As Others See Us: A Case Study in Path Analysis
Author(s): D. A. Freedman
Source: Journal of Educational Statistics, Vol. 12, No. 2, (Summer, 1987), pp. 101-128
Published by: American Educational Research Association and American Statistical Association
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Journal of Educational Statistics
Summer 1987, Vol. 12, No. 2, pp. 101-128
As Others See Us: A Case Study in Path Analysis
D. A. Freedman
University of California at Berkeley
Key words: path analysis, path models, structural equation models, regression, correlation, causation
In 1967, Blau and Duncan proposed a path model for education and
stratification. This is one of the most influential applications of statistical
modeling technique to social data. There is recent use of the same technique
in Hope's (1984) comparative study of Scotland and the United States, As
OthersSee Us: Schoolingand SocialMobilityin Scotlandand the United
States. A review of path analysis is offered here, with Hope's model used
as an example, the object being to suggest the limits of the method in
analyzingcomplexphenomena.
Argumentationcannot suffice for discoveryof any new work, since the subtlety
of natureis greatermany times than the subtletyof argument.
FrancisBacon
Introduction
The path model for education and social stratification proposed by Blau
and Duncan (1967) was one of the first applications of that method in the
social sciences, and certainly the most influential. It is often cited as a great
success story: see, for example, Adams, Smelser, and Treiman (1982, p. 46).
And it set the pattern for much subsequent research.
Indeed, path models are now widely used in the social sciences, to disen-
tangle complex cause-and-effectrelationships.Despite their popularity,I
do not believe they have in fact created much new understanding of the
I would like to thank PersiDiaconis, Otis Dudley Duncan, ArthurGoldberger,
ErichLehmann,ThomasRothenberg,JulietShaffer,and Amos Tverskyfor many
useful discussions-without implying their agreement (or disagreement)on the
issues. Neil Henry was a wonderfulreferee.
The quotationsfromKeithHope (1984), andFigure6, are reprintedwithpermission from As Others See Us: Schooling and Social Mobility in Scotland and the
UnitedStates,copyright? 1984, CambridgeUniversityPress.The researchin this
paper was partiallysupportedby NSF GrantDMS 86-01634.
101
D. A. Freedman
phenomenathey are intendedto illuminate.On the whole, they maydivert
attentionfrom the real issues, by purportingto do what cannotbe donegiven the limits on our knowledgeof the underlyingprocesses.
One problemnoticeable to a statisticianis that investigatorsdo not pay
attentionto the stochasticassumptionsbehindthe models. It does not seem
possible to derive these assumptionsfrom current theory, nor are they
easily validatedempiricallyon a case-by-casebasis. Also, the sheer technical complexityof the method tends to overwhelmcriticaljudgement.
I have no magicalsolutionto offer as a replacement.On the other hand,
repeatingwell-wornerrorsfor lack of anythingbetter to do can hardlybe
the right course of action. If I am right, it is better to abandona faulty
researchparadigmand go back to the drawingboards.
Blau and Duncan(1967)mayseem like ancienthistory,so I will illustrate
the argumentby referenceto Hope (1984). This is a comparativestudy of
education and stratificationin Scotlandand the United States. Different
chapters of the book take up different substantiveissues (which are all
interesting);it seems fair to look only at the first chapter. I ask Hope's
pardonfor usinghis workthis way, but it providesa convenientexampleof
the path-analyticparadigmin action.
At bottom, my critique is pretty simple-minded:Nobody pays much
attentionto the assumptions,and the technologytends to overwhelmcommon sense. Since the point is such an elementaryone, the argumentshould
start close to the beginning.After sayingin statisticallanguagewhat path
models assumeand whatthey do, I will outline Hope's model, and point to
the absenceof anythingconnectingit to reality-other thanthe conventions
of the paradigmand his desire. Finally, some of the metamodelingliterature will be reviewed.
A ResearchFable
Latersectionswill reviewthe foundationsof regressionmodels in formal
detail. This section presents an informalexample, to identify the issues.
Statisticalmodels are often used to make causal inferences, for example,
estimates of the impactof interventions:If we put a tariff of $10 a barrel
on imported oil, how much will that affect the price level? the Gross
National Product?If we spend another million dollars on schools, how
much will that affect test scores?
Other kinds of causal inferences, more relevantto the stratificationliterature, do not featuresuch explicitinterventions,but raisesimilarissues:
How much of a salary difference between men and women is due to sex
bias, and how much to differences in productivity?Where does ability
count for more in getting high status jobs: Scotlandor the United States?
In this section, I would like to present one highly stylized example, to
focus the issues:How muchdoes educationaffectincome?Supposewe take
a large random sample (so the distinctionbetween parametersand esti102
Case Studyin PathAnalysis
mates can be slurredover), and observe in the data that log income has a
linear regressionon education:
log (income) = a + b x (education) + error.
The errorsseem to have nearlyconstantvariance,and follow the normal
curve quite nicely. We estimate a and b by ordinaryleast squares;b turns
out to be .05, and the conclusion is that sending people to school for
anotheryear will increase their incomes on averageby 5%.
This conclusion,however,cannotrest on the way the data look, or even
on replicationacrosstime or geography.It mustdependon a theoryof how
the data came to be generated. (This is very well knownto workingsocial
scientists,and I will sketchtheirargumentin a moment;the novelty, if any,
is only in the formulation.)
In effect, the theoryhas to be (or at least have as a consequence)that we
are observingthe resultsof a controlledexperimentconductedby Nature.
Subjectsare given some numberof yearsof schooling,and then the logs of
their incomes are generatedas if by the followingtwo-stepprocedure:
(i) compute a + b x (education);
(ii) add noise.
Forwantof a better term, I will referto this procedureas a linearstatistical
law, connecting log income and education; and the whole thing can be
calledthe as-if-by-experiment
assumption.A slightlymoredramatictheory:
We mightassumethatNaturehas randomlyassignedpeople to the different
educationallevels-an as-if-randomizedassumption.(Then the log-linear
functionalform can be estimatedfrom the data.)
These sorts of theories seem to be implicitin the idea of "structural"or
"causal"equations.Of course, it is impossibleto tell just from dataon the
variablesin it whetheran equationis structuralor merelyan association.In
the latter case, all we learn is that the conditionalexpectationof the response variableshowssome connectionto the explanatoryvariables,in the
populationbeing sampled.The decisionas to whetheran equationis structural must ride either on prior theory or on close examinationof data
outside the equation. Consideringthe impact of interventionsis a useful
armchairexercise to perform in trying to reach the decision, or at least
figuringout what it means.
Comingbackto income andeducation,it maybe obviousby now thatthe
equationis not structural,even if the datalook just like textbookregression
pictures. There is a third variable, family background,which drives both
educationand income. Ourcoefficientb in the equationpicksup the effect
of the omitted variable,and is thereforea biased estimateof the impactof
the proposedintervention-sending people backto school. Well, responds
a strawmaninvestigator,here's a path model to take care of the problem:
education = c + d x (family background) + error
log (income) = e +f x (education) + g x (family background) + error.
103
D. A. Freedman
This model consistsof two equations,assumedto be structural-as if the
resultsof an experimentof nature. (These equationsare hardto interpret
on the as-if-randomizedbasis, but they do makesense-right or wrong-as
linear statisticallaws.) Anyway, says the strawman,now we've got it: f
represents the impact of education on income, with family background
controlledfor.
Unfortunately,it is not so easy. How do we knowthatit is rightthis time?
What about age, sex, or ability, just for starters?How do we know when
the equations will reliably predict the result of interventions-without
doingthe experiment?In a cartoonexample,the objectionis clear. In other
cases, the point is easily lost in the shuffle. Sad experienceshows that all
too often, the real modelers pay as little attention to justifying their
assumptionsas the strawman,and their results are no more convincing.
Of course, the methodologicalissues are extremelyperplexing,and any
attemptto bringevidenceto bearon the substantiveissuesdeservesconsiderable sympathy.Duncan (1975) and Wright(1934) stressed the assumptions andthe limitationsof the technique.Simon(1954)presenteda sophisticated defense of our strawman;and see Zellner (1984, chapter1.4) for a
critiqueof Simon. Many investigatorsseem to focus only on the statistical
calculations.This essay can be viewed as an attemptto put the spotlight
back on the assumptions.
RegressionModels
This sectionwill discussregressionmodelswhen the explanatoryvariable
X is underexperimentalcontrol, and modificationsfor observationaldata.
Some threats to validitywill be identified, and the first successfuluse of
regressionmodels will be noted.
To begin with, suppose that a variableX is underexperimentalcontrol,
and it "causes"Y in the sense that
Y = a + bX + U.
(1)
In this equation, U is a randomvariable-like a drawmade at randomfrom
a box of tickets. Y is observable,but U is not. The stochasticassumptions
are as follows:
The distributionof U is the same, no matterwhat
value of X is selected by the experimenter.
(2.1)
The expected value of U is 0.
(2.2)
The varianceof U is finite.
(2.3)
Each time the system is observed, an independent
value of U is generated.
(2.4)
(The firstandlast of these are seriousrestrictions;the two middleones have
a more technicalcharacter.)
In this model, ordinary least squares (OLS) is a sensible way to estimate
104
Case Studyin PathAnalysis
the parametersa and b. (Optimalityand robustnessare amongthe least of
our worries here.) Furthermore,the coefficient b has a straightforward
causal interpretation:It is the expected change in Y, if the experimenter
intervenes and increasesX by one unit.
Of course, many interestingvariablesare not under experimentalcontrol. On the other hand, given the experimentalmodel (1-2), it does not
matter how the data on X are collected, providedthe distributionof U is
not disturbed.(An exampleto motivatethe caveat:Selectingobservational
units on the basis of their Y-valuescan lead to systematicerrors.)
Considernext a model like (1-2), but for observationaldata. This is a
major transition;from now on, I will be discussingonly studieswhere the
investigatordoes not intervene to set the values of the explanatoryvariables. The firstpartof the model is a theoryaboutthe relationshipbetween
X and Y, the two variablesof interest. This can be stated, a bit quaintly,
as follows: Nature selects X accordingto some distribution,generates Y
accordingto the experimentalmodel (1-2), and then presents the pair
(X, Y) to the investigator,but hides U. This is a linear statisticallaw connecting Y and X.
Much(but not all) of this can be formalizedstartingfroman equationlike
(1), with a slightlydifferentinterpretation:
Y = a + bX + U.
(3)
The equation says that the random variable Y depends linearly on the
random variable X, with an unobservablerandom error or disturbance
term U. The parametersof this linear statisticallaw are a, b, and the
varianceof U. In this equation, X is random.
The stochasticassumptionson U and X are parallelto the ones in the
experimentalmodel, except that (2.4) is droppedfor now:
The disturbanceterm U is independentof the
explanatoryvariableX.
(4.1)
The disturbanceterm U has mean 0.
(4.2)
The disturbanceterm and the explanatoryvariable
have finite variance.
(4.3)
The disturbanceterm U is often interpreted as "the effect of omitted
variables."If so, interveningto changeX should not change U, and this is
perhapsthe strongestformof the independencehypothesis(4.1). Also, the
omitted variablesmust be assumedindependentof the includedvariables.
(See Pratt & Shlaifer, 1984 for discussion.)
Withpathmodels, one conventionis to standardizethe randomvariables
X and Y in Equation (3) to have mean 0 and variance1. The parametera
will then be 0. (This conventionwill be followed here for expositoryrea105
D. A. Freedman
sons, and its logic will be reviewedlater.) To make the varianceof Y come
out to 1, another conditionis needed:
var U = 1 - b2.
(4.4)
So far, X and Y are somewhat platonic. The next part of the model
introducesdata, an X-value and Y-valuefor each observationalunit;these
unitswill be indexedby i. This partof the model is intendedto connectthe
data with the assumedrelationship(3-4), and is the analog of (2.4). The
data are modeled as observedvalues of pairsof randomvariables(Xi, 17),
which are independentfrom unit to unit, and obey the assumptions(3-4).
Technically,the conditionson the data are as follows:
The triplets (Xi, Ui, Y) are independentacross units i.
(5.1)
For each i, the triplet (Xi, U1,Yi)is distributed
like (X, U, Y) in (3-4).
(5.2)
The explanatoryvariable is Xi, the response variable is 1Y,and the disturbanceterm Ui is not observable. This samplingmodel is the basis for
estimatingb fromthe databy OLS. (Of course, in the presentsetup it is the
random variablesthat are standardized,not the data: Standardizingthe
data is one step in estimatingthe parameters.Also, some investigators
prefer to condition on the X-values; this hardlyaffects the present argument, but would complicate the exposition, which attempts to define a
samplingmodel and give theoremsin terms of populationparameters.Be
the conditioningas it may, in the presentsetup the X-values are not under
the investigator'sexperimentalcontrol, and that is what matters.)
To make the assumptionsmore vivid, imaginetwo boxes of tickets, an
X-box and a U-box. In each box, the tickets have numberson them. The
tickets in the X-box averageout to 0 with a varianceof 1 (the standardization). The tickets in the U-box averageout to zero with a varianceof
1 - b2. The data are generated as follows. For each observationalunit i,
drawone ticketat random(withreplacement)fromthe X-box and another,
independently,from the U-box. The firstticket shows the value of Xi, and
the second, Ui. Now use Equation (3) to compute Y (Figure 1).
This set of assumptionslies behindthe simplepath diagramin Figure2.
The straightarrowleadingfromX to Y representsthe X-term in Equation
(3); the coefficientis shownnear the arrow.The free arrowleadinginto Y
representsthe disturbanceterm U.
The assumptions(3-4-5) are not explicitin the diagram.These assumptions are relativelystrong, but somethingratherlike them seems necessary
to justify the full range of operations made by path analysts. For some
applications,especiallywith relativelysmallsamples,normalitywouldhave
to be assumedtoo. (Of course, weaker assumptionscan be used to justify
partialconclusionsin specific cases.)
106
Case Studyin PathAnalysis
I X-box
the
I
I
the U-box
Y = bX + Ui
FIGURE 1. The observationalmodel
The division of the model into two parts
(i) the theoreticalrelationshipbetween X and Y,
(ii) the connectionwith the data on the observationalunits
is not standardbut can be helpful analytically.For example, (3-4) may
hold, but if the data are collected on a stratifiedsample then (5) fails. Or
the data could be for a simple randomsample so (5) holds, but the theoretical relationship(3-4) could be wrong.
Also, the two parts of the model play differentroles. Part (i) underlies
the causal interpretationof the parameterb, as the expected change in Y
if we intervene and change X by one unit. The idea behind (i) is that
changingX by interventionmakes no systematicimpacton U, becausethe
two are independent. So the expected change in Y is just b times the
proposed change in X. On the other hand, part (ii) enables us to learn b
from the data by OLS.
In particular,a causalinferencecan be made from observationaldatabut its validity depends on the validity of the assumptionson the relationshipbetweenX and Y for the interpretationof b, and the connection
with the data for its estimationby OLS. Together,the conditionssay that
FIGURE 2. A first path diagram
Xb--
Y
107
D. A. Freedman
the observationaldata on X and Y are generatedas if by experiment,with
Nature setting the values of X by drawingthem at randomfrom a distribution, and then generatingthe Ys from the Xs througha linear statistical
law.
This set of conditionsis somewhatmore restrictivethan assuming,for
example, that X and Y are jointly normal and (Xi, Y) are independent
replicates of (X, Y); and is not fully capturedby (3-4-5). However, the
as-if-by-experimentcondition seems to be the hallmarkof a structural
equation or causalmodel, as opposed to a mere regressionequation. One
useful way to think about this distinctionis to considerwhat the equation
says about interventions.
The technicalassumptionsin a path model involve the chancebehavior
of the disturbanceterms Ui.These termsare in principlenot observable,so
the assumptionsare difficultto verify directly.Nor is the chance behavior
of an unobservablequantitya topic that fires the imagination.Perhapsas
a result, it is hardto get path analyststo focus on the chance assumptions.
On the other hand, these assumptionsdo have empiricalcontent-and
should be tested before the model is taken seriously.
What are the main threatsto the validityof the assumptions?Three will
be mentioned now (cf. Tukey, 1954, p. 46):
(i) measurementerror in X,
(ii) nonlinearity,
(iii) omitted variables.
Problem(i) is often recognizedby workersin the field, and handledby
"latentvariable"models of the type popularizedby Joreskogand Sorbom
(1981) or Wold (1985); see Noonan and Wold (1983) for an example.
However, the purportedsolution involves the introductionof yet another
layerof assumptions,thatthere are repeatedmeasurementslinearlyrelated
to the latent variables.In my view, this only begs the question, by moving
the kind of difficultiesunderdiscussionhere to other (even less accessible)
realms. See Freedman(1985) for a discussion.
Problems(ii) and (iii) will be discusseda bit abstractlyin this section and
the next, and illustratedon Hope's model. Takenonlinearityfirst. Instead
of (1), suppose the generatingequationis
Y = a + bX + cX2 + U.
(6)
An investigatorwho fits a linear model like (1) will get an error term
uncorrelatedwith X, but dependenton it. Then, changingX must change
the error in a systematicway, and the causal inferenceis invalid.
For an extreme case, supposeX is uniformlydistributedon the interval
[-1, 1], and Y = X2. Fitting (1) gives a = 1/3 and b = 0, suggestingthat a
change in X will cause no changein Y. That is clearlywrong:The effect of
X and Y is all in the nonlinearerror U = X2 - 1/3. In particular,the pathanalytic notion of "cause" is intimately bound up with linear statistical
laws.
108
Case Studyin Path Analysis
In this example, b can be interpretedas the averagechangein Y per unit
change in X, that is, the average over X of dldxE{YIX =x}. On this
interpretation,however,b dependson the distributionof X. And the whole
"average change" idea breaks down if, for example, U = X3 - 3X/5 and
Y = U so a = b = 0. The reason is that U takes different values when X = 1
and X = -1 (cf. Tukey, 1954, p. 42).
Moving on to (iii), suppose the generatingequation is (1), but the expected value of U increases linearly with X. Then OLS will produce a
biasedestimateof b. Put anotherway, causalinferencefailsbecausechanging X makes a systematicchange in U. One source of such dependenceis
omitted variables-problem (iii).
This problemtoo is well knownto workersin the field, andtheir solution
is to expandthe systemby addingmorevariables.Thatis whatpathmodels
are all about. In sum, if the mainvariablesin the systemcan be identified,
and their causal ordering, and the form of the regressionfunctions, the
models can be more or less easily adapted.But there is a major difficulty:
Currentsocial science theory cannot deliverthat sort of specificationwith
any degree of reliability,and currentstatisticaltheory needs this information to get started.
Indeed, the methodof least squareswas developedby Gauss(1809/1963)
for use in situationswhere measurementsare well defined, where functional forms are dictatedby strongtheory, and where predictionsare routinely tested against observations.The story is worth retelling:Astronomersdiscoveredthe asteroidCereswhile makingtelescopicsweeps, but lost
it when it got too close to the Sun. FindingCeresbecameone of the major
scientificproblemsof the day.
To solve this problem,Gaussderivedthe equationsconnectingthe observations on Ceres to the parametersof the orbit, usingNewtonianmechanics. He then linearizedthe equationsof motion and estimatedthe parameters by least squares, making careful estimates of the errors due to the
linearizationand due to randomvariationin the data. Finally, he used the
equationsto predictthe currentposition of Ceres-a predictionborne out
by astronomicalobservation.
In this example, Gaussstartedfromwell-establishedtheorythatspecified
the relevantvariablesand the functionalform of their relationship.Much
carefulwork had alreadybeen done on the errorstructureof the astronomical measurements.Finally, the model was tested againstreality. These
characteristicsdifferentiatethe originalapplicationof least squaresfrom
the applicationin path models. In situationswhere theory and measurement are less well developed, simpler and more informalstatisticaltechniques might be preferable.
Path Models
This section will develop path models, as linked sets of structural equations, with the response variable from one equation being an explanatory
109
D. A. Freedman
variablein a second. The assumptionswill be highlighted,and the possibilityof testingdiscussed;threatsto validitywill be discussed.The development is parallelto the one in the previoussection, and it is convenientto
use the example of X and Y in Equation (1). Suppose that there are two
additionalvariablesin the system, Z and W. Supposethat together, Z and
W cause X; then, Z and X cause Y; and the causationis throughlinear
statisticallaws. This theory about the relationshipsof the variablescan be
expressedin a path diagram(Figure3).
The straightarrowleading from, for example, Z to X indicatesthat Z
appearsin the equationexplainingX; the free arrowleadinginto X stands
for the disturbanceterm in that equation. The path diagram,then, represents two linear statisticallaws:
X = aZ + bW + U
(7.1)
Y = cX + dZ + V.
(7.2)
The randomvariablesZ and W are "exogenous":They are viewed as
causingthe other variablesin the model, but are not themselvesexplained.
This is signaledby the curved, double-headedarrowin the diagram;next
to the arrowis the correlationbetween these two variables.The variables
X and Y are "endogenous"-explained within the model.
The randomvariables U and V in (7) are disturbanceterms. Equation
(7.1) relatesX to the exogenousvariables;then (7.2) relates Y to X andthe
exogenous variables. As is usually said, the system is "recursive"rather
than "simultaneous."(For more carefuldefinitions,see the next section.)
Informally,Natureselects (Z, W) fromsome distribution,and generates
some noise (U, V). The she or he computesX and Y from (7) and shows
(Z, W,X, & Y) to the investigator. The disturbancesU and V remain
hidden.
The parametersa, b, c, d in (7) are called "path coefficients"and are
usuallyunknown.The following are the stochasticassumptions:
FIGURE 3. A path diagram
-7
d
zw
ra
c
110
1-
Case Studyin PathAnalysis
the W-box
thc Z-box
Xi =
aZ
"Yi =
cX
+ bWi
the U-box
the, V-box
+
-t- dZi -
V
FIGURE 4. Box modelfor path diagram
The disturbanceterms U and V are independentof each other
and the exogenous variables(Z, W).
(8.1)
The disturbanceterms U and V have mean 0.
(8.2)
The disturbanceterms and exogenous variableshave
finite variance.
(8.3)
Z, W, X, and Y are all standardizedto have mean 0
and variance1.
(8.4)
Now for the connection between the theory and the data. There are n
observationalunits, indexedby i; for each unit, there are measurementson
the four variablesZ, W, X, and Y. These data are modeled as observed
values of randomvariablesZi, W, X;, and Yj,which are independentfrom
unit to unit and obey the theory expressedin (7) and (8):
The six-tuplets(Zi, W, U1, Vi,X1, Y) are
independentacrossunits i.
(9.1)
For each i, the six-tuplet(Z,, W,, U1, V, X;, Y) is
distributedlike (Z, W, U, V, X, Y) in (7-8).
(9.2)
These assumptionsare shown schematicallyin Figure4.
For a moment, come back to the omitted-variablesproblem.If (7-8) are
right,then a regressionof Y on X alone gives a biasedestimateof the effect
of X, because the regressioncoefficient picks up part of the effect of the
omitted variableZ. Specifyingthe right path model fixes this problem.
By virtue of assumption(9), the full system can be estimated by OLS.
And the parametersof the linear statistical laws do have causal interpretations,as in the previoussection. For example, suppose we intervene
by keeping W and Z fixed but increasingX by one unit: On average,this
will cause Y to increase by c units-because the disturbanceV in Y is
unrelatedto Z, W, or X, so the interventionhas no systematicimpacton
111
D. A. Freedman
V. (The usual interpretationof the coefficients does present some difficulties, to be discussedbelow in the section on directand indirecteffects.)
Again, a causal inferencehas been made from observationaldata. This
is, I think, the most attractivefeature of the methodology. However, its
validitydepends on the assumptionthat the variablesare related through
linear statisticallaws-and the path analyst got the variablesand arrows
right.
These assumptionsare embodied in the path diagramand equations
(7-8-9). They are inputsto the statisticalanalysisratherthan outputs.All
the statisticalanalysis can do (and it is no mean feat) is to estimate the
correlationsand path coefficients, or test that particularcoefficients are
zero-given the assumptions.Standardtheory does not offer any strong
tests of those assumptions;and the existingones (plottingresiduals,crossvalidation,examiningsubgroups)are seldom done by path analysts.
The point maybe a bit obscuredby the somewhattechnicalwaythe term
causal inferenceis being used. To restate matters:A theory of causalityis
assumed in the path diagram(the causal ordering,linear statisticallaws,
etc.). Within this context, what the path analysis does is to provide a
quantitativeestimateof the impactof interventions.The path analysisdoes
not derive the causal theory from the data, or test any major part of it
againstthe data. Assumingthe wrong causaltheory vitiates the statistical
calculations.
The FundamentalTheorem
One object of path analysisis to decomposecorrelationcoefficientsinto
additivecomponents.Considerfor now a generalpath diagram.The variables are still standardized,and the analogs of (7-8) are in force. The
"fundamentaltheorem" is an identity among correlationcoefficientsand
path coefficients.
First, some terminology and the assumptions.Say a variableX is a
"proximatecause" of Y if there is a straightarrowleadingfromX to Y in
the path diagram(i.e., X appearsin the regressionequationexplainingY).
By definition,an "exogenous"variablehas no proximatecausewhereasan
"endogenous"variablehas at least one proximatecause. (In this paper, I
am takingthe exogenousvariablesto be random,and will state the fundamental theorem and related decompositionsat the level of population
parameters,ratherthan sample estimates.)
There is a curved arrowjoining every pair of exogenous variables,and
one free arrowleading into each endogenousvariable.But no free arrows
lead into exogenousvariables,or curvedarrowsinto endogenousvariables.
Say X is a "remote cause" of Y if there is a sequence of straightarrows
leading from X to Y in the diagram. Assume that the path diagramis
recursive,in the followingsense:
No variablecan be a cause of itself, proximateor remote.
112
(10)
Case Studyin PathAnalysis
li~
z
W-
xaX
>?--
Y
j7
Z<
T
Y
FIGURE 5. Violatingthe conditions.In the left diagramX causesY and Y
causes X. At the right,X is a remotecause of itself.
In particular,X may be a remote cause of Y, or Y a remote cause of X, or
neither-but not both. Figure5 showstwo diagramsthat violate the conditions.
(The left hand diagramin Figure 5 provides an example of "simultaneity":X and Y are obtainedfrom Z and W by solvingthe pairof simultaneous linear equations:
X = aZ +bY + U
Y = cW + dX + V.
Such systemsare commonin econometricwork, but less commonin other
social science fields. The new difficultyis that the righthandside variables
become correlated with the errors, so OLS must be replaced by more
complex estimationprocedures.)
The "fundamentaltheorem"(Duncan, 1975,pp. 36ff or Wright,1921)is
the following:
Suppose Y is endogenous and not a cause of X,
proximateor remote. Then
(11)
rxy =
prxz,
z
zP
where
pyz is the path coefficient from Z to Y,
rxyis the correlationcoefficient between X and Y, and
the sum is over all Z that are proximatecauses of Y.
Technically,the set of exogenousvariablesis assumedindependentof the
disturbanceterms, which are independentamong themselves, as in (8.1).
113
D. A. Freedman
It follows that an endogenousvariableis independentof all the disturbance
terms-except those among its causes. The identity (11) can be applied
recursively,to expressany correlationin termsof the path coefficientsand
the correlationsamong the exogenous variables. This leads to the deep
"path tracingrule" of Wright(1921).
Direct and IndirectEffects
One object of path analysisis to measure the "direct"and "indirect"
effects of one variableon another.Take, for example, the path diagramin
Figure 3. Successiveapplicationsof (11) show
rzy= d + ac + rzwbc.
Associated with this identity is some terminology:
rzy is called the "total effect of Z on Y,"
d is the "directeffect of Z on Y,"
ac + rzwbcis the "indirecteffect of Z on Y," and
ac is the "effect of Z on Y throughX. "
The relationshipbetween the terminology and the diagram is pretty
clear, but the connectionto causal inferencemore problematic.Take the
last effect first: We interveneby holding W fixed but increasingZ by one
unit; this increasesX by a units, which in turn makes Y go up ac units. So
far, so good.
Now try the directeffect of Z on Y: We interveneby fixing W andX but
increasingZ by one unit; this should increase Y by d units. However,this
hypotheticalinterventionis self-contradictory,because fixing W and increasingZ causes an increasein X. Or is the disturbanceterm U in (7.1)
supposedto come down?How does that squarewith independence?Or the
idea that U represents"omittedvariables?"
This may seem like a pointless tease about semantics, but given the
researcheffort spent in composingand decomposingcorrelations,surely
some attention to interpretationis called for. The only possibilityfor d
seems to be the rate of changeof E (Y IX, Z) with respectto Z, and calling
this a "directeffect" is ratherstrange.
My view, stated in detail earlier, is that a path model representsthe
analysisof observationaldata as if it were the resultof an experiment.At
points such as this, it would be helpful to know more about the structure
of such hypotheticalexperiments:What is to be held constant, and what
manipulated?
Standardizingthe Variables
Should path models be given in termsof variablesin their naturalscale,
or shouldthey be standardized?The questionis hardlya new one-see, for
example, Achen (1982, p. 76), Blalock (1964, p. 51), Tukey(1954, p. 41),
or Wright (1960). Apparently, standardizing only matters when comparing
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Case Studyin PathAnalysis
path coefficients estimated from different populations, that is, different
distributionsfor the exogenousvariables.And the answerdependson what
is consideredto be invariantacrosspopulations-that is, on the formof the
social law assumedto governthe data. In other words,there is an empirical
issue.
To focus ideas, considerthe simpleregressionmodel of Equation(1). Let
X and Y denote the variablesin theirnaturalscale (dollars,yearsof schooling completed, etc.); let X and f denote the standardizedvariables.There
are two versionsof the equation, raw and standardized:
Y = a + bX + U
(1)
f = 6X+ .
(1)
Supposefirstthat equation(1) expressesa sociallaw, with parametersa,
b, andvar U, whichare invariantacrosspopulations(X-distributions).Now
the path coefficient 6 depends on the scale of X:
6 = b VvarX/[b2 varX + var U].
Twoinvestigatorswho workwith differentpopulationsandstandardizewill
get differentpath coefficients-and miss the invarianceof a and b. This is
not a good researchstrategy.
On the other hand, the law governingthe data could be in terms of
standardunits-Equation (i). That is, 6 could be invariantacross populations. Then fitting(1) in "natural"unitswill miss the invariance,because
a and b vary across populations. Of course, the situation could be more
complicatedthan (1) or (i).
Even if (1) is the rightchoice, the standarddeviationsof the disturbance
terms in more complex models need not be invariantacrosspopulations.
Consider, for example, the path model in Figure3. Suppose that the law
(7-8) underlyingthe social processis in standardunits, so the variablesZ,
W, X, and Y should be standardized.And the path coefficients will be
stable across populations(joint distributionsfor Z and W). But the standard deviations of the disturbanceterms U and V depend on the path
coefficientsand rzw,so these standarddeviationswill change from population to population, because rzwdepends on the joint distributionof the
exogenous variables.
The point can be illustratedon var U in (7.1):
1 = varX
= a2 var Z + b2 var W + var U
+ 2ab cov (Z, W) + 2a cov (Z, U) + 2b cov (W, U)
= a2 + b2 + var U + 2ab rzw.
The first line holds by the standardization.For the same reason, in the
second line, var Z = var W = 1 and cov (Z, W) = rzw. The other two covar115
D. A. Freedman
iances in line 2 vanishby (8.1). Now the equationcan be solved for var U:
var U = 1 - a2 - b2- 2abrzw.
Thus, the unexplainedvariationdependson the populationbeing studied.
Hope's Model
In this section, I will describeHope's model, andthen reviewit underthe
headings proposed earlier, stressingthe weakness of the connection between the model andthe underlyingsocialprocess.One of Hope's technical
innovations is the "autonomycoefficient," which may be viewed as an
attempt to deal with certain kinds of measurementerror. It will be discussed too. Along the way, I will try to give the flavorof the conclusions
drawnfrom the model.
First,some backgroundfor the model. Sincethe publicationof the Coleman et al. report(1966), there has been an extendeddebateconcerningthe
impact of schools on the educational attainmentsof students, and the
achievementsafterward.Jencks et al. (1972) is often cited in this connection. Chapter 1 of Hope (1984) is a contributionto this literature.It
addressesthe followingsorts of questions:Do schools matter?Does education have more of an impact in Scotland or in the United States? Of
course, to answerthese questions,the termshave to be defined, and backgroundvariablescontrolled.
Hope measures outcomes in terms of the occupationsof his subjects.
Only secondary education is considered. The backgroundvariables are
two: IQ andfather'soccupation.The Americandataare drawnfromJencks
and will not be discussedhere. For Scotland,Hope measuresoccupations
on a 9-pointscale, rangingfrom "professionalsand largeemployers"at the
top to "agriculturalworkers"at the bottom (1984, p. 17). For the path
model, occupationswere rankedaccordingto "socialstanding"by 12 college students, and the principalcomponentof the 12 rankingswas used as
the occupationvariable(p. 18).
Secondaryeducationin Scotlandwas on a tracksystem, and Hope measures this variableon a 7-point scale, accordingto the track taken by the
subjects (1984, p. 14). The top track "completedfive years of secondary
education in a general course with two foreign languages."The bottom
track"threeyears of secondaryeducationin a modifiedclass (for less able
and backwardchildrenin an ordinaryschool)."
The data are from the "ScottishMentalSurvey."The samplewas drawn
in 1947, and consists of all 11-year-oldboys born on the first day of every
other month. The childrenwere followeduntil 1964,and data are available
on nearly600 of them, includinganthropometry,scores on FormL of the
Stanford-BinetIQ test, and a "sociologicalschedule"that includedfather's
occupation.Sample attritionwas less than 10%.
Hope proposes a path model for his four variables;it is shownin Figure
6, with coefficientsestimatedfrom the Americanand Scottishdata. There
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Case Studyin PathAnalysis
are two structuralequations:
education= a x IQ + b x (father'soccupation)+ error
(12.1)
occupation= c x (education)+ d x IQ + e x (father'soccupation)
+ error.
(12.2)
(In these equations,of course,IQ, education,andoccupationare attributes
of the sons.) This model will now be reviewedunderthe headingsproposed
earlier, startingwith measurementissues.
The difficultiesin measuringintelligenceand successin life are only too
well known. But what about education?For one thing, the school variable
in the United States is quite different from the one in Scotland (years
completed ratherthan track). And this kind of differencecan have a substantial impact on the path coefficients. (A similarissue is noted in the
section on standardization,above.)
Moreover, the school variablein Scotlandincludes not only characteristics of the schools, but also characteristicsof the students. For example,
the difference between tracks 1 and 3 lies in whether the students completed the course: This is a measureof students'ability and character,as
well as of the educationreceived. The path analysisis being used to separate the inputs to the school from outputs, but the two are alreadyentangled in the school variable-before the path analysiscan do anything.
This point has been questioned by some readers. To make the issue
clearer, suppose the measuredschool variableis just anotherIQ scorelike form M instead of form L. None of the analysiswould then have anythingto do with education.Fora second examplein a similarvein, suppose
the path model in Figure 6 is right-for some variable that represents
FIGURE 6. Hope'spath model
IQ
.466
.-
\758654
Education
IQ
o.
\55
Education
.300
.357
Father's
occupation
United
147
- Occupation
35
States
Father's
occupation
.172
- Occupation
/22
Scotland
Source:Hope (1984,pp. 26-27).
117
D. A. Freedman
educationalquality.In this second hypotheticalworld, the path fromIQ to
educationrepresentsthe fact thatbrighterstudentschoose betterschooling,
on average.Supposetoo that the measuredschool variableis a linearcombinationof IQ and the underlyingeducationalqualityvariable.In this second hypothetical,the estimatedpath coefficientsare substantiallybiased;
and the bias dependson the unknownerrorstructurein the measurements.
One of Hope's technicalinnovationsmightbe viewedas an attemptto get
around the second sort of measurementproblem. He decomposes the
directeffect of educationon occupationinto an "autonomous"and a "heteronomous"component(1984, pp. 9, 24ff). To define these terms, assume
the relationshipspart of the model in Figure6, that is, the analogof (7-8).
Let E and O be the endogenousvariables,educationand occupation;let
the exogenous variablesbe I and F, IQ and father'soccupation. Hope's
decompositioncan be expressed as an equation:
rEF.
POE= POEV + POEPEJ
rEI+ POEPEF
(13)
The first term on the right is the "autonomouseffect" of education on
occupation;v is the varianceof the disturbancetermin the equationfor E.
To understandthis termgeometrically,let L and 6 be the projectionsof
E and O on the spacespannedby I and F; let E = E - Land 0- = O - 0,
so E' and O are the projectionson the orthocomplement.In otherwords,
E' and 0' represent education and occupation, net of IQ and father's
occupation:"net" means, after subtractingout the projections.The "autonomous effect" of educationon occupation,that is, the firsttermon the
rightof (13), turnsout to be the covarianceof 0' and E'. Since E' is not
standardized,the covarianceis not a regression coefficient; apparently,
division by v = varE' just gives you backPOE,as is also clear from (13).
(The remainingtwo termsin the equationseem to be part of the "heteronomous effect"; however, I see no direct geometric interpretationfor
them, nor do Hope's verbaldescriptionsin his table 1.6 make much sense
to me, let alone the identificationof these termswith the "universalistic"
and "particularistic"in education.)
In principle,it is harmlessto compute a covariance.But what is Hope's
interpretation?
Theaimof theresearchis ... to quantifytheeffectsof educationoverand
aboveanyeffectswhichit transmitsfrominputvariables[I andF]. To
accomplishthat aim, we must ask ourselvesa very simplequestion:
Towhatextentis educationactingas a transmitter
and
(heteronomously)
to whatextentis it contributing
itsownautonomous
effect,overandabove
the transmitted
effects,to the processof stratification?
(1984,p. 9)
There is at least one peculiarityin this interpretation:Supposethe path
model in Figure 6 is right. If we intervene by holding IQ and father's
occupationfixed but increase education, then it is POE that measuresthe
changein the subject'soccupation,not the "autonomouseffect." Similarly,
if the measured school variable is a linear combination of IQ and the
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Case Studyin PathAnalysis
underlyingeducationalqualityvariable,the autonomycoefficientis quite
a biased estimate of the path coefficient for educationalquality. On the
other hand, if the measuredschool variableis just anotherIQ score, the
autonomycoefficientmay say somethingabout the relationshipof IQ tests
to occupationalachievement-but nothingaboutschools,becausethese are
absent from the model. Statisticalcomputationsare no substitute for a
proper specification.
Hope cites Finney (1972) in defense of the autonomy coefficient, but
Finney'smessage is more that the "indirecteffect" of an exogenous variable on an endogenousone includesthe correlationsamongthe exogenous
variables,and is thereforenoncausal;also, this indirecteffect depends on
the population, that is, the joint distributionof the exogenous variables,
and so is not comparableacrosspopulations(cf. previoussectionson direct
effects and standardization).Finney's second criticismapplies directlyto
Hope's autonomycoefficient.
Now let me quote Hope (1984) again:
The basicidea [of the autonomycoefficient]can be quiteadequately
represented
by the followingsimpleanalogy.If we thinkof thepathsin a
pathmodelas pipesalongwhichwaterflows,andif we imagineone pipe
A andB, andanotherconnecting
B andC, thenwe naturally
connecting
wonderwhethertheflowbetweenB andC is entirelyaccounted
forbythe
flowbetweenA andB, or whethermorewatercomesintothe systemat
B. Suchincomingwatermodelsourideaof the autonomous
effectof B
of the analogy,standsfor education.(p. 8)
which,in ourapplication
This analogy does not really serve to differentiateamong disturbance
terms, direct effects, or autonomy coefficients. Indeed, since correlation
and regressioncoefficientsdescribechangesratherthanlevels, the passage
seems to be a confusion:The statisticalconstructsin the path model relate
less to the flow of water than to variationsin the flow-ripples.
This completes the discussionof measurementissues and the autonomy
coefficient. Now we come to the stochasticassumptions.Why do the variables satisfya linearstatisticallaw? Nowheredoes Hope ask this question.
With a 7-point scale for education, linearityis hardto take seriously;and
Hope himself points to noticeable skewness in the IQ data. Also, the
impact of the fast track may be larger on the brighterstudents. Heteroscedasticityis another problem: American data suggest that varianceof
intelligenceand schoolingdepend on occupationallevel (see, e.g., Crouse
& Olneck, 1979).
Omitted variables must be considered too. Is the process of social
stratificationthe same in the Highlandsas in the cities?Geographydoes not
appearin the model. Are schoolsin Edinburghsimilarto those in Glasgow?
Schools per se do not appear in the model, except through the tracks
variable. To have schools omitted is peculiarlyironic in a study of their
effects, especially when the author thinks (as Hope does, see pp. 19-20)
119
D. A. Freedman
that it is the individualcharacteristicsof differentschools that make for
strong educationaleffects.
Given such problems,what connectsthe model to reality?What makes
Hope think the assumptionshold? I could only find a few sentences responsive to these questions, and quote them in full (1984, pp. 14-15):
The theoreticalmodel thatinformsour analysisof the effectsof education
wasimplicitin the designof the ScottishMentalSurvey
on stratification
that,forboys,thesignificant
(Hope,1980).It postulates
inputstheybring
to the social system are cleverness,character,and class. Since we lack
comparativedata on character,we omit that variablefrom the current
model (it was includedin the model in Hope, 1980).
At best, this passage justifies includingIQ and father's occupationin
(12). It does not justifythe linearityor the stochasticassumptions.Nor does
it justify treating the equations in (12) as structural,rather than mere
associations;on the contrary,it flags anotherimportantmissingvariable.
And so in the end Hope does not connect Figure6 with real boys who go
to school, graduate, and get jobs. Neither does the cited article (Hope,
1980), which presents a path model rather like Figure 6, but including
variableslabeled "personality"and "qualifications."In that article, there
is much ingenuitydevoted to the geometryof factoranalysis-and none to
elucidatingthe relationshipbetween the geometryand the boys.
Hope is not alone in these respects. Indeed, I do not think there is any
reliablemethodologyin place for identifyingthe crucialvariablesin social
systems or discoveringthe functionalform of their relationships.In such
circumstances,fittingpath models is a peculiarresearchactivity:The computercan pass some planesthroughthe data, but cannotbindthe arithmetic
to the world outside.
At suchpoints as this, modelerswill often explainthat nothingis perfect,
all models are approximations,so maybe Hope's model is good enough:
How much difference can the blemishes make? In this particularcase,
blemishescould really matter, because the differencesin path coefficients
between Scotlandand the U.S. are small (the largestis for the directeffect
of IQ on education). And substantialconclusionsare drawnfrom these
differences:
Whatwe haveshownis thatScotland...is moremerit[ocratic]
thanthe
UnitedStates.Takingtransmission
of IQasuniversalistic
andtransmission
of father'soccupationas particularistic,
we may say that the ratioof
universalismto particularismin education... is 2 to 1 in the United States
andmorethan4 to 1 in Scotland.Theoveralldirecteffectof education
on
occupationis about .47 in both. Withinthis total, the ratioof autonomous
effect: universalism:particularismis 57 : 27: 15 in the United States, as
against43:47:10 in Scotland. We conclude, therefore, that data which
have previouslybeen held to manifestnegligibleautonomouseffects of
schools (in the United States) in fact ascribea strongereffect to schools
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Case Study in Path Analysis
than they do to the characteristicsof students entering those schools.
(Hope, 1984, p. 30)
This sort of finding has to be really sensitive to the rather arbitrary
specification. To illustrate the sensitivity: Adding "personality" and
"qualifications" to the model, as in Hope (1980), makes the direct effect of
education on occupation drop from .47 to .23. (This is rather close to the
autonomy coefficient, but I do not see much logical connection between
adding those two variables and projecting out the two exogenous variables.)
The differences in path coefficients between Scotland and the United
States may be largely due to the scaling, or the differential impact of
omitted variables and measurement error. In any case, the central issue is
what connects the path model and the process by which boys get jobs; this
connection is simply not established. Assumptions matter, and with path
models it is too easy to lose track of this.
Modelers may then explain that they are only doing data reduction. Let
us agree for the moment that Hope's data look more or less like a sample
from the multivariate gaussian distribution, and that standardization is
appropriate. (Otherwise, his analysis goes off the rails almost immediately.) Now the data for each country can be summarized by a table of six
correlation coefficients. However, the path model has six parameters too,
and Hope's table 1.6 analyzes these into a dozen components. Data reduction is not the game here.
Other readers may feel that nobody could be taking the models so seriously; after all, a model is just some way of looking at the data. Well, here
is Hope's view:
We begin our study by askingwhetherScotlandwas indeed the meritocracy it is often alleged to have been. In the course of answeringthis
questionwe will refine the definitionof meritocracyto the point whereits
presence or absence, or ratherthe degree to which it is present, can be
assessedin precise, quantitativeterms. Of course, no such quantification
can be final or irrevocable;nevertheless,it has distinctadvantagesover
imprecise and impressionisticstatements. In the first place, it gives us
some idea of an orderof magnitudewe did not possess before. Second it
enablesus to comparedegreesof magnitudein differentsocieties. And in
the thirdplace it is disputableon empiricalgroundsand corrigibleaccording to rationalcriteriaof evidence and rebuttal.But the reallysignificant
effect of quantificationis, or oughtto be, none of these. Ratherdoes it lie
in the effort at refiningand exploringthe meaningof analyticalconcepts
whichemploymentof a model callsfor. ... But in so constrainingthemwe
will make every effort to see to it that meaningis not warpedbeyondthe
bounds of normalusage, but ratheris tightenedup in a way which will
commandgeneral approval.(1984, p. 6)
Hope is doing something much more ambitious than data analysis. But
his statistical technique has led him astray, and he almost knows it:
121
D. A. Freedman
to thereaderthatthefollowingworkis notmodest
It willbecomeapparent
arebuilt
in its aims.Andto thosewhoobservethatextensiveconclusions
onfairlyexiguousfoundations,
theauthorcanonlyreplythatthisis indeed
the case. (1984,p. 3)
Hope's startingpoint may have been that a son's intelligence and his
father'soccupationjointlyinfluencethe boy'seducation,and then all three
factors influence the boy's choice of occupation-influence but do not
determine.Look back at Figure6. FixingIQ, fathers'occupationalstatus,
and educationstill leaves about 70% of the variationin sons' occupational
status, both in the U.S. and in Scotland-in good agreementwith Blau and
Duncan (1967, p. 170).
These seem to me to be quite interestingfacts, not muchaffectedby the
difficulties in path modeling under discussion. Scotland may be more
meritocraticthan the U.S., but on the evidenceof Figure6 neithercountry
is exactlycaste-ridden;roughinsightswhose value is considerable.(Something like this is at the core of Blau & Duncan, 1967.)
There are now some interestingquestions, and to answerthem, Hope
takes for granted that his measurementson the variablesare connected
through linear statistical laws. He can hardly be blamed for doing so,
because nearly everyonedoes the same; but it is preciselythe move from
rough insightto full-blownpath model that seems so counterproductiveto
me.
The path model mightbe usefulto predictthe resultsof interventions,or
of changing circumstances,or to provide a better understandingof the
stratificationprocess. But this sort of interpretationmustride on a theory,
because somethingis needed to connect the statisticalcalculationsto the
process. As far as I can see, this theory is exactlywhat is missing.
Meta-Arguments
For a historical discussion of path models in sociology, see Bernert
(1983). These models were developedby Wright(e.g., 1921, 1934)for use
in genetics. But later applicationseven in that field remaincontroversial:
see Karlin(1979), or Karlin,Cameron,and Chakraborty(1983), with discussionby Wrightet al. Tukey(1954, pp. 60-66) found the method attractive, but had doubts about the one specific example he presented.
Manyinvestigatorshave writtenabout the problemof drawingcausalinferencesfromobservationaldata. Some have stressedthe as-if-randomized
assumption:for example, see Holland (1986) or Prattand Shlaifer(1984).
Others have focused on the weaknesses of causal models: see Baumrind
(1983), Cliff (1983), de Leeuw (1985), and Ling's(1983) scathingreviewof
Kenny (1979). Lieberson (1985) is quite skepticalabout the possibilityof
makingstatisticaladjustmentsthat bringobservationaldata into the as-ifrandomizedcondition.
Econometricsis an interestingtest case for the modelingapproach,because the technique is extremely sophisticated,and commercialservices
122
Case Studyin PathAnalysis
usingmacro-modelsmakerealforecastswhose accuracycanbe tracked:see
Christ (1975), Litterman (1986), McNees (1979, 1986), and Zarnowitz
(1979). Basically,the majorforecastingmodels do not do at all well unless
their equationsare revisedfrequently,and the interceptsreestimatedsubjectively by the modelers. Even then, such modelinggroupsdo no better
than the forecasterswho proceed without models. Finally, the different
modeling groups tend to make quite similar forecasts; but their models
often disagreesharplyabout the projectedimpactof policy actions.
There are prominentcritiquesof standardeconometricsand the underlying data by insiders, startingwith the classic exchangebetween Keynes
(1939, 1940)andTinbergen(1940). More recentcitationsare Hausmanand
Wise (1985), Hendry (1980), Leamer (1983), Leontief (1971), Lucas and
Sargent (1979), Morgenstern(1963), and Sims (1980). In some cases, of
course, the proposed cure may be worse than the disease.
My own views have been arguedin Daggett and Freedman(1985), and
Freedman(1985), with discussionby JoreskogandFienberg;in the context
of energy models, see Freedman(1981), or Freedman,Rothenberg,and
Sutch(1983), with discussionby Hogan and Smith;in the context of census
adjustment,see Freedmanand Navidi (1986), with discussionby Kadane
et al.
One line of defense againstfoundationalattackis Bayesian, as in Sims
(1982). The disturbanceterms are held to representnot omitted variables
but a componentof subjectiveuncertainty;anothercomponentof subjective uncertaintyis expressedby a prioron the parametersof the model. The
functionalformof the modelis commonground,on whichall suchBayesian
econometriciansmeet. Given the great diversityof functionalformsin the
econometricliterature,andtheirtransience,his argumentdoes not seem to
reach an important question: What is the relevance of textbook linear
models to the economy?
Another interestingline of defense is sketched by Achen (1982), who
saysthatwe can use datato learnaboutsocialprocess.He provesthe point,
drawingon Veblen (1975) to make a charmingand primafacie persuasive
argumentthat the ManchesterUnion-Leaderinfluencedelections in New
Hampshire.A major tool is regressionequations. He and Veblen think
througha variety of qualitativepositions about the political process, and
their implicationsfor the regressions.Then they look at the data, and only
one position survives the test. But as Achen notes (p. 29), there is no
pretenseof developinga structuralmodel;the equationsare purelydescriptive.
In some circumstances,regressionequations are useful ways to look at
data; the coefficients and the standarddeviation around the regression
plane can be good summarystatistics.Thisis so, at least when the datalook
somethinglike a sample from a multivariategaussiandistribution.Achen
and Veblen have one success story along these lines. So did Blau and
Duncan (1967). And so does Hope, as noted above. But there is a real
123
D. A. Freedman
differencebetween summarizingdata and buildingpath models. To take a
simplerexample, you can reporta battingaveragewithoutadoptingcountably additiveprobabilitytheory and the stronglaw of large numbers.
The critiqueof path models in this paperis hardlyoriginal.A few words
more about some of the conventionalresponsesmay be in order. Models
are often held to be of valuein offeringguidelinesto decisionmakersrather
than descriptionsor forecastsof behavior.The test often proposedfor such
models is whether decisionmakersfind them useful. (The argument is
strongestfor normativemodels, but is often madefor descriptivemodelsas
well.) In the present context, this argumentseems circular.
Decisionmakersmay like path analysis because the models appear to
provide answersto importantquestions,but it is the use of the models by
academicinvestigatorsthat makes the process respectable.(For example,
there are plenty of chartistson Wall Street, but their technique is not
prominentin our literature.)The real issue is old-fashioned:Why are the
answersfrom the path model at all dependable?In the end, we still need
to verify the model's assumptions,by some combinationof theory and
practice, before recommendingits conclusionsto the decisionmaker.
We arrive, then, at the descriptiveaspect. Here, two points are often
made:
(i) Valid inferencescan be drawnfrom axioms known to be false.
(ii) No model is perfect;what counts is that the model shouldbe better
than the next-best alternative.
Both points are quite weak. In the present context, the first is almost
irrelevant,because the conclusionsof path models are so close to their
assumptions.More generally,argument(i) can only shift the burdenfrom
the assumptionsto the conclusions.After all, if the modelerdoesn'tbelieve
those assumptions,they can hardlybe partof his or her reasonfor believing
the conclusions.
Even in the best of theories,individualaxiomsmaynot be testable,so the
modelerhas to considerseveralaxiomsin combinationand derivetestable
implicationsfrom the set: compareBlalock (1969, chapter2). So there is a
grainof truthto argument(i). However,the street versionalmostrelies on
a parody:If A then B, and A is false, thereforeB. The anti-syllogismcan
be tracedback to Friedman(1953), but that time he had to be kiddingus.
Argument (ii) has some merit in some situations, but its relevance to
statisticalmodelingin the social sciences is slight. The reason is that there
is a wide range of imperfectionin human knowledge, and it is not so
obviouswhere to locate path modelsin that spectrum.The resultsof a path
analysisdepend for their validityon some underlyingcausaltheory. If the
theory is rejected, the interpretationshave no foundation. Why, then,
should they be held to dominatealternatives?And other modes of enquiry
do exist: De Tocquevillewas makingcomparativestudiesof the new world
and the old long before path models arrivedon the scene.
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Case Study in Path Analysis
Conclusion
Thiskindof negativearticlemayseem incomplete.Pathanalystswill ask,
not unreasonably,"Well,whatwould you do?" To this question, I have no
general answer, any more than I can say in generalhow to do good mathematicalresearch.Still, socialscienceis possible, andneeds a strongempirical component.Even statisticaltechniquemayproveuseful-from time to
time.
There are some such techniqueson the horizon,whichdo not dependon
prior specificationthe way path models do: see Asimov (1985), Breiman,
Friedman,Olshen, and Stone (1984), Fisherkeller,Friedman,and Tukey
(1975), Huber (1985), or McDonald(1984). It remainsto be seen whether
these innovationscan be used to make improvementsover path analysis.
They do offer better data-analyticcapabilities.Log linear models should
also be mentioned (and their problemsnoted).
My opinionis that investigatorsneed to thinkmore aboutthe underlying
social processes, and look more closely at the data, withoutthe distorting
prismof conventional(and largelyirrelevant)stochasticmodels. Estimating nonexistentparameterscannotbe very fruitful.And it must be equally
a waste of time to test theorieson the basisof statisticalhypothesesthat are
rooted neitherin priortheorynor in fact, even if the algorithmsare recited
in every statisticstext without caveat.
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Author
D. A. FREEDMAN, Professor of Statistics, University of California, Berkeley, CA
94720. Specializations: statistics: modeling, foundations.
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