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PubMed2005

Mixed effects versus fixed effects modelling of binary data with inter-subject variability.

The question of whether or not a mixed effects model is required when modelling binary data with inter-subject variability and within subject correlation was reported in this journal by Yano et al. (J. Pharmacokin. Pharmacodyn. 28:389-412 [2001]). That report used simulation experiments to demonstrate that, under certain circumstances, the use of a fixed effects model produced more accurate estimates of the fixed effect parameters than those produced by a mixed effects model. The Laplace approximation to the likelihood was used when fitting the mixed effects model. This paper repeats one of those simulation experiments, with two binary observations recorded for every subject, and uses both the Laplace and the adaptive Gaussian quadrature approximations to the likelihood when fitting the mixed effects model. The results show that the estimates produced using the Laplace approximation include a small number of extreme outliers. This was not the case when using the adaptive Gaussian quadrature approximation. Further examination of these outliers shows that they arise in situations in which the Laplace approximation seriously overestimates the likelihood in an extreme region of the parameter space. It is also demonstrated that when the number of observations per subject is increased from two to three, the estimates based on the Laplace approximation no longer include any extreme outliers. The root mean squared error is a combination of the bias and the variability of the estimates. Increasing the sample size is known to reduce the variability of an estimator with a consequent reduction in its root mean squared error. The estimates based on the fixed effects model are inherently biased and this bias acts as a lower bound for the root mean squared error of these estimates. Consequently, it might be expected that for data sets with a greater number of subjects the estimates based on the mixed effects model would be more accurate than those based on the fixed effects model. This is borne out by the results of a further simulation experiment with an increased number of subjects in each set of data. The difference in the interpretation of the parameters of the fixed and mixed effects models is discussed. It is demonstrated that the mixed effects model and parameter estimates can be used to estimate the parameters of the fixed effects model but not vice versa.

Algorithms↗

PubMed2001

Sample size and statistical power of randomised, controlled trials in orthopaedics.

We reviewed all 717 manuscripts published in the 1997 issues of the British and American volumes of the Journal of Bone and Joint Surgery and in Clinical Orthopaedics and Related Research, from which 33 randomised, controlled trials were identified. The results and sample sizes were used to calculate the statistical power of the study to distinguish small (0.2 of standard deviation), medium (0.5 of standard deviation), and large (0.8 of standard deviation) effect sizes. Of the 33 manuscripts analysed, only three studies (9%) described calculations of sample size. To perform post-hoc power assessments and estimations of deficiencies of sample size, the standard effect sizes of Cohen (small, medium and large) were calculated. Of the 25 studies which reported negative results, none had adequate power (beta < 0.2) to detect a small effect size and 12 (48%) lacked the power necessary to detect a large effect size. Of the 25 studies which did not have an adequate size of sample to detect small differences, the average used was only 10% of the required number Our findings suggest that randomised, controlled trials in clinical orthopaedic research utilise sample sizes which are too small to ensure statistical significance for what may be clinically important results.

Humans↗

PubMed1988

Assessment of sodium and potassium intakes.

The sodium, potassium and creatinine contents of three non-consecutive 24-h urine samples collected by 34 selected adult individuals (10 m; 24 f) living in Cork City were determined. The pooled mean 24-h excretion of sodium and potassium in collections adjudged to be complete were 152 mmol and 78 mmol, respectively. There was no significant difference between group average weekday and weekend-day excretion of Na or K, for either males or females. This suggests that weekend 24-h urinary collections, which most subjects find more convenient, are suitable for studies of sodium and potassium intakes of groups. The ratios of intra- to inter-individual variation for 24-h urinary sodium were 1.4 and 2.1 for males and females, respectively. The corresponding ratios for 24-h urinary potassium were 6.6 for males and 4.9 for females. These ratios indicated that there were large individual day-to-day variations in urinary sodium and potassium excretion in this group. It was estimated that a sample size of 35-60 individuals would be required to estimate group mean sodium and potassium intakes by means of single 24-h urine collections.

Adolescent↗

PubMed2000

Estimation of number of subjects required for comparison of drug versus control in adaptive designs.

INTRODUCTION: Adaptive designs have often been proposed as a way of using accruing data to affect future allocation scheme in clinical trials. The goal is to assign more patients to the better treatment. To implement clinical trials efficiently, sample size must be estimated in advance. In adaptive design, it is difficult to calculate the required sample size, because the allocation probabilities keep changing during the course of the trials. METHODS: We focus on the sample size of two-arm (drug versus control) clinical trials. Based on its asymptotic properties, a formula of calculating sample size is derived for the randomised play-the-winner rule. We also compare sample size and power between the randomised play-the-winner rule and equal allocation. Some simulation studies illustrate the operating characteristics of the designs. RESULTS AND DISCUSSION: The required sample size of the randomised play-the-winner rule is slightly larger than that of the equal allocation design in most cases. The randomised play-the-winner rule is recommended for ethical reasons.

Clinical Trials as Topic↗

PubMed2003

Non-parametric estimators of a monotonic dose-response curve and bootstrap confidence intervals.

In this paper we consider study designs which include a placebo and an active control group as well as several dose groups of a new drug. A monotonically increasing dose-response function is assumed, and the objective is to estimate a dose with equivalent response to the active control group, including a confidence interval for this dose. We present different non-parametric methods to estimate the monotonic dose-response curve. These are derived from the isotonic regression estimator, a non-negative least squares estimator, and a bias adjusted non-negative least squares estimator using linear interpolation. The different confidence intervals are based upon an approach described by Korn, and upon two different bootstrap approaches. One of these bootstrap approaches is standard, and the second ensures that resampling is done from empiric distributions which comply with the order restrictions imposed. In our simulations we did not find any differences between the two bootstrap methods, and both clearly outperform Korn's confidence intervals. The non-negative least squares estimator yields biased results for moderate sample sizes. The bias adjustment for this estimator works well, even for small and moderate sample sizes, and surprisingly outperforms the isotonic regression method in certain situations.

Computer Simulation↗

PubMed1981

Statistical limitations in relation to sample size.

The statistical difficulties of estimating cancer risks from low doses of a carcinogen are illustrated by examples from radiation carcinogenesis. Although more is known about dose-response relationships for ionizing radiation than for any other environmental carcinogen, estimates of cancer risk from low radiation doses have been extremely controversial; disagreements by factors of 100 or more are not uncommon. Direct estimation, based on data from populations exposed to low doses, is usually impracticable because of sample size requirements. Curve-fitting analyses, by which higher dose data determine lower dose risk estimates, require simple dose-response models if the estimates are to be statistically stable. The current level of knowledge about biological mechanisms of carcinogenesis dose not usually permit the confident assumption of a simple model, however; thus frequently the choice is between unstable risk estimates obtained using general models and statistically stable estimates whose stability depends on arbitrary model assumptions.

Adult↗

PubMed2005

Modified exact sample size for a binomial proportion with special emphasis on diagnostic test parameter estimation.

The design of epidemiologic studies for the validation of diagnostic tests necessitates accurate sample size calculations to allow for the estimation of diagnostic sensitivity and specificity within a specified level of precision and with the desired level of confidence. Confidence intervals based on the normal approximation to the binomial do not achieve the specified coverage when the proportion is close to 1. A sample size algorithm based on the exact mid-P method of confidence interval estimation was developed to address the limitations of normal approximation methods. This algorithm resulted in sample sizes that achieved the appropriate confidence interval width even in situations when normal approximation methods performed poorly.

Algorithms↗

PubMed1984

Methodological issues in case-control studies IV: Validity and efficiency of various analysis strategies for continuous variables using the unconditional logistic regression model.

Computer simulation has been used to evaluate the performance of the unconditional logistic regression model when used to analyse continuous data from case-control studies. The size of the bias in the odds ratio estimate introduced by small sample sizes and by the use of the incorrect analysis model has been estimated for various underlying population conditions, together with the power as a measure of efficiency. It is concluded that the model is robust over a wide range of exposures to the risk factor, in the sense that both small sample sizes and use of the incorrect model introduced relatively small biases. The power of the test is again little altered by use of the incorrect analysis model.

Aged↗

PubMed1998

Power and sample size for survival analysis under the Weibull distribution when the whole lifespan is of interest.

Accessible and readily utilized software, tables and approximation formulae have been developed to estimate power and sample size for studies of time to event (survival times) when the survival times are assumed to be exponential. These methods can markedly misestimate power when the distribution is Weibull and not exponential. The Weibull distribution with increasing hazard is common in aging research, especially when the whole life span of the subjects is of interest. This note considers an extension of power and sample size calculations, previously developed under the exponential distributional assumption, to the more general case of the Weibull distribution for a prospective comparative follow-up study. The hypotheses are defined in terms of the ratio of the median survival times between two groups. It is shown that the power and sample sizes are heavily dependent on the shape parameter of the Weibull distribution. Using the extensions developed, investigators can use existing software and tables to calculate power and sample size under the assumption of a Weibull distribution.

Algorithms↗

PubMed1995

Postcranial estimates of body weight in Proconsul, with a note on a distal tibia of P. major from Napak, Uganda.

A distal tibia of Proconsul major from Napak, Uganda, is described. It is morphologically similar to other Proconsul tibiae, only much larger in size. This specimen and others are used to estimate the body weight of P. major from postcrania for the first time. Body weight is predicted from articular and diaphyseal dimensions using regression equations derived from a modern comparative sample of catarrhine primates. The estimated body weight of P. major based on the Napak tibia is 86.7 kg, whereas two other P. major specimens are smaller, giving a total range of 63.4-86.7 kg and an average of 75.1 kg. The regression equations are also used to predict the body weight of specimens from Rusinga/Mfangano belonging to P. nyanzae and P. heseloni. As the body weight estimates generated here are consistent with previous postcranial-based estimates for Proconsul species, the two sets of estimates are pooled to give means of 10.9 kg for P. heseloni (n = 6) and 35.6 kg for P. nyanzae (n = 12). These findings support the traditional assignment of two species at Rusinga/Mfangano. The postcranial body weight estimates for the three species of Proconsul are compared to body weights estimated from M1 area in order to investigate possible differences in scaling between the teeth and limbs in these species. Despite being based on a smaller sample size, the postcranial estimates clearly differentiate the three taxa, whereas the dental estimates form a more continuous distribution. Molar area overestimates body weight in P. heseloni, indicating that it is megadont compared to a large sample of modern anthropoid primates. In contrast, molar area underestimates body weight in P. Nyanzae and especially P. major, suggesting relative microdonty in these taxa.

Animals↗

PubMed2003

Efficiency of single-nucleotide polymorphism haplotype estimation from pooled DNA.

The efficiency of single-nucleotide polymorphism haplotype analysis may be increased by DNA pooling, which can dramatically reduce the number of genotyping assays. We develop a method for obtaining maximum likelihood estimates of haplotype frequencies for different pool sizes, assess the accuracy of these estimates, and show that pooling DNA samples is efficient in estimating haplotype frequencies. Although pooling K individuals increases ambiguities, at least for small pool size K and small numbers of loci, the uncertainty of estimation increases <K times that of unpooled DNA. We also develop the asymptotic variance-covariance of maximum likelihood estimates and evaluate the accuracy of variance estimates by Monte Carlo methods. When the sample size of pools is moderately large, the asymptotic variance estimates are rather accurate. Completely or partially missing genotyping information is allowed for in our analysis. Finally, our methods are applied to single-nucleotide polymorphisms in the angiotensinogen gene.

Algorithms↗

PubMed1995

Imprecise exposure assessment and the sample size requirements of case-control studies of residential magnetic field exposure and cancer in adults.

A computer program simulating case-control studies is described. It is used to estimate the minimum sample size required and to assess how this is affected by imprecise exposure assessment. In particular, the consequences of neglecting measurements of nonresidential exposure in case-control studies of residentially exposed adults are investigated. According to this model, while the consequent loss of power is not as large as was predicted by algebraic methods, it would be unwise to neglect it when planning a study.

Adult↗

PubMed2006

Using design effects from previous cluster surveys to guide sample size calculation in emergency settings.

A good estimate of the design effect is critical for calculating the most efficient sample size for cluster surveys. We reviewed the design effects for seven nutrition and health outcomes from nine population-based cluster surveys conducted in emergency settings. Most of the design effects for outcomes in children, and one-half of the design effects for crude mortality, were below two. A reassessment of mortality data from Kosovo and Badghis, Afghanistan revealed that, given the same number of clusters, changing sample size had a relatively small impact on the precision of the estimate of mortality. We concluded that, in most surveys, assuming a design effect of 1.5 for acute malnutrition in children and two or less for crude mortality would produce a more efficient sample size. In addition, enhancing the sample size in cluster surveys without increasing the number of clusters may not result in substantial improvements in precision.

Afghanistan↗

PubMed1987

A simple estimator of minimum detectable relative risk, sample size, or power in cohort studies.

In planning simple cohort mortality studies, researchers need to know what size of relative risk may be confidently detected with the projected size of the cohort and length of follow-up. Although methods for the calculation of such minimum detectable risks have been devised for case-control studies and for cohort studies where internal comparisons are the focus, this has not been explicitly done for cohort studies using external comparisons. This paper describes a simple procedure designed explicitly for investigating the adequacy of cohort size at the planning stage of a study. An example is presented of a retrospective cohort study of men in a Canadian factory. A method is shown for estimating the minimum detectable underlying relative risk for lung cancer in this cohort.

Adult↗

PubMed2004

Estimating dose equivalence for new routes of drug administration.

For patient's convenience, dose administration of insulin via oral inhalation is often considered as an alternative to subcutaneous administration. An important statistical problem is to estimate dose equivalence, which is the amount of drug needed to be delivered by inhalation to generate an equivalent pharmacokinetic (PK) response produced by a therapeutic dose of subcutaneous insulin. Because of high intersubject variability, a crossover design clinical trial is typically used where data from both routes of administration are obtained from the same subject. A linear mixed effects model is proposed to describe the relationship between AK response and insulin dose for the two routes of administration. Estimation of dose equivalence in this setting has not been discussed in the statistical literature. Several competing methods for estimating dose equivalence are proposed and contrasted. A formula for calculating an approximate sample size necessary to estimate dose equivalence with a desired precision for the new route of administration is also provided.

Administration, Inhalation↗

PubMed2001

Controlling test size while gaining the benefits of an internal pilot design.

To compensate for a power analysis based on a poor estimate of variance, internal pilot designs use some fraction of the planned observations to reestimate error variance and modify the final sample size. Ignoring the randomness of the final sample size may bias the final variance estimate and inflate test size. We propose and evaluate three different tests that control test size for an internal pilot in a general linear univariate model with fixed predictors and Gaussian errors. Test 1 uses the first sample plus those observations guaranteed to be collected in the second sample for the final variance estimate. Test 2 depends mostly on the second sample for the final variance estimate. Test 3 uses the unadjusted variance estimate and modifies the critical value to bound test size. We also examine three sample-size modification rules. Only test 2 can control conditional test size, align with a modification rule, and provide simple power calculations. We recommend it if the minimum second (incremental) sample is at least moderate (perhaps 20). Otherwise, the bounding test appears to have the highest power in small samples. Reanalyzing published data highlights some advantages and disadvantages of the various tests.

Biometry↗

PubMed2000

Correcting for ascertainment bias of relative-risk estimates obtained using affected-sib-pair linkage data.

Locus-specific sibling relative risk is often estimated using affected-sib-pair lod score analysis of affected sibships and may be used to decide whether to continue or discontinue the search for additional susceptibility genes. We showed that relative-risk estimates obtained using affected-sib-pair data are asymptotically unbiased when each pair is given a weight inversely proportional to the sibship ascertainment probability. Here we show by simulation that the extent of the bias of relative risks estimated using the incorrect ascertainment weights is small for dominant models, but large for single-locus recessive models and some two-locus heterogeneity models. Since in practice the ascertainment scheme is often unknown, we investigate methods for jointly estimating ascertainment and relative risks from affected-sibship data. Given a sufficient sample size, a reasonable estimate of relative risk may always be obtained using only affected pairs from sibships with two affected and no unaffected siblings. This estimate, which has a large variance, may then be used in a three-stage procedure (which we call the alpha method) to estimate consistently both the ascertainment probabilities and the relative risks with greater precision. We additionally propose correction factors to eliminate small-sample bias of relative risks and investigate the bias due to error in the estimate of disease locus location.

Bias↗

PubMed2006

What is the extent of prokaryotic diversity?

The extent of microbial diversity is an intrinsically fascinating subject of profound practical importance. The term 'diversity' may allude to the number of taxa or species richness as well as their relative abundance. There is uncertainty about both, primarily because sample sizes are too small. Non-parametric diversity estimators make gross underestimates if used with small sample sizes on unevenly distributed communities. One can make richness estimates over many scales using small samples by assuming a species/taxa-abundance distribution. However, no one knows what the underlying taxa-abundance distributions are for bacterial communities. Latterly, diversity has been estimated by fitting data from gene clone libraries and extrapolating from this to taxa-abundance curves to estimate richness. However, since sample sizes are small, we cannot be sure that such samples are representative of the community from which they were drawn. It is however possible to formulate, and calibrate, models that predict the diversity of local communities and of samples drawn from that local community. The calibration of such models suggests that migration rates are small and decrease as the community gets larger. The preliminary predictions of the model are qualitatively consistent with the patterns seen in clone libraries in 'real life'. The validation of this model is also confounded by small sample sizes. However, if such models were properly validated, they could form invaluable tools for the prediction of microbial diversity and a basis for the systematic exploration of microbial diversity on the planet.

Archaea↗