Education 795 Class Notes Applied Research Logistic Regression Note set 10

Today’s Agenda Announcements (ours and yours) Q/A Applied Research Logistic regression

Pure vs. Applied Research Pure research ‘Pure research is that type of research which is directed towards increase of knowledge in science… where the primary aim is a fuller understanding of the subject under study rather than the application thereof’ (NSF, 1959) Applied research ‘Research carried out for the purpose of solving practical problems’ (Pedhazur & Pedhazur, 1991)

What do You Think? The pure researcher believes the applied scientists are not creative, that applied work attracts only mediocre men/women, and that applied research is like working from a cookbook. The applied researcher believes the pure scientist to be a snob, working in his/her ivory tower and afraid to put his/her findings to a real test… like Bacon’s spider, spinning webs out off his substance. (Storer, 1966, p. 108)

Sociobehavioral Research and Policy Advocacy Another endless debate over whether scientists should limit the presentation of their findings or also act as advocates for policies they presumably support…

Poem for the Day Thou shalt not answer questionnaires Or quizzes upon World-Affairs, Nor with compliance Take any test. Thou shalt not sit With statisticians nor commit A social science (Auden, 1950, p. 69)

We Turn Now to Logistic Regression When / why do we use logistic regression? Theory behind logistic regression Running logistic regression on SPSS Interpreting logistic regression analysis

When and Why? To test predictors when the outcome variable is a categorical dichotomy (yes/no, pass/fail, survive/die) Used because having a categorical dichotomy as an outcome variable violates the assumption of linearity and homogeneity in normal regression

Dichotomous Outcomes Aldrich & Nelson present two solutions to the violation of the linear regression assumptions for dichotomous outcomes Linear Probability Models Weighted Least Squares (which we will not cover) Nonlinear Probability Models Logit (Most commonly used) Probit (which we will not cover)

Nonlinear Probability Model Log (P/(1-P)=b 0 +b 1 X 1 +…+b n X n Let’s look at the left hand side P=probability of success (defined by the researcher, e.g. pass, graduate, survive) 1-P=probability of failure (1-probability of success) This ratio demands that the estimated coefficients remain positive Taking the Logarithm of this ratio restricts the range to be from 0 to 1 The left hand side is commonly referred to as the “logit” or the log(odds)

Odds Ratio P/(1-P) is called an odds. Simple Example: Hat with 5 red chips and 10 green chips. You win if you pull a red chip. Probability of winning is 1/3 Probability of not winning is 2/3 Odds of winning = 1/3 / 2/3 = ½ or 1:2 In other words, your odds of winning are 1 to 2 (there are 2 green chips for every 1 red chip, in lay terms, you are less likely to win then to lose)

Odds Ratio Let’s reverse the example Simple Example: Hat with 10 red chips and 5 green chips. You win if you pull a red chip. Probability of winning is 2/3 Probability of not winning is 1/3 Odds of winning = 2/3 / 1/3 = 2 or 2:1 In other words, your odds of winning are 2 to 1 (there are 1 green chips for every 2 red chip, in lay terms, you are more likely to win then to lose)

Morale of the Story For Odds Ratios Less than 1, success is less likely For Odds Ratios Greater than 1, success is more likely

Logistic Function Continuous Smooth S-shaped curve Takes on values between 0<=p<=1 Increases monotonically Symmetric around 0 Let’s take a look

The S-Shaped Curve P(Y=1)

Solving for P Single predictor Multiple predictors

Assumptions of Nonlinear Model Random sample Independent observations X’s are independent (minimal collinearity among the predictors) Estimation Technique called maximum likelihood estimation is used in logit models

Interpretation Issues Unlike the coefficients in a linear regression model, logistic regression results cannot be interpreted as the rate of change in the expected value the dependent variable, but the change in the probability of Y = 1 for any particular X The rate of change in the probability of Y = 1 is dependent upon the value of X (and all other Xs, if there are any in the analysis)

Interpreting Coefficients The output from SPSS will include the  ’s (the estimated log odds) and the exp (  ) the odds. How do we interpret the odds? If exp (  ) =1.6, then a one unit increase in X predicts a 60% increase in the odds of success. If exp (  ) =2.6 then a one unit increase in X predicts a 160% increase in the odds of success If exp (  ) =.7 then a one unit increase in X predicts a 30% decrease in the odds of success

A Simple Example Many natural applications of logistic regression are from medicine Understanding Coronary Heart Disease (CHD): 100 Cases 2 Variables Age (measured in years) Evidence of CHD (1 = Yes, 0 = No)

Sample Output

Sample Output 2

Output 3 Wald is similar to t-statistic Tests the null  = 0 Exp(B) indicates the change in odds resulting from a unit change in the predictor: Exp(B) > 1: as X  Probability  Exp(B) < 1: as X  Probability 

In Class Project Run a logistic regression with one predictor. Interpret the odds ratio

For Next Week Read and Discuss The Status of Women and Minorities Among Community College Faculty. Perna. L.W. (2003) Research in Higher Education 44(2) p