Showing papers on "Statistical hypothesis testing published in 1982"

Basic Statistical Analysis

Abstract: Preface I. DESCRIPTIVE STATISTICS 1. Introduction to Statistics Stumbling Blocks to Statistics A Brief Look at the History of Statistics Gertrude Cox (1900-1978) Benefits of a Course in Statistics General Fields of Statistics Summary Key Terms and Names Problems 2. Percentages, Graphs and Measures of Central Tendency Percentage Changes-Comparing Increases with Decreases Graphs Measures of Central Tendency Appropriate Use of the Mean the Median and the Mode Summary Key Terms Problems Computer Problems 3. Variability Measures of Variability Graphs and Variability Questionnaire Percentages Key Terms Computer Problems 4. The Normal Curve and z Scores The Normal Curve z Scores Carl Friedrich Gauss (1777-1855) Translating Raw Scores into z Scores z Score Translation in Practice Fun with your Calculator Summary Key Terms and Names Problems 5. z Scores Revisited: T Scores and Other Normal Curve Transformations Other Applications of the z Score The Percentile Table T Scores Normal Cure Equivalents Stanines Grade-Equivalent Scores: A Note of Caution The Importance of the z Score Summary Key Terms Problems 6. Probability The Definition of Probability Blaise Pascal (1623-1662) Probability and Percentage Areas of the Normal Curve Combining Probabilities for Independent Events A Reminder about Logic Summary Key Terms Problems II. INFERENTIAL STATISTICS 7. Statistics and Parameters Generalizing from the Few to the Many Key Concepts of Inferential Statistics Techniques of Sampling Sampling Distributions Infinite versus Finite Sampling Galton and the Concept of Error Back to z Some Words of Encouragement Summary Key Terms Problems 8. Parameter Estimates and Hypothesis Testing Estimating the Population Standard Deviation Estimating the Standard Error of the Mean Estimating the Population of the Mean: Interval Estimates and Hypothesis Testing The t Ratio The Type 1 Error Alpha Levels Effect Size Interval Estimates: No Hypothesis Test Needed Summary Key Terms Problems Computer Problems 9. The Fundamentals of Research Methodology Research Strategies Independent and Dependent Variables The Cause-and-Effect Trap Theory of Measurement Research: Experimental versus Post Facto The Experimental Method: The Case of Cause and Effect Creating Equivalent Groups: The True Experiment Designing the True Experiment The Hawthorne Effect Repeated-Measures Designs with Separate Control Groups Requirements for the True Experiment Post Facto-Research Combination Research Research Errors Experimental Errors Meta-Analysis Methodology as a Basis for More Sophisticated Techniques Summary Key Terms Problems 10. The Hypothesis of Difference Sampling Distribution of Differences Estimated Standard Error of Difference Two-Sample t Test for Independent Samples Significance William Sealy Gossett (1876-1937) Two-Tailed t Table Alpha Levels and Confidence Level The Minimum Difference Outliner One-Tail t Test Importance of Having at Least Two Samples Power Effect Size Summary Key Terms Problems Computer Problems 11. The Hypothesis of Association: Correlation Cause and Effect The Pearson r Interclass versus Intraclass Karl Pearon (1857-1936) Missing Data Correlation Matrix The Spearman r s' 293 An Important Difference between the Correlation Correlation Coefficient and t Test Summary Key Terms and Names Problems Computer Problems 12. Analysis of Variance Advantages of ANOVA The Bonferroni Test Ronald Aylmer, Fisher (1890-1962) Analyzing the Variance Applications of ANOVA The Factorial ANOVA Eta Square and d Graphing the Interaction Summary Key Terms and Names Problems Computer Problems 13. Nominal Data and the Chi Square Chi Square and Independent Samples Locating the Difference Chi Square Percentages Square and z Scores Chi Square and Dependent Samples Requirements for Using Chi Square Summary Key Terms Problems Computer Problems III. ADVANCED TOPICS IN INFERENTIAL STATISTICS 14. Regression Analysis Regression of Y on X Sir Francis Galton (1822-1911) Standard Error of Estimate Multiple R (Linear Regression with More Than Two Variables) Path Analysis The Multiple Rand Causation Partial Correlation Summary Key Terms and Names Computer Problems 15. Repeated-Measures and Matched-Subjects Designs With Interval Data Problem of Correlated or Dependent Samples Repeated Measures, Paired t Ratio Confidence Interval for Paired Differences Within-Subjects Effect Size Testing Correlated Experimental Data Summary Key Terms Problems Computer Problems 16. Nonparametrics Revisited: The Ordinal Case Mann-Whitney U Test for Two Ordinal Distributions with Independent Selection Kruskal-Wallis H Test for Three or More Ordinal Distributions with Independent Selection Wicoxon T Test for Two Ordinal Distributions with Correlated Selection Friedman ANOVA By Ranks for Three or More Ordinal Distributions with Correlated Selection Advantages and Disadvantages of Nonparametric Tests Summary Key Terms Problems 17. Tests and Measurements Norm and Criterion Referencing: Relative Versus Absolute Performance Measure The Problem of Bias Test Reliability Validityand Measurement Theory Test Validity Item Analysis Summary Key Terms Problems Computer Problems 18. Computers and Statistical Analysis Computer Literacy The Statistical Programs Ada Lovelace (nee Byron, 1815-1852) Logic Checkpoints Answers Recommended Reading 19. Research Simulations: Choosing the Correct Statistical Test Methodology: Research's Bottom Line Checklist Questions Critical Decision Points Research Simulations: From A to Z The Research Enterprise A Final Thought: The Burden of Proof Special Unit: The Binomial Case Appendix A Appendix B Glossary References Answers to Odd-Numbered Items (and Within-Chapter Exercises) Index Statistical Hall of Fame Biographies Gertrude Cox-Chapter 1 Carl Gauss-Chapter 4 Blaise Pascal-Chapter 6 William Gossett-Chapter 10 Karl Pearson-Chapter 11 Ronald Fisher-Chapter 12 Sir Francis Galton-Chapter 14 Ada Lovelace-Chapter 18

Simultaneous tests of many hypotheses in exploratory research

Abstract: Many traditional statistical approaches to data analysis assume a relatively simple situation in which the investigator is testing a single hypothesis. Most research in psychiatry, on the other hand, is exploratory in nature and involves testing many hypotheses. Exploratory research presents special problems in data analysis, which are discussed in this overview. Special statistical approaches that are available to reduce error risk, such as the Bonferroni inequality, are described. The importance of selecting confidence levels appropriate to a particularly investigation, rather than arbitrary use of the .05 level, is also discussed.

The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note

Abstract: By means of simple diagrams this note gives an intuitive account of the likelihood ratio, the Lagrange multiplier, and Wald test procedures. It is also demonstrated that if the log-likelihood function is quadratic then the three test statistics are numerically identical and have χ2 distributions for all sample sizes under the null hypothesis.

The Behavior of Number-of-Factors Rules with Simulated Data.

Abstract: issues related to the decision of the number of factors to retain in factor analysis are identified, and three widely-used decision rules -- the Kaiser-Guttman, scree, and likelihood ratio tests -- are isolated for empirical study. Using two differing structural models and incorporating a number of relevant independent variables (such as number of variables, ratio of number of factors to number of variables, variable communality levels, and factorial complexity), the authors simulated 144 population data sets and, then, from these, 288 sample data sets, each with a precisely known (or incorporated) number of factors. The Kaiser-Guttman and scree rules were applied to the population data in Part I of the study, and all three rules were applied to the sample data sets in Part II. Overall trends and interactive results, in terms of the independent variables examined, are discussed in detail, and methods are presented for assessing the quality of the number-of-factors indicated by a particular rule.

Behavioural theories of dispersion and the mis-specification of travel demand models☆

Abstract: Conventional of first generation transport models have for some time been heavily criticised for their lack of behavioural content and inefficient use of data; more recently second generation or disaggregate travel demand models based on a theory of choice between discrete alternatives have also been viewed critically. First, it has been argued that implemented structures—and particularly the Multinomial Logit model—have not been sufficiently general to accommodate the “interaction” between alternatives; and second, and perhaps more importantly, that the underpinning theory, involving a perfectly discriminating rational man (homo economicus), endowed with complete information is an unacceptable starting point for the analysis of behaviour. In this paper the potential errors in forecasting travel response arising from theoretical misrepresentation are investigated; more generally, the problems of inference and hypothesis testing in conjuction with cross-sectional models are noted. A framework is developed to examine the consequences of the divergence between the behaviour of individuals in a system, the observed, and that description of their behaviour (which is embedded in a forecasting model) imputed by an observer, the modeller. The extent of this divergence in the context of response to particular policy stimuli is examined using Monte Carlo simulation for the following examples: (i) alternative assumptions relating to the structure of models reflecting substitution between similar alternatives; (ii) alternative decision-making processes; (iii) limited information and “satisficing” behaviour; and (iv) existence of habit in choice modelling. The method has allowed particular conclusions to be nade about the importance of theoretical misrepresentation in the four examples. More generally, it highlights the problems of forecasting response with cross-sectional models and draws attention to the problem of validation which is all too often associated solely with the goodness of statistical fit of analytic functions to data patterns.

On the Origins of the .05 Level of Statistical Significance

Abstract: Examination of the literature in statistics and probability that predates Fisher's Statistical Meth- ods for Research Workers indicates that although Fisher is responsible for the first formal statement of the .05 criterion for statistical significance, the concept goes back much further. The move toward conventional lev- els for the rejection of the hypothesis of chance dates from the turn of the century. Early statements about statistical significance were given in terms of the prob- able error. These earlier conventions were adopted and restated by Fisher. It is generally understood that the conventional use of the 5% level as the maximum acceptable prob- ability for determining statistical significance was established, somewhat arbitrarily, by Sir Ronald Fisher when he developed his procedures for the analysis of variance. Fisher's (1925) statement in his book, Statistical Methods for Research Workers, seems to be the first specific mention of the p = .05 level as deter- mining statistical significance. It is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant, (p. 47) Cochran (1976), commenting on a slightly later, but essentially similar, statement by Fisher (1926), says that, "Students sometimes ask, 'how did the 5 per cent significance level or Type I error come to be used as a standard?' ... I am not sure but this is the first comment known to me on the choice of 5 per cent" (p. 15). In the 1926 article Fisher acknowledges that other levels may be used:

Tests of Linear Hypotheses and l"1 Estimation

Abstract: statistics of a linear hypothesis in the standard linear model. These test statistics, which correspond to Wald, likelihood ratio, and Lagrange multiplier tests, are shown to have the same limiting chi-square behavior under mild regularity conditions on design and the distribution of errors. The asymptotic theory of the tests is derived for a large class of error distributions; thus in Huber's [10] terminology we investigate the behavior of the likelihood ratio test under "non-standard" conditions. The asymptotic efficiency of the 11 tests involves a modest sacrifice of power compared to classical tests in cases of strictly Gaussian errors but may yield large efficiency gains in non-Gaussian situations. The Lagrange multiplier test seems particularly attractive from a computational standpoint. We derive the asymptotic distribution of the three alternative 11 test statistics for a simple linear exclusion hypothesis. Extension of these results to hypotheses of the form R,8 = r is a straightforward exercise. When the density of the error distribution is strictly positive at the median, all three test statistics have the same limiting central x2 behavior at the null and noncentral x2 behavior for local alternatives to the null. When the variance of the error distribution is bounded, analogous results are well known for classical forms of the Wald, likelihood ratio, and Lagrange multipler tests based on least-squares methods. See, for example, Silvey [18] and the discussion in Section 4 below.

Lindley's Paradox

Abstract: (1982). Lindley's Paradox. Journal of the American Statistical Association: Vol. 77, No. 378, pp. 325-334.

Learning to Use Statistical Tests in Psychology

Abstract: Praise for the first edition: "An excellent textbook which is well planned, well written, and pitched at the correct level for psychology students. I would not hesitate to recommend Greene and d'Oliveira to all psychology students looking for an introductory text on statistical methodology." Bulletin of the British Psychological Society Learning to Use Statistical Tests in Psychology third edition has been updated throughout. It continues to be a key text in helping students to understand and conduct statistical tests in psychology without panic! It takes students from the most basic elements of statistics teaching them: How psychologists plan experiments and statistical tests Which considerations must be made when planning experiments How to analyze and comprehend test resultsLike the previous editions, this book provides students with a step-by-step guide to the simplest non-parametric tests through to more complex analysis of variance designs. There are clear summaries in progress boxes and questions for the student to answer in order to be sure that they have understood what they have read. The new edition is divided into four discrete sections and within this structure each test covered is illustrated through a chapter of its own. The sections cover: The principles of psychological research and psychological statistics Statistical tests for experiments with two or three conditions Statistical tests based on ANOVA (Analysis of Variance) conditions as well as tests for multiple comparisons between individual conditions Statistical tests to analyze relationships between variables Presented in a student-friendly textbook format, Learning to Use Psychological Tests in Psychology enables readers to select and use the most appropriate statistical tests to evaluate the significance of data obtained from psychological experiments. An errata sheet detailing the Decision Chart which is referred to can be downloaded by clicking here

Use of the Binomial Theorem in Interpreting Results of Multiple Tests of Significance

Abstract: Often in the social and behavioral sciences, several individual tests of significance are used to determine whether some common or overall hypothesis should be rejected. Thus, it becomes necessary to interpret r significant results out of n tests. Many authors contend that one or more significant results should be interpreted as an overall significant result for the set of tests. The authors of this work suggest that a more appropriate approach would be to use the binomial theorem to compute the probability that r or more Type I errors would occur when all n of the null hypotheses are true, and use this result as the level of overall significance a*. It is shown that in the independent test situation, it is possible to set an action limit r for rejection of the overall hypothesis based on some required overall level of significance a*. In addition, an upper limit is obtained for a* when r significant test results are used to reject a set of n hypotheses when the tests are dependent to an unknown extent.

Some Non-Nested Hypothesis Tests and the Relations Among Them

Abstract: In this paper we discuss several statistical techniques which may be used to test the validity of a possibly non-linear and multivariate regression model, using the information provided by estimating one or more alternative models on the same set of data. We first exposit, from a different perspective, the tests proposed by us in Davidson and MacKinnon (1981a), and discuss modified versions of these tests and extensions of them to the multivariate case. We then prove that all these tests, and also the tests previously proposed by Pesaran (1974) and Pesaran and Deaton (1978), based on the work of Cox (1961, 1962), are asymptotically equivalent under certain conditions. Finally, we present the results of a sampling experiment which shows that different tests can behave quite differently in small samples.

Testing for Significance

Abstract: This chapter discusses the general strategy of significance testing. In literary contexts, significance testing can be useful in connection with authorship attribution. The first step in significance testing is to formulate a hypothesis. The hypothesis in a significance test is about the value of a population parameter. A specific hypothesis about the relation between two parameters is called the null hypothesis, and it plays a special part in significance testing. If one is testing two texts in the course of trying to decide whether they are by the same author, there are various differences between word frequencies, word and sentence length, and the like. One sets up the null hypothesis that there is no difference between these parameters in the populations from which the passages are drawn and that the variations that are observed are because of sampling error. If the chance probability of the observed divergences is less than the level of probability fixed in advance, the null hypothesis is rejected.

Use and interpretation of statistics in wildlife journals

Abstract: Use and interpretation of statistics in wildlife journals are reviewed, and suggestions for improvement are offered. Populations from which inferences are to be drawn should be clearly defined, and conclusions should be limited to the range of the data analyzed. Authors should be careful to avoid improper methods of plotting data and should clearly define the use of estimates of variance, standard deviation, standard error, or confidence intervals. Biological and statistical signif- icance are often confused by authors and readers. Statistical hypothesis testing is a tool, and not every question should be answered by hypothesis testing. Meeting assumptions of hypothesis tests is the responsibility of authors, and assumptions should be reviewed before a test is employed. The use of statistical tools should be considered carefully both before and after gathering data.

Some examples of simulation model validation using hypothesis testing

Abstract: The use of hypothesis testing with cost-risk analysis is illustrated for simulation model validation by two examples. In the first example, Hotelling's two-sample T2 test with cost-risk analysis is used for illustrating the validation of a multivariate response self-driven steady-state simulation model representing a single server M/M/1 queueing system. In the second example, the validation of a multivariate response trace-driven terminating simulation model representing an M/M/1 system is illustrated by the use of Hotelling's one-sample T2 test with cost-risk analysis.

Simultaneous Inference in Epidemiological Studies

Abstract: Some difficulties encountered in using and interpreting significance tests in both exploratory and hypothesis testing epidemiological studies are discussed. Special consideration is given to the problems of simultaneous statistical inference--how are inferences to be modified when many significance tests are performed on the same set of data? Although some partial solutions are available, greater emphasis on estimation methods and less use of and reliance on significance testing in epidemiological studies is more appropriate.

Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)

Abstract: An iterative procedure is described for reducing the Bayes cost in decisions among M>2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing among M linearly independent pure states with minimum probability of error. The method is also applied to decisions among M unimodal coherent quantum signals in thermal noise.

Score Tests in GLIM with Applications

Abstract: The most common method of hypothesis testing in GLIM is the likelihood ratio method. However, in certain biostatistical application areas, score tests are more commonly used. Mantel-Haenszel chi-squared tests provide good examples. In other cases where a large number of competing models are being entertained, score tests may also be preferable for economy in computing.

New methods for the analysis of sex ratio data independent of the effects of family limitation

Abstract: A stochastic model has been described which allows for arbitrary Poisson Lexian and family limitation effects. This model not only provides a framework for hypothesis tests concerning the determinants of sex ratio data but provides parameter estimates of considerable interest. Application of the model to Renkonens data suggests the presence of Lexian and family limitation effects but the absence of Poisson effects. (EXCERPT)

Mixtures of Continuous and Categorical Variables in Discriminant Analysis: A Hypothesis-Testing Approach

Abstract: SUMMARY In the classical case of discrimination between two multivariate normal populations, an allocation rule based on a hypothesis-testing approach (John's Z statistic) has long been in competition with the rule based on simple parameter estimation and substitution in the Bayes procedure (Anderson's W statistic). Recent evidence suggests that the Z statistic may be preferable in certain circumstances. The purpose of this paper is to extend this approach to mixed discrete and continuous variables. Accordingly, an allocation rule based on the location model is derived by use of a hypothesis-testing argument. A method for the estimation of error rates with this rule is outlined. Some examples are given, and the new rule is compared with its direct competitors; some of its possible merits are discussed briefly. Let V be a p-component random vector which has probability densities f1(v) and t2(v) in two distinct populations 7r1 and 7r2, respectively, and suppose that it is required to use V in discriminating between the two populations. More particularly, suppose that it is necessary to decide from which of the two populations an observed value v of V has come. This is the classical problem of discriminant analysis, which has been described by many authors; a standard treatment is given by Anderson (1958, Chapter 6). Anderson shows that, if the probability densities f1(v) and t2(v) are completely specified, the optimal decision rule is given by the Bayes procedure, which is to allocate v to w1 or 7r2 according to fl(v)lf2(v) s k. Ae cut-off point k is determined by the prior probabilities of an observation coming from each population, and by the costs incurred when an individual is misclassified from each population. In practice the densities t1(v) and f2(v) are rarely known, and the only information about 7r1 and 7r2 which is usually available comes from two training sets: v(ll), v21), . . ., v(l) known to come from 7r1, and v(l2), v22), . . ., v(m2) known to come from 7r2. In such circumstances, estimation is necessary; various methods have been proposed, each involving a different level of assumptions. At one extreme, no assumptions at all are made about the functional form of the densities ti(v) (i = 1, 2), and these densities are estimated from the training sets by some nonparametric method. A comprehensive bibliography of such methods is provided by Wertz and Schneider (1979). At the other extreme, exact functional forms which include unknown parameters are assumed for the densities. A classical approach to the problem involves estimation of the unknown parameters from the training sets. Replacing the unknown parameters by these estimates in fl(v)lt2(v) leads

A multi-stage Gaussian transformation algorithm for clinical laboratory data.

Abstract: We have developed a multi-stage computer algorithm to transform non-normally distributed data to a normal distribution. This transformation is of value for calculation of laboratory reference intervals and for normalization of clinical laboratory variates before applying statistical procedures in which underlying data normality is assumed. The algorithm is able to normalize most laboratory data distributions with either negative or positive coefficients of skewness or kurtosis. Stepwise, a logarithmic transform removes asymmetry (skewness), then a Z-score transform and power function transform remove residual peakedness or flatness (kurtosis). Powerful statistical tests of data normality in the procedure help the user evaluate both the necessity for and the success of the data transformation. Erroneous assessments of data normality caused by rounded laboratory test values have been minimized by introducing computer-generated random noise into the data values. Reference interval endpoints that were estimated parametrically (mean +/- 2 SD) by using successfully transformed data were found to have a smaller root-mean-squared error than those estimated by the non-parametric percentile technique.

Diagnostic Tests for Multiple Time Series Models

Abstract: This paper is concerned with the development and application of diagnostic checks for vector linear time series models. A hypothesis testing procedure based upon the score, or Lagrangean multiplier, principle is advocated and the distributions of the test statistic both under the null hypothesis and under a Pitman sequence of alternatives are discussed. Consideration of alternative models with singular sensitivity matrices when the null hypothesis is true leads to an interpretation of the score test as a pure significance test and to a notion of an equivalence class of local alternatives. Portmanteau tests of model adequacy are also investigated and are seen to be equivalent to score tests.

Combining Independent Noncentral Chi Squared or $F$ Tests

Abstract: The problem of combining several independent Chi squared or $F$ tests is considered The data consist of $n$ independent Chi squared or $F$ variables on which tests of the null hypothesis that all noncentrality parameters are zero are based In each case, necessary conditions and sufficient conditions for a test to be admissible are given in terms of the monotonicity and convexity of the acceptance region The admissibility or inadmissibility of several tests based upon the observed significance levels of the individual test statistics is determined In the Chi squared case, Fisher's and Tippett's procedures are admissible, the inverse normal and inverse logistic procedures are inadmissible, and the test based upon the sum of the significance levels is inadmissible when the level is less than a half The results are similar, but not identical, in the $F$ case Several generalized Bayes tests are derived for each problem

Treatment Implementation and Statistical Power: A Research Note.

Abstract: The authors contend that the assumption of equal treatment implementation in program evaluations is highly questionable. Through a reanalysis of data from afield evaluation of a nutrition supplemen...

Order Restricted Statistical Tests on Multinomial and Poisson Parameters: The Starshaped Restriction

Abstract: : Likelihood ratio statistics for (i) testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped (cf. Shaked, Ann. Statist. (1979)), and for (ii) testing the null hypothesis that this parameter vector is starshaped are considered. For both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi-bar-square distribution. Analogous tests on a collection of Poisson means are also found to have asymptotic chi-bar-square distributions. (Author)

The Likelihood Ratio, Wald, and Lagrange Multiplier Tests:

Abstract: Within the framework of maximum likelihood methods the basic logic of these tests is developed by means of a simple diagram, and the connection of these tests to the standard likelihood ratio (LR) test is also established by the same diagrammatic device. Using the insights generated by our diagrammatic method we show that the distribution of these test statistics is easily derived if the log-likelihood is quadratic. Our exposition can be viewed as the geometric complement to the heuristic presentation of these tests that has been given by Silvey (1970, pp. 108-122). It can also be considered as complementary to the recent survey articles by Breusch and Pagan (1980) and Engle (1981). To introduce our approach to these tests we apply it first to the LR test and give that test a visual format that we think is novel and pedagogically helpful. Consider a classroom situation in which the discussion of maximum likelihood estimation has been completed and the instructor has just explained the basic logic of the LR test. That is to say, he will have advanced the argument that taking the ratio of likelihoods with and without the restrictions of the null hypothesis imposed is a plausible basis for a test. Furthermore, the basic equation stating the asymptotic distribution of the test statistic,