Showing papers on "Sample size determination published in 1977"

An introduction to statistical methods and data analysis

Abstract: PART I: INTRODUCTION 1. WHAT IS STATISTICS? Introduction / Why Study Statistics? / Some Current Applications of Statistics / What Do Statisticians Do? / Quality and Process Improvement / A Note to the Student / Summary / Supplementary Exercises PART II: COLLECTING THE DATA 2. USING SURVEYS AND SCIENTIFIC STUDIES TO COLLECT DATA Introduction / Surveys / Scientific Studies / Observational Studies / Data Management: Preparing Data for Summarization and Analysis / Summary PART III: SUMMARIZING DATA 3. DATA DESCRIPTION Introduction / Describing Data on a Single Variable: Graphical Methods / Describing Data on a Single Variable: Measures of Central Tendency / Describing Data on a Single Variable: Measures of Variability / The Box Plot / Summarizing Data from More Than One Variable / Calculators, Computers, and Software Systems / Summary / Key Formulas / Supplementary Exercises PART IV: TOOLS AND CONCEPTS 4. PROBABILITY AND PROBABILITY DISTRIBUTIONS How Probability Can Be Used in Making Inferences / Finding the Probability of an Event / Basic Event Relations and Probability Laws / Conditional Probability and Independence / Bayes's Formula / Variables: Discrete and Continuous / Probability Distributions for Discrete Random Variables / A Useful Discrete Random Variable: The Binomial / Probability Distributions for Continuous Random Variables / A Useful Continuous Random Variable: The Normal Distribution / Random Sampling / Sampling Distributions / Normal Approximation to the Binomial / Summary / Key Formulas / Supplementary Exercises PART V: ANALYZING DATA: CENTRAL VALUES, VARIANCES, AND PROPORTIONS 5. INFERENCES ON A POPULATION CENTRAL VALUE Introduction and Case Study / Estimation of / Choosing the Sample Size for Estimating / A Statistical Test for / Choosing the Sample Size for Testing / The Level of Significance of a Statistical Test / Inferences about for Normal Population, s Unknown / Inferences about the Population Median / Summary / Key Formulas / Supplementary Exercises 6. COMPARING TWO POPULATION CENTRAL VALUES Introduction and Case Study / Inferences about 1 - 2: Independent Samples / A Nonparametric Alternative: The Wilcoxon Rank Sum Test / Inferences about 1 - 2: Paired Data / A Nonparametric Alternative: The Wilcoxon Signed-Rank Test / Choosing Sample Sizes for Inferences about 1 - 2 / Summary / Key Formulas / Supplementary Exercises 7. INFERENCES ABOUT POPULATION VARIANCES Introduction and Case Study / Estimation and Tests for a Population Variance / Estimation and Tests for Comparing Two Population Variances / Tests for Comparing k > 2 Population Variances / Summary / Key Formulas / Supplementary Exercises 8. INFERENCES ABOUT POPULATION CENTRAL VALUES Introduction and Case Study / A Statistical Test About More Than Two Population Variances / Checking on the Assumptions / Alternative When Assumptions are Violated: Transformations / A Nonparametric Alternative: The Kruskal-Wallis Test / Summary / Key Formulas / Supplementary Exercises 9. MULTIPLE COMPARISONS Introduction and Case Study / Planned Comparisons Among Treatments: Linear Contrasts / Which Error Rate Is Controlled / Multiple Comparisons with the Best Treatment / Comparison of Treatments to a Control / Pairwise Comparison on All Treatments / Summary / Key Formulas / Supplementary Exercises 10. CATEGORICAL DATA Introduction and Case Study / Inferences about a Population Proportion p / Comparing Two Population Proportions p1 - p2 / Probability Distributions for Discrete Random Variables / The Multinomial Experiment and Chi-Square Goodness-of-Fit Test / The Chi-Square Test of Homogeneity of Proportions / The Chi-Square Test of Independence of Two Nominal Level Variables / Fisher's Exact Test, a Permutation Test / Measures of Association / Combining Sets of Contingency Tables / Summary / Key Formulas / Supplementary Exercises PART VI: ANALYZING DATA: REGRESSION METHODS, MODEL BUILDING 11. SIMPLE LINEAR REGRESSION AND CORRELATION Linear Regression and the Method of Least Squares / Transformations to Linearize Data / Correlation / A Look Ahead: Multiple Regression / Summary of Key Formulas. Supplementary Exercises. 12. INFERENCES RELATED TO LINEAR REGRESSION AND CORRELATION Introduction and Case Study / Diagnostics for Detecting Violations of Model Conditions / Inferences about the Intercept and Slope of the Regression Line / Inferences about the Population Mean for a Specified Value of the Explanatory Variable / Predictions and Prediction Intervals / Examining Lack of Fit in the Model / The Inverse Regression Problem (Calibration): Predicting Values for x for a Specified Value of y / Summary / Key Formulas / Supplementary Exercises 13. MULTIPLE REGRESSION AND THE GENERAL LINEAR MODEL Introduction and Case Study / The General Linear Model / Least Squares Estimates of Parameters in the General Linear Model / Inferences about the Parameters in the General Linear Model / Inferences about the Population Mean and Predictions from the General Linear Model / Comparing the Slope of Several Regression Lines / Logistic Regression / Matrix Formulation of the General Linear Model / Summary / Key Formulas / Supplementary Exercises 14. BUILDING REGRESSION MODELS WITH DIAGNOSTICS Introduction and Case Study / Selecting the Variables (Step 1) / Model Formulation (Step 2) / Checking Model Conditions (Step 3) / Summary / Key Formulas / Supplementary Exercises PART VII: ANALYZING DATA: DESIGN OF EXPERIMENTS AND ANOVA 15. DESIGN CONCEPTS FOR EXPERIMENTS AND STUDIES Experiments, Treatments, Experimental Units, Blocking, Randomization, and Measurement Units / How Many Replications? / Studies for Comparing Means versus Studies for Comparing Variances / Summary / Key Formulas / Supplementary Exercises 16. ANALYSIS OF VARIANCE FOR STANDARD DESIGNS Introduction and Case Study / Completely Randomized Design with Single Factor / Randomized Block Design / Latin Square Design / Factorial Experiments in a Completely Randomized Design / The Estimation of Treatment Differences and Planned Comparisons in the Treatment Means / Checking Model Conditions / Alternative Analyses: Transformation and Friedman's Rank-Based Test / Summary / Key Formulas / Supplementary Exercises 17. ANALYSIS OF COVARIANCE Introduction and Case Study / A Completely Randomized Design with One Covariate / The Extrapolation Problem / Multiple Covariates and More Complicated Designs / Summary / Key Formulas / Supplementary Exercises 18. ANALYSIS OF VARIANCE FOR SOME UNBALANCED DESIGNS Introduction and Case Study / A Randomized Block Design with One or More Missing Observations / A Latin Square Design with Missing Data / Incomplete Block Designs / Summary / Key Formulas / Supplementary Exercises 19. ANALYSIS OF VARIANCE FOR SOME FIXED EFFECTS, RANDOM EFFECTS, AND MIXED EFFECTS MODELS Introduction and Case Study / A One-Factor Experiment with Random Treatment Effects / Extensions of Random-Effects Models / A Mixed Model: Experiments with Both Fixed and Random Treatment Effects / Models with Nested Factors / Rules for Obtaining Expected Mean Squares / Summary / Key Formulas / Supplementary Exercises 20. SPLIT-PLOT DESIGNS AND EXPERIMENTS WITH REPEATED MEASURES Introduction and Case Study / Split-Plot Designs / Single-Factor Experiments with Repeated Measures / Two-Factor Experiments with Repeated Measures on One of the Factors / Crossover Design / Summary / Key Formulas / Supplementary Exercises PART VIII: COMMUNICATING AND DOCUMENTING THE RESULTS OF A STUDY OR EXPERIMENT 21. COMMUNICATING AND DOCUMENTING THE RESULTS OF A STUDY OR EXPERIMENT Introduction / The Difficulty of Good Communication / Communication Hurdles: Graphical Distortions / Communication Hurdles: Biased Samples / Communication Hurdles: Sample Size / The Statistical Report / Documentation and Storage of Results / Summary / Supplementary Exercises

Group sequential methods in the design and analysis of clinical trials

Abstract: SUMMARY In clinical trials with sequential patient entry, fixed sample size designs are unjustified on ethical grounds and sequential designs are often impracticable. One solution is a group sequential design dividing patient entry into a number of equal-sized groups so that the decision to stop the trial or continue is based on repeated significance tests of the accumulated data after each group is evaluated. Exact results are obtained for a trial with two treatments and a normal response with known variance. The design problem of determining the required size and number of groups is also considered. Simulation shows that these normal results may be adapted to other types of response data. An example shows that group sequential designs can sometimes be statistically superior to standard sequential designs.

Conflict among criteria for testing hypotheses in the multivariate linear regression model

Abstract: This paper surveys alternative testing criteria in the linear multivariate regression model, and investigates the possibility of conflict among them. We consider the asymptotic Wald, likelihood ratio (LR), and Lagrange multiplier (LM) tests. These three test statistics have identical limiting chi-square distributions; thus their critical regions coincide. A strong result we obtain is that a systematic numerical inequality relationship exists; specifically, Wald , LRa LM. Since the equality relationship holds only if the null hypothesis is exactly true in the sample, in practice there will always exist a significance level for which the asymptotic Wald, LR, and LM tests will yield conflicting inference. However, when the null hypothesis is true, the dispersion among the teststatistics will tend to decrease as the sample size increases. We illustrate relationships among the alternative testing criteria with an empirical example based on the three reduced form equations of Klein's Model I of the United States economy, 1921-1941.

Some Properties of Tests for Specification Error in a Linear Regression Model

Abstract: This article considers the test reset which is intended to detect a nonzero mean of the disturbance in a linear regression model. Analysis of an approximation to the test statistic's distribution and Monte Carlo experiments reveal that the power of the test may decline as the size of the disturbance mean increases. However, the possibility is remote and declines with increasing sample size. Alternative sets of test variables are considered, and their effect on the power of the test is studied in Monte Carlo experiments. The best set seems to be composed of powers of the explanatory variables.

Jackknifing an index of diversity

Abstract: The method of jackknifing is introduced for estimating an index of diversity and several functions of it, and is illustrated with tree data. The method yields approximately normally distributed jackknife estimates and also gives estimated standard deviations, making possible tests of hypotheses and confidence interval estimates. These results apparently can be obtained under the usual conditions of field sampling, where associations within and between species or between quadrats or segments of a traverse may be found. Because of these associations there is no guarantee that the method works; hence an eyeball and a statistical test for the approximate normality of the estimates are given and illustrated with the tree data. The tree data came from 24 quadrats arranged in two blocks of contiguous quadrats. The estimated tree-species diversity, using both indices considered, showed a smooth monotonic decrease over a 19-yr interval providing striking confirmation of the reality of this ecological parameter. The normality tests on this data showed that the normal approximations to the distributions of the jackknife estimates were justified, except in three instances, only one of which seemed to be seriously in error. A conclusion is that it seems reasonable to suppose that in many, if not most, cases, only moderate sample sizes will be needed in practice for the jackknife estimate, at least for forest data.

Testing for Hardy-Weinberg equilibrium in small samples

Abstract: An investigation is made of the validity of several statistics to test whether a small sample comes from a population having the binomial proportions p2AA, 2pqAa, q2aa, where q = I p. In particular the significance levels (P-values) indicated by these tests are compared to those of a well-known exact test (Haldane 1954). It is found that, for sample sizes of 20 or greater and significance level 0.15 or less for the exact test, a useful approximation to the significance level is obtained when the X2-statistic, with Yates' correction, is averaged with a similar statistic that used conditional expectations, and the result referred to the x2-distribution. For other situations, or for greater accuracy, a recursive relation is given that reduces the amount of computation necessary to determine the exact significance level.

Cross-sectional time-series experiments: Some suggested statistical analyses

Abstract: In the past, statistical analyses for time-series experiments have usually operated with a single-case model, thereby limiting the general applicability of the designs. In this article, alternative analytical procedures are developed for cross-sectional time-series in which the sample size is large

Tests for Variance Shift at an Unknown Time Point

Abstract: SUMMARY Two tests for variance shift in a sequence of independent normal random variables, when the initial level of variance is unknown, are investigated in this article. The first is a locally most powerful test, and the second is a test based upon cusums of x2 values. Distribution functions of the two test statistics are approximated through the use of Edgeworth expansions and/or the beta distribution by matching the first few moments. Critical points of both test statistics are tabulated for various sample sizes. Powers of the two tests are compared using a Monte Carlo example. An illustration of the application of the tests to stock market price analysis is provided.

Discriminant functions when covariances are unequal and sample sizes are moderate

Abstract: A Monte Carlo study (Wahl 1971) is compared to the study of Marks and Dunn (1974) which investigated the ability of three discriminant functions, the quadratic, best linear, and Fisher's linear discriminant function, to classify individuals into two multivariate normally distributed populations with unequal covariance matrices. Parameters that were varied in all of the studies include the distance between populations, covariance matrices, number of variables, sample size and population proportion. Our results, when related to those of Marks and Dunn, indicate sample size to be a critical factor in choosing between the quadratic and linear functions.

Is the ANOVA F-Test Robust to Variance Heterogeneity When Sample Sizes are Equal?: An Investigation via a Coefficient of Variation:

Abstract: Numerous investigations have examined the effects of variance heterogeneity on the empirical probability of a Type I error for the analysis of variance (ANOVA) F-test and the prevailing conclusion has been that when sample sizes are equal, the ANOVA is robust to variance heterogeneity. However, Box (1954) reported a Type I error rate of .12, for a 5% nominal level, when unequal variances were paired with equal sample sizes. The present paper explored this finding, examining varying degrees and patterns of variance heterogeneity for varying sample sizes and number of treatment groups. The data indicate that the rate of Type 1 error varies as a function of the degree of variance heterogeneity and, consequently, it should not be assumed that the ANOVA F-test is always robust to variance heterogeneity when sample sizes are equal.

A Table of Exact Confidence Limits for Differences and Ratios of Two Proportions and Their Odds Ratios

Abstract: A table of exact 95 percent confidence limits for differences and ratios of two proportions and their odds ratio, including one-tailed P values for the Fisher-Irwin exact test, is given for a wide variety of 2 × 2 tables with sample sizes of 20(20)100. The calculations are based on Fisher's [3] conditional theory, and the computer algorithm of Thomas [10 or 11] is used to produce the tables. It is shown how the tables may be used for sample size determination without fixing the power for a given alternative hypothesis.

Thurstone's discriminal processes fifty years later

Abstract: Four issues are discussed concerning Thurstone's discriminal processes: the distributions governing the representation, the nature of the response decision rules, the relation of the mean representation to physical characteristics of the stimulus, and factors affecting the variance of the representation. A neural schema underlying the representation is proposed which involves samples in time of pulse trains on individual neural fibers, estimators of parameters of the several pulse trains, samples of neural fibers, and an aggregation of the estimates over the sample. The resulting aggregated estimate is the Thurstonian representation. Two estimators of pulse rate, which is monotonic with signal intensity, are timing and counting ratios and two methods of aggregation are averaging and maximizing. These lead to very different predictions in a speed-accuracy experiment; data indicate that both estimators are available and the aggregation is by averaging. Magnitude estimation data are then used both to illustrate an unusual response rule and to study the psychophysical law. In addition, the pattern of variability and correlation of magnitude estimates on successive trials is interpreted in terms of the sample size over which the aggregation takes place. Neural sample size is equated with selective attention, and is an important factor affecting the variability of the representation. It accounts for the magical number seven phenomenon in absolute identification and predicts the impact of nonuniform distributions of intensities on the absolute identification of two frequencies.

Determination of significant relative risks and optimal sampling procedures in prospective and retrospective comparative studies of various sizes.

Abstract: Methods are given for determining the relative risks which it is possible to demonstrate as statistically significant with given probability in prospective and retrospective studies of a particular size. Also considered is the ratio of the sample sizes in the two groups being compared which will provide the most precise estimate of relative risk for a given total sample size.

Monitoring treatment differences in long-term clinical trials.

Abstract: When the data from long-term clinical trials are reviewed continually over time for evidence of adverse or beneficial treatment effects, the classical significance tests are not appropriate. A stimulation procedure is described which provides, for different mortality rates and different patterns of patient enrollment, the correct critical regions corresponding to specified frequencies of looks at the data over the course of the study. The power of the test and the robustness of the critical regions for differences in pattern of enrollment, length of study, mortality model, and sample size are discussed. An application is made to a drug trial in coronary heart disease.

Evaluating interobserver reliability of interval data.

Abstract: Previous recommendations to employ occurrence, nonoccurrence, and overall estimates of interobserver reliability for interval data are reviewed. A rationale for comparing obtained reliability to reliability that would result from a random-chance model is explained. Formulae and graphic functions are presented to allow for the determination of chance agreement for each of the three indices, given any obtained per cent of intervals in which a response is recorded to occur. All indices are interpretable throughout the range of possible obtained values for the per cent of intervals in which a response is recorded. The level of chance agreement simply changes with changing values. Statistical procedures that could be used to determine whether obtained reliability is significantly superior to chance reliability are reviewed. These procedures are rejected because they yield significance levels that are partly a function of sample sizes and because there are no general rules to govern acceptable significance levels depending on the sizes of samples employed.

Some exact conditional tests of independence for, r × c cross-classification tables

Abstract: Exact conditional tests of independence in cross-classification tables are formulated based on theχ 2 statistic and statistics with stronger operational interpretations, such as some nominal and ordinal measures of association. Guidelines for the table dimensions and sample sizes for which the tests are economically implemented on a computer are given. Some selected sample sizes and marginal distributions are used in a numerical comparison between the significance levels of the approximate and exact conditional tests based on theχ 2 statistic.

A Nonrandomized Unconditional Test for Comparing Two Proportions in 2×2 Contingency Tables

Abstract: It is shown that the “usual” nonrandomized, conditional test for comparing proportions using independent binomial samples, is very conservative in the sense that the actual significance level attributable to an outcome is often one-fourth to one-half of the anticipated value. A nonrandomized unconditional test is proposed, and for sample sizes up to 15, tables are given in an appendix which specify one-sided critical regions of size less than or equal to the nominal values 0.05, and 0.01 (two-sided critical regions are also given). Numerical examples illustrating the use of the tables and a brief description of the algorithm used to generate the tables are included.

Sample size determinants for r X c comparative trials.

Abstract: Samiiple size determnination Jor r X c comparative trials is described based on existing tables of the nzon-central x2 distribution and the expression Jbr the non-centrality paramneter of the appropriate x2 statistic. In addition, the optimal allocation of sample elements among the treatmiient groups is explored through the evaluation oJ the expression Jor noncentrality under the comparative trial miiodel. It is shown that a general rulehJor sample allocation, such as the "square root rule," does niot applv unijborinly. The non-central procedure is then compared to existing procedures Jor samiiple size determninations Jor 2 X 2 trials and it is shown that this procedure ields results in close agreemnent to those Jor the widely employed Halperini (twp-tailed) procedure. Finiallv, samtiple size determinations Jor other contingency table mnodels are discussed.

“Large Sample-Size Distribution of Statistics Used in Testing for Spatial Correlation”: A Reply

Abstract: Test statistics for testing for spatial correlation in continuous variables have been given by both Moran and Geary and have subsequently been generalized. It has been conjectured for a long time that under the hypothesis of no spatial correlations all these statistics are normally distributed when the sample size is large. This paper proves a very general theorem on the large sample normality of quadratic forms. As corollaries to the theorem the asymptotic normality, under the hypothesis, of all the above-mentioned statistics is established. The necessary conditions are quite unrestrictive. It is also shown, by means of a counter example, that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.

Repeated significance tests for a normal mean

Abstract: SUMMARY Repeated significance tests for a normal mean, with variance known, are studied asymptotically. Approximations are given for the significance level, power and expected sample size. These approximations are shown to be accurate enough for practical numerical purposes over a wide range of parameter values. An analogous test for the case of unknown variance is suggested and an approximation to its significance level obtained.

A Comparison of Two Tests for Equality of Two Proportions

Abstract: Two procedures for testing equality of two proportions are compared in terms of asymptotic efficiency. The comparison favors use of a statistic equivalent to Goodman's Y 2 over the usual X 2 statistic in some cases including that of equal sample sizes. Numerical comparisons indicate that the asymptotic results have some relevance for moderate sample sizes.

The contouring of orientation data represented in spherical projection

Abstract: The inadequacy of the current method of contouring orientation data in terms of the concentration of points occurring in 1% of the projected area is demonstrated. In particular, the maximum concentration and the areas occupied by the different concentrations are shown to be dependent on sample size. Using a counting area of 100/n% of the projection, where n is the sample size, yields meaningful orientation patterns which are comparable for different sample sizes. This is illustrated with reference to computer generated random and non-random orientation patterns. The data obtained are compared with predicted values and form a basis for comparison with measured orientation patterns to test their significance.

Some graphical methods in statistics: A review and some extensions

Abstract: Summary A review of some graphical methods for comparing two populations is given. The methods are applied to survival times of radiated and control mice and some results concerning the model and the effect of radiation are obtained. An extension to methods for several populations is derived and a partial table for carrying it out is given. An indication of the cost (in terms of sample size) of simultaneous graphical methods vs. pointwise and methods based on means is given. Finally, some one sample plots related to probability plots are discussed.

Asymptotic expansions of the distributions of estimates in simultaneous equations for alternative parameter sequences

Abstract: The distributions of the LIML and TSLS estimates of the coefficient of an endogenous variable in a single equation can be approximated by asymptotic expansions. This paper relates the expansions in terms of the noncentrality parameter and the sample size going to infinity, the noncentrality parameter going to infinity with the sample size held fixed, and the standard deviation of the disturbance going to zero ("small-o"). 1. INTRODUCriON RECENTLY, ASYMPTOTIC EXPANSIONS of the distributions of estimates of coefficients of a single equation in a system of simultaneous equations have been made by Anderson [1], Anderson and Sawa [2], Mariano [6 and 7], and Sargan and Mikhail [11]. The expansions have usually been carried out on the basis that the sample size increases and that the effect of the exogenous variables (the noncentrality parameter) increases along with the sample size. In this paper we consider the case of the covariance matrix of the disturbances known and alternatively the case of the sample size fixed. We relate these three cases to the approach of letting the disturbance decrease (the "small-o-" approach). The estimates treated are two-stage least squares (TSLS) and limited information maximum likelihood (LIML).

Estimating the Total Overstatement Error in Accounting Populations

Abstract: Auditors wishing to estimate the total amount of errors in a set of accounts have tended to use estimation procedures which rely on approximate normality for large sample sizes. Since this reliance is often not well-founded for sample sizes used in auditing practice, efforts have been made to circumvent this difficulty. This paper will briefly describe these efforts and then present a new approach based on the multinomial probability distribution which yields confidence bounds with known confidence levels for all sample sizes.

On a Class of Partially Sequential Two-Sample Test Procedures

Abstract: A class of test procedures is proposed for the two-sample setting when one of the samples has a fixed sample size, and the second is obtained by an inverse binomial sequential sampling scheme. These procedures allow either a classical approach or a completely nonparametric attack to a wide variety of two-sample problems, Properties of the tests are discussed, Including limiting distributions as the fixed sample size tends to infinity, and the capability of obtaining asymptotic power restrictions against specified alternatives of interest. Criteria for selecting a particular procedure are discussed.

A probabilistic method of paleobiogeographic analysis

Abstract: Existing methods of quantitative paleobiogeographic analysis, based on the statistical use of binary similarity coefficients, are shown to be defective. A new method employing a formal probability frequency function which relates the number of shared taxa to sample size and inferred population size for a pair of areas, is introduced. It assumes random selection of taxonomic samples by the fossil record from larger complete populations originally occurring in the areas concerned. Quantitative assessment of population diversity as well as population similarity expressed as percentage overlap with attendant error estimates result from the method, but the final phase of analysis remains, of necessity, both manual and inexact. Utility of the method is demonstrated by working simulated data and actual data pertaining to late Cretaceous ammonites.

Power and Sample Size for Approximate Chi-Square Tests

Abstract: Approximate chi-square tests for hypotheses concerning multinomial probabilities are considered in many textbooks. In this article power calculations and sample size based on power are discussed and illustrated for the three most frequently used tests of this type. Available noncentrality parameters and existing tables permit a relatively easy solution of these kinds of problems.