Showing papers on "Sample size determination published in 1980"
TL;DR: In this article, the problem of finding consistent estimators in other models is non-trivial, however, since the number of incidental parameters is increasing with sample size, and it is well known that analysis of covariance in the linear regression model does not have this consistency property.
Abstract: This paper deals with data that has a group structure. A simple example in the context of a linear regression model is E(yitlx, 1S, ar) = P'xit + ai (i = 1, ...,9 N; t = 1, ... T), where there are T observations within each of N groups. The ai are group specific parameters. Our primary concern is with the estimation of f3, a parameter vector common to all groups. The role of the ai is to control for group specific effects; i.e. for omitted variables that are constant within a group. The regression function that does not condition on the group will not in general identify 1: E(yitlx, 13) 0 1'xit. In this case there is an omitted variable bias. An important application is generated by longitudinal or panel data, in which there are two or more observations on each individual. Then the group is the individual, and the ai capture individual differences. If these person effects are correlated with x, then a regression function that fails to control for them will not identify f. In another important application the group is a family, with observations on two or more siblings within the family. Then the ai capture omitted variables that are family specific, and they give a concrete representation to family background. We shall assume that observations from different groups are independent. Then the ai are incidental parameters (Neyman and Scott (1948)), and 0, which is common to the independent sampling units, is a vector of structural parameters. In the application to sibling data, T is small, typically T= 2, whereas there may be a large number of families. Small T and large N are also characteristic of many of the currently available longitudinal data sets. So a basic statistical issue is to develop an estimator for j that has good properties in this case. In particular, the estimator ought to be consistent as N -> ac for fixed T. It is well-known that analysis of covariance in the linear regression model does have this consistency property. The problem of finding consistent estimators in other models is non-trivial, however, since the number of incidental parameters is increasing with sample size. We shall work with the following probability model: Yit is a binary variable with
2,398 citations
TL;DR: In this paper, the effect of unequal sample sizes on the error rates of six procedures for pair-wise multiple comparisons with homogeneous variances was studied by computer simulation, and the Tukey-Kramer method was recommended for use in the unequal sample size, homogeneous variance situation.
Abstract: The effect of unequal ni on the error rates of six procedures for pair-wise multiple comparisons beteen k treatment means with homogeneous variances was studied by computer simulation. A commonly used method, attributed to Kramer (1956) but suggested also by Tukey (1953), was found to have error rates less than the nominal value α for several patterns of inequality in the sample sizes, at least when the variations in ni were moderately large. A method that substitutes the harmonic mean sample size for n in Tukey's T method had excessively high error rates. Other methods proposed more recently in the literature were conservative relative to the Tukey-Kramer method. Thus, the Tukey-Kramer method is recommended for use in the unequal sample size, homogeneous variance situation.
344 citations
TL;DR: A simple approximation is provided to the formula for the sample sizes needed to detect a difference between two binomial probabilities with specified significance level and power, showing that over fairly wide ranges of parameter values and ratios of sample sizes, the percentage error is no greater than 1%.
Abstract: A simple approximation is provided to the formula for the sample sizes needed to detect a difference between two binomial probabilities with specified significance level and power The formula for equal sample sizes was derived by Casagrande, Pike and Smith (1978, Biometrics 34 , 483-486) and can be easily generalized to the case of unequal sample sizes It is shown that over fairly wide ranges of parameter values and ratios of sample sizes, the percentage error which results from using the approximation is no greater than 1% The approximation is especially useful for the inverse problem of estimating power when the sample sizes are given
327 citations
TL;DR: In this paper, Monte Carlo techniques are used to examine the applicability of the normal approximations for moderate sample sizes with moderate numbers of cells for goodness-of-fit tests for multinomial data.
Abstract: Traditional discussions of goodness-of-fit tests for multinomial data consider asymptotic chi-squared properties under the assumption that all expected cell frequencies become large. This condition is not always satisfied, however, and another asymptotic theory must be considered. For testing a specified simple hypothesis, Morris (1975) and Hoist (1972) gave conditions for the asymptotic normality of the Pearson and likelihood ratio statistics when both the sample size and number of cells become large (even if the expected cell frequencies remain small). Monte Carlo techniques are used to examine the applicability of the normal approximations for moderate sample sizes with moderate numbers of cells.
273 citations
TL;DR: In this article, an asymptotically distribution-free procedure is proposed for testing whether a univariate population is symmetric about some unknown value versus a broad class of asymmetric distribution alternatives.
Abstract: An asymptotically distribution-free procedure is proposed for testing whether a univariate population is symmetric about some unknown value versus a broad class of asymmetric distribution alternatives. The consistency class of the test is discussed and two competing tests are described, one based on the sample skewness, and the other on Gupta's nonparametric procedure. A Monte Carlo study shows that the proposed test is superior to either competitor since it maintains the designated α levels fairly accurately even for sample sizes as small as 20, while displaying good power for detecting asymmetric distributions.
216 citations
TL;DR: In this article, it was shown that the asymptotic theory seems to be appropriate when the regularity conditions obtain and sample size is at least 30, but not satisfied in all sample sizes considered.
Abstract: The use of the likelihood ratio statistic in testing the goodness of fit of the exploratory factor model has no formal justification when, as is often the case in practice, the usual regularity conditions are not met. In a Monte Carlo experiment it is found that the asymptotic theory seems to be appropriate when the regularity conditions obtain and sample size is at least 30. When the regularity conditions are not satisfied, the asymptotic theory seems to be misleading in all sample sizes considered.
206 citations
TL;DR: Three proposed stopping rules are studied by the Monte Carlo method and are shown to be about equally effective and asymptotically optimal from both Bayesian and frequentist points of view and are markedly superior to any fixed sample size procedure.
Abstract: A model for sequential clinical trials is discussed. Three proposed stopping rules are studied by the Monte Carlo method for small patient horizons and mathematically for large patient horizons. They are shown to be about equally effective and asymptotically optimal from both Bayesian and frequentist points of view and are markedly superior to any fixed sample size procedure.
198 citations
TL;DR: In this paper, three procedures are developed for the situation in which the coefficients are determined from the same sample, which give tight control of Type I error when the sample size is 50 or greater.
Abstract: In measurement studies the researcher may wish to test the hypothesis that Cronbach's alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.
163 citations
TL;DR: Four classification algorithms-discriminant functions when classifying individuals into two multivariate populations are compared and it is shown that the classification error EPN depends on the structure of a classification algorithm, asymptotic probability of misclassification P¿, and the ratio of learning sample size N to dimensionality p:N/p.
Abstract: This paper compares four classification algorithms-discriminant functions when classifying individuals into two multivariate populations. The discriminant functions (DF's) compared are derived according to the Bayes rule for normal populations and differ in assumptions on the covariance matrices' structure. Analytical formulas for the expected probability of misclassification EPN are derived and show that the classification error EPN depends on the structure of a classification algorithm, asymptotic probability of misclassification P?, and the ratio of learning sample size N to dimensionality p:N/p for all linear DF's discussed and N2/p for quadratic DF's. The tables for learning quantity H = EPN/P? depending on parameters P?, N, and p for four classifilcation algorithms analyzed are presented and may be used for estimating the necessary learning sample size, detennining the optimal number of features, and choosing the type of the classification algorithm in the case of a limited learning sample size.
153 citations
TL;DR: In this article, the authors used computer simulations to determine the sampling distributions of four indices of overlap or similarity: the coefficient of community, Morisita's index, Horn's information theory index, and Euclidean distance.
Abstract: We have used computer simulations to determine the sampling distributions of four indices of overlap or similarity: the coefficient of community, Morisita's index, Horn's information theory index, and Euclidean distance. Estimates of overlap were systematically biased downward when sample size was small and when expected values were close to 1. The standard deviations of samples of indices were greatest when expected values were intermediate between 0 and 1, and sample sizes were small. In studies having sample sizes of 25, 50, or 100, one could expect the standard error of an estimated index of similarity to fall between 0.05 and 0.10, provided that samples were truly drawn from homogeneous populations. We suggest that simulations be used to estimate confidence limits on similarity and overlap indices where hypothesis testing is required. In addition, efforts should be made to develop indices of overlap for which statistical measures of dispersion and bias can be derived analytically.
146 citations
Book•
01 Jan 1980
TL;DR: In this paper, the authors present an analysis of single-factor and mixed-within-subjects factorial designs, using the t test to analyze single-df comparisons estimating population means and effect size.
Abstract: Part 1 Experimenta l design and preliminary data analysis: introduction to experimental design - getting started, how do psychologists conduct research?, experimental research design, summary, exercises preliminary data analysis - the mean as a measure of central tendency, the variance as a measure of variability, additional descriptive techniques, summary, exercises. Part 2 The analysis of single-factor experiments: the logic of hypothesis testing - neutralizing nuisance variables through randomization, index of the treatment effects, hypothesis testing, summary, exercises calculating the F ratio - design and notation, partitioning the total sum of squares, sums of squares - computational formulas, the analysis of variance, summary, exercises evaluating the F ratio - the sampling distribution of F, determining the critical value of F, forming the decision rule, assumptions underlying the analysis, a complete numerical example, special analysis with two treatment conditions, summary, exercises, appendix: an explanation of the correction for unequal variances analytical comparisons in the single-factor design - the nature of analytical comparisons, an example of the relationship between research hypotheses and analytical comparisons, analyzing differences between pairs of means, more complex analytical comparisons, summary, exercises, appendix: using the t test to analyze single-df comparisons estimating population means and effect size - interval estimation in experiments, the magnitude of treatment effects, summary, exercises errors of hypothesis testing and statistical power - statistical errors in hypothesis testing, cumulative type 1 error, using power to estimate sample size, summary, exercises, appendix: an alternative method for estimating sample size. Part 3 The analysis of factorial designs: introduction to the analysis of factorial experiments - the factorial experiment, main effects and interaction, identifying basic deviations, the analysis of variance, calculating sums of squares, a numerical example, summary, exercises analytical comparisons in the factorial design - interpreting F tests in the factorial design, the detailed analysis of main effects, analyzing simple effects, analyzing simple comparisons, an overall plan of analysis, summary, exercises, appendix: analyzing interaction contrasts. Part 4 The analysis of within-subjects designs: the single-factor within-subjects design - reducing error variance, logic of the analysis of within-subjects designs, computational formulas for the analysis of variance, a numerical example, analytical comparisons, planning within-subjects designs, summary, exercises, appendix: using separate error terms to evaluate analytical comparisons the mixed within-subjects factorial design - a comparison of factorial designs, the logic of the analysis, analysis of variance, a numerical example, analytical comparisons, summary, exercises part contents.
TL;DR: In the present paper, several alternative statistics, based on the use of the Fisher r-to-z transform, are proposed, and their performance (as well as that of the traditional statistics) is assessed in a Monte Carlo experiment.
Abstract: The goodness-of-fit of correlational pattern hypotheses has traditionally been assessed either with a likelihood ratio statistic (in conjunction with maximum likelihood estimation) or with a quadratic form statistic (in conjunction with generalized least squares estimates). In the present paper, several alternative statistics, based on the use of the Fisher r-to-z transform, are proposed, and their performance (as well as that of the traditional statistics) is assessed in a Monte Carlo experiment. The new statistics are shown to have Type I error rate performance at smaller sample sizes which is notably superior to their more traditional counterparts.
TL;DR: In this paper, the authors give variable sampling plans for items whose failure times are distributed as either extreme-value variates or Weibull variates (the logarithms of which are from an extreme value distribution).
Abstract: In this paper, we give variables sampling plans for items whose failure times are distributed as either extreme-value variates or Weibull variates (the logarithms of which are from an extreme-value distribution). Tables applying to acceptance regions and operating characteristics for sample size n, ranging from 3 to 18, are given. The tables allow for Type II censoring, with censoring number r ranging from 3 to n. In order to fix the maximum time on test, the sampling plan also allows for Type I censoring. Acceptance/rejection is based upon a statistic incorporating best linear invariant estimates, or, alternatively, maximum likelihood estimates of the location and scale parameters of the underlying extreme value distribution. The operating characteristics are computed using an approximation discussed by Fertig and Mann (1980).
TL;DR: In this paper, the relationship between the response, y, and the subsidiary variate, x, is linear through the origin and the variance of y is proportional to x, where the correlation coefficient p between 9' and X is positive.
Abstract: In sample surveys supplementary information is often used for increasing the precision of estimators. A good example of this is the ratio method of estimation. This is most effective when the relationship between the response, y, and the subsidiary variate, x, is linear through the origin and the variance of y is proportional to x. The method can be used with simple random sampling, stratified random sampling or other types of survey designs. Let I' and X be unbiased estimators of the parameters Y and X corresponding to the variates y and x respectively, based on any probability sampling design. Examples of such parameters are population totals and means. It is assumed that X is known. For simplicity assume all measurements to be nonnegative and X and X to be positive. Let the correlation coefficient p between 9' and X be positive. Then the traditional ratio method of estimation uses Sr= IX/I to estimate Y. Let N and n < N be the population and the sample sizes respectively. Then clearly
TL;DR: The statistical considerations for this type of study are presented and criteria for deciding whether a treatment deserves further testing are specified and the required sample size is computed.
Abstract: A pilot study is a non randomized clinical trial that is conducted to decide whether a new treatment should be tested in a large controlled trial. The statistical considerations for this type of study are presented. Criteria for deciding whether a treatment deserves further testing are specified and the required sample size is computed.
TL;DR: In this paper, it was shown that the t test is not always robust to the assumption of equal population variances even when sample sizes are equal, even when the distribution of unequal variances is known.
Abstract: It is noted that disagreements have arisen in the literature about the robustness of the t test in normal populations with unequal variances. Hsu's procedure is applied to determine exact Type I error rates for t. Employing fairly liberal but objective standards for assessing robustness, it is shown that the t test is not always robust to the assumption of equal population variances even when sample sizes are equal. Several guidelines are suggested including the point that to apply t at α = .05 without regard for unequal variances would require equal sample sizes of at least 15 by one of the standards considered. In many cases, especially those with unequal N's, an alternative such as Welch's procedure is recommended.
TL;DR: In this paper, the authors proposed the use of the likelihood ratio statistic in choosing between a Neibull or gamma model, values of the probability of correct seeiection are obtained by Monte Carlo simulation.
Abstract: This paper proposes the use of tne likelihood ratio statistic in choosing between a Neibull or gamma model, values of the probability of correct seeiection are obtained by Monte Carlo simulation. This method provides some basis for decision even when the sample size is small. The technique is applied to four sets of data from the literature.
TL;DR: A useful measure of diversity was calculated for microbial communities collected from lake water and sediment samples using the Shannon index (H') and rarefaction [E(S]], which accounts for differences in sample size inherently so that comparisons are made simple.
Abstract: A useful measure of diversity was calculated for microbial communities collected from lake water and sediment samples using the Shannon index (H′) and rarefaction [E(S)]. Isolates were clustered by a numerical taxonomy approach in which limited (<20) tests were used so that the groups obtained represented a level of resolution other than species. The numerical value of diversity for each sample was affected by the number of tests used; however, the relative diversity compared among several sampling locations was the same whether 11 or 19 characters were examined. The number of isolates (i.e., sample size) strongly influenced the value of H′ so that unequal sized samples could not be compared. Rarefaction accounts for differences in sample size inherently so that such comparisons are made simple. Due to the type of sampling carried out by microbiologists, H′ is estimated and not determined and therefore requires a statement of error associated with it. Failure to report error provided potentially misleading results. Calculation of the variance of H′ is not a simple matter and may be impossible when handling a large number of samples. With rarefaction, the variance of E(S) is readily determined, facilitating the comparison of many samples.
01 Oct 1980
TL;DR: In this article, the statistical reproducibility of the dynamic and static fatigue experiments used to measure the fatigue constants was analyzed using both statistical theory and a Monte Carlo computer simulation technique.
Abstract: : The number of test samples used to characterize the fatigue constants needed for failure predictions for ceramic materials determines the confidence in these predictions. The statistical reproducibility of the dynamic and static fatigue experiments used to measure the fatigue constants was analyzed using both statistical theory and a Monte Carlo computer simulation technique. It was found that the statistical reproducibility depended not only on the number of test samples but also on the other experimental test variables. It was shown that the uncertainty in the statistical reproducibility can be large especially for sample size less than about 100. Guidelines for selecting the optimum sample size for a given dynamic or static fatigue experiment are given. It is recommended that before meaningful conclusions can be drawn regarding the effect of a test variable on fatigue, the statistical reproducibility of the experiment be determined. (Author)
TL;DR: The equations and tables presented in this paper reflect the extent to which the number of patients required to receive the new therapy depends on thenumber of historical controls selected for comparison.
TL;DR: A general model is formulated that allows for time-dependent dropout and event rates in the determination of sample sizes for long-term medical trials when a therapy group and a control group are to be compared.
TL;DR: A partial solution to the lack of understanding by business researchers of the many statistical considerations associated with classification when discriminant analysis is employed by focusing specific attention on ten factors or items that may directly influence the reported classification results.
TL;DR: In this article, the authors investigated the robustness of Z, t, and F tests against heterogeneity in a large-scale sampling study under conditions realistic to experimentation and testing in the behavioral sciences and found that robustness was strongly influenced by all of the factors investigated, and interactions among the influencing factors were often strong and complex.
Abstract: The alleged robustness of Z, t, and F tests against nonnormality and, when sample sizes are equal, of t and F tests against heterogeneity as well was investigated in a large-scale sampling study under conditions realistic to experimentation and testing in the behavioral sciences. Factors varied were: population shape (L or bell), σ1/σ2 (1/2, 1, or 2), size N of smallest sample (2, 4, 8, 16, 32, 64, 128, 256, 512, or 1,024), N1/N2 (1/3,1/2,1, 2, or 3), α (.05,.01, or.001), and test tailedness (left, right, or two). In about 25% of the situations investigated, the test failed to meet a very lax criterion for robustness at every examined N value less than 100, and in 8% at every value less than 1,000; no test met the criterion in all of the situations studied before N=512. Robustness was strongly influenced by all of the factors investigated, and interactions among the influencing factors were often strong and complex.
TL;DR: A simple approximation is given to the sample size(s) required for the Yates-corrected chi squared test to have specified power; it is then compared with other approximations and with the exact sample size for the equal sample case.
Abstract: An investigator wishes to compare two independent proportions, based on perhaps unequal sample sizes, by means of the chi squared test with the Yates' correction. A simple approximation is given to the sample size(s) required for the Yates-corrected chi squared test to have specified power; it is then compared with other approximations and with the exact sample size for the equal sample case. In that case the proposed approximation is quite similar to the approximate formula recently put forward by Casagrande, Pike and Smith (1978, Biometrics 34, 483-486).
TL;DR: In this paper, the case-control studies where the cases are individually matched to a fixed number of controls are extended to allow the matching ratio to vary; this can occur by design, by censoring, or in an interim data analysis.
Abstract: SUMMARY Previous authors have analysed case-control studies where the cases are individually matched to a fixed number of controls. This is extended here to allow the matching ratio to vary; this can occur by design, by censoring, or in an interim data analysis. Statistics are derived to test the case-control difference for both binary and continuous variables; the power functions and required sample sizes are also given. Finally the effect of the matching ratio on cost-efficiency is discussed. Examples of the calculations are shown using data on Hodgkin's disease.
TL;DR: In this article, a characterization of the structure of optimal tests for the Koopman-Darmois family is given, and a new proof of the optimality property of these tests is obtained as a corollary.
Abstract: The testing problem is to decide on the basis of repeated independent observations which of the probability densitiesf andg is true. Given upper bounds on the probabilities of error, the object is to minimize the expected sample size if the densityp is true (allowed to differ fromf andg). A characterization of the structure of optimal tests is obtained which is particularly informative in the case wheref,g, andp belong to a Koopman-Darmois family. Ifp=f org, then the optimal tests are sequential probability ratio tests (SPRT's) and a new proof of the well-known optimality property of these tests is obtained as a corollary.
TL;DR: In this article, the exact critical values for Bartlett's test for homogeneity of variances based on equal sample sizes from several normal populations are tabulated and shown how these values may be used to obtain highly accurate approximations to the critical value for unequal sample sizes.
Abstract: The exact critical values for Bartlett's test for homogeneity of variances based on equal sample sizes from several normal populations are tabulated. It is also shown how these values may be used to obtain highly accurate approximations to the critical values for unequal sample sizes. An application is given that deals with the variability of log bids on a group of federal offshore oil and gas leases.
TL;DR: In this paper, the first four exact moments of the autocorrelations of a time series realisation from a general autoregressive moving average process of order (p, d, q) with d = 0 or 1 were computed.
TL;DR: In this paper, the deficiency of sample quantiles with respect to quasiquantiles is investigated under the assumption that the true density function has bounded derivatives, and it is shown that the sample quantile is still an efficient estimator of the true quantile but the relative deficiency tends to infinity for increasing sample sizes.
Abstract: The deficiency of sample quantiles with respect to quasiquantiles is investigated under the assumption that the true density function has bounded derivatives. Then the sample quantile is still an efficient estimator of the true quantile but the relative deficiency of sample quantiles with respect to suitably defined quasiquantiles quickly tends to infinity for increasing sample sizes. If the second derivative of the true density function is bounded, then adaptive estimators will be found which are of a better performance than quasiquantiles. Corresponding results are derived for two-sided confidence intervals which are based on quasiquantiles and adaptive estimators.
TL;DR: In this paper, the problem of estimating the bounds of random variables has been discussed and the optimality of estimates when the data is censored so that only the largest or smallest of the observations is available for estimating a bound is discussed.
Abstract: : The problem of estimating the bounds of random variables has been previously discussed. Here we discuss optimality of estimates when the data is censored so that only the r largest or smallest of the observations is available for estimating a bound. For fixed r we find the linear function of the censored data which is the optimal estimator of a bound in the sense that, when the sample size is large, the estimator has smallest mean squared error among all such linear estimators. Provided r is not close to one, these estimators are almost optimal when the entire sample is available since, for example, when estimating an upper bound and the sample size is large, the largest few observations carry most of the information about the bound. This fact is illustrated in one case.