Using the INLOGIT Program to Interpret and Present the Results of Logistic Regressions

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NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

Using the INIRGrl' Program to Interpret and Present the Results of Logistic Regressions

Douglas A. W o u

January 1987 WP-87-13

Working P a p e r s are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have r e c e i v e d only limited review. Views or opinions expressed herein d o not necessarily r e p r e s e n t those of t h e Institute or of i t s National M e m b e r Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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Foreword

Nonlinear s t a t i s t i c a l models, of which logistic r e g r e s s i o n i s one example, are often used in applied demographic analysis as w e l l as in many o t h e r social sciences. P r e s e n t a t i o n of the r e s u l t s of such models c a n be enhanced by t h e calculation of predicted probabilities, o r expected values of t h e phenomenon of i n t e r e s t , y e t s u c h calculations c a n often b e r a t h e r time-consuming. This p a p e r d e s c r i b e s t h e use of a computer program, INLOGIT, which can help t h e applied r e s e a r c h e r i n t e r p r e t and display t h e r e s u l t s of a logistic r e g r e s s i o n m o d e l .

Douglas Wolf Deputy Leader Population Program

-

iii

-

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Acknowledgement

This r e s e a r c h was supported in p a r t by Grant No. AGO5153 from t h e U.S. Na- tional Institute o n Aging.

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VARIETIES OF LOGISTIC MODELS THE INLOGIT PROGRAM

TWO EXAMPLES

AVAILABILITY OF PROGRAM REFERENCES

Page

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Using the INLOGIT Program to Interpret and Present the Results of Iagistic Regressions

Models expressing nonlinear relationships among s e v e r a l v a r i a b l e s at once are commonplace i n applied social sciences. When r e s e a r c h based upon such models includes t h e s t a t i s t i c a l estimation of p a r a m e t e r s , t h e r e s e a r c h e r must typically f a c e t h e issue of how b e s t to communicate to o t h e r s both t h e content and t h e implications of t h e estimated parameters. In t h e case of a multivariate l i n e a r model, f o r example a least-squares multiple r e g r e s s i o n , t h e "content" of t h e results-the estimated r e g r e s s i o n coefficients-and t h e "implications" of t h e results-notably, t h e p a r t i a l s l o p e of t h e f i t t e d r e g r e s s i o n s u r f a c e with r e s p e c t to one of t h e arguments, holding constant t h e o t h e r arguments--are essentially t h e same.

However, in nonlinear models-examples of which include logistic regressions, t h e Probft model, discriminant analysis, and failure-time models-the estimated p a r a m e t e r s of t h e multivariate s t a t i s t i c a l model frequently d o not c o r r e s p o n d d i r e c t l y t o a n easily-interpreted concept. Typically t h e estimated p a r a m e t e r s must b e mapped, via some nonlinear function, i n t o some o t h e r domain-the probability of s o m e event, or t h e expected value of some potentially observable quantity-in o r d e r f o r t h e i r implications or p r a c t i c a l importance to b e judged.

Such computations are often r a t h e r cumbersome, which may explain t h e f a c t t h a t t h e y frequently are omitted from r e p o r t s of r e s e a r c h findings.

This p a p e r discusses a computer program, named INLOGIT, which c a n b e used to calculate predicted probabilities, slopes, and elasticities in one of t h e nonlinear models alluded to above: t h e logistic r e g r e s s i o n model. The program c a n g e n e r a t e a l a r g e volume of such predictions r a t h e r quickly, and thus c a n a s s i s t t h e r e s e a r c h e r in developing t a b l e s of r e s u l t s , g r a p h i c a l displays, and o t h e r useful in- t e r p r e t i v e material. S e v e r a l s p e c i a l cases or unusual situations can b e handled by t h e computer program.

In t h e following section of t h e p a p e r , t h e mathematics of t h e multinomial logit model are summarized. This i s followed by a step-by-step guide to t h e use and capabilities of t h e program. Two examples from t h e published l i t e r a t u r e are given

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in t h e final section.

VARIETIES

OF

LOGISTIC MODELS

Logistic r e g r e s s i o n models a r e a p p r o p r i a t e in situations in which a realiza- tion of t h e dependent v a r i a b l e c a n assume one of two o r more d i s c r e t e (possibly o r d e r e d ) categories. The independent variables c a n assume values from d i s c r e t e and/or continuous sets. Such models have become quite popular in s e v e r a l social science disciplines. P a r t i c u l a r instances of t h e g e n e r a l c l a s s of models include binary logit, multinomial logit, and conditional logit models. Recent examples from sociology include t h e p a p e r s by DiPrete and Soule (1986) and Halaby (1986). In demography, one r e c e n t p a p e r using t h e technique i s Billy et al. (1986); s e v e r a l additional citations c a n b e found in Hoffman and Duncan (1986).

In economics, t h e logistic functional form i s one of s e v e r a l special c a s e s of what i s often termed qualitative response o r d i s c r e t e choice models. A rigorous grounding of t h e multinomial logit model in rational choice t h e o r y has been provid- e d by McFadden [see McFadden (1984) and additional s o u r c e s cited therein].

Another survey, including citations to applied work, i s t h a t of Amemiya (1981).

A g e n e r a l mathematical expression f o r t h e logistic r e g r e s s i o n i s as follows:

where A i s a set of d i s c r e t e indices, y i s t h e dependent variable, and z j (.j E A) i s a set of index functions. For convenience we will let A be t h e set [1,2,

...,

J ] , with J 2 2.

The set of index functions z I,.

. . ,

z j map a t t r i b u t e s (independent variables) into t h e probabilities given by (1). Two p o l a r c a s e s c a n b e identified. In one, t h e a t t r i - butes belong t o a n individual unit ( o r decision maker) whose attributes-age, s e x , income, education, and s o on-are r e p r e s e n t e d by t h e (column) v e c t o r X f . In this f i r s t c a s e

f o r t h e i t h individual in t h e sample. Identification of t h e J v e c t o r s in (2a) re- q u i r e s imposition of a normalizing constraint, generally Bl

=

0.

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In t h e second c a s e t h e a t t r i b u t e s a r e associated with e a c h of t h e

Ji

possible outcomes, o r values of yi (the choices o r alternatives facing t h e 5 t h decision mak- e r ) . In this c a s e

where

Xi,

are t h e a t t r i b u t e s of t h e j t h a l t e r n a t i v e facing t h e i t h individual in t h e sample--e.g. p r i c e , convenience, durability, and s o on.

Intermediate c a s e s , with both "decision-maker-specific" and "alternative- specific" v a r i a b l e s (and associated p a r a m e t e r s ) c a n a l s o be formulated. For a more complete discussion of these parameterizations, s e e McFadden (1984).

A l l of t h e probabilities in (1) depend on all of t h e p a r a m e t e r s of t h e model, as w e l l as upon t h e values of all t h e independent variables. Moreover, t h e p a r t i a l ef- f e c t of a n independent variable upon one of t h e probabilities a l s o depends upon all of t h e p a r a m e t e r s and all t h e independent variables. To see this. l e t

D =

eZCI

k € 4

and note t h a t

where zkm i s t h e m t h element of X k ,

$

being t h e a t t r i b u t e s associated with c a t e g o r y k of t h e dependent variable; in (3a) pj is s h o r t h a n d f o r p t (y

=

j ).

In t h e "decision-maker-specific" formulation given by (2a)

& = X I

thus 8zj /

az,, =

@,

.

But in t h e "alternative-specific" formulation given by (2b) 8zj/

az* = 6, ,

while 8zk / &,

=

⁰f o r k

#

j

,

because 2,-the m t h a t t r i b u t e of t h e alternative--does not a p p e a r in z k . Thus in t h e l a t t e r c a s e expression (3a) specializes to

and

Bpr(v = j )

=-p,pjpk .

^{f o r k}^{# j}

^.

8zkm

Just as t h e sum of t h e probabilities in (1) must equal one, t h e sum of t h e slopes given by (3a) o r (3b.l) and (3b.2) must equal z e r o , which c a n readily be verified by summation o v e r j in (3a).

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Finally, t h e elasticity of a given probability with r e s p e c t t o changes in t h e values of o n e of t h e independent variables i s defined as

which c l e a r l y depends upon a l l t h e p a r a m e t e r s as well as t h e values of a l l t h e X's.

THE INLOGIT PROGRAM

The INLOGIT program will calculate probabilities, a n d optionally slopes and elasticities, f o r user-supplied p a r a m e t e r s and a t t r i b u t e values. Numerous useful options are available as d e s c r i b e d below. The program i s m o s t closely g e a r e d to- wards t h e attributes-of-decision-maker specification [ r e p r e s e n t e d by (2a)J but c a n b e used f o r a n attributes-of-alternatives (or mixed) specification with a p p r o p r i a t e definitions of p r o g r a m input. An example of t h e l a t t e r type of model i s given in t h e following section.

In t h i s section t h e program i s explained step-by-step, following t h e flow of a typical application of t h e program. The f i r s t phase in s u c h a n application consists of supplying t h e s t r u c t u r e and p a r a m e t e r s of t h e model; t h e second consists of supplying values of t h e independent variables, a n d obtaining t h e d e s i r e d calculations.

When beginning execution of t h e program, t h e u s e r will b e a s k e d whether or not r e s u l t s should be written to a file (if not, only output to t h e s c r e e n will b e ob- tained), and if previously c r e a t e d and s t o r e d p a r a m e t e r values/variable names are being used as input. Initially, of c o u r s e , t h e u s e r must supply t h e s e model specifications.

Input of model specifications. The u s e r i s asked to supply, in t h e following o r d e r , t h e following model specifications:

-

number of c a t e g o r i e s of t h e dependent v a r i a b l e . The default specification, as noted e a r l i e r , i s t h e attributes-of-decision-makers specification, with t h e normalization B

=

0 imposed. This, however, i s incompatible with a p u r e a t t r i b u t e s ~ f - o u t c o m e s specification. An a t t r i b u t e s - o f ~ u t c o m e s model, with J distinct t y p e s of outcomes, c a n b e accomodated by responding t o t h i s prompt with t h e value J+1 (and by invoking c e r t a i n additional options, noted below).

Such a model i s i l l u s t r a t e d i n t h e following section.

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number of e x p l a n a t o r y v a r i a b l e s ( a t t r i b u t e s ) not including constant;

-

v a r i a b l e names (optional); if not supplied, all prompts will b e by variable number;

names a t t a c h e d t o c a t e g o r i e s of dependent v a r i a b l e (optional);

p a r a m e t e r v e c t o r s . As noted above, t h e p a r a m e t e r v e c t o r B defaults t o zero.

The term "parameter vectors" h e r e r e f e r s t o quantities specific to e a c h c a t e g o r y of the dependent variable; the t e r m "independent variables" encoun- t e r e d later r e f e r s to quantities fixed a c r o s s c a t e g o r i e s of t h e dependent variable. These usages r e f l e c t t h e f a c t t h a t t h e program i s primarily oriented to t h e attributes-of-decision-maker version of t h e logistic model; in this orientation t h e m t h attribute-say, a n individual's age-is t h e same f o r e a c h possible value of t h e dependent variable, b u t t h i s a t t r i b u t e h a s a unique coefficient o r "loading" in t h e index function of e a c h c a t e g o r y of t h e dependent variable. In o r d e r to u s e this program to evaluate probabilities in a n attributes-ofalternatives specification, i t i s n e c e s s a r y t o treat as "parameters" t h e a t t r i b u t e s of a given c a t e g o r y of t h e dependent variable, a n d as "independent variables" t h e estimated logistic r e g r e s s i o n coefficients-i.e. to r e v e r s e t h e previously-described roles of "parameter" and "independent variable".

Thus, in a prototypical discrete-choice problem, in which a single a t t r i b u t e ,

"price", i s a t t a c h e d t o e a c h possible o b j e c t of choice, t h e p r i c e of t h e f i r s t , second,

...,

J t h o b j e c t will be supplied in response t o t h e program's prompt f o r

"parameter vectors." L a t e r , when t h e user is asked to supply values of t h e independent variables, t h e estimated logistic r e g r e s s i o n coefficient r e p r e s e n t i n g t h e p r i c e e f f e c t will b e e n t e r e d i n t h e a p p r o p r i a t e location in t h e X-vector.

INLOGIT includes a c a p a c i t y t o treat some independent v a r i a b l e s as t r a n s f o r - mations of o t h e r s . For e a c h d e s i r e d transformation, t h e user m u s t supply t h e index (location in t h e a t t r i b u t e v e c t o r ) of t h e s o u r c e and t a r g e t v a r i a b l e s involved in t h e transformation. The menu of available transformations, all of which are self-explanatory, a p p e a r s on t h e s c r e e n as follows:

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e n t m r 1 f o r t a r g m t = k + s o u r c m , k r e a l 2 f o r t a r g m t = n a t u r a l l o g ( s o u r c m ) 3 f o r t a r g m t = sourcm++k, k r m a l

4 f o r t a r g m t = max [ ( s o u r c e

-

^{k )}

,

8 I, k rma l ( I i n m a r s p l l n e )

5 f o r t a r g m t = Box-Cox w i t h p a r a m a t a r k k nm 8

6 f o r i n d i c a t o r f u n c t i o n

( a ) t a r g m t = I [ s o u r c m = k1

( b ) t a r g m t = 1 1 k l G s o u r c m 6 k 2 1

In e a c h c a s e , k ( o r k l and k 2 ) a r e p a r a m e t e r s which must be subsequently supplied.

If a given independent variable e n t e r s the index functions only in nonlinear form--e.g. income a p p e a r s only in logarithmic form-it may be useful t o t r e a t both

"income" and 'In[income]" as independent variables, but t o supply coefficients equal t o z e r o wherever "income" appears. This will allow t h e u s e r t o i n t e r p r e t t h e model in terms of t h e untransformed variable.

Computational options. Having supplied the dimensions, parameter values, variable names, and s o on, t h e program displays all t h e s e input values (for pur- poses of verification), then displays t h e following menu:

Type 1 t o e n t e r / c h a n g m a 1 I x - v a l u m s 2 t o changm 1 x - v a l u e

3 t o d e l e t e / s e l e c t a l o o p v a r i a b l e and l o o p rangm 4 t o e n t e r / c h a n g e z m r o - p r o b a b i l i t y r e a t r i c t i o n ( s ) 5 t o s e l e c t a s e t o f x-a f o r w h i c h s l o p e s a n d

e l a s t i c i t i e s u i l l b e c a l c u l a t e d ( d e f a u l t r a l l ) 6 t o r u n t h e c a l c u l a t i o n s

7 t o u r i t e thm c u r r e n t p a r a m e t e r s , mtc. t o d i s k 8 t o e d i t t h e c u r r e n t p a r a m e t e r s

9 t o go t o n e x t m o d e l / e x i t

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Most of t h e s e options are self-explanatory; however, a few useful f e a t u r e s of t h e program are d e s c r i b e d in m o r e detail h e r e .

The 'looping" f e a t u r e (option 3 above) allows t h e u s e r to trace out t h e changes in a l l t h e probabilities due to changes i n t h e value of one of t h e explana- t o r y variables, while holding constant a l l o t h e r v a r i a b l e s (only one loop-variable at a time c a n be evaluated). A s a f u r t h e r option, t h e program will simultaneously produce so-called "graphics output", i.e. calculated probabilities i n a form which makes easy l a t e r input into a g r a p h i c s program. If t h i s option i s s e l e c t e d , t h e following f u r t h e r prompts will a p p e a r :

Do y o u want p r e d i c t e d p r o b a b l l i t l e s f o r c a t e g o r y 1 o u t p u t ? Do y o u want p r e d i c t e d p r o b a b i l l t l e s f o r c a t e g o r y 2 o u t p u t ?

Do y o u want p r e d i c t e d p r o b a b i l i t i e s f o r c a t e g o r y J o u t p u t ?

The g r a p h i c s output will b e written to a file called GRAFOUT.DAT, in t h e following form:

f i r s t v a l u e o f l o o p i n g v a r l a b l e s e c o n d v a l u e o f l o o p i n g v a r i a b l e

l a s t v a l u e o f l o o p i n g v a r i a b l e 1

p r o b a b i l i t y o f f i r s t s e l e c t e d c a t e g o r y , g i v e n f i r s t v a l u e o f l o o p i n g v a r i a b l e

p r o b a b i l i t y o f f i r s t s e l e c t e d c a t e g o r y , g i v e n s e c o n d v a l u e o f l o o p i n g v a r i a b l e

'up

t o a maximum of 50 v a l u e s .

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p r o b a b i l i t y o f f i r s t s e l e c t e d c a t e g o r y , g i v e n l a s t v a l u e o f l o o p i n g v a r l a b l e

p r o b a b i l i t y o f s e c o n d s e l e c t e d c a t e g o r y , g i v e n f i r s t v a l u e o f l o o p i n g v a r l a b l e

e n d s o o n

Option 4-the zero-probability restriction--can be used to s u p p r e s s any number of c a t e g o r i e s of t h e dependent variable. In t h e models of household composition r e p o r t e d in Wolf (1984) o r Wolf and Soldo (1986). f o r example, t h e r e i s variation within t h e population in t h e feasibility of some c a t e g o r i e s of t h e dependent variable: f o r example, "living with children" i s r u l e d out f o r individuals without children. In this example, a comparison of t h e probabilities of selected household types, according t o t h e existence o r nonexistence of children, c a n b e accomplished by alternatively imposing and removing a zero-probability r e s t r i c - tion on t h e r e l e v a n t c a t e g o r y of t h e dependent variable.

The zero-probability option i s a l s o used t o f o r c e t h e program t o evaluate a p u r e attributes-ofalternatives specification. A s noted above, if t h e r e a r e J a l t e r - natives in such a model, t h e u s e r should specify t h a t the model has J + l a l t e r n a - tives, t r e a t i n g what i s actually the f i r s t a l t e r n a t i v e a s the second, what i s actually t h e second a l t e r n a t i v e as t h e t h i r d , and s o on, ending by t r e a t i n g what i s actually t h e J t h a l t e r n a t i v e as t h e J + l t h . Then, a zero-probability r e s t r i c t i o n c a n be imposed on t h e f i r s t a l t e r n a t i v e , yielding a correctly-specified model. The purpose of this p r o c e d u r e i s t o remove from the set of a l t e r n a t i v e s t h e one f o r which, by default, B

=

0.

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Options 1, 2, a n d 3 are used t o change values of t h e 'X"s-that is, t h e f a c t o r s t r e a t e d as constant across alternatives. Option 8 c a n be used t o modify values of t h e "parametersn-that i s , t h e f a c t o r s t r e a t e d as alternative-specific in t h e program. This option is useful when a attributessf-alternatives specification i s being evaluated, and i t i s d e s i r e d to calculate t h e implications of changes in one or m o r e a t t r i b u t e s of one or m o r e of t h e available a l t e r n a t i v e s . In t h i s way, f o r example, one c a n assess t h e e f f e c t s on choice probabilities of changing one or m o r e p r i c e s in t h e set of a l t e r n a t i v e s comprising a discrete-choice model. A somewhat incon- venient f e a t u r e of t h e p r o g r a m i s t h e f a c t t h a t i t i s not possible to loop o v e r such a p a r a m e t e r change; e a c h s u c h change must b e e n t e r e d individually. When t h i s option i s chosen, t h e program will display all t h e c u r r e n t values of t h e p a r a m e t e r s , with column and r o w indices; the user c a n change a n y of t h e p a r a m e t e r s by e n t e r - ing, in t r i p l e s , t h e column index, r o w index, and new value of t h e d e s i r e d parameter.

Finally, option 7 c a n b e used (at a n y time during execution of t h e program) to write out t h e c u r r e n t values of t h e model specifications-variable names, parameter values, and so on-to a disk file. The f i r s t time this option i s s e l e c t e d , t h e information will b e written to t h e file PARMOUT1.DAT; t h e second time, to PARMOUT2.DAT; and so on, up to five times. These f i l e s c a n later b e used as input into t h e program, bypassing t h e need to r e e n t e r t h e r e l e v a n t information. Input files must b e named PARMINI .DAT, PARMINZDAT,

. . .

, PARMINS.DAT, allowing up to five d i f f e r e n t previously-created and s t o r e d model specifications to b e evaluated on a given r u n of t h e program.

TWO

_EXAMPLES

The f i r s t example i s t a k e n from Wolf (1984), and i s a model of t h e household composition of never-married women aged 65-69. T h e r e a r e t h r e e c a t e g o r i e s of t h e dependent variable: (1) living alone; (2) living with o t h e r s (but not with siblings a n d / o r parents); and (3) living with siblings and/or p a r e n t s . This i s a n a t t r i b u t e s - of-decisionmaker model, using t h e following five independent variables as a t t r i - butes: dummy v a r i a b l e s indicating disability s t a t u s , home ownership, being black,

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and being 68 or 6 9 y e a r s of a g e (in c o n t r a s t to 65-67 y e a r s of age); and a continuous measure of income (in $ 1 0 0 0 ~ ) . The p a r a m e t e r s of t h e model are as follows:

(a) (b)

Intercepts -2.4600 4 . 3 2 8 0 d i s a b l e d 4 . 3 3 7 0 4 . 0 1 0 0 o w n home -0.1540 0.5480 black 1.8500 4 . 0 2 6 0

age 0.8550 0.6130

income 0.0520 4 . 1 1 7 0

Column (a) contains t h e p a r a m e t e r s f o r t h e second c a t e g o r y of t h e dependent variable, "others", while column (b) contains t h e p a r a m e t e r s f o r t h e t h i r d c a t e g o r y ,

"sibs/par

".

Table 1 il l u s t r a t e s t h e use of INLOGIT to c o n t r a s t t h e p r e d i c t e d probabilities of e a c h c a t e g o r y of t h e dependent v a r i a b l e according to subgroup membership.

The f i r s t set of probabilities f i x e s t h e values of d i s a b l e d , o w n home, black, and a g e at z e r o , and the value of income at $3.8 (the sample mean). For t h i s set of values of t h e explanatory variables, the model implies t h a t t h e probability of living alone i s 0.64; of living with o t h e r s , 0.07; of living with siblings/parents i s 0.29.

Implicit in t h e s e calculations i s t h e f a c t t h a t siblings a n d / o r p a r e n t s exist. The remaining r o w s of Table 1 il l u s t r a t e t h e e f f e c t s of setting e a c h of t h e indicated dummy v a r i a b l e s to 1.

Table 2 i l l u s t r a t e s t h e looping f e a t u r e of t h e program; h e r e all dummy vari- a b l e s are again set to z e r o , while income loops from 0 to 8 (in $ 1 0 0 0 ~ ) . Note t h a t at e a c h income level a d i f f e r e n t value of t h e p a r t i a l e f f e c t of income on e a c h probability [labelled (p)] and a d i f f e r e n t value of t h e income elasticity of t h e probabilities Dabelled (e)] a p p e a r s .

Finally, Table 3 i l l u s t r a t e s t h e imposition of a zero-probability r e s t r i c t i o n . H e r e , t h e probability of c a t e g o r y 3-living with siblings and/or parents-has been fixed at zero. In o t h e r words, w e are applying t h e model to t h e situation of a n old- e r woman with no surviving p a r e n t s and/or siblings. The calculated probabilities of living with "others" r i s e only slightly; m o s t of t h e probability m a s s formerly assigned to the c a t e g o r y "sibs/parl' now i s assigned to t h e c a t e g o r y "alone."

A second example i s t a k e n from McFadden (1984), and i s a model of t h e housing market choices of e l d e r l y single men. T h e r e are t h r e e c a t e g o r i e s of t h e depen- d e n t variable: home owner; renter/household head; and renter/non-head. The

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Table 1. Illustrative INLOGIT results: predicted probabilities of t h r e e household types; s e l e c t e d values of explanatory variables.

Catmgorlms P r o b h a t

1 2 3

a lonm o t h m r s s i b s / p a r

8.6386 8.8665 8.2949

Catmgorlms 1 2 3

a lonm o t h m r s s i b s / p a r

P r o b h a t 8.6529 8.8485 8 . 2 9 8 5

d i s a b l a d = 1.888; ( p ) : 8 . 8 1 2 6 -8.8154 8 . 8 8 2 8

C a t a g o r i m s 1 2 3

alonm o t h m r s s i b s / p a r

P r o b h a t 8 . 5 2 9 6 8.8473 8 . 4 2 3 1

onn horn.= 1.888; ( p ) r - 8 . 1 1 8 9 -8.8179 8 . 1 3 6 8

a 1 onm o thmrs a i b s / p r r

P r o b h a t 8.4735 8 . 3 1 3 5 8 . 2 1 3 8

b l a c k r 1.888; ( p ) r -8.2728 8.3999 -8.1279

r lonm o t h m r s s I b s / p a r

P r o b h a t 8.4768 8.1167 8.4865

a9m = 1 . 8 8 8 ; ( p ) r -8.1664 8 . 8 5 9 1 8.1873

c a t e g o r y of housing unit, r a t h e r than t h e p a r t i c u l a r housing unit selected, i s t h e dependent variable. Four a t t r i b u t e variables a p p e a r in t h e model: opcost, o r out-of-pocket c o s t s ; r e t u r n , or n e t expected r e t u r n on equity; income, i n t e r a c t e d with ownership s t a t u s (denoted yown); a n d income i n t e r a c t e d with renter/non-head s t a t u s (denoted y n h ) . Each is considered a n attribute-of-alternative variable.

However, r e t u r n i s identically z e r o f o r both the renter/household head a n d t h e renter/non-head a l t e r n a t i v e s .

The estimated coefficients on t h e a t t r i b u t e v a r i a b l e s are as follows: opcost, -4.544; r e t u r n , 2.506; yown, -.055; and y n h , -.838. This model c a n easily b e f i t into t h e INLOGIT framework as follows:

(1) in r e s p o n s e t o t h e prompt f o r number of c a t e g o r i e s of t h e dependent v a r i a b l e , e n t e r "4"-the f i r s t will b e a dummy c a t e g o r y ;

(2) e n t e r as p a r a m e t e r s t h e following numbers-

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own rent/nonhead r e n t / h e a d

i n t e r c e p t s : 0. 0. 0.

opcost: (alternative-specific values d e s i r e d f o r p u r p o s e s of illustration; e.g. alternative-specific mean values)

r e t u r n : 2.506 0. 0.

income: -0.055 -0.838 0.

(3) e n t e r a s values of independent v a r i a b l e s t h e following numbers-

opcost: -4.544

return: (value d e s i r e d f o r p u r p o s e s of illustration) income: (value d e s i r e d f o r p u r p o s e s of illustration)

In o t h e r words. t h e t w o income v a r i a b l e s c a n b e t r e a t e d as a single a t t r i b u t e of t h e decision maker; similarly, r e t u r n , which i s multiplied by z e r o in t h e index functions f o r t h e t w o r e n t e r c a t e g o r i e s , c a n b e t r e a t e d a s a n a t t r i b u t e of t h e decision maker.' I t i s also n e c e s s a r y t o impose a zero-probability r e s t r i c - tion on c a t e g o r y 1 , t h e dummy c a t e g o r y of t h e dependent variable.

Having e n t e r e d t h e model s t r u c t u r e in t h e manner outlined above, i t i s possible t o loop o v e r both r e t u r n and income, comparing t h e probabilities of e a c h type of housing t e n u r e while holding out-of-pocket costs fixed. I t is a l s o possible to examine t h e e f f e c t s of p r i c e (opcost) changes, using t h e "parameter edit" option d e s c r i b e d e a r l i e r .

An example of a computation based upon t h i s model i s provided below; in t h e illustration, t h e values of opcost are as follows: f o r "own", opcost

=

5.48;

f o r "rent/nonhead", opcost

=

1.13; f o r " r e n t h e a d " , opcost

=

0.93. Income i s fixed a t 4.3 (in $1000s), while r e t u r n

=

5.07 (in $ 1 0 0 0 ~ ) . These values in f a c t c h a r a c t e r i z e p e r s o n number 2 from t h e d a t a set upon which t h e p a r a m e t e r es- timates are based. [see McFadden (1984), Table

3.11.

The output supplied by IN- LOGIT i s as follows:

%deed, s i n c e both t h e independent v a r i a b l e r e t u r n and its coefficient appear only i n t h e i n d e x f u n c t i o n f o r t h e a l t e r n a t i v e "own", e i t h e r can be t r e a t e d a s t h e "parameter" w i t h o u t a f f e c t i n g t h e calculation o f p r e d i c t e d p r o b a b i l i t i e s . However, t h e s l o p e and e l a s t i c i t y c a l c u l a t i o n s w i l l make s e n s e o n l y when t h e c o e f f i c i e n t is t r e a t e d a s t h e "parameter".

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C a t a g o r i a s 1 2 3 4 dummy onn r a n t n h r a n t h

P r o b h a t 8 . 8 8 8 8 8 . 8 8 8 3 8.8189 8 . 9 8 8 9

o p c o s t = - 4 . 5 4 4 ; p : 8 . 8 8 8 8 8 . 8 8 1 2 8 . 8 8 2 1 - 8 . 8 8 3 4 r a t u r n ¹ 5 . 8 7 8 ; ( p ) r 8.8888 8 . 8 8 8 7 8 . 8 8 8 8 -8.8887 incorna ^P 4 . 3 8 8 ; ( p ) r 8 . 8 8 8 8 8 . 8 8 8 8 - 8 . 8 8 9 8 8 . 8 8 9 8

The model implies that an individual facing such a n opportunity s e t would be- come a renter/non-head with probability 0.9889; this i s , in fact, the actual choice made by this observation in t h e data s e t .

Table 2. Illustrative INLOGIT results: predicted probabilities of three household types; alternative values of income.

C a t a g o r i a s P r o b h a t

Incoma =

C a t e g o r i e s P r o b h a t

Income =

P r o b h a t

income =

C a t e g o r i e s P r o b h a t I n c o ~ e =

P r o b h a t

Income =

1 2 3

a l o n a o t h a r s s i b s / p a r 8.5538 8.8473 8 . 3 9 8 9 8.8245 8 . 8 8 4 6 - 8 . 8 2 9 8 8 . 8 8 8 8 8 . 8 8 8 8 8 . 8 8 8 8

1 2 3

a l o n e o t h e r s s l b s / p a r 8.6886 8.8569 8 . 3 4 2 4 8.8223 8 . 8 8 5 1 -8.8274 8.8742 8 . 1 7 8 2 -8.1598

1 2 3

a l o n e o t h e r s s i b s / p a r

8.6425 8.8676 8.2899

8.8195 8.8856 - 8 . 8 2 5 1 8 . 1 2 1 6 8.3296 -8.3464

1 2 3

8 l o n e o t h e r s 5 I b s / p a r 8.6786 8 . 8 7 9 2 8 . 2 4 2 2 8.8164 8.8868 -8.8225 8.1453 8.4573 -8.5567

1 2 3

a l one o t h e r s a 1 b s / p a r 8.7882 8.8917 8 . 2 8 8 1 8.8132 8 . 8 8 6 5 -8.8187 8 . 1 4 9 1 8 . 5 6 5 1 -8.7869

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Table 3. Illustrative INLOGIT results: predicted probabilities of t w o household types, with zero-probability r e s t r i c t i o n imposed; a l t e r n a t i v e values of income.

P r o b h a t

income =

C a t e g o r i e s P r o b h a t

incornm =

C a t o g o r I 0 s P r o b h a t

incorn. =

P r o b h a t

incornm =

P r o b h a t

incorn. =

1 2 3

a l o n e o t h e r s 8 1 b 8 / p a r 8 . 9 2 1 3 8 . 8 7 8 7 8 . 8 8 8 8 8 . 8 8 8 ; ( p ) : - 8 . 8 8 3 8 8 . 8 8 3 8 8 . 8 8 8 8 ( e l : 8 . 8 8 8 8 8 . 8 8 8 8 8 . 8 8 8 8

1 2 3

a l o n e o t h e r s s i b 8 / p a r 8.9134 8 . 8 8 6 6 8 . 8 8 8 8 2.8881 ( p ) : - 8 . 8 8 4 1 8 . 8 8 4 1 8 . 8 8 8 8 ( e l r - 8 . 8 8 9 8 8 . 8 9 5 8 8 . 8 8 8 8

1 2 3

a Ion. o t h m r s s i b 8 / p a r 8 . 9 8 4 8 8 . 8 9 5 2 8 . 8 8 8 8 4 . 8 8 8 ; ( p ) : -8.8845 8 . 8 8 4 5 8 . 8 8 8 8

( 0 ) t -8.8198 8.1882 8.8888

1 2 3

a Ion. o t h m r s 8 i b./par 8.8955 8.1845 8 . 8 8 8 8 6 . 8 8 8 ; ( p ) : -8.8849 8.8849 8 . 8 8 8 8

( 0 ) t - 8 . 8 3 2 6 8.2794 8 . 8 8 8 8

1 2 3

a l o n o o t h m r s 8 I b s / p a r 8.8853 8.1147 8 . 8 8 8 8 8 . 8 8 8 ; ( p ) r -8.8853 8.8853 8.8888

( 0 ) : -8.8477 8.3683 8 . 8 8 8 8

Graphs based on output from t h e looping f e a t u r e of INLOGIT are displayed in Figures 1

-

^3. In Figure 1 a l l variables are fixed at t h e i r alternative-specific mean values: f o r "own", o ~ c o s t

=

3.44; f o r "rent/non head", opcost

=

0.97; f o r "rent/headn, opcost

=

1.96. Income i s fixed a t t h e sample mean, 6.4. Here, w e v a r y r e t u r n from 0.8 to 4.4 (in $ 1 0 0 0 ~ ) . P r o b a - bilities f o r t h e f i r s t and t h i r d c a t e g o r i e s of housing t e n u r e a r e plotted. The c u r v e s r e v e a l t h a t when r e t u r n s to equity are low, t h e probability of being a renter/household head i s high ( o v e r 0.7), falling rapidly as r e t u r n s rise above 2.2. Over t h i s r a n g e of r e t u r n s , t h e probability of being a homeowner r i s e s from (effectively) 0 to 1. Figure 2 uses all t h e same values as does Fig- u r e 1, with one exception: t h e value of o p c o s t f o r renter/household heads i s r e d u c e d t o 1.5, about 7 5 p e r c e n t of t h e sample mean. The e f f e c t i s t o s h i f t t h e probability-of-ownership c u r v e to t h e left, while shifting t h e l e f t t a i l of t h e c u r v e r e p r e s e n t i n g the probability of being a renter/household head up- wards. The probability of being a renter/non-head (which i s not shown in t h e

Figures) essentially vanishes i n Figure 2.

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Figure 1. Pedicted housing m a r k e t choices varying r e t u r n s to home ownership.

Returns

9:

¹⁼probability of being a homeowner

2 = probability of being a renter/household head

Finally, Figure 3 i s based upon values of opcost which are fixed a t t h e i r alternative-specific sample mean values, while w e loop o v e r values of income from SO t o $15,000. A t z e r o income, t h e m o s t probable housing-market choice i s r e n t e r / n o n head. This i s quickly s u r p a s s e d by t h e probability of being a homeown- e r , which peaks at a b o u t $6000 annual income, falling slightly t h e r e a f t e r .

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Figure 2. Pedicted housing market choices varying r e t u r n s to home ownership.

Returns ( i n $10005:1

key: 1 = probability of being a homeowner -

2 = probability of being a renter/household head

AVAILABILITY OF

PROGRAM

INLOGIT i s written in F o r t r a n , and i s available t o any i n t e r e s t e d u s e r who sends t h e a u t h o r a blank diskette. The s o u r c e code, a n executable program module, and input files allowing t h e u s e r t o r e c r e a t e t h e t w o examples discussed above will b e supplied. The executable module h a s been compiled using t h e

IBM

P r o f o r t compiler, and r e q u i r e s a n 8087 c o p r o c e s s o r in o r d e r to execute. Some s o u r c e code statements may have to b e changed in o r d e r to have t h e program suc- cessfully compile with o t h e r compilers.

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Figure 3. Pedicted housing market choices varying income.

key: 1 = probability of being a homeowner -

2 = probability of being a renter/household head 3 = probability of being a renter/non head

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REFERENCES

Amemiya, T. 1981. Qualitative response models: A survey. Journal of Economic L i t e r a t u r e 19: 1483-1536.

Billy, John 0. G., Nancy S. Landale, and Steven D. McLaughlin. 1986. The effect of marital s t a t u s at f i r s t birth on marital dissolution among adolescent mothers.

Demography 23: 329-349.

DiPrete, Thomas A. and Whitman T. Soule. 1986. The organization of c a r e e r lines:

Equal employment opportunity and status advancement in a Federal bureau- c r a c y . American Sociological Review 51: 295-309.

Halaby, Charles N . 1986. Worker attachment and workplace authority. American Sociological Review 51: 634-49.

Hoffman, Saul D. and Greg J. Duncan. 1986. Discrete choice models in demographic r e s e a r c h . Unpublished manuscript.

McFadden, Daniel L. 1984. Econometric analysis of qualitative response models. In Z. Griliches and M. D. Intriligator (eds.), Handbook of Econometrics, Volume 11. N e w York: North-Holland.

Wolf, Douglas A. 1984. Kin availability and t h e living arrangements of o l d e r women. Social Science Research 13: 72-89.

Wolf. Douglas A. and Beth J. Soldo. 1986. The households of o l d e r unmarried women: Micro-decision models of s h a r e d living arrangements. Presented at t h e annual meetings of t h e Population Association of America, San Francisco.