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Poisson Regression

Nega)ve Binomial

Zero Inﬂated ial Mul)nom n Regressio

AGENDA

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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 • Understand the impact of cost & 'me on the various modes of transport

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1

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

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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

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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

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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

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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

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Goodness of fit Linear

GLM

Analysis of Variance

Analysis of Deviance

Residual Deviance

Residual Sum of Squares

OLS

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”

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