Rasch Meta-Metres of Growth for Some Intelligence and Attainment Tests: A Meta-metre for some Intelligence and Attainment Tests

This book adapts Rasch’s approach for quantifying growth on physiological variables, where growth decelerates, to intellectual variables. To apply this approach, it is necessary to construct measurements in a constant unit over the relevant range of the variable. With such measurements, the book illustrates the approach to quantifying growth on six intellectual variables - two intelligences tests and two each of tests of proficiencies in reading comprehension and mathematics. The book discusses how it is not immediately obvious that deceleration on a quantitative scale should also hold for the growth in intellectual variables. It goes on to show that this is indeed the case with all six tests analysed and considers some implications of this feature for understanding intellectual development, in particular the centrality of the growth trajectory set in early life. Foreword Preface Reference Acknowledgements References Contents About the First Author Part I Preface to Part I 1 The Formulation of a Meta-Metre of Growth 1.1 The Concept of a Meta-Metre and a Growth Law 1.2 Rationale for the Equations of Linear Growth in a Meta-Metre of Time 1.3 A Meta-Metre for the Control of the Rate of Growth 1.4 A Meta-Metre for Measurements of Variables with an Arbitrary Origin 1.5 Dynamic Consistency and Self-replication 1.6 Estimates of the Relative Rate of Growth: Eliminating the Meta-Metre 1.6.1 Estimates of Bn ,An 1.7 Estimate of the Meta-Metre τ(t) = ln(t + λ) 1.8 Estimates of Individual Rates of Growth in the Meta-Metre 1.9 The Roles of λ in the Meta-Metre τ(t) = ln(t + λ) References 2 Application of a Meta-Metre to Physiological Examples of Growth 2.1 Example 1: Growth in the Weight of Babies 2.1.1 Comparisons of Relative Rates of Growth of Boys and Girls 2.1.2 Comparisons of Boys and Girls in an Estimated Meta-Metre of Time 2.2 Example 2: Dental Development in Pima Indian Children 2.2.1 IRT Analysis for the Definition of the Continuum of Example 2 and Dental Development 2.2.2 SLM Analysis for the Definition of the Continuum of Example 2 and Dental Development 2.2.3 SLM Analysis for the Definition of the Continuum of Example 2 References 3 Meta-Metres of Growth on Two Intelligence Tests 3.1 Growth on a Stanford-Binet Test of General Intelligence 3.1.1 Bock’s Critique of Thurstone’s Study of Growth 3.1.2 Bock’s Analysis of Growth on the Stanford-Binet Test 3.1.3 Mental Growth on the Stanford-Binet Test Analysed with the SLM 3.1.4 Estimates and Fit of Item Difficulties of the Stanford-Binet Test with the SLM 3.2 Growth on the Test of Raven’s Progressive Matrices 3.2.1 Data and Analysis from the Standard Progressive Matrices 3.2.2 Adjusting for Bias in the Original Estimates of Measurements 3.2.3 An Anomalous Measurement and the Relevance of λ in the Meta-Metre 3.2.4 Estimates of Measurements and an Adjustment for Bias 3.3 Implications of a Quantitative Constant Unit and the Interpretation of Growth References 4 Meta-Metres of Growth on Two Reading Attainment Tests 4.1 The Data Sets 4.1.1 Vertical Scaling 4.2 Growth on the Variable of Reading from the ECLS-K Study 4.2.1 Estimates of Relative Growth and Growth in the Meta-Metre 4.2.2 Statistical Analyses of Rates of Growth and Intercepts from Individuals 4.2.3 Relative Rates of Growth from an Eliminated and Estimated Meta-Metre 4.2.4 Individual Growth Curves 4.2.5 Resilience of the Meta-Metre 4.3 Growth on the Variable of Reading from the NLSY Study 4.3.1 Estimates of Relative Growth and Growth in the Meta-Metre 4.3.2 Statistical Analyses of Comparisons of Rates of Growth and Intercepts 4.3.3 Relative Rates of Growth from an Eliminated and Estimated Meta-Metre 4.3.4 Individual Growth Curves 4.3.5 Resilience of the Meta-Metre References 5 Meta-Metres of Growth on Two Mathematics Attainment Tests 5.1 The Data Sets 5.1.1 Vertical Scaling 5.2 Growth on the Variable of Mathematics from the ECLS-K Study 5.2.1 Estimates of Relative Growth and Growth in the Meta-Metre 5.2.2 Statistical Analyses of Rates of Growth and Intercepts from Individuals 5.2.3 Relative Rates of Growth from an Eliminated and Estimated Meta-Metre 5.2.4 Individual Growth Curves 5.2.5 Resilience of the Meta-Metre 5.3 Growth on the Variable of Mathematics from the NLSY Study 5.3.1 Estimates of Relative Growth and Growth in the Meta-Metre 5.3.2 Statistical Analyses of Comparisons of Rates of Growth and Intercepts 5.3.3 Relative Rates of Growth from an Eliminated and Estimated Meta-Metre 5.3.4 Individual Growth Curves 5.3.5 Resilience of the Meta-Metre References 6 Some Implications of a Decelerating Meta-Metre for Attainment Variables 6.1 Growth on a Quantitative Scale Relative to Grade Equivalents 6.1.1 Grade Equivalents and Deceleration 6.1.2 Comparisons in Growth of Groups in Terms of Grade Equivalents 6.1.3 Implications of Setting an Initial Status and Trajectory of Growth References Part II Preface to Part II 7 Probabilistic Models for Measurement: Invariance of Comparisons 7.1 Measurement 7.1.1 The Elementary Measurement of Mass 7.2 Measurement of Non-physical Variables: Thurstone 7.2.1 A Distinction Between the Variable and Its Manifestation 7.2.2 A Measurable Variable—Unidimensionality 7.2.3 The Law of Comparative Judgement 7.2.4 The Error Variance and Case V of the LCJ 7.2.5 Estimates of the Contrast αi - αj 7.2.6 Estimates of the Scale Values αi ,αj ,αk 7.2.7 Invariance of Estimates αi ,i = 1,2,3, ,I 7.2.8 Person Attitude Measurement 7.2.9 Thurstone and Person Attainment Measurement 7.2.10 The Apparent Subversion of Thurstone’s Requirements 7.2.11 Substantive Theory and Thurstone’s Measurement of Attitude 7.3 Invariance of Comparisons: Rasch 7.3.1 Requirement of Invariant Comparisons 7.3.2 Invariant Comparisons Inferred from Observed Responses 7.3.3 Sufficient Statistics and the Resultant Model 7.3.4 Individual Measure Estimates and Information 7.3.5 Sufficient Statistics for the Instrument Parameters 7.3.6 Applications of the PRM 7.4 Relationship Between Thurstone’s LCJ and Rasch’s Invariance of Comparisons 7.4.1 The Dichotomous Probabilistic Rasch Model 7.4.2 Invariant Comparisons of Items Relative to Proficiencies of Persons 7.4.3 Evidence of a Constant Unit—The Comparison of Three Items 7.4.4 Some Similarities and Differences Between the LCJ and the SLM 7.4.5 Numerical Equivalence Between the Cumulative Normal and the SLM 7.4.6 The SLM and the Guttman Structure 7.5 Instruments Composed of Dichotomous Items 7.5.1 Elaboration of the Simple Logistic Model for Instruments Composed of Multiple Dichotomous Items 7.5.2 Some Person Measurement Properties of the SLM 7.5.3 Item Response Theory 7.5.4 The Two-Parameter Logistic Model 7.5.5 A Note on Classical Test Theory and the Measurement of Change References 8 Measurement, Quantitative Laws, and the Study of Growth in an Alternate Meta-Metre 8.1 Quantitative Physical Laws and Measurement 8.1.1 Physical Laws and Generality 8.1.2 Substantive Theory, Measurement, and Lawful Relationships 8.1.3 Measurement Leads to a Law 8.1.4 Scientific Theory Directs Measurement 8.1.5 Measurement and Theoretical Understanding Are Concomitant 8.1.6 Control of Extraneous Variables 8.2 An Alternate Meta-Metre that is a Function of Status at Any Time 8.2.1 Rate of Growth Inversely Proportional to the Square of the Current Status 8.2.2 The Construction of a Meta-Metre as a Function of Current Status 8.2.3 The Cube-Root Meta-Metre and Its Relation to the Logarithmic Meta-Metre 8.2.4 Characterisation of Growth in a Cube-Root Meta-Metre 8.3 Implications of the Equivalence of Cube-Root and Logarithmic Meta-Metres 8.3.1 Basis for the Choice of a Meta-Metre 8.3.2 Choice of a Meta-Metre and Further Research 8.3.3 A Possible Decelerating Law and Theory Development 8.4 The Lexile Framework as a Case Towards a Law 8.4.1 The Measurement of Reading Comprehension in the Lexile Framework 8.4.2 Potential Explanation for Decelerating Growth References 9 Overview of Selected Intelligence and Attainment Tests 9.1 Intelligence Tests 9.1.1 The Stanford-Binet Test of Intelligence 9.1.2 Raven’s Progressive Matrices 9.1.3 Characteristics of the Stanford-Binet and Raven’s Progressive Matrices Which Lend Themselves to Conforming to Rasch’s Simple Logistic Model 9.1.4 Mapping Piaget’s Qualitative Stages on to a Scale of Raven’s Progressive Matrices 9.1.5 Continuity and Different Meanings of the Same Distance at Different Points on the Continuum 9.1.6 Learning to Perform on Intelligence Tests—Examples of the UMAT and RPM 9.2 Attainment Tests—The Variable of Reading 9.2.1 The Variable of Reading 9.2.2 Qualitative Distinctions in Reading Mapped onto a Single Measurement Continuum 9.3 Attainment Tests—The Variable of Mathematics 9.3.1 School Mathematics as a Reflective and Formative Variable 9.3.2 The Frame of Reference and the Development of Mathematical Attainment 9.3.3 Qualitative Differences on the Same Continuum 9.4 Potential Explanations for Decelerating Growth References