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1. Advanced Regression Nega)ve Binomial Poisson Regression Zero Inflated Mul)nomial Regression AGENDA © 2013 ExcelR Solutions. All Rights Reserved

2. Multinomial Regression •  Logis'c regression (Binomial distribu'on) is used when output has ‘2’ categories •  Mul'nomial regression (classifica'on model) is used when output has > ‘2’ categories •  Extension to logis'c regression •  No natural ordering of categories Mode of transport Car Carpool Bus Rail All modes Count 218 32 81 122 453 •  Response variable has > ‘2’ categories & hence we apply mul'logit Probability 0.48 0.07 0.18 0.27 1 •  Understand the impact of cost & 'me on the various modes of transport © 2013 ExcelR Solutions. All Rights Reserved

3. Multinomial Regression Whether we have ‘Y’ (response) or ‘X’ (predictor), which is categorical with ‘s’ categories ü  Lowest in numerical / lexicographical value is chosen as baseline / reference ü  Missing level in output is baseline level ü  We can choose the baseline level of our choice based on ‘relevel’ func'on in R ü  Model formulates the rela'onship between transformed (logit) Y & numerical X linearly ü  Modeling quan'ta've variables linearly might not always be correct •  © 2013 ExcelR Solutions. All Rights Reserved

4. Multinomial Regression - Output Itera'on History: •  Itera've procedure is used to compute maximum likelihood es'mates •  # itera'ons & convergence status is provided •  -2logL = 2 * nega've log likelihood •  -2logL has χ2 distribu'on, which is used for hypothesis tes'ng of goodness of fit # parameters = 27 © 2013 ExcelR Solutions. All Rights Reserved

5. Multinomial Regression - Output ‘car’ has been chosen as baseline •  x = vector represen'ng the values of all inputs •  Log(P(choice = carpool | x) / P(choice = car | x) = β20 + β21 * cost.car + β22 * cost.carpool + ……………. This equa'on compares the log of probabili'es of carpool to car The regression coefficient 0.636 indicates that for a ‘1’ unit increases the ‘cost.car’, the log odds of ‘carpool’ to ‘car’ increases by 0.636 •  Intercept value does not mean anything in this context •  •  If we have a categorical X also, say Gender (female = 0, male = 1), then regression coefficient (say 0.22) indicates that rela've to females, males increase the log odds of ‘carpool’ to ‘car’ by 0.22 © 2013 ExcelR Solutions. All Rights Reserved

6. Probability •  Let p = p(x | A) be the probability of any event (say airi'on) under condi'on A (say gender = female) Odds •  Then p(x | A) ÷ (1 - p(x | A) is called the odds associated with the event Odds Ratio •  If there are two condi'ons A (gender = female) & B (gender = male) then the ra'o p(x | A) ÷ (1 - p(x | A) / p(x | B) ÷ (1 - p(x | B) is called as odds ra'o of A with respect to B Relative Risk •  p(x | A) ÷ p(x | B) is called as rela've risk hips://en.wikipedia.org/wiki/Rela've_risk © 2013 ExcelR Solutions. All Rights Reserved

7. Odds Ratio •  Odds ra'o is computed from the coefficients in the linear model equa'on by simply exponen'a'ng •  Exponen'ated regression coefficients are odds ra'o for a unit change in a predictor variable •  The odds ra'o for a unit increase in cost.car is 1.88 for choosing carpool vs car © 2013 ExcelR Solutions. All Rights Reserved

8. Goodness of fit Linear Analysis of Variance Residual Deviance OLS GLM Analysis of Deviance Residual Sum of Squares Maximum Likelihood •  Residual Deviance is -2 log L •  Adding more parameters to the model will reduce Residual Deviance even if it is not going to be useful for predic'on •  In order to control this, penalty of “2 * number of parameters” is added to to Residual deviance •  This penalized value of -2 log L is called as AIC criterion •  AIC = -2 log L + 2 * number of parameters Note: “Mul'logit Model with Interac(on” © 2013 ExcelR Solutions. All Rights Reserved