Week 10: Ordinal Outcomes (Scott Long Chapter 5 Part 2)economics-course.weebly.com/uploads/2/...10_ordinal... · Ordinal Outcomes Interpretation the probability of an outcome. Discrete

Week 10: Ordinal Outcomes (Scott Long Chapter 5Part 2)

Tsun-Feng Chiang*

*School of Economics, Henan University, Kaifeng, China

June 2, 2014

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Ordinal Outcomes Interpretation

Interpretation

The better way to interpret the estimated results of ordinal outcomesmodel is by reporting predicted probabilities of the choice of a categorym given X ,

Pr(y = m|X ) = F (τm − X β)− F (τm−1 − X β)

These probabilities can be used in a variety of ways to show therelationships between the independent variables and the dependentcategories.

Mean and RangeIt is useful to begin by examining the mean, minimum andmaximum predicted probabilities over the sample:

meanPr(y = m|X ) =1N

N∑i=1

Pr(yi = m|x i)

min (max)Pr(y = m|X ) = mini (maxi)Pr(yi = m|x i)

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After running an ordered logit called ord_logit, use predict to calculatepredicted probabilities.> logit_prob <- predict(ord_logit, data = WARM, type="probs")> logit_prob

Figure Predicted Probabilities of m = 1, · · · ,4 for all sample

To get the max, min and mean for a category, 2 for example, usemax(probit_prob[ ,2]), min(probit_prob[ ,2]), mean(probit_prob[ ,2]),respectively.

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Plotting Predicted ProbabilitiesWe can also discuss how the predicted probabilities change bygraphs when one independent variable changes. However, thevalue of of the change in probabilities depends on the levels of allof the independent variables that are not changing, thosevariables should be fixed in given values.

For example, we want to discuss how age affects women’s attitudein 1989 so we allow age to vary, and gender is fixed in woman(MALE = 0), year is fixed in 1989 (YR89 = 1), and take themeans for other independent variables. Then the predictedprobabilities of outcome (or category) m is

Pr(WARM = m|AGE ,MALE = 0,YR89 = 1,X rest = X rest ) =

F (τm − X ∗β)− F (τm−1 − X ∗β)

where X ∗ is the data where the elements in the column for age isfrom 20 to 80; for male is all 0; for year 1989 is all 1, and for therest of the independents variables are sample means.

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Figure 5.3 A Predicted Probabilities for Women in 1989

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Plotting Cumulative ProbabilitiesThe cumulative probability is the probability that the outcome(category) is less than or equal to some value. The cumulativeprobability of being less than or equal to m is

Pr(y ≤ m|X ) =m∑

j=1

Pr(y = j |X ) = F (τm − Xβ)(eq.4)

ex . Pr(y ≤ 2|X ) = Pr(y = 1|X ) + Pr(y = 2|X )

= F (τ1−Xβ)−F (τ0−Xβ)+F (τ2−Xβ)−F (τ1−Xβ) = F (τ2−Xβ)

In our example, first calcaulte Pr(y ≤ 1|X ), the possibility of SD,then calculate Pr(y ≤ 2|X ), the possibility of SD and D, and so on.These probabilities can be plotted to uncover overall trend.

Table of Predicted ProbabilitiesThe table can clearly compare the changes and differences inpredicted probabilities for various groups of sample.

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Figure 5.3 B Cumulative Probabilities for Women in 1989

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Table 5.5 Predicted Probabilities by Gender and Year for the Ordered Logit

Model (keep other independent variables at the sample means)

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Marginal Effect (Partial Change)It is not of interest to see how y∗ changes in any independentvariable xl , instead the change of predicted probabilities thaty = m is of interest. Recall that

Pr(y = m|X ) = F (τm − Xβ)− F (τm−1 − Xβ)

Taking the partial derivative with respect to xl ,

∂Pr(y = m|X )

∂xl=∂F (τm − Xβ)

∂xl− ∂F (τm−1 − Xβ)

∂xl

= βl f (τm−1 − Xβ)− βl f (τm − Xβ)

= βl [f (τm−1 − Xβ)− f (τm − Xβ)]

The marginal effect is the slope of the curve relating to xl toPr(y = m|X ). The sign of the marginal effect is not necessarilythe same sign of βl since [f (τm − Xβ)− f (τm−1 − Xβ)] can beeither negative or positive.

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Since the marginal effect depends on the levels of all variables, wemust decide on which values of the variables to use when computingthe effect. One method is to compute the average over allobservations,

mean∂Pr(y = m|X )

∂xl=

1N

N∑i=1

βl [f (τm−1 − X iβ)− f (τm − X iβ)]

More commonly, the marginal effect is computed at the mean values ofall variables:

∂Pr(y = m|X )

∂xl= βl [f (τm−1 − Xβ)− f (τm − Xβ)](eq.5)

Or the marginal effect can be evaluated at other values, especiallywhen there are dummy variables in X . Generally, the marginal effectdoes not indicate the change in the probability that would be observedfor a unit change in xl . However, if an independent variable varies overa region of the probability curve that is nearly linear, the marginal effectcan be used to summarize the effect of a unit change in the variable on

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the probability of an outcome.Discrete ChangeInterpretation using marginal effects can be misleading when theprobability curve is changing rapidly or when an independentvariable is a dummy variable. Discrete change is the change in thepredicted probability for a change in xl from the start value xS tothe end value xE , holding all other variables at X :

∆Pr(y = m|X )

∆xl= Pr(y = m|X rest , xl = xE )−Pr(y = m|X rest , xl = xS)

when xl is continuous, xE = xS + δ where δ is a finite constantwhose value depends on the purpose of the analysis; when xl is adummy variable, xE = 0(1) while xS = 1(0). There are someuseful ways for summarizing discrete change,

Range

∆Pr(y=m|X)∆xl

= Pr(y = m|X rest , xl = max xl )− Pr(y = m|X rest , xl =

min xl )

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Discrete Change (Cont.)a unit change centered around the mean

∆Pr(y=m|X)∆xl

= Pr(y = m|X , xl = xl + 1/2)− Pr(y = m|X , xl =

xl − 1/2)

a standard deviation (sl ) change centered around the mean

∆Pr(y=m|X)∆xl

= Pr(y = m|X , xl = xl + sl/2)− Pr(y = m|X , xl =

xl − sl/2)

The effects of an independent variable xl can be summarized bycomputing the average of the absolute values of the changesacross all of the outcome categories. The average absolutediscrete change equals

∆ =1J

J∑j=1

∣∣∣∣∆Pr(y = j |X )

∆xl

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Discrete Change (Cont.)It is recommended that we can use representative observation asa base and see how the predicted probabilities of the outcomecategory m change when the independent variable deviates fromits value of this observation’s. Being a representative, thisobservation has continuous variables equal to the sample meansand dummy variables equal to the sample mode. However, if theindependent variables are highly skewed, ex. income or financialassets, assessing change relative to the mean can be misleadingand the changes relative to the median might be more useful.

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Ordinal Outcomes The Parallel Regression Assumption

The Parallel Regression Assumption

From (eq. 4), the cumulative probability of being less than or equal tom for an observation i is

Pr(yi ≤ m|x i) =m∑

j=1

Pr(yi = j |x i) = F (τm − x i β)

where β represents the parameters estimated from the ordered probitor the ordered logit models. Combine τm and the intercept β0, rewritethe cumulative probability,

Pr(yi ≤ m|x i) = F ((τm − β0)− β1xi1 − β2xi2 − · · · − βkxik )

For the case of 4 categories (J = 4), there are three cumulativeprobabilities,

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Pr(yi ≤ 1|x i) = F ((τ1 − β0)− β1xi1 − β2xi2 − · · · − βkxik )



The ordered model does not allow coefficients other than the interceptto change across the equations. In graph (see next page), it means thecdf or the cumulative probability curve shift left or right parallelly whenm changes. In other words, given the value of the cumulativeprobability, the slope with respect to any independent variable x is thesame when m changes. This property in the ordered logit and probitmodels is called the Parallel Regression Assumption.

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Table 5.4 Illustration of the Parallel Regression Assumption

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Now we calculate the cumulative probabilities using binary outcomemodels. Suppose there are 4 outcomes, we can create 4− 1 = 3binary variables B. First, let B1 = 1 be the choice of outcome 1 (y = 1),B1 = 0 be the choices of outcomes 2, 3 ,and 4 (y = 2,3,4). Run thebinary variable model and calculate predicted probability of B1 = 1,

Pr(B1 = 1|x i) = Pr(yi ≤ 1|x i) = F (β10 − β11xi1 − β12xi2 − · · · − β1kxik )

Second, let B2 = 1 be the choices of outcomes 1 and 2 (y = 1,2),B2 = 0 be the choices of outcomes 3 ,and 4 (y = 3,4). Run the binaryvariable model and calculate predicted probability of B2 = 1,


Third, let B3 = 1 be the choices of outcomes 1, 2, and 3 (y = 1,2,3),B3 = 0 be the choice of outcome 4 (y = 4). Run the binary variablemodel and calculate predicted probability of B3 = 1,


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Compared with cumulative probability functions where all coefficients(except the thresholds τ ) are identical in all equations the ordinaloutcome models, the coefficients βj1 are not necessarily different for agiven j in the binary outcome models. The parallel regressionassumption is like imposing constraints that βj1 = β1, βj2 = β2 , · · ·,βjk = βk for all j = 1,2, · · · , J − 1 on the binary outcome models. Moreformally, the parallel regression assumption says that

β1

β2...βk

=

β11β12

...β1k

=

β21β22

...β2k

· · · =

βJ−11βJ−12

...βJ−1k

(eq.6)

(eq. 6) or if the parallel regression assumption holds can be testedinformally and formally.

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Informal TestRun an ordered model to obtain β. Then run the first binary modelwhere the binary outcome is defined as 1 if y ≤ 1, and 0 else, toobtain β1. Second, Then run the first binary model where thebinary outcome is defined as 1 if y ≤ 2, and 0 else, to obtain β2.And so on up to the outcome equal to 1 if y ≤ J − 1. Directlycompare the values of β, β1, · · ·, and βJ−1, if they are verydifferent, then the parallel regression assumption is violated,otherwise it is not.

Formal TestsThe parallel regression assumption can also be tested by twoformal tests: score test and the Brant’s Wald test. The Wald test isa more general method which test whether assumption is violatedfor all independent variables or only for some.

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Table 5.8 Informal Test of the Parallel Regression Assumption

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The parallel regression assumption is frequently violated based oneither an informal or the formal tests. When the assumption isrejected, alternative models should be considered that do not imposethe constraint of parallel regression. The multinomial logit model is oneof them.

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Nominal Outcomes

Nominal Outcomes

When a variable is nominal, the categories cannot be ordered. Butmodels for nominal outcomes are often used when the dependentvariable is ordinal.

OccupationCommuting: driving a car, taking the bus and walking to the busstop, taking the bus and driving to the bus stop, carpooling, etc.The choice of a new vehicle: sports car, van, compact, Jeep,convertible, etc.

Sometime, it is done to avoid the parallel regression assumption of theordered model. Other times there may be uncertainty as to whetherthe dependent variable should be considered as ordinal. For example,the types of cars can be ordered by their prices, but we are not surewhether consumers’ preference move in the order of cars’ prices.

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Nominal Outcomes Introduction to the Multinomial Logit Model

Introduction to the Multinomial Logit Model

The multinomial logit model (MNLM) is one of the most frequently usedmodels for nominal outcomes. The effects of the independentvariables are allowed to differ for each outcome. The MNLM can bethought of as simultaneously estimating binary logits for all possiblecomparison among the outcome categories.

Suppose there is a nominal outcome y three categories, A, B, and Cwith NA, NB, and NC observations in each category. First we use theobservations NA and NB, and let A be y = 1 and B be y = 0. In thebinary logit with only one independent variable, the ratio of probabilityfor A to B is

Pr(A|X )

Pr(B|X )=

Pr(A|X )

1− Pr(A|X )=

exp(β0,A|B+β1,A|Bx)

1+exp(β0,A|B+β1,A|Bx)

11+exp(β0,A|B+β1,A|Bx)

= exp(β0,A|B + β1,A|Bx)

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The coefficient β1,A|B can be interpreted as a unit increases in x , thelog of the probabilities of A versus B changes β1,A|B.

How to justify this statement?

Let Ω(X ) =Pr(A|X )

Pr(B|X ), so

∂lnΩ(X )

∂xl= βl,A|B

Similarly, we can use NB and NC observations to estimate the binarylogit and the probabilities of B versus C:

Pr(B|X )

Pr(C|X )= exp(β0,B|C + β1,B|Cx)

Then select observations NA and NC for the logit:

Pr(A|X )

Pr(C|X )= exp(β0,A|C + β1,A|Cx)

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However, this a binary logit redundant. If you know the probabilities ofA versus B, and the probabilities of B versus C, then you can derivethe probabilities of A versus C without doing the estimation. To seethis, multiply two relative probabilities,

Pr(A|X )

Pr(B|X )× Pr(B|X )

Pr(C|X )=

Pr(A|X )

Pr(C|X )

⇒ exp(β0,A|B + β0,B|C + (β1,A|B + β1,B|C)x) = exp(β0,A|C + β1,A|Cx)

Therefore,

β0,A|B + β0,B|C = β0,A|C ; β1,A|B + β1,B|C = β1,A|C(eq.1)

(eq. 1) describes necessary relationships among the parameters in thepopulation. They will not hold with sample estimates from the threebinary logits because the observations used in three models aredifferent. In the MNLM, all of the logits are estimated simultaneously,so do the probabilities of one category versus another category.

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Nominal Outcomes The Multinomial Logit Model

The Multinomial Logit Model (MNLM)

There are several methods to derive the MNLM.

The MNLM as a Probability ModelSuppose y = j and j = 1,2, · · · ,m, · · · , J (the number is a notation,these categories are not assumed to be ordered). Let Pr(y = m|X ) bethe probability of observing outcome m given X . Assume thatPr(y = m|X ) is a function of the linear combination Xβm where thevector βm = (β0m, · · · , βlm, · · · , βkm)′ includes the intercept β0m and thecoefficient βl for the effect of xl on outcome m. In contrast to theordered logit model, βm differs for each outcome.

To ensure that the probabilities Pr(y = m|X ) are nonnegative, we takethe exponential of Xβm: exp(Xβm) > 0. The sum of all exponentialfunctions from exp(Xβ1) to exp(XβJ) is

∑Jj=1 exp(Xβj).

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The MNLM as a Probability Model (Continued)

For every observation i , the probability of a category m is exp(x iβm)divided by the sum,

Pr(yi = m|x i) =exp(x iβm)∑Jj=1 exp(x iβj)

(eq.2)

Now the probabilities sum to 1, i.e.∑J

m=1 Pr(yi = m|x i) = 1.

Identification(eq. 2) is unidentified since more than one set of parametersgenerates the same probabilities of the observed outcomes. To seethis, multiply (eq. 2) by exp(x iδ)/exp(x iδ),


× exp(x iδ)

exp(x iδ)

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Identification (Continued)

=exp(x iβm + x iδ)∑Jj=1 exp(x iβj + x iδ)

=exp(x i [βm + δ])∑Jj=1 exp(x i [βj + δ])

While the value of the probabilities are unchanged, the originalparameters βm have been replaced by different parameters βm + δ.Because δ can be any nonzero vector of parameters, βm + δ can beany nonzero vector of parameters too.

To identify the model, we must impose constraints on the β’s. Themost common constraint is

β1 = 0

Clearly, if we add a nonzero δ to β1, the assumption that β1 = 0 isviolated.

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Identification (Continued)Adding this constraint to the model and rewrite (eq. 2),


, where β1 = 0

Since exp(x iβ1) = exp(x i0) = 1, the model is commonly written as

Pr(yi = 1|x i) =1

1 +∑J

j=2 exp(x iβj)(eq.3)

Pr(yi = m|x i) =exp(x iβm)

1 +∑J

j=2 exp(x iβj)for m > 1(eq.4)

The MNLM as a Discrete Choice ModelThe discrete choice model is based on the principle that an individualchooses the outcome that maximizes utility gained from that choice.

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The MNLM as a Discrete Choice Model (Continued)For an individual i , the utility uim derived from choice m equals:

uim = µim + εim

where µim is the predictable utility associated with choice m forindividual i , and εim is the random error associated with that choice.µim is assumed to be a linear combination of the characteristics(independent variables) of an individual:

µim = x iβm = β0 + β1xi1 + β2xi2 + · · ·+ βkxik

When we observe the individual i chooses a category m, it must bethat

Pr(yi = m) = Pr(uim > uij for all j 6= m)⇒ Pr(yi = m) = Pr(µim + εim > µij + εij for all j 6= m)⇒ Pr(yi = m) = Pr(εij < µim − µij + εim for all j 6= m)

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The MNLM as a Discrete Choice Model (Continued)It is proved that if and only if the random error ε’s are independent andhave a type I extreme- value distribution:

f (εij) = exp(−εij)exp(−exp(−εij))

then the probabilities of the observed outcomes satisfy the MNLMresults in (eq. 3) and (eq. 4).

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Week 10: Ordinal Outcomes (Scott Long Chapter 5 Part 2)economics-course.weebly.com/uploads/2/...10_ordinal... · Ordinal Outcomes Interpretation the probability of an outcome. Discrete