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OutlineIntroduction
BasicsApplication
Typical analysis
Ordinal Regression
LISA short course
July 22, 2009
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
IntroductionCategorical Variable
BasicsMultinomial VariableLogit Models for Multinomial Variable
ApplicationExample 1Example 2Example 3
Typical analysis
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Categorical Variable
Categorical Variable
A Categorical Variable has a measurement scale consisting of a setof categories.
I In politics: Liberal, Moderate, or Conservative.
I In medicine: Benign, Probably Benign, Suspicious, orMalignant.
I Opinion: Extremely Disagree, ..., Extremely Agree.
In general it measures attitudes, opinions, effectiveness, etc.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Categorical Variable
Response variable distinction
I Nominal. Measures having categories without a naturalordering.
I Catholic, Protestant, Jewish, Muslims.I Automobile, Bicycle, Bus, Subway, Walk.I Classical, Country, Folk, Jazz, Rock.I Apartment, Condominium, House, other.
I Ordinal. Measures having categories with natural ordering.I Size: Small, Medium, Large.I Social class: Upper, Middle, Lower.I Medical Condition: Good, Fair, Serious, Critical.I Intervals: < 10 years, 10-12 years, >12 years.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Multinomial Variable
Let Y=j be a random variable having more than two possibleoutcomes (j = 1, 2, . . . , J). The probability P(Y = j) = πj forj = 1, . . . , JEx. Suppose the following question with 5 categories response (Y):The current health plan is awesome...
1. I completely disagree
2. I disagree
3. Neutral (It does not bother me at all)
4. I agree
5. I completely agree
And suppose the respondent chooses number 3, then Y = 3Further, you assume all the categories have the same probability ofbeing selected, then πj(=3) = 1/5
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Probability and its cumulative
Let Y be the response and setting x, the probability can beexpressed as P(Y = j |x) = πj(x). This equation relates theprobability with the actual response and the regressors expressedby x.For ordinal variables, we are more interested in modeling the
P(Y ≤ j |x) = π1(x) + . . .+ πj(x)
Ex. From the last question:
1. P(Y ≤ 1|x) = 1/5
2. P(Y ≤ 2|x) = 1/5 + 1/5 = 2/5
3. P(Y ≤ 3|x) = 1/5 + 1/5 + 1/5 = 3/5
4. P(Y ≤ 4|x) = 1/5 + 1/5 + 1/5 + 1/5 = 4/5
5. P(Y ≤ 5|x) = 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Odds and Logit
Let’s P(Y = j |x) be the probability defined above. The Odds aredefined as:
P(Y = j |x)
1− P(Y = j |x), for j=1,. . . ,J
The Logit is defined as:
log[P(Y = j |x)] = logP(Y = j |x)
1− P(Y = j |x)
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
The Cumulative Logits
The cumulative odds are defined as:
P(Y ≤ j |x)
1− P(Y ≤ j |x)
and the cumulative logits are defined as:
logit[P(Y ≤ j |x)] = logP(Y ≤ j |x)
1− P(Y ≤ j |x)
logit[P(Y ≤ j |x)] = logπ1(x) + . . .+ πj(x)
πj+1(x) + . . .+ πJ(x)
for j = 1, . . . , J − 1.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Proportional odds model
logit[P(Y ≤ j |x)] = αj + x′β, for j = 1, . . . , J − 1.
Notice that each cumulative logit has its own intercept, αj .The usual model assumes that the vector β has the same effect foreach logit. The same impact on the logit regardless the category.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Proportional odds model
The cumulative logit model satisfies:
logit[P(Y ≤ j |x1)]− logit[P(Y ≤ j |x2)] = β′(x1 − x2)
The odds of making Y ≤ j at x = x1 are
exp[β′(x1 − x2)]
times the odds at x = x2
So, it is clear that the log of odds ratio is proportional to thedistance. This property gives the name of this model, proportionalodds model.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Multinomial VariableLogit Models for Multinomial Variable
Other cumulative link functions
1. Probit Model:
Φ−1(P(Y ≤ j |x)) = αj + βx
2. Complementary log-log link (Proportional Hazards Model)
log{-log [1− P(Y ≤ j |x)]} = αj + βx.
A property of this link is:
P(Y > j |x1) = [P(Y > j |x2)]exp[β′(x1−x2)]
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example
(Agresti, 2002). It is a study of mental health for a random sampleof adult residents of Alachua County, Fl. It relates mentalimpairment to two explanatory variables.
I Mental (response). Mental impairment is an ordinal responsewith categories well, mild symptom formation, moderatesymptom formation, and impaired.
I Life (regressor 1). The life events index is a measure of thenumber of important life events: birth of child, new job,divorce, or death in family, within the past 3 years.
I SES (regressor 2). Socioeconomic status is measured here asbinary: 0-low and 1-high.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example
For this first example, let’s considered only one regressor, Life.Then the models is:
logit[P(Mental ≤ j |Life)] = αj + β ∗ Life
The task is to find values for αj and β on the model, for j =Well,Mild, Moderate.A code on the response outcomes scale is exercised: 1:Well, 2:Mild, 3: Moderate, 4: Impaired.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example procedure
In JMP:I Download the data set from the following link: http:
//filebox.vt.edu/users/cvelasco/statwww/MentalImpairment.jmp
I Make sure that the response variable has the ORDINAL data type: rightclick on the response’s column and select Column info, then change toOrdinal.
I Go to JMP’s main menu, choose ANALYZE99KFIT MODEL. Then, selectMind from the list of columns, and click on the Y’s icon. Also select Lifeand click on the Add’s icon.
I Make sure that on the Personality box is written Ordinal Logistic(otherwise, go back to the second step)
I Click on the Run Model icon.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example output
The intercept in the analysis is in the plot the following:
e0.2613
1 + e0.2613= 0.565
e1.656
1 + e1.656= 0.84
The final model for the probability under the proportional Oddslogit model is:
P(Y ≤ j |Life) =eα̂j−0.2879∗Life
1 + eα̂j−0.2879∗Life
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example 2
From the last data set, adding other variable to the model
logit[P(Mental ≤ j |Life,SES)] = αj + β1 ∗ Life + β2 ∗ SES
We also consider the model with a interaction of those twovariables:
logit[P(Mental ≤ j |Life,SES)] = αj+β1∗Life+β2∗SES+β3∗Life∗SES
At this point there is not a plot as part of the output. This isbecause there are two variable into the model, a surfacerepresentation is hard to present. Thus, a summary table pop upas output.
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example 3
Male and female subjects received an active or placebo treatmentfor their arthritis pain, and the subsequent extent of improvementwas recorded as marked, some, or none*. (To download the data:http://filebox.vt.edu/users/cvelasco/statwww/Arthritis.jmp)
* Stokes, Maura. 2000. Categorical data analysis using SAS system
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Example 1Example 2Example 3
Example 3
LISA short course Ordinal Regression
OutlineIntroduction
BasicsApplication
Typical analysis
Questionnaire, having 10 questions (Yij , for i = 1, . . . , n andj = 1, . . . , 10). All of them on the Likert scale (coded from 1 -5).People create scores for each person interviewed:
Zi =10∑j=1
Yij , for i =, . . . , n
This new variable Zi will have a different range. It is not longerordinal. Associating this one to a set of regressor turns to be aregular regression procedure.
LISA short course Ordinal Regression