Roger B. Hammer Assistant Professor Department of Sociology Oregon State University Conducting Social Research Logistic Regression Categorical Data Analysis

Random Variable A variable whose numerical value is determined by chance, the outcome of a random phenomenon.A variable whose numerical value is determined by chance, the outcome of a random phenomenon. Discrete has a countable number of values.Discrete has a countable number of values. Continuous can take on any value in an interval.Continuous can take on any value in an interval. Conducting Social Research Is “statistical anxiety” continuous or discrete?

Continuous or Discrete Actual measurement is always discrete.Actual measurement is always discrete. Distinction between variables that can take on lots versus many values.Distinction between variables that can take on lots versus many values. Not always clear or unambiguous.Not always clear or unambiguous. Conducting Social Research Is income continuous or discrete?

Binary. Ordinal/Ranked. Nominal/Unranked. Count. Censored. Conducting Social Research Categorical/Limited Variables

Normal distribution of dependent variable and independent variables will increase likelihood that the residuals are normally distributed, however this is not necessary for residuals to be normally distributed. Conducting Social Research Normally Distributed Errors

Conducting Social Research Binomial Distribution Fireplace

Ordinary Least Squares and the Linear Probability Model Conducting Social Research

Ordinary Least Squares Linear Probability Model

If we interpret the predicted value of Y as a probability. Conducting Social Research Ordinary Least Squares Model fire = total_living_area By definition, the observed values are 1 or 0, but the probability is continuous, so explained variation will be lower than might be expected.

If we interpret the predicted value of Y as a probability. Conducting Social Research Ordinary Least Squares Model fire = total_living_area By definition, probabilities cannot be greater than 1 (or less than 0), which is inevitable with any linear model.

Only two possible values of resdiuals for any value of X. Conducting Social Research Ordinary Least Squares Model fire = total_living_area If Y is a function of X and the errors depend on the mean of Y, then the assumption of constant variance is invalid.

A flexible, general-purpose modeling strategy with straight-forward interpretation for dichotomous Y variables. Similar to the Linear Regression Model but with different underlying mathematical basis. Estimated using maximum likelihood techniques, an iterative technique. Conducting Social Research Logistic Regression

P(Y = 1) denotes the probability that Y equals 1. Conducting Social Research Probabilities and Odds Finding a function that approaches but never exceeds the 0 and 1 probability limits. Odds range from 0 when P(Y = 1) to ∞ when P(Y = 0). Thus, we have a floor of 0, but not yet a ceiling of 1.

P(Y = 1) = 0.6 Conducting Social Research Probabilities and Odds P(Y = 1) = 0.4

Conducting Social Research Odds and the Log (Odds) Logit Logits range from - ∞ when P(Y = 0) to ∞ when P(Y = 1). If the logit is a linear function of X variables then P is a nonlinear sigmoid (s-shaped) curve that ranges from 0 to 1.

Binary Logistic Regression Model Conducting Social Research

Binary Logistic Regression Model Conducting Social Research If the linear part of the equation is as large as possible, that is infinity, then:

Binary Logistic Regression Model Conducting Social Research If the linear part of the equation is as small as possible, that is negative infinity then:

Conducting Social Research Logit and Logistic Regression

Conducting Social Research Logistic Regression The slope of the logit changes as the probability ranges from 0 to 1.

Conducting Social Research Interpreting Logistic Regression Coefficients The slope of the logit changes as the probability ranges from 0 to 1.

Conducting Social Research Logistic Regression Model fire = total_living_area

Easily accessible regression coefficient. Not easily interpretable. Conducting Social Research Interpreting Results Log Odds

Use the mean of all independent variables in the equation and then increase one variable one unit. Meaning of an average dummy variable is ambiguous. Conducting Social Research Interpreting Results Average Observation Change

At a probability of.5, multiplying a logit coefficient by.25, approximates a linear probability model coefficient. Conducting Social Research Interpreting Results Rough Estimate

More intuitive interpretation. Takes advantage of the tractable form of the logit. Assesses the change in the odds based on a change in X. Conducting Social Research Interpreting Results Odds Ratios

Conducting Social Research Interpreting Results Odds Ratios Taking the exponential: Compare the odd before and after changing the quantity of an X variable:

For a unit change in x the odds are expected to change by a factor of the odds ratio. If the odds ratio is greater than 1, the odds are larger. If the odds ratio is less than 1, the odds are smaller. The effect is not dependent on the level of any variable. Conducting Social Research Interpreting Results