A Gaussian Process Arising from Likert-Type Scaling
01 Jun 2017-Vol. 18, Iss: 1, pp 77-87
TL;DR: In this paper, the convergence of a sequence of stochastic processes obtained from Likert-type scaling with increasing number of categories with increasing sample sizes is considered. And conditions are exhibited under which the sequence constructed from cumulative class frequencies by linear interpolation with respect to the class boundaries converges to a Gaussian process in law when the support of latent distribution is a bounded interval.
Abstract: Convergence of a sequence of stochastic processes obtained from Likert-Type scaling with increasing number of categories with increasing sample sizes is considered. Conditions are exhibited under which the sequence constructed from cumulative class frequencies by linear interpolation with respect to the class boundaries converges to a Gaussian process in law when the support of latent distribution is a bounded interval. We then extend the result to more general unbounded support. Quadratic variation of the limiting process is derived.
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12 Mar 2003
TL;DR: In this article, the authors present a survey on dimensionality, reliability, and validity of Latent Constructs in the context of scale development in the social sciences, focusing on the dimensionality of construct, items, and a set of items.
Abstract: About the Authors Chapter One: Introduction and Overview Purpose of the Book. Perspectives on Measurement in the Social Sciences. Latent Constructs Overview of dimensionality, reliability, and validity Overview of recommended procedures and steps in scale development. Chapter Two: Dimensionality Introduction. Dimensionality of construct, items, and a set of items. Does uni-dimensionality of a set of items imply uni-dimensionality of items or construct? Relevance of uni-dimensionality. How to assess dimensionality of constructs. Chapter Three: Reliability Introduction The true-score model Coefficient alpha Generalizability Theory Chapter Four: Validity Overview of Construct Validity Translation validity Criterion validity Convergent and discriminant validity Known-group validity Nomological validity Chapter Five: Steps 1 and 2: Construct Definition and Generating and Judging Measurement items Chapter 5: Steps 1 and 2: Construct Definition and Judging Measurement Items Introduction Step 1: Construct definition and content domain Step 2: Generating and judging measurement items Applications of Steps 1 and 2. Chapter Six: Step 3: Designing and Conducting Studies to Develop the Scale Introduction Pilot testing Conducting multiple studies for initial development and validation Initial item analyses: Exploratory factor analysis (EFA) Initial item and reliability analyses A final caveat EFA and item and reliability analyses examples from the literature Chapter 7: Step 4: Finalizing the Scale Introduction EFA and additional item analyses Confirmatory Factor Analyses (CFA) Additional evaluations of validity Establishing norms Applying generalizability theory Chapter Eight: Concluding Remarks Index
2,330 citations
TL;DR: In this article, negative association is defined as the property that a random variable is negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of Ω(1, 2, \cdots, k, k) of a function f(X, i, i = 1, 3, 4, k), f(G, g), g, g, j, j \in A_1), g(G), g), rbrack \leq 0, for all nondecreasing
Abstract: Random variables, $X_1, \cdots, X_k$ are said to be negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of $\{1, 2, \cdots, k\}, \operatorname{Cov}\lbrack f(X_1, i \in A_1), g(X_j, j \in A_2) \rbrack \leq 0$, for all nondecreasing functions $f, g$. The basic properties of negative association are derived. Especially useful is the property that nondecreasing functions of mutually exclusive subsets of NA random variables are NA. This property is shown not to hold for several other types of negative dependence recently proposed. One consequence is the inequality $P(X_i \leq x_i, i = 1, \cdots, k) \leq \prod^k_1P(X_i \leq x_i)$ for NA random variables $X_1, \cdots, X_k$, and the dual inequality resulting from reversing the inequalities inside the square brackets. In another application it is shown that negatively correlated normal random variables are NA. Other NA distributions are the (a) multinomial, (b) convolution of unlike multinomials, (c) multivariate hypergeometric, (d) Dirichlet, and (e) Dirichlet compound multinomial. Negative association is shown to arise in situations where the probability measure is permutation invariant. Applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures. In a somewhat striking example, NA and positive association representing quite strong opposing types of dependence, are shown to exist side by side in models of categorical data analysis.
1,410 citations
TL;DR: Using individual (not summated) Likert-type items (questions) as measurement tools is common in agricultural education research The Journal of Agricultural Education published 188 research articles in Volumes 27 through 32.
Abstract: Using individual (not summated) Likert-type items (questions) as measurement tools is common in agricultural education research The Journal of Agricultural Education published 188 research articles in Volumes 27 through 32 Responses to individual Likert-type items on measurement instruments were analyzed in 95, or more than half, of these articles After reviewing the articles analyzing individual Likert-type items, 5 1 (54%) reported only descriptive statistics (eg, means, standard deviations, frequencies/percentages by category) Paired Likert-type items or sets of items were compared using nonparametric statistical techniques (eg, chi-square homogeneity tests, Mann-Whitney-Wilcoxon U tests, Kruskal-Wallis analysis of variance tests) in 12 (13%) of the articles Means for paired Likert-type items were compared using parametric statistical procedures (eg t-tests or analysis of variance F-tests) in 32 (34%) of the articles
676 citations
TL;DR: The comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population as discussed by the authors.
Abstract: Let {X
i, 1≤i≤n} be a negatively associated sequence, and let {X*
i
, 1≤i≤n} be a sequence of independent random variables such that X*
i
and X
i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(∑
n
i=1
X
i)≤Ef(∑
n
i=1
X*
i
) for any convex function f on R
1 and that Ef(max1≤k≤n
∑
n
i=k
X
i)≤Ef(max1≤k≤n
∑
k
i=1
X*
i
) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.
322 citations
01 Apr 1997
TL;DR: Albawn et al. as mentioned in this paper examined the effect of alternative scale formats on reporting of intensity of attitudes on Likert scales of agreement and found that the two-stage format generated the greatest percentage of extreme position (Le. most intense) responses across scales.
Abstract: Gerald Albawn This study examined the effect of alternative scale formats on reporting of intensity of attitudes on Likert scales of agreement. A standard one-stage format and an alternate two-stage format were tested in three separate studies on samples of university students in three countries. In general, the two-stage format generated the greatest percentage of extreme-position (Le. most intense) responses across scales. A test of predictive ability showed that the two-stage format was a better predictor of product preferences. Underlying data structures did not differ much between the two.
231 citations