Blinded sample size recalculation in multicentre trials with normally distributed outcome

The internal pilot study design enables to estimate nuisance parameters required for sample size calculation on the basis of data accumulated in an ongoing trial. By this, misspecifications made when determining the sample size in the planning phase can be corrected employing updated knowledge. According to regulatory guidelines, blindness of all personnel involved in the trial has to be preserved and the specified type I error rate has to be controlled when the internal pilot study design is applied. Especially in the late phase of drug development, most clinical studies are run in more than one centre. In these multicentre trials, one may have to deal with an unequal distribution of the patient numbers among the centres. Depending on the type of the analysis (weighted or unweighted), unequal centre sample sizes may lead to a substantial loss of power. Like the variance, the magnitude of imbalance is difficult to predict in the planning phase. We propose a blinded sample size recalculation procedure for the internal pilot study design in multicentre trials with normally distributed outcome and two balanced treatment groups that are analysed applying the weighted or the unweighted approach. The method addresses both uncertainty with respect to the variance of the endpoint and the extent of disparity of the centre sample sizes. The actual type I error rate as well as the expected power and sample size of the procedure is investigated in simulation studies. For the weighted analysis as well as for the unweighted analysis, the maximal type I error rate was not or only minimally exceeded. Furthermore, application of the proposed procedure led to an expected power that achieves the specified value in many cases and is throughout very close to it.     

The genetic structure of a tribal population, the Yanomama indians. VII. Anthropometric differences among Yanomama villages

Anthropometric data on 12 variables in 19 villages of the Yanomama Indians demonstrate significant heterogeneity in physique among villages of this tribe. Mahalanobis' distances (D²) calculated from the data lead to the tentative conclusion of a general correspondence between anthropometric and geographic distances separating villages. The mean stature of the Yanomama is smaller than that of most other South American tribes which have been measured, and the Yanomama are genetically distinct from the other small Indians as shown by genetic distances based on allele frequencies for a variety of genetic markers. Since some subjects were measured more than once by the same and by different observers, it was possible to calculate approximate estimates of variance within and between observers. Univariate analysis indicates that face height and nose height are especially susceptible to systematic differences in technique between observers. The variances obtained in this field study compare favorably with those of some classical laboratory studies described in the literature. It was found that measurement error nevertheless probably makes a substantial contribution to anthropometric distance between villages. The median error variance as a fraction of that of Herskovits ('30) is 0.62 for the seven measurements in common with this study. The median value of the error variance for the 12 variables in this study is between 16% and 17% of the total variance.     

Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments.

Optimal design of experiments as well as proper analysis of data are dependent on knowledge of the experimental error. A detailed analysis of the error structure of kinetic data obtained with acetylcholinesterase showed conclusively that the classical assumptions of constant absolute or constant relative error are inadequate for the dependent variable (velocity). The best mathematical models for the experimental error involved the substrate and inhibitor concentrations and reflected the rate law for the initial velocity. Data obtained with other enzymes displayed similar relationships between experimental error and the independent variables. The new empirical error functions were shown superior to previously used models when utilized in weighted non-linear-regression analysis of kinetic data. The results suggest that, in the spectrophotometric assays used in the present study, the observed experimental variance is primarily due to errors in determination of the concentrations of substrate and inhibitor and not to error in measuring the velocity.     

Rosetta error model for gene expression analysis

MOTIVATION: In microarray gene expression studies, the number of replicated microarrays is usually small because of cost and sample availability, resulting in unreliable variance estimation and thus unreliable statistical hypothesis tests. The unreliable variance estimation is further complicated by the fact that the technology-specific variance is intrinsically intensity-dependent. RESULTS: The Rosetta error model captures the variance-intensity relationship for various types of microarray technologies, such as single-color arrays and two-color arrays. This error model conservatively estimates intensity error and uses this value to stabilize the variance estimation. We present two commonly used error models: the intensity error-model for single-color microarrays and the ratio error model for two-color microarrays or ratios built from two single-color arrays. We present examples to demonstrate the strength of our error models in improving statistical power of microarray data analysis, particularly, in increasing expression detection sensitivity and specificity when the number of replicates is limited.     

Self-designing two-stage trials to minimize expected costs

In the design of clinical trials, the sample size for the trial is traditionally calculated from estimates of parameters of interest, such as the mean treatment effect, which can often be inaccurate. However, recalculation of the sample size based on an estimate of the parameter of interest that uses accumulating data from the trial can lead to inflation of the overall Type I error rate of the trial. The self-designing method of Fisher, also known as the variance-spending method, allows the use of all accumulating data in a sequential trial (including the estimated treatment effect) in determining the sample size for the next stage of the trial without inflating the Type I error rate. We propose a self-designing group sequential procedure to minimize the expected total cost of a trial. Cost is an important parameter to consider in the statistical design of clinical trials due to limited financial resources. Using Bayesian decision theory on the accumulating data, the design specifies sequentially the optimal sample size and proportion of the test statistic's variance needed for each stage of a trial to minimize the expected cost of the trial. The optimality is with respect to a prior distribution on the parameter of interest. Results are presented for a simple two-stage trial. This method can extend to nonmonetary costs, such as ethical costs or quality-adjusted life years.     

Nonparametric analysis of covariance for comparing change in randomized studies with baseline values subject to error

Change from baseline to a follow-up examination can be compared among two or more randomly assigned treatment groups by using analysis of variance on the change scores. However, a generally more sensitive (powerful) test can be performed using analysis of covariance (ANOVA) on the follow-up data with the baseline data as a covariate. This approach is not without potential problems, though. The assumption of ordinary ANCOVA of normally distributed errors is speculative for many variables employed in biomedical research. Furthermore, the baseline values are inevitably random variables and often are measured with error. This report investigates, in this situation, the validity and relative power of the ordinary ANCOVA test and two asymptotically distribution-free alternative tests, one based on the rank transformation and the other based on the normal scores transformation. The procedures are illustrated with data from a clinical trial. Normal and several nonnormal distributions, as well as varying degree of variable error, are studied by Monte Carlo methods. The normal scores test is generally recommended for statistical practice.     

An alternative derivation of Harville's restricted log likelihood function for variance component estimation

Estimation of variance components in linear mixed models is important in clinical trial and longitudinal data analysis. It is also important in animal and plant breeding for accurately partitioning total phenotypic variance into genetic and environmental variances. Restricted maximum likelihood (REML) method is often preferred over the maximum likelihood (ML) method for variance component estimation because REML takes into account the lost degree of freedom resulting from estimating the fixed effects. The original restricted likelihood function involves a linear transformation of the original response variable (a collection of error contrasts). Harville's final form of the restricted likelihood function does not involve the transformation and thus is much easier to manipulate than the original restricted likelihood function. There are several different ways to show that the two forms of the restricted likelihood are equivalent. In this study, I present a much simpler way to derive Harville's restricted likelihood function. I first treat the fixed effects as random effects and call such a mixed model a pseudo random model (PDRM). I then construct a likelihood function for the PDRM. Finally, I let the variance of the pseudo random effects be infinity and show that the limit of the likelihood function of the PDRM is the restricted likelihood function.     

Blinded versus unblinded estimation of a correlation coefficient to inform interim design adaptations

Regulatory authorities require that the sample size of a confirmatory trial is calculated prior to the start of the trial. However, the sample size quite often depends on parameters that might not be known in advance of the study. Misspecification of these parameters can lead to under‐ or overestimation of the sample size. Both situations are unfavourable as the first one decreases the power and the latter one leads to a waste of resources. Hence, designs have been suggested that allow a re‐assessment of the sample size in an ongoing trial. These methods usually focus on estimating the variance. However, for some methods the performance depends not only on the variance but also on the correlation between measurements. We develop and compare different methods for blinded estimation of the correlation coefficient that are less likely to introduce operational bias when the blinding is maintained. Their performance with respect to bias and standard error is compared to the unblinded estimator. We simulated two different settings: one assuming that all group means are the same and one assuming that different groups have different means. Simulation results show that the naïve (one‐sample) estimator is only slightly biased and has a standard error comparable to that of the unblinded estimator. However, if the group means differ, other estimators have better performance depending on the sample size per group and the number of groups.     

Validation of a nested error component model to estimate damage caused by corn rootworm larvae

The corn rootworm complex (Coleoptera: Chrysomelidae) constitutes a significant threat to maize production in the United States, and more recently, in Europe. We conducted an analysis of readily available field trial data to validate an existing damage function for corn rootworm larvae. We used a nested error component model with unbalanced panel data to describe the relationship between yield loss and root injury caused by these insects. These data were collected by personnel with the Insect Management and Insecticide Evaluation Programme (Department of Crop Sciences, University of Illinois) and represent 19 location‐years. To our knowledge, this is the largest data set used to estimate a damage function for corn rootworm larvae. Unlike many experiments examining the relationship between root injury and yield loss caused by corn rootworm larvae, the data set used for our analysis includes many Bt maize hybrids. Our model suggests that for each node of roots injured by corn rootworm larvae, a yield loss of approximately 15% can be expected. Statistically significant variance components included an effect of location and experimental error. We speculate that variation in weather across experimental sites was the principal factor contributing to the significant effect of location. The substantial experimental error observed for our model highlights the limitations of utilizing a multi‐year, geographically diverse damage function for predicting yield loss because of root injury on a small scale. We discuss major factors contributing to the variance components estimated by our model and suggest techniques for improving future analyses of the damage function for corn rootworm larvae.     

Statistical analysis of the direct count method for enumerating bacteria.

The direct count method for enumerating bacteria in natural environments is widely used. This paper analyzes the sources of variation contributed by the various levels of the method: subsamples, filters, and microscope fields. Based on a nested analysis of variance, we show that most of the variance (less than 80%) is caused by the fields and that the filters contributed nearly all of the remaining variance. The replication at each of the levels determines the total cost and error of a measurement. We compared several sampling schemes, including an optimal strategy which gives the lowest possible variance for a given cost. We recommend that preparing one filter from one subsample is adequate only if the samples are closely spaced in time or distance; otherwise, one filter should be prepared from two or preferably three subsamples. This sampling scheme emphasizes the importance of the highest level of replication. Our analysis shows that the accuracy of the direct count method can be substantially improved (by 20 to 50%) without a large increase in cost when the proper degree of replication at each level is performed.     

Estimation of heritability using variety trials data from incomplete blocks

An assessment of the heritability of a trait is useful in formulating a breeding strategy for crop improvement. We have considered the estimation of broad-sense heritability from a single-location trial and from multi-locational trials conducted in incomplete blocks. Using residual maximum likelihood estimates of variance components, we estimated the heritability and obtained expressions for the estimate of its bias and its standard error. The estimation procedure is illustrated for 25 barley genotypes evaluated at four locations in West Asia and North Africa during 1992.     

Precision of the daily pollen count. Identifying sources of variation using variance component models

Summary During the pollen season, a daily account of airborne pollen is reported by radio and newpapers in Denmark as «Today's pollen count». The Aerobiological Group under the Danish Asthma and Allergy association is responsible for the daily identification and counting of pollen. In 1984 the Group set up a trial to investigate the reproducibility of the pollen count.Three trained pollen counters independently examined the same specimens from two Burkard pollen traps. The traps were located on the roof of the Meterological Institute in Copenhagen 15 meters above ground level in the northern outskirts of central Copenhagen. This is the usual sampling site. Specimens from each trap from 20 days during the grass pollen season were selected. Four species groups were identified in sufficient numbers for statistical analysis. These were, in order of occurrence, spruce/pine (Picea/Pinus), grass (Poaceae), nettle (Urtica) and mugwort (Artemisia).Identification of sources of variation and assessment of their relative importance were carried out using variance components models. A detailed account of the method is given.Qualitatively equivalent conclusions were reached for each group: i) The expected seasonal variation was identified; ii) The counters contributed significantly to the variation of pollen counts; iii) The traps did not contribute to the variance, but a day-trap interaction was identified. This interaction was interpreted. as a problem of instrumental variation over time of traps of variation in meteorological conditions; iv) The total variance was larger than originally expected. The relative uncertainty was greater than 50 per cent.The variance of the daily pollen count cannot be reduced much by reducing counter variation or day-trap interaction. The variation among pollen counters is small, and the day-trap interaction is difficult to fully comprehend. The basic measurement error can be reduced by examining a larger area of each specimen. Although this implies a greater cost, it is probably the most effective method of reducing the uncertainty of «Today's pollen count».     

Statistical analyses of enzyme inhibitor kinetics: a hierarchical model-dependent approach.

Inference of the nature of an enzyme inhibitor and calculations of inhibitor constants from linear plots are prone to be unreliable when experimental error is significant. This may be minimised by a method of statistical analysis using nonlinear regression methods and modelling using the variance ratio test of a hierarchy of models of inhibitor behaviour.     

Estimation of heritability from varietal trials data

We present the estimation of heritabilities of an observed trait in situations where evaluation of several pure breeding lines is performed in a trial at a single location and in trials from several locations. For the single location situation, we evaluate exact confidence intervals, the probability of invalid estimates, and the percentage points of the distribution of heritability. Simulations were performed to numerically verify the results. Additionally, approximations to the bias and standard error of the estimate were obtained and are presented along with their simulated values and coefficients of skewness and kurtosis. For trials in several locations, explicit expressions for exact values of confidence limits are not available. Further, one would require knowledge of one more parameter, represented by the ratio of genotype x environment (G x E) interaction variance to error variance, in addition to the number of genotypes, replication and true heritability value. Approximations were made for bias and the standard error of estimates of heritability. The evaluation of the distribution of heritability and its moments was recognized as a problem of the linear function of an independent chi-square. The methods have been illustrated by data from experiments on grain and straw yield of 64 barley genotypes evaluated at three locations.     

Use of molecular markers for estimating breeding parameters: a case study in a Pinus pinaster Ait. progeny trial

The management of a genetic improvement program is based on the knowledge of the genetic parameters and their relationships to determine the genetic gains. Knowledge of the coefficient of coancestry (θ) is a requirement for efficient progeny testing scheme and for estimating additive variance components for any quantitative trait. When using open-pollinated families, most authors assume that the seedlings are related as half-sibs, but this is not always true. Our aim was to estimate a mean value of the coancestry coefficient of the families present in a maritime pine Pinus pinaster Ait. (maritime or cluster pine) progeny trial originating from seed collected in a clonal seed orchard and to study how deviations from the standard assumption of θ = 0.125 affect heritability estimations. Five highly polymorphic microsatellite markers were scored in 125 offspring from a subsample of five families from the progeny trial. The mean value of the coancestry coefficient of the families present in this progeny trial was 0.130. Differences between the unadjusted and adjusted heritability estimates were more pronounced in wood density (0.609 and 0.586, respectively) than in diameter (0.166 and 0.154, respectively). We conclude that in the trial, the associated error in heritability estimates due to the inclusion of full-sibs, when assuming a standard coefficient of relationship among open-pollinated sibs of 0.250, was low and that this result is robust with respect to the number of families sampled, given unbiased estimates of average relationship among offspring within sib families.     

Two-way minimization: a novel treatment allocation method for small trials

Randomization is a hallmark of clinical trials. If a trial entails very few subjects and has many prognostic factors (or many factor levels) to be balanced, minimization is a more efficient method to achieve balance than a simple randomization. We propose a novel minimization method, the 'two-way minimization'. The method separately calculates the 'imbalance in the total numbers of subjects' and the 'imbalance in the distributions of prognostic factors'. And then to allocate a subject, it chooses--by probability--to minimize either one of these two aspects of imbalances. As such, it is a method that is both treatment-adaptive and covariate-adaptive. We perform Monte-Carlo simulations to examine its statistical properties. The two-way minimization (with proper regression adjustment of the force-balanced prognostic factors) has the correct type I error rates. It also produces point estimates that are unbiased and variance estimates that are accurate. When there are important prognostic factors to be balanced in the study, the method achieves the highest power and the smallest variance among randomization methods that are resistant to selection bias. The allocation can be done in real time and the subsequent data analysis is straightforward. The two-way minimization is recommended to balance prognostic factors in small trials.     

Analysis of covariance in randomized trials: More precision and valid confidence intervals,without model assumptions

“Covariate adjustment” in the randomized trial context refers to an estimator of the average treatment effect that adjusts for chance imbalances between study arms in baseline variables (called “covariates”). The baseline variables could include, for example, age, sex, disease severity, and biomarkers. According to two surveys of clinical trial reports, there is confusion about the statistical properties of covariate adjustment. We focus on the analysis of covariance (ANCOVA) estimator, which involves fitting a linear model for the outcome given the treatment arm and baseline variables, and trials that use simple randomization with equal probability of assignment to treatment and control. We prove the following new (to the best of our knowledge) robustness property of ANCOVA to arbitrary model misspecification: Not only is the ANCOVA point estimate consistent (as proved by Yang and Tsiatis, 2001) but so is its standard error. This implies that confidence intervals and hypothesis tests conducted as if the linear model were correct are still asymptotically valid even when the linear model is arbitrarily misspecified, for example, when the baseline variables are nonlinearly related to the outcome or there is treatment effect heterogeneity. We also give a simple, robust formula for the variance reduction (equivalently, sample size reduction) from using ANCOVA. By reanalyzing completed randomized trials for mild cognitive impairment, schizophrenia, and depression, we demonstrate how ANCOVA can achieve variance reductions of 4 to 32%.     

Quantifying reproducibility for differential proteomics: noise analysis for protein liquid chromatography-mass spectrometry of human serum

SUMMARY: Using replicated human serum samples, we applied an error model for proteomic differential expression profiling for a high-resolution liquid chromatography-mass spectrometry (LC-MS) platform. The detailed noise analysis presented here uses an experimental design that separates variance caused by sample preparation from variance due to analytical equipment. An analytic approach based on a two-component error model was applied, and in combination with an existing data driven technique that utilizes local sample averaging, we characterized and quantified the noise variance as a function of mean peak intensity. The results indicate that for processed LC-MS data a constant coefficient of variation is dominant for high intensities, whereas a model for low intensities explains Poisson-like variations. This result leads to a quadratic variance model which is used for the estimation of sample preparation noise present in LC-MS data.     

Cementum annulation and age determination in Homo sapiens. I. Tooth variability and observer error

In order to test the feasibility of cementum annulations to estimate age in humans, observer error and tooth variability in cementum ring counts were evaluated in a sample of 42 mandibular canine and first premolar pairs. Additionally, two sectioning techniques were evaluated. Demineralized thin sections (7 micron) stained with hematoxylin are the preferred technique since their age related variance is greater than 75% for all tooth types examined. In contrast, less than 50% of the total variance was accounted for among individuals when mineralized sections (80 micron) stained with alizarin red were used. Intertooth variability in ring counts of demineralized sections was large between canines and premolars (43%). Premolars provide counts with lower interobserver error and are the preferred tooth. In an expanded sample (N = 51) of demineralized premolars, intraobserver and interobserver error accounted for 2% and 5% of the total variance, respectively. Evaluation of several experimental designs showed that increasing the number of slides per tooth has the greatest effect on reducing variance followed by increasing the number of observers. Increasing the number of observations has little effect. Cementum ring counts are measurable to a highly repeatable extent and provide a level of repeatability greater than that reported for the pubic symphysis and auricular surface aging techniques.     

Quantitative Genetics of DROSOPHILA MELANOGASTER. I. Sexual Dimorphism in Genetic Parameters for Wing Traits

Sexual dimorphism in genetic parameters is examined for wing dimensions of Drosophila melanogaster. Data are fit to a quantitative genetic model where phenotypic variance is a linear function of additive genetic autosomal variance (common to both sexes), additive genetic X-linked variances distinct for each sex, variance due to common rearing environment of families, residual environmental variance, random error variance due to replication, and variance due to measurement error and developmental asymmetry (left vs. right sides). Polygenic dosage compensation and its effect on genetic variances and covariances between sexes is discussed. Variance estimates for wing length and other wing dimensions highly correlated with length support the hypothesis that the Drosophila system of dosage compensation will cause male X-linked genetic variance to be substantially larger than female X-linked variance. Results for various wing dimensions differ, suggesting that the level of dosage compensation may differ for different traits. Genetic correlations between sexes for the same trait are presented. Total additive genetic correlations are near unity for most wing traits; this indicates that selection in the same direction in both sexes would have a minor effect on changing the magnitude of difference between sexes. Additive X-linked correlations suggest some genotype x sex interactions for X-linked effects.