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Phd in Transportation / Transport Demand Modelling 1/60

Phd Program in Transportation

Transport Demand Modeling

João de Abreu e Silva

Session 10 Binary and Ordered Choice Models

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Uses of binary and ordered models

Binary and ordered dependent variables are quite common in

transportation planning:

Binary

Mode choice (when only two alternatives exist);

Ordered

Car ownership levels in households;

Quantitative rankings;

Satisfaction levels (using likert scales);

Accident severity levels;




Discrete choice and Utility theory

It is useful to link the statistical model to an underlying theoretical

construct.

Discrete outcome models have been tied to utility theory.

Traditional approaches from microeconomic theory have decision

makers choosing among a set of alternatives such that their utility

(satisfaction) is maximized subject to the prices of the alternatives

and an income constraint.

Problem – any purchase affects other purchases (they are not

independent)

It is possible to assume that the consumption among different groups

of products is independent - Separability. E.g. choice of cereal brand

and car purchase.




Random utility and ordered models

Random Utility for an ordered choice might be formulated in the

following way:

Which could be easily transformed in a binary choice




Binary choice models

Replacing b’x-m6 by g’xi (the constant term is included in both).

We could have the following regression in which y*i is a latent

(unobserved variable)





The probabilities are bounded between 0 and 1

The probability increases when g’xi increases

The distribution for the error term defines the type of model

Binary probit – the distribution for ei is normal

Binary logit – the distribution for ei is logistic





F is the cumulative distribution function

Both the normal and the logistic distributions are symmetric




Identification

We are assuming that the error term has a zero mean and variance equal to 1 or

p2/3 (logit) – Normalizations of the model

Considering explicitly the constants and considering that the average for the error

term is q

As long as we have a constant term the model doesn´t loose its generality

Because the results in the model (yi) are only zeros and ones there is an

indeterminacy. To remove it is conventional to use the above referred values for the

variance




Maximum Likelihood estimation

For the Probit Model the LL function is




Marginal Effects

Marginal effects

The model parameters could be used to estimate partial changes in

the probabilities, altough not directly as in the linear regression

The marginal effects for

Probit

Logit




Marginal Effects

The magnitude of the marginal effects depends also on the other

variables and coefficients. They depend were the calculation is made




Odds ratios

The odds in favor of a certain response are for the logit

If z changes in one unit

For a change of 1 in z the odds are expected to change by a factor of

exp(q), holding all other variables constant.




Elasticities

Elasticities are common in transportation studies

Because they a ratio of percentage change elasticities cannot be

computed for dummy variables. For these semi-elasticities should be

computed (the denominator is equal to 1).

The denominator computation removes the asymmetry in the computation (not dependent

on whether the change is from di = 1 to 0 or from 0 to 1).




Inference about the parameters

Inference about coefficients is based on the standard “z” test:




Wald test

When the objective is to test more than one coefficient

simultaneously the Wald test is used.

E.g. test if two coefficients are simultaneously equal to zero

The result is a chi squared statistic with degrees of freedom equal to

the number of restrictions.

Where R is a matrix of coefficients in the linear restrictions, q is a vector

of constants




Likelihood ratio test

The likelihood ratio test is used to test if the model is a statistical

improvement over a base model (null or restricted model).

The statistic has a limiting chi squared distribution with degrees of

freedom equal to the number of restrictions being tested.

The likelihood ratio test that all the slope coefficients in the probit or

logit model are zero. “Similar” to the F test.

Where P1 is the proportion of the observations that have dependent variable equal to 1.





The likelihood ratio test provides a more convenient approach for

testing homogeneity of strata in the data. E.g. if it makes sense to

build different models for both genders.


freedom equal to g-1 times the number of parameters in the model.

The null hypothesis states that the same ordered choice model

applies to both strata.




Fit measures

Pseudo R2

Adjusted Pseudo R2

M is the number of parameters in the model




Binary model - example

Data from Tomar 2003/2004, 250 observations.

Evaluate Car use in home based trips as a function of socioeconomic and land use

patterns

TI – dependent variable (1 if car was used, 0 otherwise)

IDADE – age of the respondent

SEXO – gender (1 if man)

FDIM – Household dimension

DENS – population density in the residence zone

DEFEST – parking deficit (1 if yes)

NADUL – number of adults in the household

AIDM – average age of the adults

EMP – employed (1 if respondent was employed)





+----------------------------------------+

| Fit Measures for Binomial Choice Model |

| Probit model for variable TI |

+----------------------------------------+

| Proportions P0= .344000 P1= .656000 |

| N = 250 N0= 86 N1= 164 |

| LogL= -142.031 LogL0= -160.913 |

| Estrella = 1-(L/L0)^(-2L0/n) = .14844 |

+----------------------------------------+

| Efron | McFadden | Ben./Lerman |

| .13421 | .11734 | .61239 |

| Cramer | Veall/Zim. | Rsqrd_ML |

| .14114 | .23318 | .14020 |

+----------------------------------------+

| Information Akaike I.C. Schwarz I.C. |

| Criteria 1.20825 1.33502 |

+----------------------------------------+

+---------------------------------------------------------+

|Predictions for Binary Choice Model. Predicted value is |

|1 when probability is greater than .500000, 0 otherwise.|

|Note, column or row total percentages may not sum to |

|100% because of rounding. Percentages are of full sample.|

+------+---------------------------------+----------------+

|Actual| Predicted Value | |

|Value | 0 1 | Total Actual |

+------+----------------+----------------+----------------+

| 0 | 32 ( 12.8%)| 54 ( 21.6%)| 86 ( 34.4%)|

| 1 | 20 ( 8.0%)| 144 ( 57.6%)| 164 ( 65.6%)|

+------+----------------+----------------+----------------+

|Total | 52 ( 20.8%)| 198 ( 79.2%)| 250 (100.0%)|

+------+----------------+----------------+----------------+

=======================================================================

Analysis of Binary Choice Model Predictions Based on Threshold = .5000

-----------------------------------------------------------------------

Prediction Success

-----------------------------------------------------------------------

Sensitivity = actual 1s correctly predicted 87.805%

Specificity = actual 0s correctly predicted 37.209%

Positive predictive value = predicted 1s that were actual 1s 72.727%

Negative predictive value = predicted 0s that were actual 0s 61.538%

Correct prediction = actual 1s and 0s correctly predicted 70.400%

-----------------------------------------------------------------------

Prediction Failure

-----------------------------------------------------------------------

False pos. for true neg. = actual 0s predicted as 1s 62.791%

False neg. for true pos. = actual 1s predicted as 0s 12.195%

False pos. for predicted pos. = predicted 1s actual 0s 27.273%

False neg. for predicted neg. = predicted 0s actual 1s 38.462%

False predictions = actual 1s and 0s incorrectly predicted 29.600%

=======================================================================





+----------------------------------------+


| Probit model for variable TI |

+----------------------------------------+

| Proportions P0= .344000 P1= .656000 |

| N = 250 N0= 86 N1= 164 |

| LogL= -142.031 LogL0= -160.913 |

| Estrella = 1-(L/L0)^(-2L0/n) = .14844 |

+----------------------------------------+


| .13421 | .11734 | .61239 |


| .14114 | .23318 | .14020 |

+----------------------------------------+


| Criteria 1.20825 1.33502 |

+----------------------------------------+

+---------------------------------------------------------+





+------+---------------------------------+----------------+



+------+----------------+----------------+----------------+

| 0 | 32 ( 12.8%)| 54 ( 21.6%)| 86 ( 34.4%)|

| 1 | 20 ( 8.0%)| 144 ( 57.6%)| 164 ( 65.6%)|

+------+----------------+----------------+----------------+

|Total | 52 ( 20.8%)| 198 ( 79.2%)| 250 (100.0%)|

+------+----------------+----------------+----------------+

=======================================================================


-----------------------------------------------------------------------

Prediction Success

-----------------------------------------------------------------------






-----------------------------------------------------------------------

Prediction Failure

-----------------------------------------------------------------------






=======================================================================





Binary Logit Logit


;Rhs = One,independent variables (separated by commas)$

+---------------------------------------------+

| Binary Logit Model for Binary Choice |


| Model estimated: Oct 26, 2010 at 00:48:18PM.|

| Dependent variable TI |














| Prob[ChiSqd > value] = .1103005E-04 |

| Hosmer-Lemeshow chi-squared = 15.75682 |

| P-value= .04600 with deg.fr. = 8 |

+---------------------------------------------+





+--------+--------------+----------------+--------+--------+----------+


+--------+--------------+----------------+--------+--------+----------+

---------+Characteristics in numerator of Prob[Y = 1]

Constant| 3.59537669 1.13332753 3.172 .0015

IDADE | .01325360 .01562042 .848 .3962 43.0720000

SEXO | .66921981 .29950572 2.234 .0255 .45200000

FDIM | .43930887 .21328635 2.060 .0394 2.97200000

DENS | -.98707555 .38885378 -2.538 .0111 .45024000

DEFEST | -.69848440 .33199452 -2.104 .0354 .22400000

NADUL | -1.00721380 .26843054 -3.752 .0002 2.39600000

AIDM | -.03992124 .02069419 -1.929 .0537 43.0200000

EMP | -.45643740 .37192597 -1.227 .2197 .65200000

+--------------------------------------------------------------------+

| Information Statistics for Discrete Choice Model. |

| M=Model MC=Constants Only M0=No Model |

| Criterion F (log L) -142.36317 -160.91327 -173.28680 |

| LR Statistic vs. MC 37.10019 .00000 .00000 |

| Degrees of Freedom 8.00000 .00000 .00000 |

| Prob. Value for LR .00001 .00000 .00000 |

| Entropy for probs. 142.36317 160.91327 173.28680 |

| Normalized Entropy .82155 .92860 1.00000 |

| Entropy Ratio Stat. 61.84724 24.74705 .00000 |

| Bayes Info Criterion 1.31559 1.46399 1.56298 |

| BIC(no model) - BIC .24739 .09899 .00000 |

| Pseudo R-squared .11528 .00000 .00000 |

| Pct. Correct Pred. 70.40000 .00000 50.00000 |

| Means: y=0 y=1 y=2 y=3 y=4 y=5 y=6 y>=7 |

| Outcome .3440 .6560 .0000 .0000 .0000 .0000 .0000 .0000 |

| Pred.Pr .3440 .6560 .0000 .0000 .0000 .0000 .0000 .0000 |

| Notes: Entropy computed as Sum(i)Sum(j)Pfit(i,j)*logPfit(i,j). |

| Normalized entropy is computed against M0. |

| Entropy ratio statistic is computed against M0. |

| BIC = 2*criterion - log(N)*degrees of freedom. |

| If the model has only constants or if it has no constants, |

| the statistics reported here are not useable. |

+--------------------------------------------------------------------+




Binary model - example +----------------------------------------+


| Logit model for variable TI |

+----------------------------------------+

| Proportions P0= .344000 P1= .656000 |

| N = 250 N0= 86 N1= 164 |

| LogL= -142.363 LogL0= -160.913 |

| Estrella = 1-(L/L0)^(-2L0/n) = .14587 |

+----------------------------------------+


| .13396 | .11528 | .61110 |


| .13831 | .22961 | .13791 |

+----------------------------------------+


| Criteria 1.21091 1.33768 |

+----------------------------------------+

+---------------------------------------------------------+





+------+---------------------------------+----------------+



+------+----------------+----------------+----------------+

| 0 | 31 ( 12.4%)| 55 ( 22.0%)| 86 ( 34.4%)|

| 1 | 19 ( 7.6%)| 145 ( 58.0%)| 164 ( 65.6%)|

+------+----------------+----------------+----------------+

|Total | 50 ( 20.0%)| 200 ( 80.0%)| 250 (100.0%)|

+------+----------------+----------------+----------------+





=======================================================================


-----------------------------------------------------------------------

Prediction Success

-----------------------------------------------------------------------






-----------------------------------------------------------------------

Prediction Failure

-----------------------------------------------------------------------






=======================================================================





Marginal effects - Logit Logit


;Rhs = One,independent variables (separated by commas)

;Marginal Effects$

+-------------------------------------------+

| Partial derivatives of probabilities with |

| respect to the vector of characteristics. |

| They are computed at the means of the Xs. |

| Observations used are All Obs. |

+-------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]|Elasticity|

+--------+--------------+----------------+--------+--------+----------+

---------+Marginal effect for variable in probability

Constant| .78476651 .24198166 3.243 .0012

IDADE | .00289288 .00341363 .847 .3967 .18374445


SEXO | .14358532 .06248443 2.298 .0216 .09570572

FDIM | .09588839 .04613818 2.078 .0377 .42024662

DENS | -.21544998 .08474301 -2.542 .0110 -.14304739


DEFEST | -.16067168 .07877697 -2.040 .0414 -.05307338

NADUL | -.21984558 .05768361 -3.811 .0001 -.77677267

AIDM | -.00871365 .00451725 -1.929 .0537 -.55278958


EMP | -.09686083 .07636889 -1.268 .2047 -.09312906




Ordered and Multinomial models

multinomial discrete models does not account for the ordinal nature

of the discrete data and thus the ordering information is lost. If an

unordered model is used to model ordered data, the model

parameter estimates remain consistent but there is a loss of

efficiency.




Ordered Response Models

Ordered probability models are derived by defining an unobserved

variable, z. This unobserved (latent) variable is typically specified as

a linear function for each observation

Observed ordinal data, y, are defined as:

m are estimable parameters (thresholds)

that define y, and are estimated jointly

with the model parameters (b)





Estimation problem: determining the probability of I specific

ordered responses for each observation. It is accomplished by

making an assumption on the distribution of e.

m0 is set to zero

mi is the upper threshold

mi-1 is the lower threshold

-





The choice for a Ordered Logit or Ordered Probit depends largely on

the assumptions about the error term and is strongly related with the

convenience of use

Ordered Probit

Ordered Logit




Maximum Likelihood estimation

The likelihood function is

Where din is 1 if the observed discrete outcome for observation n is i,

(zero otherwise)

The log-likelihood is

The maximization of the LL fuction is subject to 0≤m1 ≤m2 ≤… ≤mI-2




Thresholds

IIf bX increases




Marginal Effects

Evaluating the effect of individual estimated parameters in ordered

probability models, might be misleading for the intermediate

thresholds

Depending on the location of the thresholds, it is not necessarily

clear what effect a positive or negative bk has on the probabilities of

these “interior” categories

Marginal effects provide the direction of the probability for each

category.




Marginal Effects

Marginal effects can be misleading:

When the probability curve changes rapidly

When and independent variable is a dummy variable

It could be substituted by Discrete Change:

Change in the predicted probability for a change in xk from the start

value xs to the end value xe

The value of discrete change depends on:

The values assumed for the variables that are not changing;

The value at which xk starts;

The amount of change in xk




Marginal Effects

For continuous variables the change in the probability could be

estimated for:

a unit change around the mean;

a standard deviation change centered on the mean

Change from minimum to maximum

It is possible to calculate the average absolute discrete change which

is relative to the effects of one variable across all of the outcome

categories.




Inference about the parameters

Inference about coefficients is based on the standard “z” test:

Inference about the threshold parameters would be meaningless,

and is not generally carried out (although NLogit presents it).




Wald test

When the objective is to test more than one coefficient

simultaneously the Wald test is used.

E.g. test if two coefficients are simultaneously equal to zero

The result is a chi squared statistic with degrees of freedom equal to

the number of restrictions.

Where R is a matrix of coefficients in the linear restrictions, q is a vector

of constants and V is the estimated asymptotic covariance matrix of the

coefficients





The likelihood ratio test is used to test if the model is a statistical

improvement over a base model (null or restricted model).

The test statistic is simply twice the difference between the log

likelihoods for the null and alternative models.

Is usually simpler than the Wald test. Asymptotically, the two

statistics have the same characteristics when the assumptions of the

model are met. But the two tests could conflict for a particular

significance level (LR is the preferred one)





The likelihood ratio test provides a more convenient approach for

testing homogeneity of strata in the data. E.g. if it makes sense to

build different models for both genders.


freedom equal to g-1 times the number of parameters in the model.

The null hypothesis states that the same ordered choice model

applies to both strata.




Fit measures

Pseudo R2

Adjusted Pseudo R2

M is the number of parameters in the model




Measures of fit based on predictions

nj* is the count of the most frequent outcome




Fit measures

Fit measures could be used to compare models to each other, not

only to baseline (null) models.

The following measures are not normalized to the unit interval, but

are based on the log likelihood function

M – number of parameters, n-sample size

They reward parsimony and small samples. A better model is one

with a smaller information criterion.




Parallel slopes assumption

Important assumption in an Ordered Model

This is the definition of a set of binary models with different intercepts





Changing the intercepts only shifts the probability curves right or left

It doesn´t changes the slope





One way to test the parallel slopes assumption is to use the Brant

test, which is formulated for ordered logit

The logit formulation implies the proportional odds assumption

This could be informally tested using i-1 binary models and compare

their coefficients




Brant test

The following hypothesis are tested

» or

The Wald statistic used is:

Where Asy.Var is the assymptotic Covariance matrix

is obtained by stacking the individual binary logit estimates of b

The Brant test could be calculated both the global model and for individual

coefficients




Example – Ordered Response Model

A survey made in Seattle tried to assess the opinion of drivers about

HOV lanes being opened to all users regardless of number of

occupants ( disagree strongly, disagree, neutral, agree, agree

strongly)

Build an ordered probit regressing this variable against the following:

Drive alone (1, 0 otherwise)

Flexible working time (1, 0 otherwise)

Commuter household income

Old age (1 is 50 years old or more, 0 otherwise)

Number of times in the past five commutes that changed route or

departure time

N= 322





Syntax using Nlogit/Limdep

ORDERED

;Lhs= y label

;Rhs=one,xlabels(separated by commas)$





+---------------------------------------------+

| Ordered Probability Model |


| Model estimated: Oct 26, 2010 at 11:24:32AM.|

| Dependent variable OPINION |














| Prob[ChiSqd > value] = .0000000 |

| Underlying probabilities based on Normal |

+---------------------------------------------+

Log Likelihood ratio

Pseudo Rho2





+---------------------------------------------+


| Cell frequencies for outcomes |

| Y Count Freq Y Count Freq Y Count Freq |

| 0 99 .307 1 85 .263 2 26 .080 |

| 3 36 .111 4 76 .236 |

+---------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+


+--------+--------------+----------------+--------+--------+----------+


Constant| -.64741849 .20409362 -3.172 .0015

DALONE | 1.07608334 .15683714 6.861 .0000 .77018634

FLEXIBLE| .20336728 .12544314 1.621 .1050 .54037267

INCOME | .198148D-05 .139144D-05 1.424 .1544 75900.4876

OAGE | .24365556 .15049386 1.619 .1054 .20807453

CHANGE | .06698714 .05038731 1.329 .1837 .73913043

---------+Threshold parameters for index

Mu(1) | .76959111 .06536482 11.774 .0000

Mu(2) | .99994690 .07005703 14.273 .0000

Mu(3) | 1.35038329 .08101695 16.668 .0000

Thresholds

Z test for the coefficients

P-values





+---------------------------------------------------------------------------+

| Cross tabulation of predictions. Row is actual, column is predicted. |

| Model = Probit . Prediction is number of the most probable cell. |

+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| 0| 99| 47| 34| 0| 0| 18|

| 1| 85| 18| 34| 0| 0| 33|

| 2| 26| 4| 8| 0| 0| 14|

| 3| 36| 3| 12| 0| 0| 21|

| 4| 76| 13| 24| 0| 0| 39|

+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

|Col Sum| 322| 85| 112| 0| 0| 125| 0| 0| 0| 0| 0|

+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+





Display marginal effects ORDERED

;Lhs= y label

;Rhs=one,xlabels(separated by commas)

;Marginal Effects$

For the dummy variables the marginal effects don´t make sense. Discrete change should

estimated instead using excel.

+-------------------------------------------------------------------------+

| Summary of Marginal Effects for Ordered Probability Model (probit) |

+-------------------------------------------------------------------------+

Variable| Y=00 Y=01 Y=02 Y=03 Y=04 Y=05 Y=06 Y=07 |

--------------------------------------------------------------------------+

*DALONE -.3979 .0355 .0450 .0809 .2365

*FLEXIBL -.0703 -.0085 .0061 .0146 .0581

INCOME .0000 .0000 .0000 .0000 .0000

*OAGE -.0804 -.0154 .0058 .0160 .0740

CHANGE -.0231 -.0030 .0020 .0048 .0193





Brant test ORDERED

;Lhs= y label

;Rhs=one,xlabels(separated by commas)

;Brant test$

+------------------------------------------------+

| Brant specification test for equal coefficient |

| vectors in the ordered probit model. The model |

| implies that normit[Prob(y>j|x)]=mj - beta(j)*x|

| for all j = 0,..., 3. The chi squared test is |

| H0:beta(0) = beta(1) = ... beta( 3) |

| Chi squared test statistic = 13.74008 |

| Degrees of freedom = 15 |

| P value = .54533 |

+------------------------------------------------+

===========================================================================

Specification Tests for Individual Coefficients in Ordered Logit Model

(Note, Coefficients for values beyond y = 5 are not reported.)

Degrees of freedom for each of these tests is 3

===========================================================================

| Brant Test | Coefficients in implied model Prob(y > j). |

Variable| Chi-sq P value | 0 | 1 | 2 | 3 | 4 | 5 |

DALONE | 2.17 .53843 | 1.1814| 1.0059| .9745| .8310|

FLEXIBLE| 2.28 .51544 | .2276| .1178| .2419| .2443|

INCOME | 1.13 .76890 | .0000| .0000| .0000| .0000|

OAGE | 4.00 .26188 | .4150| .2768| .2385| -.0196|

CHANGE | 3.10 .37591 | .0904| .0853| .0272| .0644|

What can we conclude?





Ordered Logit ORDERED

;Logit

;Lhs= y label


+---------------------------------------------+



















| Underlying probabilities based on Logistic |

+---------------------------------------------+

+---------------------------------------------+



















| Underlying probabilities based on Normal |

+---------------------------------------------+

Ordered Logit Ordered Probit





+--------+--------------+----------------+--------+--------+----------+


+--------+--------------+----------------+--------+--------+----------+


Constant| -.68463225 .27072789 -2.529 .0114

DALONE | 1.83872892 .26601268 6.912 .0000 .77018634

FLEXIBLE| .27397570 .20566894 1.332 .1828 .54037267


Mu(1) | 1.25747687 .10533638 11.938 .0000

Mu(2) | 1.62939558 .11316012 14.399 .0000

Mu(3) | 2.21141414 .13503295 16.377 .0000

+--------+--------------+----------------+--------+--------+----------+


+--------+--------------+----------------+--------+--------+----------+


Constant| -.38909587 .15856689 -2.454 .0141

DALONE | 1.08069300 .15572418 6.940 .0000 .77018634

FLEXIBLE| .17114231 .12347051 1.386 .1657 .54037267


Mu(1) | .75749605 .06465232 11.716 .0000

Mu(2) | .98507310 .06944107 14.186 .0000

Mu(3) | 1.33379198 .08050137 16.569 .0000

Ordered Logit

Ordered Probit




Recommended readings

Washington, Simon P., Karlaftis, Mathew G. e Mannering (2003)

Statistical and econometric Methods for Transportation Data

Analysis, CRC

Greene, William and Hensher, David A. (2010) Modelling Ordered

Choice. A primer, Cambridge University Press

Long, J. Scott (1997) Regression Models for categorical and limited

dependent variables, Sage



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