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Statistical theories of mental test scores

Author: Lord, Frederic M. ; Novick, Melvin R. ; Birnbaum, Allan Series: Addison-Wesley series in behavioral science: quantitative methods Publisher: Information Age Publishing (IAP) 2008.Language: EnglishDescription: 568 p. ; 24 cm.ISBN: 9781593119348Type of document: BookBibliography/Index: Includes bibliographical references and index
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Book Europe Campus
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Print BF431 .L67 2008
(Browse shelf)
001198294
Available 001198294
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Includes bibliographical references and index

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Statistical Theories of Mental Test Scores Contents PART 1 The Foundations of Mental Measurement Theory Chapter 1 Measurement in Psychology and Education 1.1 The need for a theory of mental testing ............................................................................. 13 1.2 Psychological theory and its function ............................................................................... 15 1.3 Measurement as a basis of model construction ................................................................. 16 1.4 The place of measurement in psychology ......................................................................... 19 1.5 Levels of measurement ...................................................................................................... 20 1.6 The specification of interval scales ................................................................................... 21 1.7 Deterministic and probabilistic models 23 1.8 The assumptions underlying test theory models ............................................................... 24 Chapter 2 The Construction of True and Error Scores 2.1 Introduction ....................................................................................................................... 27 2.2 The distribution of measurements on a fixed person 29 2.3 True score as an expectation ............................................................................................. 30 2.4 The construction of the error random variable for a fixed person . 31 2.5 The random selection of persons ....................................................................................... 32 2.6 Construction of the linear model ....................................................................................... 34 2.7 Derivation of the usual assumptions of the classical model .............................................. 36 2.8 What is error? 2.9 The many concepts of true score ....................................................................................... 2.10 Experimental independence .............................................................................................. 2.11 Linear experimental independence ................................................................................... 2.12 Replicate measurements 2.13 Parallel measurements and parallel forms 38 39 44 45 46 47 PART 2 The Classical Test Theory Model Chapter 3 Basic Equations of the Classical Model for Tests of Fixed Length 3.1 The classical linear model: restatement of assumptions ..................................................... 55 3.2 Expectations, variances, and correlations ........................................................................... 56 3.3 Relationships based on parallel measurements ................................................................... 58 3.4 Definitions, interpretations, and applications ...................................................................... 60 3.5 The validities of a test ......................................................................................................... 61 3.6 An alternative statement of the classical model .................................................................. 63 3.7 Regression theory for the classical model .......................................................................... 64 3.8 Errors of measurement, estimation, and prediction ............................................................ 66 3.9 Attenuation formulas ........................................................................................................... 69 3.10 Elementary models for inferring true change ..................................................................... 74 Chapter 4 Composite Tests 4.1 Introduction ......................................................................................................................... 82 4.2 Composite measurements with two components ................................................................ 83 4.3 Composite measurements with n components .................................................................... 85 4.4 Coefficient a and the reliability of composite measurements ............................................. 87 4.5 The internal structure of tests .............................................................................................. 85 4.6 Expectations, variances, and covariances of weighted composites . . ................................ 96 4.7 The correlation between two composite measurements ..................................................... 97 Chapter 5 Basic Equations of the Classical Model for Homogeneous Tests of Variable Length 5.1 Test length as a test parameter ............................................................................................ 103 5.2 The classical model with a continuous test length parameter . . . ....................................... 104 5.3 Statement of the assumptions .............................................................................................. 105 5.4 The true score as the observed score of a person on a test of infinite length .............................................................................................................. 108 5.5 The fundamental theorem ................................................................................................... 108 5.6 Expectations and variances ................................................................................................. 109 5.7 Covariances ......................................................................................................................... 110 5.8 Correlations among observed, true, and error scores ........................................................... 111 5.9 Expectations, variances, and correlations of lengthened tests ............................................ 111 5.10 The Spearman-Brown formula ........................................................................................... 112 5.11 The effect of test length on validity .................................................................................... 114 5.12 Comparing reliabilities and validities of tests of differing lengths ...................................... 118 5.13 The most reliable composite with a specified true score .................................................... 119 5.14 Maximizing the reliability of the composite when component lengths are fixed ............................................................................................................ 123 5.15 Maximizing the validity of a test battery as a function of relative test lengths for a fixed total testing time........................................................... 124 Chapter 6 Factors Affecting Measurement Precision, Estimation, and Prediction 6.1 Introduction ..................................................................................................................... 129 6.2 Effect of group heterogeneity on test reliability ............................................................. 129 6.3 Speed and power tests ..................................................................................................... 131 6.4 Conditions of measurement affecting reliability ............................................................. 133 6.5 Experimental problems in correcting for attenuation 137 6.6 Accuracy of the Spearman-Brown prophecy formulas ................................................... 139 6.7 Reliability as a generic concept ...................................................................................... 139 6.8 Effect of explicit and incidental selection on test validity: the two-variable case ...................................................................................................... 140 6.9 The effect of selection on test validity: the three-variable case ....................................... 144 6.10 The effect of selection on test validity: the general case ................................................. 146 6.11 Accuracy of the selection formulas................................................................................... 147 Chapter 7 Some Estimates of Parameters of the Classical Model 7.1 Introduction ..................................................................................................................... 151 7.2 Estimating true score ...................................................................................................... . 152 7.3 An unbiased estimate of the specific error variance from parallel measurements ................................................................................................... 153 7.4 The use of estimated error variances ............................................................................... 159 7.5 Specific true-score variance estimated from an analysis of variance components ................................................................................................. 160 7.6 A general formulation of the estimation problem as an analysis of variance components .............................................................................. 162 7.7 An estimate of an upper bound on the specific error variance from measurements that are not strictly parallel ............................................................ 166 PART 3 Other Weak True-Score Models Chapter 8 Some Test Theory for Imperfectly Parallel Measurements 8.1 Defining true score .......................................................................................................... 173 8.2 The generic error of measurement .................................................................................. 176 8.3 The generic error variance .............................................................................................. 177 8.4 Basic properties of generic errors of measurement ........................................................... 180 8.5 Generic true-score variance ............................................................................................... 8.6 The relation between generic and specific true-score variances . . . ................................. 8.7 Estimating generic parameters describing a single test form . . . ...................................... 8.8 Comparisons of estimates of error variance ...................................................................... 8.9 Substantive considerations regarding choice among estimates . . . ................................... Chapter 9 Types of Reliability Coefficients and Their Estimation 9.1 Introduction ........................................................................................................................ 198 9.2 Estimating the specific reliability coefficient .................................................................. 200 9.3 Statistical properties of an estimated variance ratio ........................................................ 201 9.4 Specific reliability theory for composite tests .................................................................. 203 9.5 Maximum likelihood estimation of reliability for normally distributed scores 204 9.6 The frequency distribution of the estimated reliability .................................................... 206 9.7 The generic reliability coefficient .................................................................................... 208 9.8 Generic reliability for a single test ................................................................................... 209 9.9 Use and interpretation of reliability coefficients .............................................................. 211 9.10 The reliability of ordinal measurements ............................................................................. 214 9.11 Use of factor loadings as reliability coefficients ................................................................. 216 9.12 Estimating reliability without using parallel forms ............................................................. 216 Chapter 10 Some Test Theory for r-Equivalent Measurements, Including Estimation of Higher-Order Moments 10.1 Introduction and definitions ........................................................................................... 224 10.2 An assumption of linear experimental independence .................................................... 225 10.3 Immediate implications 226 10.4 Basic theorem for r-equivalent measurements ............................................................... 227 10.5 Third-order moments ...................................................................................................... 228 10.6 Higher-order moments and cumulants ........................................................................... 230 10.7 Regression of true score on observed score ................................................................... 230 10.8 Implications, applications, and limitations ..................................................................... 232 Chapter 11 Item Sampling in Test Theory and in Research Design 11.1 Introduction .................................................................................................................... 234 11.2 Matrix sampling 236 11.3 Generalized symmetric means ....................................................................................... 238 11.4 First- and second-degree gsm's 241 11.5 Estimating true-score moments ...................................................................................... 245 11.6 Estimating the relation of observed score to true score .................................................. 248 11.7 Estimating the relation between scores on parallel test forms........................................ 248 184 185 187 191 194 11.8 Estimating the observed-score statistics for lengthened tests ...................................... 249 11.9 Frequency distribution of errors of measurement for binary items . ........................... 250 11.10 Item sampling as a technique in research design ......................................................... 252 11.11 Estimating a mean from a single item sample ............................................................. 253 11.12 Estimating a mean by multiple matrix sampling ......................................................... 255 11.13 Estimating group mean differences ............................................................................. 257 11.14 Estimating observed-score variances by item sampling 259 PART 4 Validity and Test Construction Theory Chapter 12 Validity 12.1 Introduction .................................................................................................................... 261 12.2 Regression and prediction .............................................................................................. 262 12.3 Linear regression functions 264 12.4 Multiple and partial correlation ..................................................................................... 265 12.5 Partial and multiple correlation and regression in n variables ....................................... 267 12.6 The screening of predictor variables ............................................................................. 269 12.7 Suppressor variables, moderator variables, and differential predictability ................................................................................................................. 271 12.8 Incremental validity 273 12.9 Validity and the selection ratio ...................................................................................... 275 12.10 Some remarks on the explication of the concept of validity as a correlation coefficient ........................................................................................... 277 12.11 Construct validity .......................................................................................................... 278 Chapter 13 The Selection of Predictor Variables 13.1 Introduction .................................................................................................................... 284 13.2 Some sampling problems ............................................................................................... 284 13.3 Formal procedures for selecting predictor variables 288 13.4 Prediction in future samples .......................................................................................... 289 13.5 The effect of relative test lengths on reliability and validity: the multiple predictor case .......................................................................................... 293 13.6 The determination of relative test lengths to maximize the multiple correlation ............................................................................................... 295 Chapter 14 Measurement Procedures and Item-Scoring Formulas 14.1 Introduction ................................................................................................................... 302 14.2 Guessing and omitting ................................................................................................... 303 14.3 A simple formula score ................................................................................................... 305 14.4 Properties of the simple formula score ........................................................................... 307 14.5 A simple regression model for scoring items ................................................................. 310 14.6 The regression method with a simple model that assumes partial knowledge ......................................................................................................... 312 14.7 Other item-scoring formulas ........................................................................................... 313 14.8 An evaluation of partial knowledge ................................................................................ 314 14.9 Methods for discriminating levels of partial knowledge concerning a test item .................................................................................................. 315 14.10 Assumptions underlying the personal probability approach to item scoring .............................................................................................................. 319 14.11 Reproducing scoring systems ........................................................................................... 321 Chapter 15 Item Parameters and Test Construction 15.1 Introduction ..................................................................................................................... 327 15.2 Item difficulty ................................................................................................................. 328 15.3 Item discriminating power 329 15.4 Item validity 332 15.5 Product moment correlations for dichotomous items ..................................................... 335 15.6 Biserial correlation .......................................................................................................... 337 15.7 Comparison of biserial and point biserial coefficients ................................................... 340 15.8 Tetrachoric correlation .................................................................................................... 345 15.9 A comparison of tetrachoric and phi coefficients ........................................................... 346 15.10 Considerations in the choice of test construction techniques . . . . .................................. 350 15.11 Formula scoring and corrections for chance success ...................................................... 352 15.12 Invariant item parameters ................................................................................................. 353 Chapter 16 Latent Traits and Item Characteristic Functions 16.1 Introduction ..................................................................................................................... 358 16.2 Latent variables 359 16.3 Local independence ......................................................................................................... 360 16.4 Item-test regression ......................................................................................................... 363 16.5 The normal ogive model ................................................................................................. 365 16.6 Conditions leading to the normal ogive model ............................................................... 370 16.7 Correlation matrix with one common factor .................................................................. 371 16.8 A sufficient condition for normal ogive item characteristic curves ................................ 374 16.9 Normal ogive parameters: item difficulty ...................................................................... 376 16.10 Normal ogive parameters: item discriminating power ................................................... 377 16.11 Practical use of normal ogive item parameters .............................................................. 379 16.12 Conditional distribution of test scores ............................................................................ 384 16.13 A relation of latent trait to true score .............................................................................. 386 16.14 Typical distortions in mental measurement .................................................................... 387 PART 5 Some Latent Trait Models and Their Use in Inferring an Examinee's Ability (Contributed by Allan Birnbaum) Chapter 17 Some Latent Trait Models 17.1 Introduction ................................................................................................................. 397 17.2 The logistic test model ................................................................................................ 399 17.3 Other models ............................................................................................................... 402 17.4 The test as a measuring instrument: examples of classification and estimation of ability levels by use of test scores ................................................ 405 17.5 Information structure of a test and transformations of scale of scores ........................................................................................................... 410 17.6 Transformations of scales of ability ........................................................................... 411 17.7 Calculations of distributions of test scores ................................................................. 414 17.8 Quantal response models in general ........................................................................... 420 17.9 Estimation of item parameters .................................................................................... 420 17.10 Validity of test models ................................................................................................. 422 Chapter 18 Test Scores, Sufficient Statistics, and the Information Structures of Tests 18.1 Sufficient statistics: definition and interpretation ...................................................... 425 18.2 Conditions for sufficiency of a statistic ..................................................................... 428 18.3 Test scores and sufficient statistics ............................................................................. 429 18.4 Sufficiency and the logistic test model ....................................................................... 431 18.5 Sufficiency and the information structures of tests .................................................... 434 Chapter 19 Classification by Ability Levels 19.1 Classification rules for distinguishing two levels of ability ........................................ 436 19.2 Two-point classification problems ................................................................................ 437 19.3 Locally best weights and classification rules .............................................................. 442 19.4 More general classification rules, composite scores, and statistical efficiency in general ................................................................................... 444 19.5 Quantitative appraisal and efficient design of classification rules ............................. 446 Chapter 20 Estimation of an Ability 20.1 Introduction .................................................................................................................. 453 20.2 Some algebra of information functions ........................................................................ 453 20.3 More general methods of estimation: maximum likelihood ........................................ 455 20.4 The information functions of various test items .......................................................... 460

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