1 G Lect 11M Binary outcomes in psychology Can Binary Outcomes Be Studied Using OLS Multiple Regression? Transforming the binary outcome Logistic Regression G Multiple Regression Week 11 (Monday)

2 G Lect 11M Binary outcomes in psychology Psychologists tend to study quantitative processes such as reaction time, strength of attitudes, achievement scores, and severity of depression In some cases the outcome is binary or categorical »Decisions (e.g. to sit next to certain person) »Passing some threshold to a new qualitative state (such as major depression) »Death, illness, stressful event »Promotion, retirement, selection

3 G Lect 11M Example: Diagnosis of Major Depression in Puerto Rican Adolescents Adolescents were assigned diagnoses on the basis of either their or parents’ reports The binary outcome makes a bad plot.

4 G Lect 11M Binary Outcomes and OLS Multiple Regression Why not regress a (0,1) binary outcome Y on X? »Y=B 0 +B 1 X 1 +B 2 X 2 +e »Interpret E(Y|X) as Prob(Y=1|X) E(Y|X)= B 0 +B 1 X 1 +B 2 X 2 Puerto Rican Example »Y is major depression in past year (0 absent, 1 present) »X 1 is centered age »X 2 dummy code for female gender »Perhaps the risk of depression goes up linearly with each year

5 G Lect 11M Numerical Results The probability of depression among male 15 year olds appears to be.026 in the community sample. The probability of depression in the clinic sample appears to be.125 higher than in the community sample. For each year of age, the probability seems to go up.013. The probability for females is.031 higher than males. The probability for 10 year old community males is negative.

6 G Lect 11M Formal Objections to OLS Analysis Linear formula for E(Y|X) may lead to values that exceed the logical (0,1) interval. Var(Y|X) is not constant »If binary variable is thought to be a binomial random variable, E(Y|X)=p X, Var(Y|X)= p X (1- p X ) »We expect (0,1) variation around region.4< p X <..6 »We don't expect much variation when p X approaches 0 or 1 (e.g. p X =.01 suggests Y=1 will almost never be seen.) »OLS is not efficient

7 G Lect 11M Residual distribution from depression example This distribution is not normal. Variation of residuals depend on probability of outcome.

8 G Lect 11M Transforming the binary outcome We need a transformation of E(Y|X) that is unbounded and that is easy to interpret. »The odds, p X /(1- p X ), is used by gaming fans. "The odds are 2 to 1 that our team will win" odds are unbounded in the positive direction, but still bounded by 0 »p X =.5 gives odds of 1.0 »p X =>  gives odds=  »p X =>0  gives odds=0 Taking the ln of the odds makes it unbounded in both directions

9 G Lect 11M Ln odds:  The transformation ln[p X /(1- p X )] gives a metric that is centered around p X =.5 and ranges from  to . »ln[p X /(1- p X )] =  X »When p X =.5,  X = 0.

10 G Lect 11M Logistic Regression The ln(odds)=  is called a logit. A regression model with  as the outcome is called logistic regression »  = B 0 + B 1 X The parameters of logistic regression have a natural interpretation when their antilog (exp) is taken »The constant is simply the odds of the outcome when X=0 »The slope becomes an odds ratio

11 G Lect 11M Example with Logistic Regression Constant indicates the odds of depression among 15 yr male community is.026 »Probability is similar,.025 The odds increase by a factor of for each year of age. The odds increase by a factor of 1.65 for females. The odds increase for clinic patients

12 G Lect 11M Plot of Fitted Probabilities with Age Females in the clinic sample have substantially higher risk when they are older. The interaction of sex by age is not significant.

13 G Lect 11M Details of Interpreting Logistic Coefficients Consider a simple model with one X dummy (0,1) variable »  = B 0 + B 1 X when X=0 »E(  X=0) =  0 = B 0 B 0 is the ln odds in the reference group exp(B 0 ) is the odds of Y=1 in group 0 when X=1 »E(  X=1) =  1 = B 0 + B 1

14 G Lect 11M Transforming Logits for Interpretation exp(   ) is the odds of Y=1 in group 0 exp(   ) is the odds of Y=1 in group 1 »B 1 is the odds ratio.