-value. Ours is therefore an a posteriori testing approach, in which no dichotomous topology is specified in advance. We implement the test for the case in which all sites behave identically and the substitution model has a single parameter. Under these conditions, the P-value can be easily calculated on the basis of the probabilities of change on the branches of the most likely star topology, because under these assumptions, each site can become informative independently from every other site; accordingly, the total number of informative sites of each kind is binomially distributed. We explore the test's type I error by applying it to data produced in star topologies having all branches equally long, or having two short and two long branches, and various degrees of homoplasy. The test is conservative but we demonstrate, by means of a discreteness correction and progressively assumption-free calculations of the P-values, that (1) the conservativeness is mostly due to the discrete nature of informative sites and (2) the P-values calculated empirically are moreover mostly quite accurate in absolute terms. Applying the test to data produced in dichotomous topologies with increasing internal branch length shows that, despite the test's "conservativeness," its power is much higher than that of the bootstrap, especially when the relevant informative sites are few.     

GOLDmineR: improving models for classifying patients with chest pain

The laboratory is dealing with reporting tests as information needed to make clinical decisions. The traditional statistical quality control measures which assigns reference ranges based on 95 percent confidence intervals is insufficient for diagnostic tests that assign risk. We construct a basis for risk assignment by a method that builds on the 2 x 2 contingency table used to calculate the C2 goodness-of-fit and Bayesian estimates. The widely used logistic regression is a subset of the regression method, as it only considers dichotomous outcome choices. We use examples of multivalued predictor(s) and a multivalued as well as dichotomous outcome. Outcomes analyses are quite easy using the ordinal logit regression model.     

Segregation analysis of restless legs syndrome: possible evidence for a major gene in a family study using blinded diagnoses

OBJECTIVE: The objective of this study was to ascertain the most likely inheritance pattern of restless legs syndrome (RLS) using segregation analysis. METHODS: Probands were RLS patients presenting to the Neurology and Sleep clinics of the Johns-Hopkins Bayview medical center with willing first and second degree relatives. Blinded diagnosis was made in those who exhibited the four diagnostic features of RLS. Analysis was performed on RLS as a dichotomous trait and considering age of onset models on 590 phenotyped subjects from 77 pedigrees. RESULTS: All non-genetic models were rejected considering RLS as a dichotomous trait. A single locus Mendelian dominant model with gender as a covariate had best fit with allele frequency of 0.077 and complete penetrance. RLS frequency in non-carrier subjects was estimated to be 0.14. Two underlying distributions of age of onset, with a possible dichotomy at 26.3 years, were identified. Contrary to the results for RLS as a dichotomous trait, age of onset models did not indicate single major gene inheritance. CONCLUSION: This segregation analysis suggests that the pattern of segregation is consistent with that of a single major locus, when RLS is treated as a dichotomous trait without considering age of onset. The high rate of phenocopies matches known population frequencies and taken with significant residual familial effects and the lack of evidence for a major gene controlling age of onset, indicates that non-genetic causes of RLS may exist and RLS is a complex disorder.     

Diagnostic schemes for fine needle aspirates of breast masses

A comparison was made of four statistically based schemes for classifying epithelial cells from 243 fine needle aspirates of breast masses as benign or malignant. Two schemes were computer-generated decision trees and two were user generated. Eleven cytologic characteristics described in the literature as being useful in distinguishing benign from malignant breast aspirates were assessed on a scale of 1 to 10, with 1 being closest to that described as benign and 10 to that described as malignant. The original computer-generated dichotomous decision tree gave 6 false negatives and 12 false positives on the data set; another tree generated from the current data improved performance slightly, with 5 false negatives and 10 false positives. Maximum diagnostic overlap occurred at the cut-point of the original dichotomous tree. The insertion of a third node evaluating additional parameters resulted in one false negative and seven false positives. This performance was matched by summing the scores of the eight characteristics that individually were most effective in separating benign from malignant. We conclude that, while statistically designed, computer-generated dichotomous decision trees identify a starting sequence for applying cytologic characteristics to distinguish between benign and malignant breast aspirates, modifications based on human expert knowledge may result in schemes that improve diagnostic performance.     

Genetic linkage analysis of a dichotomous trait incorporating a tightly linked quantitative trait in affected sib pairs

Many complex diseases are usually considered as dichotomous traits but are also associated with quantitative biological markers or quantitative risk factors. For such dichotomous traits, although their associated quantitative traits may not directly underly the diagnosis of the disease status, if the associated quantitative trait is also linked to the chromosomal regions linked to the dichotomous trait, then joint analysis of dichotomous and quantitative traits should be more efficient than consideration of them separately. Previous studies have focused on the situation when a dichotomous trait can be modeled by a threshold process acting on a single underlying normal liability distribution. However, for many complex disorders, including most psychiatric disorders, diagnosis is generally based on a set of binary or discrete criteria. These traits cannot be modeled on the basis of a threshold process acting on an underlying continuous trait. We propose a likelihood-based method that efficiently combines such a discrete trait and an associated quantitative trait in the analysis, using affected-sib-pair data. Our simulation studies suggest that joint analysis increases the power to detect linkage of dichotomous traits. We also apply the proposed new method to an asthma genome-scan data set and incorporate the total serum immunoglobulin E level in the analysis.     

Mnemonics are an Effective Tool for Adult Beginners Learning Plant Identification

Most beginners are introduced to plant diversity through identification keys, which develop differentiation skills but not species memorisation. We propose that mnemonics, memorable ‘name clues’ linking a species name with morphological characters, are a complementary learning tool for promoting species memorisation. In the first of two experiments, 64 adults in a group-learning environment were taught species identification using mnemonics, an educational card game and a text-based dichotomous key. In the second experiment, 43 adults in a self-directed learning environment were taught species identification using mnemonics and a pictorial dichotomous key. In both experiments, mnemonics produced the highest retention rates of species identification based on vegetative characters. The educational value of these findings is discussed for vegetative plant identification and broader applications.     

Variance Component Estimation for Dichotomous Characters and Its Use for Estimating Heritability

By transforming dichotomous characters (for instance the presence or absence of a certain disease) to zero-one characters it is possible to estimate the variance and variance components. This paper explains the peculiarities occurring in the linear model and estimation functions and the application of the method for heritability estimation.     

A Note on Heritability of a Dichotomous Trait

In a random effects model for a dichotomous trait, we prove that the heritability of the dichotomous trait given in Dempster & Lerner (1950) is strictly smaller than the intraclass correlation. A numerical comparison is given.     

Complexity and power in case-control association studies

A general method is described for estimation of the power and sample size of studies relating a dichotomous phenotype to multiple interacting loci and environmental covariates. Either a simple case-control design or more complex stratified sampling may be used. The method can be used to design individual studies, to evaluate the power of alternative test statistics for complex traits, and to examine general questions of study design through explicit scenarios. The method is used here to study how the power of association tests is affected by problems of allelic heterogeneity and to investigate the potential role for collective testing of sets of related candidate genes in the presence of locus heterogeneity. The results indicate that allele-discovery efforts are crucial and that omnibus tests or collective testing of alleles can be substantially more powerful than separate testing of individual allelic variants. Joint testing of multiple candidate loci can also dramatically improve power, despite model misspecification and inclusion of irrelevant loci, but requires an a priori hypothesis defining the set of loci to investigate.     

A scaling model for dichotomous branching processes

We derive an affine scaling law for dichotomous branching processes in biological systems. An application to the human bronchial tree demonstrates good agreement with experimental results.