ORDERED MULTINOMIAL LOGISTIC REGRESSION ANALYSIS

Pooja Shivraj Southern Methodist University

 Linear Regression

 Logistic Regression   Dichotomous dependent variable (yes/no, died/

didn’t die, at risk/not at risk, etc.)   Predicts the probability of a person belonging

in that category.

KINDS OF REGRESSION ANALYSES

QUICK REVIEW: LOGISTIC REGRESSION

 Values calculated from linear regression are continuous – need to be transformed on a 0-1 scale to represent probability since 0 ≤ p ≤ 1

 Logistic regression probability calculated by:

p ^ = e

(B1x + B0)

e (B1x + B0) 1 +

CLASS EXAMPLE: LOGISTIC REGRESSION

 Probability of a person complying for a mammogram, based on whether or not they get a physician’s recommendation

CLASS EXAMPLE: LOGISTIC REGRESSION

p ^ = e (B1x + B0)

e (B1x + B0) 1 +

Probability of complying if NOT recommended by physician:

Probability of complying if recommended by physician:

p ^ = e (2.29(0) - 1.84)

e (2.29(0) - 1.84) 1 + p ^ =

e (2.29(1) - 1.84)

e (2.29(1) - 1.84) 1 +

= 0.14 = 0.61

ORDERED MULTINOMIAL LOGISTIC REGRESSION ANALYSIS

 Type of logistic regression that allows more than two discrete outcomes

 Outcomes are ordinal:   Yes, maybe, no   First, second, third place   Gold, silver, bronze medals   Strongly agree, agree, neutral, disagree,

strongly disagree

ASSUMPTION

 No perfect predictions – one predictor variable value cannot solely correspond to one dependent variable value – check using crosstabs.

ORDERED LOGISTIC REGRESSION EXAMPLE

 Load libraries:

 Load data: pooj<-read.csv("http://www.ats.ucla.edu/stat/r/dae/ologit.csv")

library(arm) library(psych)

ORDERED LOGISTIC REGRESSION EXAMPLE

 Variables:   apply – college juniors reported likelihood

of applying to grad school (0 = unlikely, 1 = somewhat likely, 2 = very likely)

  pared – indicating whether at least one parent has a graduate degree (0 = no, 1 = yes)

  public – indicating whether the undergraduate institution is a public or private (0 = private, 1 = public)

  gpa – college GPA

> str(pooj) 'data.frame': 400 obs. of 4 variables: $ apply : int 2 1 0 1 1 0 1 1 0 1 ... $ pared : int 0 1 1 0 0 0 0 0 0 1 ... $ public: int 0 0 1 0 0 1 0 0 0 0 ... $ gpa : num 3.26 3.21 3.94 2.81 2.53 ... > table(pooj$apply) 0 1 2 220 140 40 > table(pooj$pared) 0 1 337 63 > table(pooj$public) 0 1 343 57

> xtabs(~pooj$pared+pooj$apply) pooj$apply pooj$pared 0 1 2 0 200 110 27 1 20 30 13 > xtabs(~pooj$public+pooj$apply) pooj$apply pooj$public 0 1 2 0 189 124 30 1 31 16 10

CHECK ASSUMPTION – CROSS-TABS

 Why is this important?

SINGLE PREDICTOR MODEL - GPA > library(arm) > summary(m1<-bayespolr(as.ordered(pooj$apply)~pooj$gpa)) Call: bayespolr(formula = as.ordered(pooj$apply) ~ pooj$gpa) Coefficients: Value Std. Error t value pooj$gpa 0.7109 0.2471 2.877 Intercepts: Value Std. Error t value 0|1 2.3306 0.7502 3.1065 1|2 4.3505 0.7744 5.6179 Residual Deviance: 737.6921 AIC: 743.6921

0|1 1|2

CUMULATIVE DISTRIBUTION FUNCTION

LABELING COEFFICIENTS Coefficients:

Value Std. Error t value

pooj$gpa 0.7109 0.2471 2.877

Intercepts:

Value Std. Error t value

0|1 2.3306 0.7502 3.1065

1|2 4.3505 0.7744 5.6179

Coefficient of the model coef<- m1$coef

Intercepts of the model intercept <- m1$zeta

 Let us look at the likelihood of students with an average GPA applying to graduate school.

> x<-mean(pooj$gpa)

[1] 2.998925

TRANSFORMING OUTCOMES TO PROBABILITIES

prob<-function(input){exp(input)/(1+exp(input))}

(p0<-prob(intercept[1]-coef*x))

0.5493198

(p1<-prob(intercept[2]-coef*x)-p0)

0.3525213 (p2<-1-(p0+p1))

0.0981589

WHY NOT USE LINEAR REGRESSION? > summary(linreg<-lm(pooj$apply~pooj$gpa))

Call:

lm(formula = pooj$apply ~ pooj$gpa)

Residuals:

Min 1Q Median 3Q Max

-0.7917 -0.5554 -0.3962 0.4786 1.6012

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -0.22016 0.25224 -0.873 0.38329

pooj$gpa 0.25681 0.08338 3.080 0.00221 **

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6628 on 398 degrees of freedom

Multiple R-squared: 0.02328, Adjusted R-squared: 0.02083

F-statistic: 9.486 on 1 and 398 DF, p-value: 0.002214

AND OUR ASSUMPTIONS AREN’T MET…

LINEAR REGRESSION VERSUS ORDERED LOGISTIC REGRESSION

The decision between linear regression and ordered multinomial regression is not always black and white. When you have a large number of categories that can be considered equally spaced simple linear regression is an optional alternative (Gelman & Hill, 2007).

Moral of story: Always start by checking the

assumptions of the model.

USING MULTIPLE PREDICTORS summary(m2 <- bayespolr(as.ordered(apply)~gpa + pared + public ,pooj)) Call: bayespolr(formula = as.ordered(apply) ~ gpa + pared + public, pooj) Coefficients: Value Std. Error t value gpa 0.6041463 0.2577039 2.3443424 pared 1.0274106 0.2636348 3.8970973 public -0.0528103 0.2931885 -0.1801240 Intercepts: Value Std. Error t value 0|1 2.1638 0.7710 2.8064 1|2 4.2518 0.7955 5.3449 Residual Deviance: 727.002 AIC: 737.002

TRANSFORMING OUTCOMES TO PROBABILITIES (coef<- m2$coef)

gpa pared public

0.6041463 1.0274106 -0.0528103

(intercept<-m2$zeta)

0|1 1|2

2.163841 4.251774

(x1<-cbind(0:4, 0 , .14))

[,1] [,2] [,3]

[1,] 0 0 0.14

[2,] 1 0 0.14

[3,] 2 0 0.14

[4,] 3 0 0.14

[5,] 4 0 0.14

(x2<-cbind(0:4, 1 , .14))

[,1] [,2] [,3]

[1,] 0 1 0.14

[2,] 1 1 0.14

[3,] 2 1 0.14

[4,] 3 1 0.14

[5,] 4 1 0.14

TRANSFORMING OUTCOMES TO PROBABILITIES prob<-function(VAR){exp(VAR)/(1+exp(VAR))}

> (p1<-prob(intercept[1]-x1 %*% coef))

[,1]

[1,] 0.9119769

[2,] 0.8498732

[3,] 0.7556908

[4,] 0.6282669

[5,] 0.4801055

> (p2<-prob(intercept[2]-x1 %*% coef)-p1)

[,1]

[1,] 0.07538029

[2,] 0.12722869

[3,] 0.20318345

[4,] 0.29895089

[5,] 0.39428044

> (p3<-1-(p1+p2))

[,1]

[1,] 0.01264281

[2,] 0.02289816

[3,] 0.04112575

[4,] 0.07278223

[5,] 0.12561404

TRANSFORMING OUTCOMES TO PROBABILITIES > (p4<-prob(intercept[1]-x2 %*% coef))

[,1]

[1,] 0.7876055

[2,] 0.6695483

[3,] 0.5254116

[4,] 0.3769123

[5,] 0.2484150

> (p5<-prob(intercept[2]-x2 %*% coef)-p1)

[,1]

[1,] 0.05348287

[2,] 0.08867445

[3,] 0.13730004

[4,] 0.19186675

[5,] 0.23347632

> (p6<-1-(p4+p5))

[,1]

[1,] 0.1589117

[2,] 0.2417772

[3,] 0.3372883

[4,] 0.4312209

[5,] 0.5181087

PLOTTING THE RESULTS Undergrad.GPA <-0:4 plot(Undergrad.GPA, p1, type="l", col=1, ylim=c(0,1)) lines(0:4, p2, col=2) lines(0:4, p3, col=3) lines(0:4, p4, col=1, lty = 2) lines(0:4, p5, col=2, lty = 2) lines(0:4, p6, col=3, lty = 2) legend(1.5, 1, legend=c("P(unlikely)", "P(somewhat likely)", "P(very likely)", "Line Type when Pared = 0", "Line Type when Pared = 1"), col=c(1:3,1,1), lty=c(1,1,1,1,2))

PRACTICE Read in the following table (Quinn, n.d.): practice <- read.table("http://www.stat.washington.edu/quinn/classes/536/data/nes96r.dat", header=TRUE)

Task: Run a regression using the ordered multinomial logistic model to predict the variation in the dependent variable ClinLR using the independent variables PID and educ.

ClinLR = Ordinal variable from 1-7 indicating ones view of Bill Clinton’s political

leanings, where 1 = extremely liberal, 2 = liberal, 3 = slightly liberal, 4 = moderate, 5= slightly conservative, 6 = conservative, 6 = extremely conservative.

PID = Ordinal variable from 0-6 indicating ones own political identification, where 0 = Strong Democrat and 6 = Strong Republican

educ = Ordinal variable from 1-7 indicating ones own level of education, where 1 = 8 grades or less and no diploma, 2 = 9-11 grades, no further schooling, 3 = High school diploma or equivalency test, 4 = More than 12 years of schooling, no higher degree, 5 = Junior or community college level degree (AA degrees), 6 = BA level degrees; 17+ years, no postgraduate degree, 7 = Advanced degree

REFERENCES

 Gelman, A. & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. New York: Cambridge University Press.

 Quinn, K. (n.d.). Retrieved from http://www.stat.washington.edu/quinn/classes/536/data/nes96r.dat  UCLA: Academic Technology Services. (n.d.). Retrieved from http://www.ats.ucla.edu/stat/r/dae/ologit.csv