categorical - Outline 1.1 Categorical Response Data Chapter...

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Chapter 1. Introduction: Distributions and Inference for Categorical Data Deyuan Li School of Management, Fudan University Feb. 28, 2011 1 / 67 Outline 1.1 Categorical Response Data 1.2 Distributions for Categorical data 1.3 Statistical Inference for Categorical Data 1.4 Statistical Inference for Binomial Parameters 1.5 Statistical Inference for Multinomial Parameters 2 / 67 1.1 Categorical Response Data A Categorical variable has a measurement scale consisting of a set of categories. For example: (1) liberal, moderate or conservative in political philosophy; (2) normal, benign, probably benign, suspicious or malignant for diagnoses of breast cancer based on a mammogram. 1.1.1 Response-explanatory variable distinction Y : Response variable = dependent variable X : Explanatory variable = independent variable This book focuses on methods for categorical response variables . Explanatory variables can be of any type. 3 / 67 1.1.2 Nominal-ordinal scale distinction Categorical variables have two primary types of scales: Nominal variables : categories without a natural ordering; e.g., religious affiliation (Catholic, Protestant, Jewish, Muslim, other). The order of listing the categories is irrelevant. The statistical analysis does not depend on that ordering. Ordinal variables : having ordered categories; e.g., social class (upper, middle, lower). The categories are ordered, but the distances between categories are unknown; i.e., the distances cannot be measured by exact numerical values. 4 / 67
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Interval variables have numerical distances between any two values; e.g., blood pressure level, annual income. The way that a variable is measured determines its classification. For example, “education” is nominal when measured as public or private school; ordinal when measured by highest degree attained (none, high school, bachelor’s, master’s or doctorate); interval when measured by number of years of education (0 , 1 , 2 , . . . ). 5 / 67 A variable’s measurement scale determines which statistical methods are appropriate. In the measurement hierarchy , interval > ordinal > nominal. Statistical methods for variables of one type can also be used for variables at higher levels, but not at lower levels. E.g., (1) statistical methods for nominal variables can be used with ordinal variables by ignoring the ordering of categories. (2) Methods for ordinal variables cannot be used with nominal variables. However, it is usually best to apply methods appropriate for the actual scale. 6 / 67 1.1.3 Continuous-discrete variable distinction Continuous variables: take lots of values. Discrete variables: take few values. This book deals with certain types of discretely measured responses: nominal variables; ordinal variables; discrete interval variables having relatively few values; continuous variables grouped into a small number of categories.
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