1 Logistic Regression Political Analysis 2 Week 8

Logistic Regression Political Analysis 2, Week 8, Lab 7 Lab instructor Jeff Wright (Nuffield)

Recap • Linear regression models are useful for approximating many relationships between variables. •

Transforming variables Recall from last week… • Non-linear relationships between X and Y can

Changing the model • Another option at our disposal is to change the statistical

Changing the model: LPM Some options: 1. OLS Linear Probability Model Assumptions of ordinary

Changing the model: Logit Some options: 2. Logistic regression While OLS models the relationship

Linear to logistic Some drawbacks of LPM - Potential for out-ofbound predictions (recall Olympic

Probability and odds Logistic regression allows us to determine the likelihood of an outcome

Probability and odds Range: 0 to +∞ Range: 0 to 1 The relationship between

Logistic regression • The left side of our logistic regression equation represents the “logit.

Logistic regression cont. How to interpret logit coefficients? Log of odds ratio Means a

Interpreting logit coefficients To interpret logit coefficients, we have two options: 1. Log of

Interpreting logit coefficients cont. Critically… …whether coefficients are interpreted using odds or probabilities…. .

Evaluation Please take a few minutes at the end of class to evaluate this

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Logistic Regression Political Analysis 2, Week 8, Lab 7 Lab instructor Jeff Wright (Nuffield) jeffrey. wright@politics. ox. ac. uk 2

Recap • Linear regression models are useful for approximating many relationships between variables. • Ex: 1. The effect of foreign aid on GDP p. c. (Easterly 2003) Ex: 2. The effect of democracy on trade liberalization (Milner & Kubota 2005) • But some relationships violate the assumption of linearity, i. e. , the marginal effect of X (our IV) on Y (our DV) is not constant. Exponential Logarithmic Cubic 3 Quadratic

Transforming variables Recall from last week… • Non-linear relationships between X and Y can be modelled by transforming X. • Ex: 1. • Ex: 2. Y= B 0 + B 1*X + B 2*X 3 Y= B 0 + B 1*X + B 2*X 2 + B 3*loge. Z 4

Changing the model • Another option at our disposal is to change the statistical model to better estimate the relationship between Xs and Y. • The model we choose will depend on the nature of our variables • Focus for today: the dependent variable is binary (Y=0 or Y=1) Variable Y=1 Y=0 Legislation bill passes bill fails Conflict war peace IO participation member not a member Climate change support signatory to Paris non-signatory to Paris Legislative influence party in majority party not in majority Civil war onset no onset 5

Changing the model: LPM Some options: 1. OLS Linear Probability Model Assumptions of ordinary least squares apply e. g. , DV is a linear function of the IVs; normal distribution of errors; etc. But DV is modelled differently! Y = Pr(event occurring) 6

Changing the model: Logit Some options: 2. Logistic regression While OLS models the relationship btw IVs and the DV as a straight line (with predicted Ys btw -∞ and +∞)… logistic regression models the relationship as a smooth curve (with predicted Ys btw 0 and 1). Logit models allow us to assume a certain king of nonlinear relationship btw the IVs and the Pr(Y=1) 7

Linear to logistic Some drawbacks of LPM - Potential for out-ofbound predictions (recall Olympic 100 m sprint example) - Non-normality of errors (binomial instead), which can invalidate t-tests - Heteroscedasticity (variance of residuals is not constant 8

Probability and odds Logistic regression allows us to determine the likelihood of an outcome as measured by the probability or odds it occurs Probability (Pr) • Range is 0 to 1 • Pr(Y=1) + Pr(Y=0)) = 1 Ex: If the probability of Art. 50 being invoked in 2017 is 20% (. 2) then the probability of Art. 50 not being invoked is 80% (. 8) Odds • Range is 0 to +∞ • Odds is equal to Pr(Y=1)/Pr(Y=0) or Pr(Y=1)/1 -Pr(Y=1) Ex: Odds of invoking Art. 50 in 2017 are. 2/. 8 =. 25 = the odds of Art. 50 being invoked are 1: 4. 9

Probability and odds Range: 0 to +∞ Range: 0 to 1 The relationship between odds and probability is monotonic. As odds , probability , and vice versa 10

Logistic regression • The left side of our logistic regression equation represents the “logit. ” • The logit is the transformation of Y from odds to the log of odds. (This log transformation is also monotonic. ) 11

Logistic regression cont. How to interpret logit coefficients? Log of odds ratio Means a one-unit increase in β (warl) reduces the log of odds ratio for civil war onset by. 95437. 12

Interpreting logit coefficients To interpret logit coefficients, we have two options: 1. Log of odds ratio (conditional) odds ratio • Take the coefficient’s exponential and subtract from 1 • Gives us the percentage change in odds following a one-unit increase in X 2. Generate predicted probabilities (focus of today) • • Specify range of values for an IV of interest Specify constant values for non-predicted IVs Store in a dataframe predict() Probabilities from 0 to 1 13

Interpreting logit coefficients cont. Critically… …whether coefficients are interpreted using odds or probabilities…. . . the marginal effect of any independent variable depends on the value of that independent variable as well as the values of other covariates in the model. 14

Evaluation Please take a few minutes at the end of class to evaluate this term’s lab! https: //www. surveymonkey. co. uk/r/QS 2_MT 16 15