Regression Analysis with the Ordered Multinomial Logistic Model

Braden HoelzleSouthern Methodist University

December 2009

Situating the Model

GLM – Generalized Linear Model

Linear Regression Logistic Regression

Ordered Multinomial

Logistic Regression

Unordered Multinomial

Logistic Regression

Review: Logistic Regression

• Dichotomous Dependent Variable• Independent Variables can be dichotomous,

integral, categorical…etc.• We are trying to predict the probability that a

person does or doesn’t have a trait• Example: At risk of dropping out or Not at risk• Others??

Transform to Probability

• Probability range = (0 ≤ p ≥ 1)• Therefore we must transform continuous

values to the range 0-1 by using the formula:

Or expanded to:

A Quick Example> m1 <- glm(comply ~ physrec, family = binomial(link = "logit"))> summary(m1)Call:glm(formula = comply ~ physrec, family = binomial(link = "logit")) Coefficients:

Estimate Std. Error z value Pr(>|z|)(Intercept) -1.8383 0.4069 -4.518 6.26e-06physrec 2.2882 0.4503 5.081 3.75e-07

The probability of complying if NOT recommended by physician: exp(-1.8383)/(1 + exp(-1.8383))0.1372525The probability of complying if recommended by physician: exp(-1.8383 + 2.2882)/(1 + exp(-1.8383 + 2.2882))0.6106392

Ordered Multinomial Logistic Model

Four Types of Scales1. _________ - mutually exclusive categories w/ no logical order.2. _________ - mutually exclusive categories w/ logical rank order.3. _________ - ordered data w/ equal distance between each point (no absolute zero).4. _________ - ordered data w/ equal distance between each point (w/ a “true” zero).

What type of data would you expect our ordered multinomial regression to model?

Definition

• The ordered multinomial logistic model enables us to model ordinally scaled dependent variables with one or more independent variables.

• These IV(s) can take many different forms (ie. real numbers values, integers, categorical, binomial, etc.).

Does this Occur Much?“Ordinal data are the most frequently encountered type of data in the social

sciences” (Johnson & Albert, 1999, p. 126). • Examples

– Yes, maybe, no– Likert scale (Strongly Agree – Strongly Disagree)– Always, frequently, sometimes, rarely, never– No hs diploma, hs diploma, some college, bachelor’s degree, master’s degree,

doctoral degree– Free school lunch, reduced school lunch, full price lunch– 0-10k per year, 10-20K per year, 20-30K per year, 30 – 60K per year, > 60K per

year– Low, medium, high– Basic math, regular math, pre-AP math, AP math– Nele’s dancing ability, Meg’s dancing ability, Saralyn’s dancing ability, Jose’s

dancing ability, Kyle’s dancing ability, Braden’s dancing ability, a rock

Running Regression using the Ordered Multinomial Logistic Model in R

Load/Install Libraries:library(arm)library (psych)

Load data (UCLA – Academic Technology Services, n.d.)

mydata <- read.csv(url("http://www.ats.ucla.edu/st at/r/dae/ologit.csv"))

attach(mydata)

Definitions

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 agraduate degree (0 = no, 1 = yes)public – indicating whether the undergraduate institution is a pubic or private (0 = private, 1 = public)gpa – college gpa

Which variable will likely be our dependent variable?

Description of Data> str(mydata)'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 ...

> summary(mydata$gpa) Min. 1st Qu. Median Mean 3rd Qu. Max. 1.900 2.720 2.990 2.999 3.270 4.000

> table(apply)apply 0 1 2 220 140 40

> table(pared)pared 0 1 337 63

> table(public)public 0 1 343 57

Crosstabs> xtabs(~ pared + apply) applypared 0 1 2 0 200 110 27 1 20 30 13

> xtabs(~ public + apply) applypublic 0 1 2 0 189 124 30 1 31 16 10

Why would this information be important for running our ordered multinomial logistic model?

Assumptions

No perfect predictions – one predictor variable value cannot solely correspond to one dependent variable value. (ex. – Every student w/ parents who went to graduate school cannot indicate that they are very likely to attend graduate school) – check using crosstabs ( see slide 12).

No empty or very small cells – see crosstabs.Sample Size – always requires more cases than OLS

regression.

Running a Single Predictor Model> summary(m1 <- bayespolr(as.ordered(apply)~gpa,data=mydata))Call:bayespolr(formula = as.ordered(apply) ~ gpa, data = mydata)

Coefficients: Value Std. Error t valuegpa 0.7109826 0.247078 2.877563

Intercepts: Value Std. Error t value0|1 2.3308 0.7502 3.1068 1|2 4.3508 0.7744 5.6183

Residual Deviance: 737.6921 AIC: 743.6921

Transforming Outcomes to Probabilities (beta <- coef(m1)) gpa 0.7109826 (tau <- m1$zeta) 0|1 1|2 2.330831 4.350816 x<- 3 ##### Note: mean = 2.999

logit.prob <- function(eta){exp(eta)/(1+exp(eta))} (p1 <- logit.prob(tau[1] - x * beta))0.54931 (p2<- logit.prob(tau[2] - x * beta) - logit.prob(tau[1] - x * beta))0.3525327 (p3<- 1 - logit.prob(tau[2] - x * beta))0.09815732 p1+p2+p31

Adding Multiple Predictors> summary(m2 <- bayespolr(as.ordered(apply)~gpa + pared +

public ,data=mydata))Call:bayespolr(formula = as.ordered(apply) ~ gpa + pared + public, data = mydata)

Coefficients: Value Std. Error t valuegpa 0.6041463 0.2577039 2.3443424pared 1.0274106 0.2636348 3.8970973public -0.0528103 0.2931885 -0.1801240

Intercepts: Value Std. Error t value0|1 2.1638 0.7710 2.80641|2 4.2518 0.7955 5.3449

Residual Deviance: 727.002 AIC: 737.002

Transforming Outcomes to Probabilities

(beta <- coef(m2)) gpa pared public 0.6041463 1.0274106 -0.0528103 (tau <- m2$zeta) 0|1 1|2 2.163841 4.251774 (x<- cbind(0:4, 0 , .15)) [,1] [,2] [,3][1,] 0 0 0.15[2,] 1 0 0.15[3,] 2 0 0.15[4,] 3 0 0.15[5,] 4 0 0.15 (x2<-cbind(0:4, 1 , .15)) [,1] [,2] [,3][1,] 0 1 0.15[2,] 1 1 0.15[3,] 2 1 0.15[4,] 3 1 0.15[5,] 4 1 0.15

Transforming Outcomes to Probabilities (cont.)

logit.prob <- function(eta){exp(eta)/(1+exp(eta))} (p1 <- logit.prob(tau[1] - x %*% beta)) [,1][1,] 0.8976849[2,] 0.8274435[3,] 0.7238159[4,] 0.5888766[5,] 0.4390981 (p2<- logit.prob(tau[2] - x %*% beta) - logit.prob(tau[1] - x %*% beta)) [,1][1,] 0.08838526[2,] 0.14736050[3,] 0.23102713[4,] 0.33148400[5,] 0.42421801 (p3<- 1 - logit.prob(tau[2] - x %*% beta)) [,1][1,] 0.01392982[2,] 0.02519605[3,] 0.04515695[4,] 0.07963941[5,] 0.13668388

Transforming Outcomes to Probabilities (cont.)

(p4 <- logit.prob(tau[1] - x2 %*% beta)) [,1][1,] 0.7584777[2,] 0.6318601[3,] 0.4840202[4,] 0.3389252[5,] 0.2188751 (p5<- logit.prob(tau[2] - x2 %*% beta) - logit.prob(tau[1] - x2 %*% beta)) [,1][1,] 0.2035536[2,] 0.3007906[3,] 0.3992730[4,] 0.4663890[5,] 0.4744476 (p6<- 1 - logit.prob(tau[2] - x2 %*% beta)) [,1][1,] 0.03796871[2,] 0.06734929[3,] 0.11670683[4,] 0.19468576[5,] 0.30667730

Plotting the Results

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Undergrad.GPA

p1

P(unlikely)P(somewhat likely)P(very likely)Line Type when Pared = 0Line Type when Pared = 1

Why Not Use Linear Regression?> summary(m1.2<-lm(apply~gpa, data=mydata))

Call:lm(formula = apply ~ gpa, data = mydata)

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 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 freedomMultiple R-squared: 0.02328, Adjusted R-squared: 0.02083 F-statistic: 9.486 on 1 and 398 DF, p-value: 0.002214

What Do Our Results Mean?

Plug in a gpa of 3:> (y.hat<-(-.2201 + (.2568 * 3)))[1] 0.5503This means that we expect someone w/ a 3.0 gpa

to fall about half way between unlikely (0) and slightly likely (1) to apply to grad school. But what is half way between these two points (a little unlikely?, neither likely nor unlikely?, very slightly likely?) This is somewhat vague.

Our Graph using Linear Regression

200 400 600 800 1000 1200 1400

400

600

800

100

01

200

Graph 1:TAKS Lexile Predicted by MAP Lexile

MAP Lexile

TA

KS

Le

xile

A Normal OLS Line An OLS Line on Our Data

We Royally Violate our Assumptions

0.3 0.4 0.5 0.6 0.7 0.8

-1.0

0.0

1.0

Fitted values

Res

idua

ls

Residuals vs Fitted1859486

-3 -2 -1 0 1 2 3

-10

12

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als Normal Q-Q

1859486

0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.5

1.0

1.5

Fitted values

Sta

ndar

dize

d re

sidu

als Scale-Location

1859486

0.000 0.010 0.020

-10

12

Leverage

Sta

ndar

dize

d re

sidu

als

Cook's distance

Residuals vs Leverage

13

185

78

However…

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).

** But check your assumptions!!

PracticeRead in the following table (Quinn, n.d.):nes96 <-

read.table("http://www.stat.washington.edu/quinn/classes/536/data/nes96r.dat", header=TRUE)

Run a regression using the ordered multinomial logistic model to predict the variation in the dependent variable ClinLR using the dependent 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. NewYork: Cambridge University Press.

• Johnson, V. E. & Albert, J. H. (1999). Statistics for the social sciences and public policy: Ordinal datamodeling. New York: Springer.

• 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/st at/r/dae/ologit.csv