An Introduction to Psychometric Theory with Applications in R

Written by: William Revelle (Northwestern University)

This site offers access to a yet-to-be-published book as the author refines it for release. It covers that aspect of personality psychology called psychometrics, which studies how to measure what we talk and think about. It also offers several articles and links to information about using the R software platform.

This book is aimed for beginners in psychometrics (and perhaps in R) who want to use the basic principals of psychometric theory in their substantive research. As an introduction to psychometrics some major philosophical issues about the meaning of measurement (e.g, Barrett (2005), Borsboom and Mellenbergh (2004), and Michell (1997)) will not be discussed in the detail they deserve, nor will many of the basic models be derived from first principles in the manner of Guilford (1954), McDonald (1999), or Nunnally (1967). It is hoped, however, that the reader will become interested enough in the theory and practice of psychometrics to delve into those much deeper texts.

The author of this psychometrics textbook suggests that most scientists read books backwards. That is, they start at the later chapters and if they understand them, stop reading. When understanding is not achieved, only then might they go to an earlier chapter. For that reason, according to the author, “the appendix on R is meant to allow the eager reader to start running programs in R without reading anything else. However, the introductory chapters are meant to be useful as they consider the meaning of our observations, the inferences we are able to draw from observations and the inferences we can not make.”

Table of Contents of this Pyschometrics Textbook

  1. Introduction
    1. Constructs and measures
      1. Observational and Experimental Psychology
      2. Data = Model + Residual
    2. A theory of data
    3. Basic summary statistics – problems of scale
    4. Covariance, regression, and correlation
    5. Multiple and partial correlation and regression
    6. Factor, Principle Components and Cluster Analysis
    7. Classical theory and the Measurement of Reliability
    8. Latent Trait Theory – The “New Psychometrics”
    9. Validity
      1. Decision Theory
    10. Structural Equation Modeling
  2. A Theory of Data
    1. A Theory of Data: Objects, People, and Comparisons
      1. Modeling the comparison process
    2. Models and model fitting
    3. A brief diversion: Functions in R
    4. Tournaments: Ordering people
      1. Scaling of People
      2. Alternative approaches to scaling people
      3. Assigning numbers to people – the problem of rewarding merit
    5. Social Networks: Proximities of People
      1. Rectangular data arrays of similarity
      2. Square arrays of similarity
    6. The Scaling of Objects
      1. Weber-Fechner scales of subjective experience
      2. Thurstonian Scaling
      3. Alternative solutions to the ranking of objects
      4. Why emphasize Thurstonian scaling?
    7. Multiple Dimensional Scaling: Distances between Objects
    8. Preferential Choice: Unfolding Theory
      1. Individual Preferences – the I scale
      2. Joint Preferences – the J scale
      3. Partially ordered metrics
      4. Multidimensional Unfolding
    9. Measurement of Attitudes and Abilities
      1. Measurement of abilities
        1. Guttman scales
        2. Normal and logistic trace line models
      2. Measurement of attitudes
    10. Theory of Data: some final moments
  3. The problem of scale
    1. Four broad classes of scales
      1. Factor levels as Nominal values
      2. Integers and Reals: Ordinal or Metric values?
    2. Graphical and numeric summaries of the data
      1. Sorting data as a summary technique
    3. Numerical estimates of central tendency
      1. Mode: the most frequent
      2. Median: the middle observation
      3. 3 forms of the mean
      4. Comparing variables or groups by their central tendency
    4. The effect of non-linearity on estimates of central tendency
      1. Circular Means
    5. Whose mean? The problem of point of view
      1. Average length of time in psychotherapy
      2. Average class size
    6. Non-linearity and interpretation of experimental effects
      1. Linearity, non-linearity and the properties of measurement
    7. Measures of dispersion
      1. Measures of range
      2. Average distance from the central tendency
        1. Median absolute deviation from the median
        2. Sums of squares and Euclidean distance
      3. Deviation scores and the standard deviation
      4. Coefficient of variation
    8. Geometric interpretations of Variance and Covariance
    9. Variance, Covariance, and Distance
    10. Standard scores as unit free measures
    11. Assessing the higher order moments of the normal and other distributions
    12. Generating commonly observed distributions
    13. Mixed distributions
    14. Robust measures of dispersion
    15. Monotonic transformations of data and “Tukey’s ladder”
    16. What is the fundamental scale?
  4. Covariance, Regression, and Correlation
    1. Correlation as the geometric mean of regressions
    2. Regression and prediction
    3. A geometric interpretation of covariance and correlation
    4. The bivariate normal distribution
      1. Confidence intervals of correlations
      2. Testing whether correlations differ from zero
      3. Testing the difference between correlations
        1. Testing independent correlations: r12 is different from r34
        2. Testing dependent correlations: r12 is different from r23
        3. Testing dependent correlations: r12 is different from r34
    5. Other estimates of association
      1. Pearson correlation equivalents
        1. Spearman p: a Pearson correlation of ranks
        2. Point biserial: A Pearson correlation of a continuous variable with a dichotomous variable
        3. Phi: A Pearson correlation of dichotomous data
        4. Tetrachoric and polychoric correlations
        5. Biserial and polyserial correlations: An estimated Pearson correlation of a continuous variable with an ordered categorical variable
        6. Correlation and comorbidity
    6. Other measures of association
      1. Naturally dichotomous data
        1. Odds ratios, risk ratios, and the problem of base rates
        2. Yule’s Q and Y
        3. Even more measures of association for dichotomous data
        4. Base rates and inference
      2. Measures of association for categorical data
      3. Intraclass Correlation
      4. Quantile Regression
      5. Kendall’s Tau
      6. Circular-circular and circular-linear correlations
    7. Alternative estimates of effect size
    8. Sources of confusion
      1. Restriction of range
      2. Spurious correlations
        1. The misuse of ratios, sums and differences
        2. Correlation induced by ipsatization and other devices
        3. Simpson’s Paradox ad the within versus between correlation problem
        4. Correlations of means (doesn’t equal) correlations of observations
        5. Base rates and skew
      3. Non-linarity, outliers and other problems: the importance of graphics
  5. Multiple correlation and multiple regression
    1. The variance of composites
    2. Multiple regression
      1. Direct and indirect effects, suppression and other surprises
      2. Interactions and product terms: the need to center the data
      3. Confidence intervals of the regression and regression weights
      4. Multiple regression from the covariance/correlation matrix
      5. The robust beauty of linear models
    3. Partial and semi-partial correlation
      1. Alternative interpretations of the partial correlation
    4. Alternative regression techniques
      1. Logistic regression
      2. Poisson regression, quasi-Poisson regression, and negative-binomial regression
      3. Using multiple regression for circular data
      4. Robust regression using M estimators
  6. Constructs, Components, and Factor models
    1. Principle Components: an observed variable model
      1. Eigenvalues and Eigenvectors
      2. Principle components
    2. Exploratory Factor Analysis: a latent variable model
      1. Principle Axes Factor Analysis as an eigenvalue decomposition of a reduced matrix
      2. Maximum Likelihood Factor Analysis and its alternatives
        1. Minimal Residual Factor Analysis
        2. Factor analysis by using optim function
      3. Comparing extraction techniques
      4. Exploratory analysis with more than one factor/component
      5. Comparing factors and components – part 1
    3. Rotations and Transformations
      1. Orthogonal rotations
      2. Oblique transformations
      3. Non-Simple Structure Solutions: The Simplex ands Circumplex
      4. Hierarchical and higher order models
      5. comparing factor solutions
    4. The number of factors/components problem
    5. The number of subjects problem
    6. The problem of factoring items
    7. Confirmatory Factor Analysis
    8. Alternative procedures for reducing the complexity of the data
      1. MDS solutions remove the general factor
      2. Cluster analysis – poor man’s factor analysis?
        1. Non-hierarchical clustering
        2. Hierarchical clustering
    9. Estimating factor scores, finding component and cluster scores
      1. Extending a factor solution to new variables
      2. Comparing factors and components – part 2
  7. Classical Test Theory and the Measurement of Reliability
    1. Reliability and True Scores
      1. Parallel Tests, Reliability, and Corrections for Attenuation
      2. Tau equivalent and congeneric tests
    2. Reliability and internal structure
      1. Split half reliability
      2. Domain sampling
        1. Correlation of an item with a domain
        2. Correlation of a test with the domain
      3. The internal structure of a test. Part 1: coefficent α
      4. The internal structure of a test. Part 2: Guttman’s lower bounds of reliability
      5. The internal structure of a test. Part 3: coefficients α, β, ωh and ωt
    3. A comparison of internal consistency estimates of reliability
    4. Estimation of reliability
      1. Test-retest reliability: Stability across time
      2. Intraclass correlations and the reliability of ratings across judges
      3. Generalizability theory: reliability over facets
      4. Reliability of a composite test
      5. Reliability of a difference score
    5. Using reliability to estimate true scores
  8. The “New Psychometrics” – Item Response Theory
    1. Dichotomous items and monotonic trace lines: the measurement of ability
      1. Rasch Modeling – one parameter IRT
      2. The normal ogive – another one parameter models
      3. Parameter estimation
      4. Item information
      5. Two parameter models
      6. Three parameter models
      7. Four parameter models
    2. Polytomous items
      1. Ordered response categories
      2. Multiple choice ability items
    3. IRT and factor analysis of items
    4. Test bias and Differential Item Functioning
    5. Non-monotone trace lines – the measurement of attitudes (incomplete when I visited on 6/15/15)
    6. IRT and adaptive testing
    7. Item banking and item comparison
    8. Non-parametric IRT (incomplete on 6/15/15)
    9. Classical versus IRT models – does it make a difference?
  9. Validity
  10. Reliability + Validity = Structural Equation Models

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An Introduction to Psychometric Theory with Applications in R

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