The Analysis of Covariance and Alternatives

Statistical Methods for Experiments, Quasi-Experiments, and Single-Case Studies

Bradley Huitema

• Publisher: Wiley
• Year: 2011
• Print ISBN: 9780471748960
• eBook ISBN: 9781118067468
• Edition: 2
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

A complete guide to cutting-edge techniques and best practices for applying covariance analysis methods

The Second Edition of Analysis of Covariance and Alternatives sheds new light on its topic, offering in-depth discussions of underlying assumptions, comprehensive interpretations of results, and comparisons of distinct approaches. The book has been extensively revised and updated to feature an in-depth review of prerequisites and the latest developments in the field.

The author begins with a discussion of essential topics relating to experimental design and analysis, including analysis of variance, multiple regression, effect size measures and newly developed methods of communicating statistical results. Subsequent chapters feature newly added methods for the analysis of experimentswith orderedtreatments, including two parametric and nonparametric monotone analyses as well as approaches based on the robust general linear model and reversed ordinal logistic regression. Four groundbreaking chapters on single-case designs introduce powerful new analyses for simple and complex single-case experiments. This Second Edition also features coverage of advanced methods including:

• Simple and multiple analysis of covariance using both the Fisher approach and the general linear model approach
• Methods to manage assumption departures, including heterogeneous slopes, nonlinear functions, dichotomous dependent variables, and covariates affected by treatments
• Power analysis and the application of covariance analysis to randomized-block designs, two-factor designs, pre- and post-test designs, and multiple dependent variable designs
• Measurement error correction and propensity score methods developed for quasi-experiments, observational studies, and uncontrolled clinical trials

Thoroughly updated to reflect the growing nature of the field, Analysis of Covariance and Alternatives is a suitable book for behavioral and medical scineces courses on design of experiments and regression and the upper-undergraduate and graduate levels. It also serves as an authoritative reference work for researchers and academics in the fields of medicine, clinical trials, epidemiology, public health, sociology, and engineering.

PART I

Basic Experimental Design and Analysis

CHAPTER 1

Review of Basic Statistical Methods

1.1 INTRODUCTION

Statistical methods are often subsumed under the general headings “descriptive” and “inferential.” The emphasis in behavioral and medical science statistics books is frequently on the inferential rather than the descriptive aspects of statistics. This emphasis sometimes occurs because descriptive statistics such as means, variances, standard deviations, correlation coefficients, regression coefficients, and odds ratios can be described and explained in relatively few chapters. The foundations of inferential statistics generally require longer explanations, a higher level of abstract thinking, and more complex formulas. Associated with the large amount of space devoted to inference is a failure on the part of many professionals to appreciate that description is usually the most informative aspect of a statistical analysis.

A perusal of many current journals reveals that the overemphasis on inference (especially tests of significance) and the underemphasis on simple description is widespread; the inferential tail is frequently allowed to wag the descriptive dog. Descriptive statistics are not only underemphasized, they are sometimes completely ignored. One frequently encounters research outcomes reported in terms of only probability values or statements of “significant” or “nonsignificant” with no mention of the size of the difference (e.g., a difference between sample means) associated with the inferential results. The sources of this perversity go beyond statistical training; editorial policies of journals, demands of funding agencies, and criteria established by governmental agencies are also involved. An exploration of the development of this problem is interesting but tangential to this review; it will not be pursued here.

The remaining sections of this chapter begin with a review of conventional hypothesis testing (including elementary statistical decision theory) and interval estimation procedures for the simple randomized two-group experiment. Issues associated with standardized effect sizes, measures of association, generalization of results, and the control of nuisance variation are also presented.

1.2 ELEMENTARY STATISTICAL INFERENCE

Research workers generally employ inferential statistics in dealing with the problem of generalizing results based on sample data to the populations from which the subjects were selected. Suppose we (1) randomly select N mentally retarded patients from a Michigan institution, (2) randomly assign these patients to treatments 1 and 2, (3) apply the treatments, and (4) obtain an outcome measure Y on each patient. A useful descriptive measure of the differential effectiveness of the two treatments is the difference between the two sample means. This difference is an unbiased point estimate of the difference between the corresponding population means.

If it turns out that the sample mean difference is large enough to be of clinical or practical importance, the investigator may want to make a statement about the difference between the unknown population means μ1 and μ2. Population mean μ1 is the mean score that would have been obtained if treatment 1 had been administered to all mentally retarded patients in the whole institutional population. Population mean μ2 is the mean score that would have been obtained if instead the second treatment had been administered to the whole institutional population. Inferential tests and confidence intervals are widely used to evaluate whether there are sufficient sample data to state that there is a difference between unknown population means and (in the case of the confidence interval) to provide an interval within which it can be argued that the population mean difference will be found. A summary of hypothesis testing and confidence interval methods for the two-group design is presented in Table 1.1. An understanding of the conceptual foundation for these methods requires that several crucial distinctions be made among different types of measures and distributions; these are reviewed next.

Table 1.1 Summary of the Computation and Interpretation of the Independent Samples t-Test and 95% Confidence Interval for the Case of a Randomized Two-Group Experiment

A. Hypothesis Test

Null hypothesis: H0: .

Test: Independent samples t (assume homogeneous population variances): where are the within-group sum of squared deviation (centered) scores for groups 1 and 2, respectively; are sample sizes associated with groups 1 and 2; and is the estimate of the standard error of the difference.

When the t-ratio is obtained from sample data it is called the obtained value of t (denoted as tobt). After it is computed a decision rule is invoked in order to decide whether to reject the null hypothesis. Two forms of decision rules are described below; before either one can be used it is necessary to specify the level of α. Recall that the level of significance α is set before the experiment is carried out. If a directional (one-tailed) test is desired, the level of α may be denoted as α1. If a nondirectional (two-tailed) test is involved, then the level of alpha may be denoted as α2.

Decision rule for a nondirectional test using t. If the absolute value of tobt is ≥ tcv (where tcv is the critical value), then reject H0; otherwise retain. The critical value of t is often based on α2 set at .05; the degrees of freedom = N − 2 (where N = n1 + n2).

Decision rule for a nondirectional test using the p-value. Because most current statistics computer programs provide p-values in addition to obtained t-values, the modern approach is to simply inspect the p-value and compare it with the level of α that has been specified. The decision rule for a nondirectional test is as follows: if the p-value is ≤ α2, reject H0; otherwise retain.

B. Confidence Interval

In the case of a simple two-group independent samples experiment the relevant 95% confidence interval is computed as follows: where is the estimate of the standard error of the difference; and is the critical value of the t statistic bas...

Table of contents

• Cover
• Series
• Title Page
• Copyright
• Dedication
• Preface
• PART I: Basic Experimental Design and Analysis
• PART II: Essentials of Regression Analysis
• PART III: Essentials of Simple and Multiple ANCOVA
• PART IV: Alternatives for Assumption Departures
• PART V: Single-Case Designs
• PART VI: ANCOVA Extensions
• PART VII: Quasi-Experiments and Misconceptions
• Appendix: Statistical Tables
• References
• Index