Search Results

This chapter offers an in-depth discussion of the three statistical assumptions of item response theory (IRT), namely unidimensionality, local independence, and correct model specification. Each of these assumptions is described more fully, with a focus on procedures for testing each assumption. For each assumption, a number of statistical tests is proposed and explored in the literature. In addition, three common methods for testing unidimensionality are discussed: analysis of the eigenvalues of the inter-item correlation matrix, Stout's test of essential unidimensionality, and indices based on the residuals from a unidimensional solution. Weaknesses of some common approaches and indices is noted, and newer alternative procedures are described. Unfortunately, some of these procedures are quite complex and not easily implemented.

Chapter 1 defines important terms including basic statistics, multivariate analysis, and inferential statistics, as a way to introduce readers to the book’s premise—that a thorough understanding of basic statistics is critical in the successful application of more advanced statistical methods. Readers are also introduced to assumptions and other requirements necessary for inferential statistical testing, making predictions about relationships between variables, and making causal inferences. The chapter then provides a description of each of the multivariate methods—multiple linear regression, Analysis of variance and Covariance, and path analysis—covered in the book. The chapter concludes with an overview of subsequent chapters and a description of the National Educational Longitudinal Study of 1988 (NELS: 88), which is the data set used for the book’s examples of each method.

Becoming a well-rounded data analyst requires more than the skills covered in this book. This postlude sketches some ways in which the types of thinking covered here can be extended to real problems in data analysis. Different ways of evaluating the assumptions of linear regression are considered, including plotting, hypothesis tests, and out-of-sample prediction. If the assumptions are not met, simple linear regression can be extended in various ways, including multiple regression, generalized linear models, and mixed models (among many other possibilities). This postlude concludes with a short discussion of the themes of the book: probabilistic models, methodological pluralism, and the value of elementary statistical thinking.