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PubMed1999

Methods for combining rates from several studies.

When several independent groups have conducted studies to estimate a procedure's success rate, it is often of interest to combine the results of these studies in the hopes of obtaining a better estimate for the true unknown success rate of the procedure. In this paper we present two hierarchical methods for estimating the overall rate of success. Both methods take into account the within-study and between-study variation and assume in the first stage that the number of successes within each study follows a binomial distribution given each study's own success rate. They differ, however, in their second stage assumptions. The first method assumes in the second stage that the rates of success from individual studies form a random sample having a constant expected value and variance. Generalized estimating equations (GEE) are then used to estimate the overall rate of success and its variance. The second method assumes in the second stage that the success rates from different studies follow a beta distribution. Both methods use the maximum likelihood approach to derive an estimate for the overall success rate and to construct the corresponding confidence intervals. We also present a two-stage bootstrap approach to estimating a confidence interval for the success rate when the number of studies is small. We then perform a simulation study to compare the two methods. Finally, we illustrate these two methods and obtain bootstrap confidence intervals in a medical example analysing the effectiveness of hyperdynamic therapy for cerebral vasospasm.

Confidence Intervals↗
PubMed1999

Continual reassessment methods in phase I trials of the combination of two drugs in oncology.

Most phase I trials in oncology use standard methods for treating successive groups of patients with increasing doses in order to determine the maximum tolerated dose (MTD). These methods have been criticized because they treat many patients at suboptimal dose levels, and do not provide an accurate estimation of the best dose level. Continual reassessment methods for the study of toxicity in single agent phase I trials have recently been advocated since they present many advantages over traditional methods. Although the advantages of these methods are recognized by most clinical investigators, their use is not widespread and their advantages have not yet been universally accepted. A maximum likelihood continual reassessment method was conducted retrospectively and compared to the originally planned standard method in a two drug combination phase I trial in order to study its applicability in this setting. Calculations from the binomial distributions and simulations were used for identifying the MTD, for the proportion of patients treated at the MTD or at one dose level just below, and for the proportion of patients treated at doses above the MTD. If the new method had been applied in this study, the MTD would have been reached much earlier, since, most of the time, higher dose levels were recommended. This result shows the feasibility of the new method in a two-drug setting and its use should be encouraged since fewer patients are treated at suboptimal dose levels or at dose levels above the MTD.

Antineoplastic Agents, Phytogenic↗
PubMed2000

A new method for the accurate determination of the isotopic state of single amide hydrogens within peptides using Fourier transform ion cyclotron resonance mass spectrometry.

A new method is presented to accurately determine the probability of having a deuterium or hydrogen atom on a specific amide position within a peptide after deuterium/hydrogen (D/H) exchange in solution. Amide hydrogen exchange has been proven to be a sensitive probe for studying protein structures and structural dynamics. At the same time, mass spectrometry in combination with physical fragmentation methods is commonly used to sequence proteins based on an amino acid residue specific mass analysis. In the present study it is demonstrated that the isotopic patterns of a series of peptide fragment ions obtained with capillary-skimmer dissociation, as observed with a 9.4 T Fourier transform ion cyclotron resonance (FTICR) mass spectrometer, can be used to calculate the isotopic state of specific amide hydrogens. This calculation is based on the experimentally observed isotopic patterns of two consecutive fragments and on the isotopic binomial distributions of the atoms in the residue constituting the difference between these two consecutive fragments. The applicability of the method is demonstrated by following the sequence-specific D/H exchange rate in solution of single amide hydrogens within some peptides.

Amides↗
PubMed2000

Sample size calculations with compliance information.

In randomized clinical trials, non-compliance will lead to the loss of power in the standard intention-to-treat analysis and one should account for this in sample size calculations. In this paper a new sample size formula for a binary outcome is proposed in which compliance information is considered. The proposed method is based purely on the treatment randomization. We compare it to the conventional sample size calculation method based on the two independent binomial distributions assumption. We examine 3100 combinations of risks of control group (baseline risks), treatment effects (risk differences), compliances in the test treatment group and the control treatment group, test sizes and powers. We found that, compared to the conventional method, the proposed method gives similar sample sizes for baseline risks 0.4-0.6, larger sample sizes for baseline risks less than 0.4, and smaller sample sizes for baseline risks greater than 0.6 when the true risk difference is negative.

Coronary Disease↗
PubMed1978

Genetic and evolutionary fitness.

The advantages and disadvantages of evolutionary fitness (probability that a single mutant line will not become extinct) and genetic fitness (mean fecundity) are compared. For deterministic processes the two are equivalent, but for stochastic branching processes they may be totally unrelated except that an absolute genetic fitness of unity or less implies an evolutionary fitness of zero. To know the variance as well as the mean family size does not in general uniquely determine the evolutionary fitness. Except where genetic fitness is close to unity, the impact of selection is shown to be rapid for the binomial, Poisson, negative binomial, and truncated negative binomial distributions. Evolutionary fitness, though somewhat cumbersome, has greater relevance to evolution, genetic counseling, and voluntary population control; but genetic fitness which is much easier to handle is the more appropriate measure where a large number of mutants is involved. Some empirical data on the transmission of various types of characters from parent to child are analyzed to allow comparison of genetic fitness, Crow's index, and a Malthusian parameter, with evolutionary fitness. There is a fair, but far from perfect, agreement among them. Multiple correlation of evolutionary fitness with mean and variance of family size taken jointly suggests a much more satisfactory approximation. It thus appears that, at the least, the population geneticist cannot afford to ignore the variance (which is not adequately represented in Crow's index). These relationships, based on two sets of data only may be accidental and should be invoked with caution. It seems more than likely that other aspects of the distribution of family size (eg, even higher moments) may contain relevant information in certain cases.

Biological Evolution↗
PubMed1984

Meckel syndrome in different populations.

We report on 18 infants from 13 families where the infant was affected with the Meckel syndrome. The parents belong to various national groups--Russians, Byelorussians, Poles, Ukranians, Letts, and Tatars. One child was from an incestuous union (half-sister and half-brother), in 4 families the parents were natives of the same or neighboring villages; other parents apparently were not related. Excluding 3 couples from Central Russia, the Ukraine, and Tatary, the other 10 families were the inhabitants of the Moscow region, Byelorussia, and Latvia. In 3 of these families at least one grandparent was of Tatar descent. At the same time the frequency of Tatars in these regions is less than 1%. Using the Newton binomial distribution it was shown that the hypothesis about equal frequency of the Meckel syndrome gene among Tatars and other national groups under study may be excluded completely, and therefore the alternative hypothesis about an unusually high frequency of this gene among Tatars must be accepted. Such analysis may be useful for comparative evaluation of gene frequencies in populations which cannot be studied directly.

Abnormalities, Multiple↗
PubMed2005

C-fiber (Remak) bundles contain both isolectin B4-binding and calcitonin gene-related peptide-positive axons.

Unmyelinated nerve fibers (Remak bundles) in the rodent sciatic nerve typically contain multiple axons. This study asked whether C-fiber bundles contain axons arising from more than one type of neuron. Most small neurons of the lumbar dorsal root ganglion (DRG) are either glial cell line-derived neurotrophic factor dependent or nerve growth factor dependent, binding either isolectin B4 (IB4) or antibodies to calcitonin gene-related peptide (CGRP), respectively. Injection of IB4-conjugated horseradish peroxidase into a lumbar DRG resulted in intense labeling of IB4 axons, with very low background. Visualized by confocal fluorescence, IB4-binding and CGRP-positive nerve fibers originating from different DRG neurons came together and remained closely parallel over long distances, suggesting that these two types of axon occupy the same Remak bundle. With double-labeling immunogold electron microscopy (EM), we confirmed that IB4 and CGRP axons were distinct and were found together in single Remak bundles. Previous studies indicate that some DRG neurons express both CGRP and IB4 binding. To ensure that our immunogold results were not a consequence of coexpression, we studied large populations of unmyelinated axons by using quantitative single-label EM. Tetramethylbenzidine, a chromogen with strong intrinsic signal amplification of IB4-horseradish peroxidase, labeled as many as 52% of unmyelinated axons in the dorsal root. Concomitantly, 97% of the Remak bundles with more than one axon contained at least one IB4-labeled axon. Probabilistic modeling using binomial distribution functions rejected the hypothesis that IB4 axons segregate into IB4-specific bundles (P < 0.00001). We conclude that most Remak bundle Schwann cells simultaneously support diverse axon types with different growth factor dependences.

Animals↗
PubMed1985

A statistical examination of historical controls for mouse bone marrow cytogenetic assays.

Data from 1,111 controls from assays run over 11 years are examined to determine a most powerful statistical procedure for detecting a mutagenic effect. It is concluded that the data do not show a constant probability of chromosomal abnormalities and that the data do not fit a simple Poisson or binomial distribution. Empirical Bayes techniques are used to derive a test that declares an effect if three or more cells in a group of 50 tested contain abnormalities.

Analysis of Variance↗
PubMed1987

Sporadic vs familial classification given etiologic heterogeneity: I. Sensitivity, specificity, and positive and negative predictive value.

Environmental factors are etiologically important in many non-Mendelian familial disorders in man. Because such disorders often occur as "sporadic" cases, (ie, an affected individual with no affected relatives), it is tempting to assume that such cases represent an "environmental" form of the disorder. This paper presents an evaluation of the sensitivity, specificity, and positive and negative predictive power (PPV and NPV) of this "sporadic vs familial classification." The model assumes etiologic heterogeneity with a subpopulation of cases due to a "major" environmental event acting independent of genotype and the remaining cases resulting from a generalized single major locus (SML). Sibship size is modeled by a truncated negative binomial distribution. For rare disorders, this classification has high sensitivity and NPV but low specificity and PPV. As the disorder becomes more common, sensitivity and NPV fall while specificity and PPV rise. The power of the method increases substantially with increasing sibship size up to four or five, but further increases in power are minimal. MZ twins add considerable power to the method but aunts and uncles add little if anything. Both a correlational (phi) and an agreement-based (kappa) statistic indicate that, under most realistic circumstances, the relationship between etiology and family history is modest.

Alleles↗
PubMed2003

MMPI-2: Confidence intervals for random responding to the F, F Back, and VRIN scales.

Several studies have investigated random responding to the F, F Back, and VRIN scales. Only one study attempted to provide practical cutoff scores for these scales, but was unable to reach definitive cutoffs. This study uses the normal approximation to the binomial distribution and provides confidence interval bounds for random responding at the 95, 90, and 85% levels for the F, F Back, and VRIN scales. The possibility that humans asked to respond randomly produce F, F Back, and VRIN scores different from computer-generated random scores was investigated. The results show nonsignificant differences between human and computer responses for the F and F Back scales and mixed results for the VRIN scale.

Computer Simulation↗
PubMed1992

Novel approach to estimate quality of binary random powder mixtures: samples of constant volume. I: Derivation of equation.

An equation of the quality of binary random mixtures that applies to powder samples of constant bulk volume was derived from the binomial distribution. In contrast to the Stange-Poole equation for samples of constant mass, this approach can also be used for constituents with large differences in particle size and in bulk density. The validity of this equation was verified with tablets directly compressed from mixtures composed of equal mass proportions of sucrose [volume-weighted/volume-number mean diameter (dv), 504 microns] as the coarse ingredient (A) and of a microcrystalline cellulose (Avicel PH 101)-talc mixture (dv, 60 microns) as the fine constituent (B). Because of the difference in the bulk densities of A and B, the equation estimated a coefficient of the random content variation, which for A was double that for B. The content variations found with the tablets were in excellent agreement with the calculated values and reflected the differences predicted from theory. The Stange-Poole equation yielded identical values of the content variations of A and B, which is in contrast to the experimental results.

Chemistry, Pharmaceutical↗
PubMed1993

Confidence limits on the branching order of phylogenetic trees.

We describe a confidence test for branching order that can aid protein phylogeny reconstruction as well as the evaluation of the optimal tree. It is proposed that the process resulting in the observed amino acid residue differences, which is the basis for the identification of the order and relative times of divergence events, is appropriately described by a modification of the negative binomial distribution. The relative total numbers of mutations (accepted and nonaccepted), which result in a given number of amino acid differences, may be obtained as the expectation of this distribution. The associated variances enable significant differences in tree branching order to be established. If the total rates of mutation of the genes encoding the compared proteins are equal, the expected total mutations and their associated variances map identically to their relative times of divergence. In addition, significantly different rates of change (due to differences in total mutation rate and/or acceptance rate) may be identified without the requirement of outlying reference group. The method is equally applicable to phylogenies derived from DNA or RNA sequence information.

Amino Acid Sequence↗
PubMed2002

Bayesian random effects meta-analysis of trials with binary outcomes: methods for the absolute risk difference and relative risk scales.

When conducting a meta-analysis of clinical trials with binary outcomes, a normal approximation for the summary treatment effect measure in each trial is inappropriate in the common situation where some of the trials in the meta-analysis are small, or the observed risks are close to 0 or 1. This problem can be avoided by making direct use of the binomial distribution within trials. A fully Bayesian method has already been developed for random effects meta-analysis on the log-odds scale using the BUGS implementation of Gibbs sampling. In this paper we demonstrate how this method can be extended to perform analyses on both the absolute and relative risk scales. Within each approach we exemplify how trial-level covariates, including underlying risk, can be considered. Data from 46 trials of the effect of single-dose ibuprofen on post-operative pain are analysed and the results contrasted with those derived from classical and Bayesian summary statistic methods. The clinical interpretation of the odds ratio scale is not straightforward. The advantages and flexibility of a fully Bayesian approach to meta-analysis of binary outcome data, considered on an absolute risk or relative risk scale, are now available.

Anti-Inflammatory Agents, Non-Steroidal↗
PubMed2004

On the estimation of the binomial probability in multistage clinical trials.

Due to the optional sampling effect in a sequential design, the maximum likelihood estimator (MLE) following sequential tests is generally biased. In a typical two-stage design employed in a phase II clinical trial in cancer drug screening, a fixed number of patients are enrolled initially. The trial may be terminated for lack of clinical efficacy of treatment if the observed number of treatment responses after the first stage is too small. Otherwise, an additional fixed number of patients are enrolled to accumulate additional information on efficacy as well as on safety. There have been numerous suggestions for design of such two-stage studies. Here we establish that under the two-stage design the sufficient statistic, i.e. stopping stage and the number of treatment responses, for the parameter of the binomial distribution is also complete. Then, based on the Rao-Blackwell theorem, we derive the uniformly minimum variance unbiased estimator (UMVUE) as the conditional expectation of an unbiased estimator, which in this case is simply the maximum likelihood estimator based only on the first stage data, given the complete sufficient statistic. Our results generalize to a multistage design. We will illustrate features of the UMVUE based on two-stage phase II clinical trial design examples and present results of numerical studies on the properties of the UMVUE in comparison to the usual MLE.

Clinical Trials, Phase II as Topic↗
PubMed2005

Covariate-adjusted adaptive randomization in a sarcoma trial with multi-stage treatments.

We present a Bayesian design for a multi-centre, randomized clinical trial of two chemotherapy regimens for advanced or metastatic unresectable soft tissue sarcoma. After randomization, each patient receives up to four stages of chemotherapy, with the patient's disease evaluated after each stage and categorized on a trinary scale of severity. Therapy is continued to the next stage if the patient's disease is stable, and is discontinued if either tumour response or treatment failure is observed. We assume a probability model that accounts for baseline covariates and the multi-stage treatment and disease evaluation structure. The design uses covariate-adjusted adaptive randomization based on a score that combines the patient's probabilities of overall treatment success or failure. The adaptive randomization procedure generalizes the method proposed by Thompson (1933) for two binomial distributions with beta priors. A simulation study of the design in the context of the sarcoma trial is presented.

Antineoplastic Agents↗
PubMed2005

Careful use of pseudo R-squared measures in epidemiological studies.

Many epidemiological research problems deal with large numbers of exposed subjects of whom only a small number actually suffers the adverse event of interest. Such rare events data can be analysed by employing an approximate Poisson model. The objective of this study is to challenge the interpretability of the corresponding Poisson pseudo R-squared measure. It will lack sensible interpretation whenever the approximate Poisson outcome is generated by counting the number of events within covariate patterns formed by cross-tabulating categorical covariates. The failure is caused by the immanent arbitrariness in the definition of the covariate patterns, that is, independent Bernoulli events, B(1,pi), are arbitrarily combined into binomially distributed ones, B(n,pi), which are then approximated by the Poisson model.

Adult↗
PubMed2006

Intraclass correlation coefficients and bootstrap methods of hierarchical binary outcomes.

Intraclass correlation coefficients are designed to assess consistency or conformity between two or more quantitative measurements. When multistage cluster sampling is implemented, no methods are readily available to estimate intraclass correlations of binomial-distributed outcomes within a cluster. Because statistical distribution of the intraclass correlation coefficients could be complicated or unspecified, we propose using a bootstrap method to estimate the standard error and confidence interval within the framework of a multilevel generalized linear model. We compared the results derived from a parametric bootstrap method with those from a non-parametric bootstrap method and found that the non-parametric method is more robust. For non-parametric bootstrap sampling, we showed that the effectiveness of sampling on the highest level is greater than that on lower levels; to illustrate the effectiveness, we analyse survey data in China and do simulation studies.

Biometry↗
PubMed2007

Robustness of a multivariate normal approximation for imputation of incomplete binary data.

Multiple imputation has become easier to perform with the advent of several software packages that provide imputations under a multivariate normal model, but imputation of missing binary data remains an important practical problem. Here, we explore three alternative methods for converting a multivariate normal imputed value into a binary imputed value: (1) simple rounding of the imputed value to the nearer of 0 or 1, (2) a Bernoulli draw based on a 'coin flip' where an imputed value between 0 and 1 is treated as the probability of drawing a 1, and (3) an adaptive rounding scheme where the cut-off value for determining whether to round to 0 or 1 is based on a normal approximation to the binomial distribution, making use of the marginal proportions of 0's and 1's on the variable. We perform simulation studies on a data set of 206,802 respondents to the California Healthy Kids Survey, where the fully observed data on 198,262 individuals defines the population, from which we repeatedly draw samples with missing data, impute, calculate statistics and confidence intervals, and compare bias and coverage against the true values. Frequently, we found satisfactory bias and coverage properties, suggesting that approaches such as these that are based on statistical approximations are preferable in applied research to either avoiding settings where missing data occur or relying on complete-case analyses. Considering both the occurrence and extent of deficits in coverage, we found that adaptive rounding provided the best performance.

Adolescent↗