Estimators and tests in the analysis of multiple nonindependent 2 x 2 tables with partially missing observations

We present methods for the analysis of a K-variate binary measure for two independent groups where some observations may be incomplete, as in the case of K repeated measures in a comparative trial. For the K 2 X 2 tables, let theta = (theta 1,..., theta K) be a vector of association parameters where theta k is a measure of association that is a continuous function of the probabilities pi ik in each group (i = 1, 2; k = 1,..., K), such as the log odds ratio or log relative risk. The asymptotic distribution of the estimates theta = (theta 1,..., theta K) is derived. Under the assumption that theta k = theta for all k, we describe the maximally efficient linear estimator theta of the common parameter theta. Tests of contrasts on the theta are presented which provide a test of homogeneity Ha: theta k = theta l for all k not equal to l. We then present maximally efficient tests of aggregate association Hb: theta = theta 0, where theta 0 is a given value. It is shown that the test of aggregate association Hb is asymptotically independent of the preliminary test of homogeneity Ha. These methods generalize the efficient estimators of Gart (1962, Biometrics 18, 601-610), and the Cochran (1954, Biometrics 10, 417-451), Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748), and Radhakrishna (1965, Biometrics 21, 86-98) tests to nonindependent tables. The methods are illustrated with an analysis of repeated morphologic evaluations of liver biopsies obtained in the National Cooperative Gallstone Study.     

The large-sample relative efficiency of the Mantel-Haenszel estimator in the fixed-strata case

The large-sample relative efficiency of the Mantel-Haenszel estimator psi MH of an odds ratio is investigated for the case in which the number of strata is fixed and the sample sizes within each stratum increase indefinitely. The results show that psi MH is very efficient over a wide range of designs likely to occur in practice. However, conditions are identified under which the relative efficiency of this widely used estimator can be unusually low.     

Generalized Mantel-Haenszel estimators for K 2 x J tables

Mickey and Elashoff (1985, Biometrics 41, 623-635) gave an extension of Mantel-Haenszel estimation to log-linear models for 2 x J x K tables. Their extension yields two generalizations of the Mantel-Haenszel odds ratio estimator to K 2 x J tables. This paper provides variance and covariance estimators for these generalized Mantel-Haenszel estimators that are dually consistent (i.e., consistent in both large strata and sparse data), and presents comparisons of the efficiency of the generalized Mantel-Haenszel estimators.     

Methods of estimation in log odds ratio regression models

McCullagh's (1984, Journal of the Royal Statistical Society, Series B 46, 250-256) approximation to the conditional maximum likelihood estimator in log odds ratio regression models is shown to have negligible asymptotic bias unless the odds ratios are large and the sample sizes in individual 2 X 2 tables are very small. In application to two sets of case-control data, it yields results virtually indistinguishable from those of the conditional analysis. A generalization of the Mantel-Haenszel estimator proposed by Davis (1985, Biometrics 41, 487-495) does not approximate the conditional results nearly as well.     

A Monte Carlo evaluation of five interval estimators for the relative risk in sparse data

The relative risk (RR) is one of the most frequently used indices to measure the strength of association between a disease and a risk factor in etiological studies or the efficacy of an experimental treatment in clinical trials. In this paper, we concentrate attention on interval estimation of RR for sparse data, in which we have only a few patients per stratum, but a moderate or large number of strata. We consider five asymptotic interval estimators for RR, including a weighted least-squares (WLS) interval estimator with an ad hoc adjustment procedure for sparse data, an interval estimator proposed elsewhere for rare events, an interval estimator based on the Mantel-Haenszel (MH) estimator with a logarithmic transformation, an interval estimator calculated from a quadratic equation, and an interval estimator derived from the ratio estimator with a logarithmic transformation. On the basis of Monte Carlo simulations, we evaluate and compare the performance of these five interval estimators in a variety of situations. We note that, except for the cases in which the underlying common RR across strata is around 1, using the WLS interval estimator with the adjustment procedure for sparse data can be misleading. We note further that using the interval estimator suggested elsewhere for rare events tends to be conservative and hence leads to loss of efficiency. We find that the other three interval estimators can consistently perform well even when the mean number of patients for a given treatment is approximately 3 patients per stratum and the number of strata is as small as 20. Finally, we use a mortality data set comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the estimators discussed in this paper.     

Estimating effective population size or mutation rate with microsatellites

Microsatellites are short tandem repeats that are widely dispersed among eukaryotic genomes. Many of them are highly polymorphic; they have been used widely in genetic studies. Statistical properties of all measures of genetic variation at microsatellites critically depend upon the composite parameter theta = 4Nmicro, where N is the effective population size and micro is mutation rate per locus per generation. Since mutation leads to expansion or contraction of a repeat number in a stepwise fashion, the stepwise mutation model has been widely used to study the dynamics of these loci. We developed an estimator of theta, theta; (F), on the basis of sample homozygosity under the single-step stepwise mutation model. The estimator is unbiased and is much more efficient than the variance-based estimator under the single-step stepwise mutation model. It also has smaller bias and mean square error (MSE) than the variance-based estimator when the mutation follows the multistep generalized stepwise mutation model. Compared with the maximum-likelihood estimator theta; (L) by, theta; (F) has less bias and smaller MSE in general. theta; (L) has a slight advantage when theta is small, but in such a situation the bias in theta; (L) may be more of a concern.     

Estimation of a common odds ratio in case-control studies of familial aggregation

A new estimator of a common odds ratio is proposed for case-control studies of familial aggregation. The proposed estimator is a modification to the usual Mantel-Haenszel estimator that relies on an empirical adjustment for the within-family clustering which is typical of such designs. A simulation study shows that the estimator tends to have smaller mean squared error than the unmodified Mantel-Haenszel estimator under conditions likely to arise in practice. The construction of confidence intervals is also discussed.     

A Sometimes-Pool Estimator of the Mean Life

The paper deals with the problem of estimating the mean life θ in an exponential model \documentclass{article}\pagestyle{empty}\begin{document}$f(x/\theta ) = \frac{1}{\theta }e - x/\theta$\end{document}. It is assumed that in addition to the current ordered sample, we have a sample collected sometime in the recent past when the mean life might have been β. We have proposed a Sometimes—Pool procedure which is based on the outcome of a preliminary test of H₀: θ=β and obtained the expressions of the bias and MSE. An attempt has been made to locate that region in the parameter space in which the proposed estimator does better (in MSE sense) than the usual estimator based only on the current sample.     

Estimating population density of the Formosan subterranean termite, Coptotermes formosanus(Isoptera: Rhinotermitidae) using the effective sampling area of in-ground monitoring stations

The effective sampling area of a monitoring station, alpha, was calculated for several Coptotermes formosanus Shiraki colonies in Broward County, Florida, USA. A simple mark-recapture protocol provided data on termite station catch within a foraging range of a colony. Average recapture probability was 0.005 close to the release point (< 5 m) and declined to 0.0008 at a distance of 51 to 60 m. The relation between the log % termites recaptured was fitted with log distance, to determine P(x), the average proportion of captured termites that started at distance x from the release point. The effective sampling area was estimated by using P(x) and the equation, alpha 2 pi integral [ x P(x)] dx. Integrating this equation, an average estimate alpha that ranged from 0.607 to 14.5 m2 was obtained. Factors influencing the variation of alpha among the colonies are discussed. The effective sampling area estimated should be taken as a reliable estimator that translates subterranean termite catches into termite population density.     

Estimation of a common effect parameter from sparse follow-up data

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.     

Asymptotically efficient two-step estimators of the hazards ratio for follow-up studies and survival data

Asymptotically efficient estimators of a common hazard rate ratio (for follow-up studies) and the proportional hazards ratio (for survival studies) are obtained by a single iteration of the "Mantel-Haenszel" estimator appropriate for each setting. Estimators of their variance are also developed. The two-step estimator for survival data and its variance estimator are shown by simulation to be minimally biased and the estimator is shown to be efficient relative to the Cox partial likelihood estimator in small samples.     

Human liver alcohol dehydrogenases catalyze the oxidation of the intermediary alcohols of the shunt pathway of mevalonate metabolism

Human liver alcohol dehydrogenase (ADH) catalyzes the oxidation of 3,3-dimethylallyl alcohol, the intermediary alcohol of the shunt pathway of mevalonate metabolism. ADH isozymes differ in their activities toward this alcohol in the order gamma 1 gamma 1 greater than gamma 2 gamma 2 approximately alfa alfa greater pi pi approximately beta 2 beta 2 approximately beta 1 beta 1 much greater than chi chi; kcat/Km values are 1.4 x 10(8), 1.9 x 10(7), 1.4 x 10(7), 5.6 x 10(6), 3.6 x 10(6), 1.6 x 10(6) and 2.5 x 10(3) M-1 min-1, respectively. The intermediary alcohols geraniol and farnesol of the proposed branch pathways of mevalonate metabolism are also oxidized by these isozymes with similar relative efficiencies. The genetic determinants of ADH isozymes may contribute to the observed differences in serum cholesterol levels among and within various populations.     

Estimation of the hazards ratio in two grouped samples

A simple estimator of the hazards ratio of two grouped samples is proposed. If the number of time grouping intervals is fixed, the following asymptotics hold: unbiasedness, and full efficiency when the true hazards ratio is 1 and the probability of failure in each interval is small. Under the latter condition, the estimator is equivalent to "MHP" estimator (Mantel-Haenszel estimator for a Poisson model). Simulations show that this estimator performs better than others when grouping is coarse. An asymptotically unbiased estimator of its variance is proposed.     

Linkage analysis in familial adenomatous polyposis: order of C11P11 (D5S71) and pi 227 (D5S37) loci at the apc gene

Linkage analysis with DNA probes C11P11 and pi 227 is reported in six Scottish families with familial adenomatous polyposis. Two families were informative for C11P11 and all six were at least partly informative for pi 227. Two C11P11-apc and two pi 227-apc recombinants were identified and one of these was recombinant for both C11P11-apc and pi 227-apc. A further possible combined C11P11-apc and pi 227-apc recombination event was also identified. Peak lod score for linkage of C11P11 to apc was 5.80 at a recombination fraction (theta) of 0.069 (95% probability limits 0.012-0.191) and for linkage of pi 227 to apc was 3.19 at theta = 0.110 (95% probability limits 0.023-0.286). Peak lod score for linkage of C11P11 to pi 227 was 1.79 at theta = 0.00. The data support a gene order of pi 227-C11P11-apc.     

A comparative study of conditional maximum likelihood estimation of a common odds ratio

The finite-sample properties of various point estimators of a common odds ratio from multiple 2 X 2 tables have been considered in a number of simulation studies. However, the conditional maximum likelihood estimator has received only limited attention. That omission is partially rectified here for cases of relatively small numbers of tables and moderate to large within-table sample sizes. The conditional maximum likelihood estimator is found to be superior to the unconditional maximum likelihood estimator, and equal or superior to the Mantel-Haenszel estimator in both bias and precision.     

Estimating Genetic Variability with Restriction Endonucleases

The estimation of the amount of sequence variation in samples of homologous DNA segments is considered. The data are assumed to have been obtained by restriction endonuclease digestion of the segments, from which the numbers and frequencies of the cleavage sites in the sample are determined. An estimator, p, of the proportion of sites that are polymorphic in the sample is derived without assuming any particular population genetic model for the evolution of the population. The estimator is very close to the EWENS, SPIELMAN and HARRIS (1981) estimator that was derived with the symmetric WRIGHT-FISHER neutral mode. ENGELS (1981) has also recently proposed an estimator of the same quantity, and he arrived at his estimator without assuming a particular population genetic model. The sampling variance of p and ENGELS' estimator are derived. It is found that the sampling variance of p is lower than the sampling variance of ENGELS' estimator. Also, the sampling variance of theta, an estimate of theta (=4Nu) is obtained for the symmetric WRIGHT-FISHER neutral model with free recombination and with no recombination.     

Distributions of Potential in Cylindrical Coordinates and Time Constants for a Membrane Cylinder

A mathematical problem relating to membrane cylinders is stated and solved; its implications are illustrated and discussed. The problem concerns the volume distribution, in cylindrical coordinates, of the electric potential inside and outside a membrane cylinder of finite length (with sealed ends), during passive decay of an initially nonuniform membrane potential. The time constants for equalization with respect to the angle, theta, are shown to be typically about ten thousand times smaller than the time constant, tau(m) = R(m)C(m), for uniform passive membrane potential decay. The time constants for equalization with respect to length are shown to agree with those from one-dimensional cable theory; typically, they are smaller than tau(m) by a factor between 2 and 10. The relation of the membrane current density, I(m)(theta, x, t), to the values (at the outer membrane surface) of the extracellular potential phi(e)(r, theta, x, t) and of partial differential(2)phi(e)/ partial differentialx(2), is examined and it is shown that these quantities are not proportional to each other, in general; however, under certain specified conditions, all three of these quantities are proportional with each other and with phi(i)(r, theta, x, t) and partial differential(2)phi(i)/ partial differentialx(2) (at the inner membrane surface). The relation of these results to those of one-dimensional cable theory is discussed.     

Estimators of the Mantel-Haenszel variance consistent in both sparse data and large-strata limiting models

This paper proposes a new estimator of the variance of the Mantel-Haenszel estimator of common odds ratio that is easily computed and consistent in both sparse data and large-strata limiting models. Monte Carlo experiments compare its performance to that of previously proposed variance estimators.     

Estimating the unbiased estimator theta for population genetic survey data

We consider a method of approximating Weir and Cockerham's theta, an unbiased estimator of genetic population structure, using values readily available from published studies using biased estimators (Wright's F(ST) or Nei's G(ST)). The estimation algorithm is shown to be useful for both model populations and real-world avian populations. However, the correlation between Wright's F(ST) and Weir and Cockerham's theta is strong when compared among 39 empirical avian datasets. Thus, the advantage of approximating an unbiased estimator is unclear considering the small actual effect of theta's bias-removing power on empirical datasets.     

Nucleotide variation at the runt locus in Drosophila melanogaster and Drosophila simulans.

Intra- and interspecific nucleotide variation for the major developmental gene runt in Drosophila was studied in D. melanogaster and D. simulans. The 1.5-kb protein-coding region and the 0.4-kb intron of the runt gene were sequenced for 11 alleles in each species. The D. melanogaster alleles originated from east Africa. Estimated parameters of intraspecific variation in D. melanogaster (exons: theta = 0.018, pi = 0.018; intron: theta = 0.014, pi = 0.014) and D. simulans (exons: theta = 0.007, pi = 0.005; intron: theta = 0.008, pi = 0.005) were below average for other X-linked genes, while divergence between species (exons: D = 0.094; intron: D = 0.069) fell within the normal range for both silent and replacement changes. This estimate for runt, along with published values for three other genes in regions of normal recombination, show east African D. melanogaster to be roughly twice as polymorphic as D. simulans. The majority of nucleotide variation, silent and replacement, in both species was found to be selectively neutral using various statistical tests (HKA, McDonald-Kreitman, Tajima, and Fu and Li tests). Monte Carlo simulations of the coalescent process significantly rejected a Wright-Fisher model with respect to an amino acid polymorphism and the distribution of polymorphic sites among the D. simulans lines. This indicated an old lineage and may reflect ancestral population substructuring in D. simulans.