# Intermediate Statistics: A Modern Approach

By James P. Stevens | Go to book overview
 Confidence intervals and measures of associations (variance accounted for) are mentioned as two ways of determining the practical significance of a study. Several cautions to be observed in using measures of association are mentioned, and an example is given to illustrate that a "small" amount of variance accounted for could indeed be practically significant. Planned comparisons are presented as an alternative way to analyze the k group problem. Here the researcher a priori is setting up comparisons among the group means corresponding to his or her hypotheses. An overall F test is not required in this approach. Planned comparisons are a more powerful approach, and overall α can be controlled through use of the Bonferroni Inequality.

APPENDIX

Theorem. The estimated variance for a contrast L + ̂ is given by

where the ni are the groups sizes for the k groups.

Proof. The estimated contrast is
L + ̂=c1x + ̄1+ c2x + ̄2+ . . . ckx + ̄k

Now we take the variance for both sides
var(L + ̂)=var(c1x + ̄1+ c2x + ̄2+ . . .+ ckx + ̄k)

Since the means are sampled from independent groups, the random variables c1x + ̄1, . . ., ckx + ̄k, are uncorrelated. But for uncorrelated variables the variance of a sum is equal to the sum of the variances (since all the covariance terms are 0), Thus,
var(L + ̂) = var(c1x + ̄1) + var(c2x + ̄2) + ⋯ + var(ckx + ̄k)

Now recall from introductory statistics that if c is a constant, then the variance of cx is var(cx) = c2var(x).

Using this result the above may be rewritten as
var(L + ̂) = c12 var(x + ̄1) + c22 var(x + ̄2) + ⋯ + ck2 var(x + ̄k)

Now, it is well known that the variance of a sample mean based on a sample of size n is given by var(x + ̄) = σ2/n, where σ2 is the variance of the population. (See Glass & Hopkins, 1984, pp. 188-190, for a proof.) Applying this result to x + ̄1, x + ̄2, etc., we obtain: var(L + ̂)= c12σ2/n1+ c22σ2/n2+ ⋯ + ck2σ2/nk

-111-

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Intermediate Statistics: A Modern Approach

• Title Page iii
• Contents vii
• Preface xi
• 1 - Introduction 1
• Appendix 1: Obtaining the Mean and Variance on the Ti-30xa and Casio Fx-300sa Calculators 52
• Appendix 2: Obtaining the Z Scores Within Each Group for a Variable Using Split File, Descriptives Procedure, and Case Summaries in Spss 53
• 2 - One Way Analysis of Variance 63
• 2.1 Introduction 63
• 2.20 Summary 110
• Appendix 111
• 3 - Power Analysis 121
• 3.1 Introduction 121
• 3.10 Summary 139
• 4 - Factorial Analysis of Variance 145
• 4.1 Introduction 145
• 4.8 Summary 193
• Appendix: Doing a Balanced Two Way Anova with a Calculator 194
• 5 - Repeated Measures Analysis 203
• 5.1 Introduction 203
• 5.15 Summary 234
• 6 - Simple and Multiple Regression 239
• 6.1 Simple Regression 239
• 6.18 Summary of Important Points 292
• Appendix: the Press Statistic 294
• 7 - Analysis of Covariance 305
• 7.1 Introduction 305
• 7.15 Summary 335
• Appendix a Data Sets 341
• Appendix B Statistical Tables 371
• Appendix C Power Tables 383
• References 393
• Answers to Selected Exercises 399
• Author Index 419
• Subject Index 421
• Sas Statview Supplement for Intermediate Statistics a Modern Approach Second Edition 427
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