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^{1}

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Edited by: Pietro Cipresso, IRCCS Istituto Auxologico Italiano, Italy

Reviewed by: Juergen Heller, Universität Tübingen, Germany; Tomer Fekete, KU Leuven, Belgium

*Correspondence: Clintin P. Davis-Stober

This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (

Guttman scales form the conceptual foundation for modern Item Response Theory (IRT). For example, Guttman scales underlie the Rasch model (e.g., Andrich,

In this brief note, we extend the results of Zwick (

The core idea of a Guttman scale is that the set of items under consideration forms a unidimensional scale, i.e., if a person obtains a correct response to an item then this person would obtain a correct response to all “easier” items. Table

1 | 0 | 0 | 0 | 0 | 0 |

2 | 1 | 0 | 0 | 0 | 0 |

3 | 1 | 1 | 0 | 0 | 0 |

4 | 1 | 1 | 1 | 0 | 0 |

5 | 1 | 1 | 1 | 1 | 0 |

6 | 1 | 1 | 1 | 1 | 1 |

As in Zwick (

Under our assumptions, the covariance between any items

As one would expect, Equation (1) is closely related to the Pearson product-moment correlation, which, as described by Zwick (

Parallel to Zwick (

^{cov}, with entries given by Equation

The proof is in the Appendix.

Note how

The eigenvectors of the covariance matrix also have an elegant, closed-form expression.

^{cov} defined by Equation_{i} of eigenvalue

The proof is in the Appendix.

Guttman (_{i, m} as a value _{i, j}) ≠ _{i, j+1}),

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
---|---|---|---|---|

1 | ||||

0 | ||||

1 | 0 | –1 | 0 | 1 |

0 | ||||

1 |

_{1} is the eigenvector corresponding to the maximal eigenvalue

In this section, we illustrate how our analytic results could be used to evaluate responses conforming to modern IRT models. We consider the well-known

where _{i} ∈ ℝ is the item difficulty parameter and θ ∈ ℝ is the person-specific ability parameter.

From the perspective of the 2PL model, Guttman items are obtained by letting the _{i} (item discrimination) parameter values become arbitrarily large (e.g., van Schuur,

In the next section, we compare our results to simulated data that relax the assumption of a uniform distribution over response patterns. In the first simulation study, we compare our results to data generated from a Rasch model (Equation 5 with _{i} = 1, _{i} for each item, i.e., high discrimination among items.

For this study, we considered six conditions comprised of: 4, 6, 8, 16, 32, and 64 test items. For each condition, the difficulty of the items, _{i}, was equally spaced along the interval [−1, 1]. For each condition, we randomly sampled 5000 values of θ from a standard normal distribution (e.g., Anderson et al., _{i}, to Equation (5), with _{i} = 1 for all test items, i.e., a Rasch model. Thus, for each condition, we have 5000 simulated responses to the test items.

For each condition, we computed the covariance matrix of the items using the 5000 simulated responses, i.e., we calculated the sample covariance of the 5000 responses. We then numerically calculated the eigenvalues of this covariance matrix. Figure

In this simulation study, we consider nearly identical conditions to the first, with the exception that the item discrimination parameters, _{i}, are large in size, indicating excellent item discrimination. As in the previous study, we considered six conditions comprised of: 4, 6, 8, 16, 32, and 64 test items. For each condition, the difficulty of the items, _{i}, was equally spaced along the interval [−1, 1]. As before, for each condition, we randomly sampled 5000 values of θ from a standard normal distribution. We obtained simulated responses to the items by applying the sampled θ values, and item difficulties, _{i}, to Equation (5), with _{i} = 3,

For each condition, we computed the covariance matrix of the 5000 simulated responses and numerically calculated the eigenvalues of the generated covariance matrix for each condition. Figure

This study illustrates that our analytic results, which are derived under the strong assumption of uniformly distributed response patterns, may be useful as an approximation even when the ability parameter is normally distributed. This approximation is best when the difficulty range of the items are within a single standard deviation of the mean and the items have excellent discriminability. As the range of the item difficulty increases and/or the variance of the ability parameter distribution shrinks, the approximation becomes much poorer. Our Matlab code for generating these graphs and exploring other configurations is available as an online supplement.

We derived closed-form solutions for the eigenvalues and eigenvectors of the covariance matrix of dichotomous Guttman items, under a uniform sampling assumption. We demonstrated that these eigenvalues and eigenvectors are simple trigonometric functions of the number of items,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank Edgar Merkle, Jay Verkuilen, and David Budescu for comments on an earlier draft. Davis-Stober was supported by National Science Foundation grant (SES-1459866, PI: Davis-Stober).

The Supplementary Material for this article can be found online at:

To prove the main results of the paper, we first derive the general inverse of the covariance matrix of Guttman items under our uniformity assumptions. This inverse has a special tridiagonal form. From this tridiagonal form, we apply known algebraic results to obtain the required eigenvalues and eigenvectors.

Define

We denote by ^{(i)} the ^{n + 1}, and by ^{(i)} the ^{(0)} = (1, 1, …, 1)′ and ^{(n + 1)} = (0, 0, …, 0)′. Thus in ℝ^{n + 1} we have for

and then also, for

Obtaining the eigenvalues and eigenvectors of the covariance matrix via the columns ^{(i)} of

centering each vector ^{(i)} (for ^{(i)} from each component; let us denote by ṽ^{(i)} the resulting vector;

computing the element _{ij} of the matrix ^{(i)} · ṽ^{(j)};

deriving the inverse of

inferring the eigenvalues and eigenvectors of ^{−1} (then also of ^{−1}, a tridiagonal matrix;

finally, observing that the covariance matrix equals

Let us rephrase these steps in a more geometric fashion. In Step 1, ṽ^{(i)} is the image of ^{(i)} by the orthogonal projection from ℝ^{n + 1} to the hyperplane ^{(i)} ∈ ^{(i)} − ^{(i)}, a constant vector, is orthogonal to ^{(i)} − ^{(j)} belongs to

In Step 2, we compute _{i, j} as the scalar product of ṽ^{(i)} with ṽ^{(j)}. Taking the scalar product of both sides of the previous equation with ṽ^{(j)}, we get for

Now because

we see that row _{i, •} is the mean of rows _{i − 1, •} and _{i + 1, •} except for its diagonal element which is _{0, •} = (0, 0, …, 0) and _{n + 1, •} = (0, 0, …, 0), we can also allow ^{(0)} = ṽ^{(n + 1)} = (0, 0, …, 0)′]. This special property of

(indeed, the product of the above matrix with

The form of ^{−1} follows a particular tridiagonal form that has been extensively studied in the mathematics literature. Elliott (_{i} of ^{−1}, and (selected) corresponding eigenvectors _{i}, for

Because the covariance matrix σ^{cov} equals ^{cov} are equal to ^{−1}.

Proposition 1 now follows from the fact that the eigenvalues of σ^{cov} are equal to ^{−1}, and Proposition 2 from the fact the matrices σ^{cov}, ^{−1} have the same eigenvectors. This completes the proof. □