Logistic Regression

Conceptual Framework - LR Dependent variable: two categories with underlying propensity (yes/no) (absent/present) Independent variables: Continuous Categorical (Treat the same way as linear regression)

Maximum Likelihood Equations to difficult to solve Have to “plug in” numbers until you minimize the error Minimizing error is the same thing as maximizing the likelihood Can solve equations that cannot be algebraically solved, but have some kind of best solution in theory.

Two Categories Natural Categories vs. Underlying Propensity Categories and Attenuation Logistic infers the variance that is removed by shrinking the categories to two

The Odds Ratio

The Logistic Model

Logistic Equation

Coefficients Same as linear regression: 1.Make the best weight for each variable. We call these B coefficients 2.Figure out how good your guess for B is based on the data. We call this the standard error. Different from linear regression: 1.Interpretation of B: need to take the exp(B) [because we are using a different relationship to Y. 2.We are asking whether exp(B) is different than 1. (And we have to pick between the likelihood ratio test and Wald)

Test of Coefficients Wald Test: Just using the Maximum Likelihoood standard error to ask questions about the parameters. Very analogous to the tests used on coefficients in linear regression Likelihood ratio test This is a generalized way of testing an arbitrary hypothesis in maximum likelihood using a chi-squared distribution. These two tests are asymptotically equivalent (i.e. as you sample size goes to infinity). Both are really just testing if exp(B) is different from 1 and they usually agree

Interpretation of LR Omnibus test: Chi-square “Do these variables, collectively, do a better job than the mean only model?” Explain variance: Nagelkerke R 2 Maximum likelihood version of R-squared: “What percentage of the model misfit have I explained?” Model fit: Hosmer & Lemeshow test “Is this model correctly classifying the number of cases you would expect?” Classification: Percentage Correct This will make sense to a lot of people, but might not be the statistical measure you are looking for.

Interpretation of Odds Ratio Odds Ratio (OR) or Exp(B) This is the odds of the event occurring divided by the odds of the event not occurring: For example, and OR of 1.2 means that for every unit increase in the independent variable, the dependent variable is 1.2 times more likely to occur. (or 20% more likely to occur) An OR more than one means the DV is more likely to occur with increases in the IV An OR less than one means the DV is less likely to occur with increases in the IV

Some Variations Binary LR: DV has 2 attributes IVs can be categorical or continuous Multinomial LR: DV has 3 or more attributes (not ordered) IVs can be categorical or continuous Ordinal LR: DV has 3 or more ordered attributes IVs can be categorical or continuous

Example of LR Presentation Research question: What is the effect of goal attainment, social integration, academic integration, and sex on extrinsic motivation?

Introduction Extrinsic motivation – going to college to receive external rewards, such as increased finances & status (high=1; low=2). Goal attainment – the development of educational goals, such as grade attainment (scale 1-7). Academic integration – college integration related to academic success, such as doing homework, studying, asking professors questions (scale 1-7). Social integration – college integration related to social success, such as making friends or ‘hanging out’ at school (scale 1-7).

Binary Logistic Regression Binary Logistic Regression of Goal Attainment, Social & Academic Integration, & Sex on Extrinsic Motivation VariableBWaldExp(B) Goal Social * Academic ** Sex *** (M=1; F=0) Constant4.142 χ 2 = 50.65, df = 4, p <.001; n=337; Nagelkerke R 2 =.213 High Extrinsic Motivation = 1; Low Extrinsic Motivation = 2; p<.001; *p<.05; **p<.01; ***p<.001

Discussion Multivariate analyses suggests that college students who: were socially integrated were academically integrated and male were more likely of being highly extrinsically motivated