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PubMed2004

Who was student and why do we care so much about his t-test?

Statistics is the study of populations, how they relate to one another and what effect sampling has in terms of representing the original population. Use of statistical tools has been facilitated by modern computer technology; however, given the ease in which results are obtained, it is easy to overlook potentially incorrect use of the various statistical tests. Statistical tools are invaluable for investigators because they make it possible to determine if scientific results are important. When a test is used, it is important to know that the result of any statistical test is valid to avoid erroneous conclusions. To do so, the investigator must have a basic understanding of the test's assumptions and limitations. Modern statistical packages often include a variety of results relating to the test's applicability to the data analyzed. Not uncommonly, biologists are unfamiliar with these analyses. This review intends to improve the reader's understanding of t-tests by providing a history of Student's t-test, and its assumptions, applications, and limitations. Part 1 of the series ("The Mean and Standard Deviation: What Does It All Mean?") reviewed basic aspects of distributions, measures of central tendency, and dispersion assessment. Small sample size effects on accurate estimation of a population mean and group comparisons for continuous data are presented in this review.

Data Interpretation, Statistical↗
PubMed1984

Estimating parameters of the family-size distribution in ascertainment sampling schemes: numerical results.

It is argued that, in any ascertainment sampling scheme using data from families of various sizes, there is never any need to assume a particular form for the (unknown) family-size distribution. There exists a simple conditional method, making no assumptions about the family-size distribution, that is always preferable to the assumption of any particular distributional form. Furthermore, the simplicity of the conditional method gives insights into properties of estimates of genetic and ascertainment parameters, which are not available when a particular form for the family-size distribution is assumed.

Biometry↗
PubMed2003

Genetic and demographic dynamics of small populations of Silene latifolia.

Small local populations of Silene alba, a short-lived herbaceous plant, were sampled in 1994 and again in 1999. Sampling included estimates of population size and genetic diversity, as measured at six polymorphic allozyme loci. When averaged across populations, there was very little change between samples (about three generations) in population size, measures of within-population genetic diversity such as number of alleles or expected heterozygosity, or in the apportionment of genetic diversity within and among populations as measured by F(st). However, individual populations changed considerably, both in terms of numbers of individuals and genetic composition. Some populations doubled in size between samples, while others shrank by more than 75%. Similarly, expected heterozygosity and allele number increased by more than two-fold in individual populations and decreased by more than three-fold in others. When population-specific change in number and change in measures of genetic diversity were considered together, significant positive correlations were found between the demographic and genetic variables. It is speculated that some populations were released from the demographic consequences of inbreeding depression by gene flow.

Alleles↗
PubMed1992

Ethnic differentiation at VNTR loci, with special reference to forensic applications.

Allele-rich VNTR loci provide valuable information for forensic inference. Interpretation of this information is complicated by measurement error, which renders discrete alleles difficult to distinguish. Two methods have been used to circumvent this difficulty--i.e., binning methods and direct evaluation of allele frequencies, the latter achieved by modeling the data as a mixture distribution. We use this modeling approach to estimate the allele frequency distributions for two loci--D17S79 and D2S44--for black, Caucasian, and Hispanic samples from the Lifecodes and FBI data bases. The data bases are differentiated by the restriction enzyme used: PstI (Lifecodes) and HaeIII (FBI). Our results show that alleles common in one ethnic group are almost always common in all ethnic groups, and likewise for rare alleles; this pattern holds for both loci. Gene diversity, or heterozygosity, measured as one minus the sum of the squared allele frequencies, is greater for D2S44 than for D17S79, in both data bases. The average gene diversity across ethnic groups when PstI (HaeIII) is used is .918 (.918) for D17S79 and is .985 (.983) for D2S44. The variance in gene diversity among ethnic groups is greater for D17S79 than for D2S44. The number of alleles, like the gene diversity, is greater for D2S44 than for D17S79. The mean numbers of alleles across ethnic groups, estimated from the PstI (HaeIII) data, are 40.25 (41.5) for D17S79 and 104 (103) for D2S44. The number of alleles is correlated with sample size. We use the estimated allele frequency distributions for each ethnic group to explore the effects of unwittingly mixing populations and thereby violating independence assumptions. We show that, even in extreme cases of mixture, the estimated genotype probabilities are good estimates of the true probabilities, contradicting recent claims. Because the binning methods currently used for forensic inference show even less differentiation among ethnic groups, we conclude that mixture has little or no impact on the use of VNTR loci for forensics.

Alleles↗
PubMed2003

Gene genealogies when the sample size exceeds the effective size of the population.

We study the properties of gene genealogies for large samples using a continuous approximation introduced by R. A. Fisher. We show that the major effect of large sample size, relative to the effective size of the population, is to increase the proportion of polymorphisms at which the mutant type is found in a single copy in the sample. We derive analytical expressions for the expected number of these singleton polymorphisms and for the total number of polymorphic, or segregating, sites that are valid even when the sample size is much greater than the effective size of the population. We use simulations to assess the accuracy of these predictions and to investigate other aspects of large-sample genealogies. Lastly, we apply our results to some data from Pacific oysters sampled from British Columbia. This illustrates that, when large samples are available, it is possible to estimate the mutation rate and the effective population size separately, in contrast to the case of small samples in which only the product of the mutation rate and the effective population size can be estimated.

DNA, Mitochondrial↗
PubMed2002

Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm.

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.

Algorithms↗
PubMed2002

A comparison of methods to handle skew distributed cost variables in the analysis of the resource consumption in schizophrenia treatment.

BACKGROUND: Transformation of the dependent cost variable is often used to solve the problems of heteroscedasticity and skewness in linear ordinary least square regression of health service cost data. However, transformation may cause difficulties in the interpretation of regression coefficients and the retransformation of predicted values. AIMS OF THE STUDY: The study compares the advantages and disadvantages of different methods to estimate regression based cost functions using data on the annual costs of schizophrenia treatment. METHODS: Annual costs of psychiatric service use and clinical and socio-demographic characteristics of the patients were assessed for a sample of 254 patients with a diagnosis of schizophrenia (ICD-10 F 20.0) living in Leipzig. The clinical characteristics of the participants were assessed by means of the BPRS 4.0, the GAF, and the CAN for service needs. Quality of life was measured by WHOQOL-BREF. A linear OLS regression model with non-parametric standard errors, a log-transformed OLS model and a generalized linear model with a log-link and a gamma distribution were used to estimate service costs. For the estimation of robust non-parametric standard errors, the variance estimator by White and a bootstrap estimator based on 2000 replications were employed. Models were evaluated by the comparison of the R2 and the root mean squared error (RMSE). RMSE of the log-transformed OLS model was computed with three different methods of bias-correction. The 95% confidence intervals for the differences between the RMSE were computed by means of bootstrapping. A split-sample-cross-validation procedure was used to forecast the costs for the one half of the sample on the basis of a regression equation computed for the other half of the sample. RESULTS: All three methods showed significant positive influences of psychiatric symptoms and met psychiatric service needs on service costs. Only the log- transformed OLS model showed a significant negative impact of age, and only the GLM shows a significant negative influences of employment status and partnership on costs. All three models provided a R2 of about.31. The Residuals of the linear OLS model revealed significant deviances from normality and homoscedasticity. The residuals of the log-transformed model are normally distributed but still heteroscedastic. The linear OLS model provided the lowest prediction error and the best forecast of the dependent cost variable. The log-transformed model provided the lowest RMSE if the heteroscedastic bias correction was used. The RMSE of the GLM with a log link and a gamma distribution was higher than those of the linear OLS model and the log-transformed OLS model. The difference between the RMSE of the linear OLS model and that of the log-transformed OLS model without bias correction was significant at the 95% level. As result of the cross-validation procedure, the linear OLS model provided the lowest RMSE followed by the log-transformed OLS model with a heteroscedastic bias correction. The GLM showed the weakest model fit again. None of the differences between the RMSE resulting form the cross- validation procedure were found to be significant. DISCUSSION: The comparison of the fit indices of the different regression models revealed that the linear OLS model provided a better fit than the log-transformed model and the GLM, but the differences between the models RMSE were not significant. Due to the small number of cases in the study the lack of significance does not sufficiently proof that the differences between the RSME for the different models are zero and the superiority of the linear OLS model can not be generalized. The lack of significant differences among the alternative estimators may reflect a lack of sample size adequate to detect important differences among the estimators employed. Further studies with larger case number are necessary to confirm the results. IMPLICATIONS: Specification of an adequate regression models requires a careful examination of the characteristics of the data. Estimation of standard errors and confidence intervals by nonparametric methods which are robust against deviations from the normal distribution and the homoscedasticity of residuals are suitable alternatives to the transformation of the skew distributed dependent variable. Further studies with more adequate case numbers are needed to confirm the results.

Adolescent↗
PubMed1994

Use of small-sample-based reference limits on a group basis.

The evaluation of a biochemical or hematological quantity measured in a study group of employees during occupational health assessments involves a comparison with a reference sample group. Part of this evaluation consists of checking whether the percentage of values larger than a predetermined upper reference limit is significantly larger than the percentage normally expected (2.5%, if the 97.5 percentile is used as the upper reference limit). The reference limit, however, is estimated from a random reference sample, the size of which, for many reasons, may be relatively small; as a consequence, the reference limit estimate will be imprecise. In situations in which the reference sample size is smaller than or not much larger than the study sample size, this imprecision results in the usual binomial test of significance being highly inappropriate. We provide an exact nonparametric test valid for all reference sample sizes.

Adult↗
PubMed2001

Estimating the total number of alleles using a sample coverage method.

Previously reported methods for estimating the number of different alleles at a single locus in a population have not described a useful general result. Using the number of alleles observed in a sample gives an underestimate for the true number of alleles. The similar problem of estimating the number of species in a population was first investigated in 1943. In this article we use the sample coverage method proposed by Chao and Lee in 1992 to estimate the number of alleles in a population when there are unequal allele frequencies. Simulation studies under the recurrent mutation model show that, for reasonable sample sizes, a significantly better estimate of the true number can be obtained than that using only the observed alleles. Results under the stepwise mutation model and infinite-allele model are presented. Possible applications include improving the characterization of the prior distribution for the allele frequencies, adjusting the estimates of genetic diversity, and estimating the range of microsatellite alleles.

Alleles↗
PubMed1989

Sample sizes for constructing confidence intervals and testing hypotheses.

Although estimation and confidence intervals have become popular alternatives to hypothesis testing and p-values, statisticians usually determine sample sizes for randomized clinical trials by controlling the power of a statistical test at an appropriate alternative, even those statisticians who recommend the use of confidence intervals for inference. There is merit in achieving consistency in the techniques for data analysis and sample size determination. To that end, this paper compares sample size determination with use of the length of the confidence interval with that obtained by control of power.

Angina Pectoris↗
PubMed1991

The influence of sampling chamber dimensions on aerosol particle size measurement by cascade impactor and twin impinger.

The influence of the sampling chamber dimensions upon particle size estimation by cascade impaction has been investigated and compared with measurements by the twin impinger. Aerosols of salbutamol and disodium fluorescein (DF) were generated from pressurized metered dose inhalers. The mass median aerodynamic diameter (MMAD) by the Andersen Impactor for salbutamol ranged from 2.0 to 2.8 microns with geometric standard deviations (g.s.d.) of 1.7 to 2.4. These observations were independent of the distance travelled to the first impaction surface of the impactor and volume of the sampling chamber. The DF MMAD ranged from 5.0 to 6.9 microns with g.s.d. values of 1.7 to 1.8. Changes in droplet size within the sampling chamber may cause significant differences in particle size estimates as indicated by the cascade impaction data for DF. The respirable fraction of the salbutamol samples was similar whether determined by impaction or using the impinger. The latter device has previously been indicated to give clinically relevant estimates of respirable fraction for commercial inhalation aerosol devices.

Aerosols↗
PubMed1999

Estimating sexual dimorphism and size differences in the fossil record: a test of methods.

Investigations of size variation in fossil and archaeological skeletal assemblages may be complicated by incomplete skeletons, biased representation of sexes, and the lack of morphological features that identify sex. In order to refine our ability to evaluate size variation, we test the accuracy of three methods that are currently used to estimate size differences in unsexed (pooled) samples: the means method, the median method, and a newly applied technique, the method of moments. Using body mass data from 42 primate species, we calculated actual levels of sexual dimorphism for each species and compared these values to estimates produced by each method. Multivariate regression was used to examine the effects of sample distribution characteristics, including sample size, kurtosis, skewness, sample variance, sex ratio, and intrasexual variance on the performance of the methods. None of the methods appears to be especially accurate. However, one of the simplest methods, the means method, performs relatively well. Factors that lead to inaccuracies in estimation are not readily evident based on multiple regression analysis. We urge caution in the utilization of these techniques, and advocate further analysis of simulated data. Am J Phys Anthropol 110: 95-104, 1999.

Animals↗
PubMed1997

The coalescent process with selfing.

A method of estimating the selfing rate using DNA sequence data was recently proposed by Milligan. Unfortunately, a number of errors make interpretation of his results problematic. In the present paper we first show how the usual coalescent process can be adapted to models that include selfing, and then use this result to find moment estimators as well as the likelihood surface for the selfing rate, s, and the scaled mutation rate, theta. We conclude that, regardless of the method used, large sample sizes are necessary to estimate s with any degree of certainty, and that the estimate is always highly sensitive to recent changes in the true value.

Computer Simulation↗
PubMed2000

Variability in airborne and biological measures of exposure to mercury in the chloralkali industry: implications for epidemiologic studies.

Exposure assessment is a critical component of epidemiologic studies, and more sophisticated approaches require that variation in exposure be considered. We examined the intra- and interindividual sources of variation in exposure to mercury vapor as measured in air, blood, and urine among four groups of workers during 1990-1997 at a Swedish chloralkali plant. Consistent with the underlying kinetics of mercury in the body, the variability of biological measures was dampened considerably relative to the variation in airborne levels. Owing to the effects of intraindividual variation, estimating workers' exposures from a few measurements can attenuate measures of effect. To examine such effects on studies relating long-term exposure to a continuous health outcome, we evaluated the utility of each exposure measure by comparing the necessary sample sizes required for accurate estimation of a slope coefficient obtained from a regression analysis. No single measure outperformed the others for all groups of workers. However, when workers were evaluated together, creatinine-corrected urinary mercury better discriminated workers' exposures than airborne or blood mercury levels. Thus, pilot studies should be conducted to examine variability in both air and biomonitoring data because quantitative information about the relative magnitude of the intra- and interindividual sources of variation feeds directly into our efforts to design an optimal sampling strategy when evaluating health risks associated with occupational or environmental contaminants.

Adult↗
PubMed2004

Statistical issues in multisite effectiveness trials: the case of brief strategic family therapy for adolescent drug abuse treatment.

The statistical development of the multisite Brief Strategic Family Therapy (BSFT) Trial of the National Institute on Drug Abuse's Clinical Trials Network provides a useful, real example of how an effectiveness trial can differ from an efficacy trial. In particular, two design elements distinguish this effectiveness trial from an efficacy trial. First, because the goal of the trial is to show that the use of BSFT would be an improvement on current practice, it was decided to compare BSFT to treatment as usual at each location. This decision ensures that the trial has the most ecological validity to the participating community treatment providers. Second, the desire to generalize the results to general clinical practice dictates that variability (in effect) across community treatment providers be estimated using a random effects model. These two decisions jointly influence the sample size calculations. Allowing variation in treatment as usual, will increase the variability in effect sizes across sites and estimation of this variability as a random effect necessitates a larger sample size (both number of community treatment providers and participants per community treatment provider), than is the case for a fixed site effect estimate. Details of these effects and their implications for the statistical design are presented.

Adolescent↗
PubMed1989

Single-sperm typing: determination of genetic distance between the G gamma-globin and parathyroid hormone loci by using the polymerase chain reaction and allele-specific oligomers.

The frequency of recombination between the G gamma-globin (HBG2) and parathyroid hormone (PTH) loci on the short arm of human chromosome 11 was estimated by typing greater than 700 single-sperm samples from two males. The sperm-typing technique employed involves the polymerase chain reaction and allele-specific oligonucleotide hybridization. Our maximum likelihood recombination fraction estimate of 0.16 (95%) confidence interval, 0.13-0.19) falls well within previous estimates based on family studies. With current technology and a sample size of 1000 sperm, recombination fractions down to approximately 0.009 can be estimated with statistical reliability; with a sample size of 5000 sperm, this value drops to about 0.004. Reasonable technological improvements could result in the detection of recombination frequencies less than 0.001.

Alleles↗
PubMed2006

The accuracy of statistical estimates in genetic studies of aging can be significantly improved.

The sample size of the data used in genetic studies is often a factor limiting the accuracy of statistical estimates. In this paper we suggest a new approach to evaluation of genetic influence on risk of development of aging-related health disorders. The approach results in substantial improvement of the accuracy of statistical estimates without an increase in the size of the genetic sample. The approach is based on the joint analysis of data from the genetic samples and easily accessible non-genetic data, such as data collected in epidemiological, demographic, and longitudinal studies of human aging and aging-related pathologies.

Aging↗
PubMed2006

A measure of the performance of biomarkers for disease.

The statistical properties required for effective biomarkers for disease are examined. It is shown that an "effectiveness parameter" D can be calculated that summarises the performance of a given biomarker and can distinguish between effective and ineffective biomarkers. D can be readily calculated from published summaries of biomarker levels and provides a simpler alternative to the commonly used "Area under the Curve" statistic. The impact of within-individual and between-individual variation in biomarker levels is also evaluated. An approach to the choice of sample size for experiments to estimate D is described.

Biomarkers↗