Lecture 18Ordinal and Polytomous Logistic Regression

BMTRY 701

Biostatistical Methods II

Categorical Outcomes

Logistic regression is appropriate for binary outcomes

What about other kinds of categorical data?• >2 categories• ordinal data

Standard logistic is not applicable unless you ‘threshold’ the date or collapse categories

BMTRY 711: Analysis of Categorical Data This is just an overview

Ordinal Logistic Regression

Ordinal Dependent Variable• Teaching experience• SES (high, middle, low)• Degree of Agreement• Ability level (e.g. literacy, reading)• Severity of disease/outcome• Severity of toxicity

Context is important Example: attitudes towards smoking

Proportional Odds Model

One of several possible regression models for the analysis of ordinal data, and also the most common.

Model predicts the ln(odds) of being in category j or beyond.

Simplifying assumption: “proportional odds”• Effect of covariate assumed to be invariant across splits• Example: 4 categories

0 vs 1,2,3 0,1 vs 2,3 0,1,2 vs 3

• Assumes that each of these comparisons yields the same odds ratio

Motivating Example: YTS

The South Carolina Youth Tobacco Survey (SC YTS) is part of the National Youth Tobacco Survey program sponsored by the Centers for Disease Control and Prevention. The YTS is an annual school-based survey designed to evaluate youth-related smoking practices, including initiation and prevalence, cessation, attitudes towards smoking, media influences, and more. The SC YTS is coordinated by the SC Department of Health and Environmental Control and has been administered yearly since 2005. Data for this report are based on years 2005-2007. The SC YTS uses a two-stage sample cluster design to select a representative sample of public middle (grades 6-8) and high school (grades 9-12) students.

Ordinal Outcomes

. tab cr44

“do you think | smoking | cigarettes | makes young | people look | cool or fit | in?” | Freq. Percent Cum.---------------+-----------------------------------definitely yes | 460 6.07 6.07 probably yes | 818 10.79 16.86 probably not | 1,329 17.53 34.38definitely not | 4,975 65.62 100.00---------------+----------------------------------- Total | 7,582 100.00

.

“do you think | young people | risk harming | themselves if | they smoke | from 1 - 5 | ciga | Freq. Percent Cum.---------------+-----------------------------------definitely yes | 5,387 70.98 70.98 probably yes | 1,283 16.91 87.89 probably not | 360 4.74 92.63definitely not | 559 7.37 100.00---------------+----------------------------------- Total | 7,589 100.00

What factors are related to these attitudes?

Gender? Grade? Race? parental education (surrogate for SES)? year? (2005, 200, 2007) have tried cigarettes? school performance? smoker in the home?

Tabulation of gender vs. look cool

“do you think | smoking | cigarettes | makes young | people look | cool or fit | gender in?” | 0 1 | Total---------------+----------------------+----------definitely yes | 278 177 | 455 probably yes | 446 364 | 810 probably not | 692 628 | 1,320 definitely not | 2,158 2,797 | 4,955 ---------------+----------------------+---------- Total | 3,574 3,966 | 7,540

Possible “breaks”

OR = 1.81

male female

def yes

278 177

else 3296 3789

OR = 1.59

male female

yes 724 541

no 2850 3425

OR = 1.57

male female

else 1416 1169

def no 2158 2797

Proportional Odds Assumption

How to implement this? Model the probability of ‘cumulative’ logits Instead of

Here, we have

)1(1

)1(

yP

yPodds

)(1

)(

kyP

kyPodds

The (simple) ordinal logistic model

1,...,1;)|(-1

)|(ln 1

Kkx

xkyP

xkyPk

Warning! different packages parameterize it different ways!Stata codes it differently than SAS and R.

Notice how this differs from logistic regression:there is a ‘level’ specific intercept.

But, there is just ONE log odds ratio describing the association between x and y.

Example

. ologit lookcool gender

Iteration 0: log likelihood = -7465.0108Iteration 1: log likelihood = -7418.1251Iteration 2: log likelihood = -7418.0256

Ordered logistic regression Number of obs = 7540 LR chi2(1) = 93.97 Prob > chi2 = 0.0000Log likelihood = -7418.0256 Pseudo R2 = 0.0063

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gender | .4605529 .0476442 9.67 0.000 .367172 .5539338-------------+---------------------------------------------------------------- /cut1 | -2.525572 .0529346 -2.629322 -2.421823 /cut2 | -1.375663 .0380258 -1.450193 -1.301134 /cut3 | -.4159722 .0336987 -.4820204 -.349924------------------------------------------------------------------------------

R estimation

Different parameterization

Makes you think about what the model is doing!

KkxxkyP

xkyPk ,...,2;

)|(-1

)|(ln 1

> library(Design)> oreg <- lrm(lookcool ~ gender, data=data)> oreg

Logistic Regression Model

lrm(formula = lookcool ~ gender, data = data)

Frequencies of Responses 1 2 3 4 455 810 1320 4955

Frequencies of Missing Values Due to Each Variablelookcool gender 196 50

Obs Max Deriv Model L.R. d.f. P C Dxy 7540 2e-12 93.97 1 0 0.552 0.104 Gamma Tau-a R2 Brier 0.206 0.054 0.014 0.056

Coef S.E. Wald Z Py>=2 2.5256 0.05293 47.71 0y>=3 1.3757 0.03803 36.18 0y>=4 0.4160 0.03370 12.34 0gender 0.4606 0.04764 9.67 0

MLR. ologit lookcool gender evertried smokerhome grade school_perf

Iteration 0: log likelihood = -2123.9232Iteration 1: log likelihood = -2052.0964Iteration 2: log likelihood = -2051.2897Iteration 3: log likelihood = -2051.2895

Ordered logistic regression Number of obs = 2125 LR chi2(5) = 145.27 Prob > chi2 = 0.0000Log likelihood = -2051.2895 Pseudo R2 = 0.0342

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gender | .214247 .09052 2.37 0.018 .0368309 .391663 evertried | 1.048804 .0999844 10.49 0.000 .852838 1.24477 smokerhome | .1350945 .0931715 1.45 0.147 -.0475182 .3177072 grade | .0646475 .0253649 2.55 0.011 .0149332 .1143618school_per~e | -.0407656 .0591738 -0.69 0.491 -.1567441 .0752128-------------+---------------------------------------------------------------- /cut1 | -.9447746 .301191 -1.535098 -.3544511 /cut2 | .3491469 .2940768 -.2272331 .9255269 /cut3 | 1.386131 .2950286 .8078857 1.964377------------------------------------------------------------------------------

It is a pretty strong assumption

How can we check? Simple check as shown in 2x2 table. Continuous variables: harder

• need to consider the model• no direct ‘tabular’ comparison

multiple regression: does it hold for all? Tricky! It needs to make sense and you need to

do some ‘model checking’ for all of your variables

Worthwhile to check each individually.

There is another approach

There is a test of proportionality. Implemented easily in Stata with an add-on

package: omodel• Ho: proportionality holds• Ha: proportionality is violated

Why? violation would require more parameters and would be a larger model

What does small p-value imply?• but be careful of sample size!• large sample sizes will make it hard to ‘adhere’ to

proportionality assumption

Estimation in Stata. omodel logit lookcool gender

Iteration 0: log likelihood = -7465.0108Iteration 1: log likelihood = -7418.1251Iteration 2: log likelihood = -7418.0256

Ordered logit estimates Number of obs = 7540 LR chi2(1) = 93.97 Prob > chi2 = 0.0000Log likelihood = -7418.0256 Pseudo R2 = 0.0063

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gender | .4605529 .0476442 9.67 0.000 .367172 .5539338-------------+---------------------------------------------------------------- _cut1 | -2.525572 .0529346 (Ancillary parameters) _cut2 | -1.375663 .0380258 _cut3 | -.4159722 .0336987 ------------------------------------------------------------------------------

Approximate likelihood-ratio test of proportionality of oddsacross response categories: chi2(2) = 2.43 Prob > chi2 = 0.2964

. omodel logit lookcool grade

Iteration 0: log likelihood = -7425.0617Iteration 1: log likelihood = -7424.7193Iteration 2: log likelihood = -7424.7193

Ordered logit estimates Number of obs = 7505 LR chi2(1) = 0.68 Prob > chi2 = 0.4079Log likelihood = -7424.7193 Pseudo R2 = 0.0000

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- grade | -.0106001 .0128062 -0.83 0.408 -.0356997 .0144995-------------+---------------------------------------------------------------- _cut1 | -2.784359 .0678301 (Ancillary parameters) _cut2 | -1.640955 .0567613 _cut3 | -.6923403 .0534291 ------------------------------------------------------------------------------

Approximate likelihood-ratio test of proportionality of oddsacross response categories: chi2(2) = 22.31 Prob > chi2 = 0.0000

What would the ORs be?

Generate three separate binary outcome variables from the ordinal variable• lookcool1v234• lookcool12v34• lookcool123v4

Estimate the odds ratio for each binary outcome

Stata Code

gen lookcool1v234=1 if lookcool==2 | lookcool==3 | lookcool==4replace lookcool1v234=0 if lookcool==1gen lookcool12v34=1 if lookcool==3 | lookcool==4replace lookcool12v34=0 if lookcool==1 | lookcool==2gen lookcool123v4=1 if lookcool==4replace lookcool123v4=0 if lookcool==2 | lookcool==3 | lookcool==1

logit lookcool1v234 gradelogit lookcool12v34 gradelogit lookcool123v4 grade

Results

For a one grade difference (range = 6 – 12)

• lookcool1v234 vs. grade: OR = 1.002 (0.93)

• lookcool12vs34 vs. grade: OR = 1.04 (p=0.03)

• lookcool123v4 vs. grade: OR = 0.98 (p=0.11)

Another approach: Polytomous Logistic Regression

Polytomous (aka Polychotomous) Logistic Regression

Fits the regression model with all contrasts. Can be used as an inferential model Or, can be used to estimate odds ratio to see if

they look ‘ordered” Model is different though

1,...,1;)|(

)|(ln

KkxxKyP

xkyPkk

. mlogit lookcool gender

Iteration 0: log likelihood = -7465.0108Iteration 1: log likelihood = -7417.1379Iteration 2: log likelihood = -7416.8737Iteration 3: log likelihood = -7416.8737

Multinomial logistic regression Number of obs = 7540 LR chi2(3) = 96.27 Prob > chi2 = 0.0000Log likelihood = -7416.8737 Pseudo R2 = 0.0064

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------definitely~s | gender | -.7108369 .1003382 -7.08 0.000 -.9074962 -.5141776 _cons | -2.049316 .0637222 -32.16 0.000 -2.174209 -1.924423-------------+----------------------------------------------------------------probably yes | gender | -.4625306 .0762255 -6.07 0.000 -.6119298 -.3131314 _cons | -1.576618 .0520148 -30.31 0.000 -1.678565 -1.474671-------------+----------------------------------------------------------------probably not | gender | -.3564113 .0621157 -5.74 0.000 -.4781559 -.2346668 _cons | -1.137351 .0436861 -26.03 0.000 -1.222974 -1.051728------------------------------------------------------------------------------(lookcool==definitely not is the base outcome)

Interpretation

For gender, notice the ordered nature of the odds ratio

Suggests that it may be appropriate to use an ordinal model

This model is more general, less restrictive but, sort of a mess to interpret

. mlogit lookcool grade

Iteration 0: log likelihood = -7425.0617Iteration 1: log likelihood = -7414.6932Iteration 2: log likelihood = -7414.6755

Multinomial logistic regression Number of obs = 7505 LR chi2(3) = 20.77 Prob > chi2 = 0.0001Log likelihood = -7414.6755 Pseudo R2 = 0.0014

------------------------------------------------------------------------------ lookcool | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------definitely~s | grade | .0051085 .0264996 0.19 0.847 -.0468299 .0570468 _cons | -2.407127 .1089328 -22.10 0.000 -2.620632 -2.193623-------------+----------------------------------------------------------------probably yes | grade | -.0390659 .0207178 -1.89 0.059 -.0796721 .0015402 _cons | -1.672094 .0825905 -20.25 0.000 -1.833968 -1.510219-------------+----------------------------------------------------------------probably not | grade | .0627357 .0166685 3.76 0.000 .030066 .0954055 _cons | -1.562533 .0709374 -22.03 0.000 -1.701568 -1.423499------------------------------------------------------------------------------(lookcool==definitely not is the base outcome)

In R?

mlogit library requires a data transformation step