sociological methods & research...

http://smr.sagepub.com/Sociological Methods & Research

http://smr.sagepub.com/content/1/2/147The online version of this article can be found at:

DOI: 10.1177/004912417200100201

1972 1: 147Sociological Methods & ResearchDavid R. Heise

Employing Nominal Variables, Induced Variables, and Block Variables in Path Analyses

Published by:

http://www.sagepublications.com

can be found at:Sociological Methods & ResearchAdditional services and information for

http://smr.sagepub.com/cgi/alertsEmail Alerts:

http://smr.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://smr.sagepub.com/content/1/2/147.refs.htmlCitations:

What is This?

- Nov 1, 1972Version of Record >>

at NORTHWESTERN UNIV LIBRARY on January 17, 2013smr.sagepub.comDownloaded from

http://smr.sagepub.com/

http://smr.sagepub.com/content/1/2/147

http://www.sagepublications.com

http://smr.sagepub.com/cgi/alerts

http://smr.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.com/journalsPermissions.nav

http://smr.sagepub.com/content/1/2/147.refs.html

http://smr.sagepub.com/content/1/2/147.full.pdf

http://online.sagepub.com/site/sphelp/vorhelp.xhtml


[147]

EMPLOYING NOMINAL

VARIABLES, INDUCED

VARIABLES, AND BLOCK

VARIABLES IN PATH

ANALYSES

David R. Heise

Department of SociologyUniversity of North Carolina

ABSTRACT

In a variety of problems, it is desirableto have a single coefficient summarizethe causal effects of a set of variables

when other variables are controlled.

The "sheaf coefficient" presentedhere does this and can be employedmeaningfully in the context of path

analysis models.

AUTHOR’S NOTE: For helpfulcomments and suggestions concern-ing an earlier version of this paper, I

am grateful to Arthur S. Goldbergerand also to Hubert M. Blalock, Jr., T.Michael Carter, Morgan Lyons, andLauren Seiler. Inadequacies that re-main are my own. Drafting of dia-grams was funded by NIMH grant 5T01 MH12414.



[148]

ometimes indicators are determinants of an abstract

construct, and observations on the indicators must be usedto assess the relations between the construct and othervariables. Psychometric theory is of little value in such problemsbecause it is based largely on the assumption that indicators aredetermined by a latent construct instead of being determinantsof it. Blalock (1971) considered the matter in detail andconcluded that serious analytic problems may arise whenindicators are determinants of the construct they are meant tomeasure (see also Land, 1970). In fact, Blalock gave evidencethat when the problem exists in a completely uncompromisedform, meaningful analyses cannot be done at all.Some special models of analysis are being devised to deal

with such problems. Blalock (1971) showed that solutionssometimes are possible when at least one traditional indicator isused to measure the latent construct (i.e., an indicator that isdetermined by the construct rather than a determinant of theconstruct). Hauser and Goldberger ( 1971 ) developed a solutionfor problems in which an unmeasured variable is related to a

dependent variable (or to several dependent variables), and it

can be assumed that the unmeasured variable is uncorrelatedwith any of the other determinants of the dependent vari-

able(s). The presentation here extends some existing ideaswhich are reviewed in a later section to give another solutionapplicable under special conditions to three common analyticproblems, illustrated by the following examples.

NOMINAL VARIABLES

Suppose a theory proposed that attitude toward education isa function of socioeconomic status and the nominal variable, .ethnicity, and an empirical study is to be conducted to assessthe impact of each of these variables. In some respects, this islike a common path analysis problem as diagrammed in Figure1. The arrows pointing from SES and ethnicity to attitudetoward education reflect the causal postulates of the theory; the



[ 149]

Figure 1.

curved line between SES and ethnicity suggests that these

variables may be related, and the arrow from the term labeled uindicates that the attitude variable is a function of unexplaineddisturbances in addition to the two independent variables. Thegoal of an empirical analysis would be to estimate the values ofthe two path coefficients, p and q. Yet, despite its ordinaryappearance, the model in Figure 1 is unsolvable. Ethnicity is anominal variable rather than an interval variable, so directcorrelations between ethnicity and the other variables in theproblem cannot be calculated, and these correlations are

required to estimate p and q.This problem can be recast into the form indicated in Figure

2. Here ethnicity stands for an unmeasured set of numericalvalues, chosen so that each value corresponds to one ethniccategory, while the set as a whole relates linearly with attitudetoward education. A unit’s position on the ethnicity variable isdefined by the unit’s profile on a set of indicators, where eachindicator is simply one of the original ethnic categories scoredon a zero-one basis. In other words, the nominal variable hasbeen decomposed into a set of dummy variables which are

conceived to be determinants of position on a hypotheticalethnicity scale that relates linearly to the attitude variable. Inthe example, the dummy variables are labeled &dquo;English&dquo; and



[150]

Figure 2.

&dquo;French&dquo; (with, say, an &dquo;Other&dquo; category being implicit), asmight be appropriate in studying the Canadian population.A possibility exists that a unit’s position on the ethnicity

scale may be inaccurately defined because of errors in classifi-cation. A hypothetical gamma factor is presented as contrib-uting to ethnicity to represent the idea that the particular set ofethnic codes may not accurately define variations for the

ethnicity variable. Gamma stands for the unknown numericalcorrections needed to adjust each person’s position on the scaleto correspond to the true position for that person’s ethnicgroup.’ I

Given the model in Figure 2, the goal is to estimate the

parameters p and q. Note that in this problem:

(1 ) p is the effect parameter linking an unmeasured construct to adependent variable.

(2) p must be estimated from observations on the dummy-variable&dquo;indicators&dquo; of the unmeasured construct;

(3) These indicators are interpretable as determinants of the unmeas-



[151]

ured construct. The curved lines in the diagram point up otherimportant aspects of the problem.

(4) The indicators themselves may be intercorrelated. (In the case ofdummy variables, one would expect negative correlations.) Further-more,

(5) each indicator variable also may be correlated with the other majorindependent variable in the analysis, in this case SES. (Thecorrelations with SES represent the possibility that membership in

’

a particular ethnic group may be associated in some way with socialrank.) Finally,

(6) the presence of curved lines in the diagram implies that the

parameters p and q must be estimated using some sort of

multivariate partialling procedure. That is, p must represent theeffect of ethnicity controlled for SES, and likewise q must beestimated controlling for ethnicity.

The specific variable names in Figure 2 should not obscurethe fact that this model is paradigmatic of a general class ofproblems involving nominal variables other than ethnicity (e.g.,religion or geographic region) and interval variables other thanSES and attitude toward education. The model in Figure 3drops specific variable names but represents the same structure.Further, the nature of the problem is basically unchanged ifthere are more than two dummy variable indicators representingthe nominal construct and even if there are several additional

independent variables like SES that need to be controlled inestimation procedures. Thus Figure 2 (or Figure 3) simplypresents one instance of a rather general class of problemsinvolving the use of nominal variables in path analysis.

INDUCED VARIABLES

A number of important social constructs arise as extrapola-tions from specific observations on humans or other social unitsand are unmeasurable except in terms of the original observa-tions. Actually, SES would be an instance though it was treated



[152]

in the example above as a directly measured variable. As

typically used, SES is a construct induced from observablevariations in income, education, occupational prestige, and soon; yet it has no measurable reality apart from these variableswhich are conceived to be its determinants (assuming thatvariables like deference are considered functions of SESrather than identical to it). Level of sociocultural develop-ment would be another example: the construct represents aninduction from concrete variations among societies in such

things as stratification, division of labor, technology, artisticproductions, and the like. Yet there is no obvious way tomeasure the construct except in terms of these things that giverise to the notion.

Suppose a theory proposes that sociocultural developmentand population are two factors that determine a nation’s levelforeign adventurism, and sociocultural development is to bemeasured in terms of variations in stratification and technology.The model in Figure 3 now can be applied to this problem. Thetwo indicators, WI and W2, stand for stratification and

technology, and x stands for a hypothetical scale of sociocul-tural development that relates linearly to the measure of foreignadventurism. Since the two indicators may not define a unit’strue position on this scale with perfect accuracy, a gamma

Figure 3.



[153]

correction factor again is relevant to represent additionaldeterminants of position on the sociocultural developmentscale. The population variable is represented by y, and foreignadventurism (the dependent variable) is represented by z. Thearrows labeled p and q indicate that both sociocultural

development and population are conceived as determinants offoreign adventurism. The curved lines in the diagram indicatethat the indicators may be related to each other, and also thateach indicator variable may be related to population.

Here again, it should be clear that the example is para-

digmatic of a general class of problems involving hypothetical,induced variables and other directly measured variables. Thenature of the problem would be unchanged if there wereadditional indicator variables for the hypothetical construct orif there were additional independent variables like population.

BLOCK EFFECTS

z

In thinking about social processes, it is not uncommon to

group variables that have some unifying feature distinguishingthem from other variables. For example, suppose that a theoryproposes that political liberalism is a function of current ageand also of a whole series of early family influences likemother’s liberalism, father’s liberalism, parental disciplinarypractices, and so on. It would not be implausible in this case toconceive of the dependent variable as a function of age and theblock concept, family socialization, and given this way of

thinking, it might be of considerable interest to develop a .

summary statement concerning the relative impact of age ascompared with family socialization, even though the latter

concept really comprises a multitude of different influences.One way of doing this is to conceptualize a hypothetical familysocialization scale, defining it in terms of all the influences inthe family block in such a way that the single compositevariable relates linearly to the measure of political liberalism.Again Figure 3 is relevant. Here, for simplicity, consider only



[154]

mother’s liberalism and father’s liberalism as determinants (WIand W2) of family socialization (x); the Gamma factor indicatesthese two indicators do not by themselves adequately representthe family socialization block. The straight line paths labeled pand q indicate that family socialization and age (y) both areviewed as determinants of a person’s current political liberalism(z). The curved lines indicate that the family variables may berelated among themselves and also that they may be related tothe other independent variable in the analysis. (In this case,correlations between the parental liberalism variables and agemight stand for cohort variations in family socialization.)

Again, this example is paradigmatic of a general class of

problems in which the impact of a block of variables is to besummarized by a single coefficient that can be compared withother effect parameters. It makes no difference how manyvariables are in the block, and it also would not affect theessential nature of the problem if the analysis included morethan one directly measured independent variable like age.

DEFINING A SOLVABLE MODEL

The examples above suggest that the model diagrammed inFigure 3 arises ubiquitously in social research, so the model anda procedure for estimating its parameters are of practicalinterest. Unfortunately, the parameters cannot be estimated inall such problems, though a solution is possible when certainrestrictive conditions are met.

The model as diagrammed has seven parameters: one corre-lation between the indicator variables; two correlations betweenthe indicators and the independent variable y; two pathcoefficients linking the indicator variables to the unmeasuredconstruct; one path coefficient linking the unmeasured variableto z; and one path coefficient linking y to z. (The path linkingthe correction factor gamma to the unmeasured construct x

and the path linking the disturbance factor u to the dependentvariable z both represent residual effects, and coefficients for



[155]

these paths could be obtained easily once the other parametersare known.) These parameters are the basic unknowns to beestimated from empirical data.’A standard path analysis approach to estimating these

parameters would involve writing a series of equations whichuse products and sums of the parameters to predict the value ofeach observable correlation among variables in the model. These

equations then would be turned around to solve for the

parameters in terms of the observed correlations. However, for asolution to exist, there must be at least as many equations asthere are unknowns-otherwise, the parameters are unidenti-fiable, and thus generally there must be at least as manyobserved correlations as there are unknowns. It is at this pointthat this model presents problems.

The variable x in this system is not directly measured at all;therefore it contributes no observed correlations for use in

analysis. The remaining four variables are measured, and theyprovide six observed correlations, which could be used to set upsix equations involving the model parameters. However, thesewould be six equations in seven unknowns, and therefore theset would be unsolvable.

This conclusion that the parameters are unidentifiable is

generally valid, and it is possible to define a solution only underrestricted conditions. The parameters are identifiable when theset of indicators perfectly defines the unmeasured construct.This is because total determination of the unmeasured construct

provides an additional equation stating that all of the variancein the unmeasured construct is a function of the correlations

among the indicators and of the path coefficients linking the .

indicators to the latent construct (Hauser and Goldberger,1971). (This statement refers to the situation in which all

variables are standardized; if the variables are not standardized,the equation involves the variances and covariances of the

indicators.) With the one additional equation, there are as manyequations as there are unknowns, and so the system of

equations is solvable (see Appendix).The parameters in Figure 3 are unidentifiable because that



[156]

Figure 4.

model includes the correction factor gamma, implying that x isnot strictly a function of the indicator variables being used. Anew model that deletes the gamma factor is shown in Figure 4,and the parameters in this model are identifiable because here xis totally a function of the w’s.

In substantive terms, this condition for identifiability meansthat: ( 1 ) units must be accurately coded or measured withrespect to each indicator; (2) the analysis must include everyindicator variable that logically is a component of the latentconstruct and that empirically is an independent source ofvariation in the dependent variable.

ESTIMATION PROCEDURES

General formulas for estimating the parameters are derived inthe appendix. This section shows the procedure to be followedto obtain estimates of the parameters in situations like thoseillustrated by Figures 4, 5, and 6. It is assumed that a solution isto be obtained for the case where all variables are scaled instandardized form.

Step 1. To estimate the parameters in Figure 4, first calculate



[157]

the multiple regression of the dependent variable z on all othermeasured variables (i.e., the combined set of w~ , w2 , and y) inorder to obtain the standardized partial regression coefficientsfor each variable. The variable y is assumed to be measured

directly and accurately, and the obtained regression coefficientfor that term therefore is the estimate of its effect parameter, q.

Step 2. The parameter p henceforth is called a &dquo;sheafcoefficient&dquo; to convey the idea that it is a single measure ofmultiple effects. It is estimated as the square root of the sum onthe right side of equation I below. (This formula uses anabbreviated notation for the partial regression coefficient; e.g.,ol2 ordinarily would be written 13zW2. ~~~ .)

P = 13wI + 13W2 + 213wl13w2rwI W2 I i I

The first term on the right is the square of the standardized

regression coefficient for w,, the second term is the square ofthe standardized regression coefficient for w2 , and the thirdterm is two times the product of the regression coefficient forWj I the regression coefficient for w2 , and the observedcorrelation between w, and w2 .

Generally it would be natural to choose the positive squareroot in this step. However, the negative value could be used, inwhich case the sense of the unmeasured construct is inverted,and all the signs of the coefficients obtained in the next stepwould be reversed.

Step _3. Once the sheaf coefficient is obtained, the a’s areobtained as follows.

13w I 13w 2al = p-; a2 = p- I

p p

That is, a, 1 is the standardized regression coefficient for w,divided by p, and a2 is the standardized regression coefficientfor W2 divided by p.



[158]

If the latent construct is measured by three indices ratherthan two, as indicated in Figure 5, the same basic procedure isfollowed except that the formula for estimating p is expandedas follows.

P - 2 - 132 WI +. 132 W2 + 132 W3 + 2(13 WI 13 W2 r WI W2 +I3w )l3w / WI W 3 + I3w 213w 3 r W 2 w) [31

Obviously, even more indicators could be dealt with byexpanding the formula further.

There theoretically is no limit to the number of differentunmeasured constructs that can be included in the same

analysis, and the inclusion of one or more directly measuredindependent variables is not a necessary aspect of the model.For example, the problem in Figure 6 is solvable in terms of thesame paradigm. First, the standardized partial regression coeffi-cients are obtained for w,, w2 , vi, and V2 when z is regressedon them. Then formulas I and 2 are applied two times: onceusing the regression coefficients for w, and W2 to obtain p,

a, , and a2 ; and once again using the regression coefficients forvi and v2 to obtain q, d, , and d2.

Figure 5.



[159]

Figure 6.

A sheaf coefficient is always to be interpreted as a standard-ized coefficient. However, if it is desirable to view the indicatorvariables as unstandardized, the values of the appropriate a’s canbe obtained by substituting the unstandardized regressioncoefficients (b’s) for the standardized regression coefficients

(/3’s) in formula 2.

SIGNIFICANCE TESTS

Formulas for testing whether a sheaf coefficient has a specificvalue other than zero could be derived by estimating thestandard error of the coefficient, following Klein’s (1953: 258)procedure. However, such formulas would be complex and havenot been obtained as yet.

Goldberger (1964: 177) provides a comparatively simpleprocedure for testing the null hypothesis that adding a subset ofvariables to a multiple regression produces no increase in thecoefficient of determination R2. The significance of a singleregression coefficient routinely is tested in terms of how muchit increases R2 (see Johnston, 1963: 124), and so in a parallelway, Goldberger’s test provides a test for the null hypothesisthat a sheaf coefficient is zero. The test statistic would be:



[160]

F= (N_k_1). Rk_Rh 4F=

(k - h) ’

1 -Rk h

[41

where h is the number of regressors before adding the indicatorvariables; k is the total number of regressors, including thesubset; N is the number of sample observations; Rk is thecoefficient of determination obtained when the regression is

carried out on all k variables; and Rh is the coefficient ofdetermination obtained when the regression is carried out on

only h variables, the subset of indicators being excluded.The statistic in formula 4 is evaluated by referring to a table

of the F distribution, using (k-h) degrees of freedom for thenumerator and (N~-k-1) degrees of freedom for the denomi-nator.3 3

One can test the significance of a single indicator variable as acontributor to variance in the dependent variable (e.g., z in

Figure 4) by applying the routine test to its partial regressioncoefficient obtained in the full regression of k variables.

PRECAUTIONARY NOTES

The form of a latent variable always is dependent on theproblem in which it appears. The latent variable, as defined bythe parameters of the model, is not just a composite formedfrom its indicators; it is the composite that best predicts thedependent variable in the analysis when the other independentvariables in the analysis are controlled. Thus the meaning of thelatent construct is as much a function of the dependent variableas it is a function of its indicators, and the results of any singleanalysis cannot be used to create a generalized scale of

ethnicity, SES, sociocultural development, and so on. While itsometimes may be possible to create a generalized scale, it

would have to represent the optimal solution for many differentdependent variables, and the appropriate analysis in such a casewould be based on an adaptation of Hauser and Goldberger’s( 1971 ) model.



[161]

The sheaf coefficient is subject to the usual rules of pathanalysis, but ordinarily it cannot be given the manipulatoryinterpretation that is sometimes possible with path coefficients.This is simply because it does not make sense to talk about

units of change in constructs like ethnicity or childhoodsocialization. Like all path coefficients (as opposed to unstand-ardized path-regression coefficients), the value of the sheafcoefficient depends on the standard deviations of the variablesin the given population. Hence, its value cannot be generalized .

across populations.All of the assumptions usually involved in a recursive path

analysis (Heise, 1969) also are involved here. In particular, it isworth stressing that the estimation procedure is meaningfulonly if the dependent variable is indeed causally dependent onthe independent variables, rather than vice versa, and only if theanalysis includes each variable that has a significant effect onthe dependent variable and that also is related to independent

variables. being considered. The latter assumption is likely to beviolated when one misses a key indicator variables for theunmeasured construct, so missing an indicator can violate twoassumptions of the model rather than just one.When the model is used to deal with nominal variables, a

number of special statistical issues arise concerning the use ofdummy variables in regression analysis. These issues are dis-cussed by Cohen (1968) and Lyons and Carter (1971);Goldberger (1968: 112-118) has provided a discussion at a

higher technical level.

Measurement imprecision. The validity of the parameterestimation procedure depends on the assumption that theunmeasured construct is completely specified by its indicators,and this assumption implies that two conditions must be metfor perfectly valid analyses: ( 1 ) enough indicators are used todefine all the logically and empirically significant variations inthe latent construct; (2) perfectly reliable indicators are used, sothat all variations in the latent variable are accurately assessed.To get some notion of the effects of violating these



[162]

requirements in an induced variable or block variable analysis,4a hypothetical system was created by assigning values to theparameters in Figure 5 (the values are given in Table 1 ). This setof parameters and the rules of path analysis were employed togenerate a set of &dquo;observed&dquo; correlations which then were

subjected to the following three analyses:

(a) It was assumed that the indicator variables could not be measured

precisely, but could be measured with a reliability of .64. The&dquo;observed&dquo; correlations were attenuated to reflect this level of

measurement, and then these attenuated correlations were analyzedaccording to the process outlined previously in order to see howestimates of model parameters would be affected by imprecision inmeasuring the indicator variables. The results are presented in thesecond row of Table 1.

(b) It was assumed that all measurements were made without errors,’

but that the analysis failed to include one of the variables

determining x. Then an attempt was made to estimate the majorparameters using just the correlations among wl, w2, y, and z.Results are presented in row three of Table 1.

(c) Finally, it was assumed that the problems of measurement errorand of an incomplete set of indicators existed simultaneously. Thatis, parameter estimates were made from attenuated correlations andfrom the reduced set of correlations resulting when indicator W3 isignored. These results are presented in the bottom row of Table 1.

The figures in Table 1 reveal several characteristics of theestimation errors produced by inadequate measures of thelatent construct. First, regardless of the source of the measure-ment error, the estimate of the sheaf coefficient, p, is biaseddownward while the estimates of the coefficients as, a2, and a3are biased upward. Second, the errors in estimating the a’s aregreater than the error in estimating p. Third, the estimate of theparameter q also is influenced by inadequate measurement ofthe latent construct even though the variables y and z werepresumed to be measured without error.

Taking into account the sizes of the estimation errors, it canbe seen that the estimates of the key parameters in the system,



[163]

TABLE 1

EFFECTS OF IMPRECISE MEASUREMENTS ON ESTIMATESOF PARAMETERS IN A HYPOTHETICAL SYSTEM

NOTE: The structure of the system is indicated in Figure 5. True values of thecorrelations among independent variables were set as follows:

PWl w2 = PWl w3 = /~w2w3 = 0.30; pwly = Pw2y = /~w3y ° 0.10.

p and q, are informative even when measurements of the latentconstruct are fairly poor. In all three of the cases presented inTable 1, the estimates of these parameters would suggest thatboth x and y influence z and that the amount of influence foreither variable is in the low to moderate range.

Overall, the results suggest that minor violations of the

complete determination assumption need not hinder applicationof the model providing that (1) concern focuses on the relationsbetween major variables in the system rather than on therelations between indicators and the latent construct, and (2)the exact values of the parameter estimates are not taken tooseriously but rather are used to assess a probable range ofquantitative effect. However, this conclusion in no way impliesthat measurement and definition problems are inconsequential.The greater the errors in defining and measuring the latentvariable (and other variables), the greater the errors in param-eter estimation, and excessive errors will lead to uninterpretableor misleading results.

RELATIONSHIP TO OTHER SUMMARY COEFFICIENTS

A multiple-partial correlation coefficient (see Blalock, 1960:350) sometimes is used to summarize the effects of a set of



[164]

variables while controlling for others, and Sullivan (1971) hasargued that the multiple-partial coefficient is appropriate forcausal analysis with blocks of variables. The general reasoninginvolved in deriving the sheaf coefficient is in many ways similarto Sullivan’s arguments. However, the multiple-partial cor-

relation coefficient has the same disadvantages for causal

analysis as the ordinary partial correlation coefficient. It

estimates a causal parameter different from the one ordinarilyintended, since its use implies a model in which the dependentvariable is conceived to be a function of an extra, unmeasuredvariable (Linn and Werts, 1969) that researchers rarely have inmind when they conduct their analysis. Duncan (1970) hasdiscussed the advantages of regression coefficients over partialcorrelations in causal analyses, and his remarks apply here,keeping in mind that multiple-partial correlations are beingcompared with sheaf coefficients, which alternatively might becalled &dquo;multiple-partial beta coefficients.&dquo;

The strength of a relation between an experimental treatmentwith two or more levels and the dependent variable in an

analysis of variance can be assessed by a statistic called c.~ 2

(Hays, 1968: 406-407), and this statistic parallels the sheafcoefficient presented here. Calculation of ca 2 proceeds byestimating the variance in a dependent variable that is due to aparticular treatment from the means within treatment levels,and then dividing this estimate by the total variance to indicatethe proportion of total variance that is due to the treatment. Nocorrections for correlated treatments are necessary since thebalanced design in standard experiments artificially sets alltreatment correlations at zero. To illustrate the relation of c.~ 2

to the presentation here, suppose that Figure 6 represents anexperiment in which a treatment W has two levels, w and w2 ,and a treatment V also has two levels, vl and v2. Assuming abalanced design is used to collect data, the treatment correla-tions (p W 1 ~l , PWl W2’ and so on) are all zero. Then calculationof the omegas leads to statistics that have the followingproperty: c.~ z , W = P2 and (.,)2 z.w = q2 .

Morgan et al. (1962: 510-511 ) followed a similar logic to



[165]

obtain a measure of importance for categorical independentvariables when the variables (treatments in Hays’ terms) arecorrelated. Their procedure involves estimating the means of thedependent variable within each category of the independentvariable of interest, using dummy-variable regression analysis inorder to correct for the relations among the different independ-ent variables. (The regression procedure is the same procedureas has been recommended here.) Then, as with Hays’ procedure,an estimate of variance is obtained from these means anddivided by the total variance to give a ratio indicating the

relationship between the categorical variable and the dependentvariable. Morgan et al. use the square root of this ratio, notingthat it can be viewed as a partial regression coefficient. It hasthe same value the sheaf parameter presented here would havewhen estimated for the same data.

Hays’ LV and the importance measure of Morgan et al. bothanticipate the sheaf coefficient. But these prior developmentswere not integrated with more general problems of structuralequations nor were the procedures extended to deal with

induced variables or block variables.

RELATIONSHIP TO OTHER MODELS

The sheaf coefficient was inspired by a number of recentdiscussions concerning the treatment of &dquo;difficult&dquo; variables inpath analysis, and the present model is related to some of theother models that have been proposed. These other models arediscussed briefly below in order to point out how the presentmodel differs from them in emphasis and in functions.

Boyle (1970) and the critiques of Boyle’s article by Wertsand Linn ( 1971 ) and Lyons and Carter (1971) have dealt withthe problem of introducing ordinal variables into path analyses.Boyle proposed that an ordinal variable can be decomposed intoa set of dummy variables representing its measurement cate-

gories and that this set of dummy variables can be enteredanalytically between the original variable and the dependent



[166]

variable to estimate effect parameters without assuming intervalmeasurement. The model presented here can be used in place ofthe Boyle model to deal with ordinal variables, and the presentmodel perhaps has some advantage in parsimony. Again, theordinal variable is decomposed into its measured categorieswhich are represented as a set of dummy variables. However,then the ordinality of these categories is ignored, and theanalysis proceeds as in the case of nominal variables. Once theanalysis is complete, one could define the values or the latentscale that correspond to each category of the ordinal variable,allowing an examination of how these values relate to the

original assumption of ordinality.Blalock ( 1971 ) suggested the use of compromise models in

which a latent construct is assessed not only by its determinantsbut also by one or more traditional indicators (i.e., variablesthat are determined by the latent variable rather than vice

versa). Blalock’s solution to the identification problem is

essentially to modify one’s research design so as to include sometraditional indicators of every unmeasured construct. There isno faulting this as a desirable research strategy. But situationsdo arise in which the strategy cannot be implemented, and it isin these situations that the model presented here becomesrelevant.

Hauser and Goldberger (1971) focused on situations thatinvolve a number of determinants of an unmeasured construct

and also a number of variables that are determined by the latentconstruct, as indicated by the diagram in Figure 7. To solve forthe unknowns in this situation, they introduced the assumptionthat the latent construct is totally determined by its indicators,and then they further assumed that none of the variables at thetop of the diagram is related to those at the bottom exceptthrough the latent variable. Superficially, it may seem that themodel presented here is encompassed by the Hauser-Goldbergermodel as a special case when there is only one dependentvariable.

The distinction between the models can be seen by compar-ing Figures 4 and 6 with Figure 9 and imagining that the latter



[167]

Figure 7.

involves just one dependent variable so that it is in a formrelevant to the situations dealt with in this paper. The

comparison emphasizes that the model presented here allowsone to assess how several possibly correlated variables affect adependent variable when some or all of these variables are

implicit and measured only in terms of their determinants. TheHauser-Goldberger model provides a solution only for onelatent construct, and this solution is defined only if there are noadditional independent variables in the analysis. On the otherhand, the model here presents a solution for only one

dependent variable at a time, whereas the Hauser-Goldbergermodel provides the optimal solution when there are several

dependent variables.The two models could be combined as illustrated in Figure 8.

In such a case, one could apply the Hauser-Goldberger modelfirst, treating the w’s and y’s in Figure 8 as the w’s in Figure 7(the z’s are comparable in both diagrams), and treating x inFigure 7 as analogous to v in Figure 8. Estimates of f, and f2 inFigure 8 correspond to estimates of PI and P2 in Figure 7.Estimates of the a’s in Figure 7 then could be substituted forthe regression coefficients in equations I and 2 leading to



[168] S

Figure 8.

estimates of at , a2, p, and q in Figure 8. Such a procedurecould be a powerful approach to data analysis in cases where aproblem involves multiple dependent variables and multipleindependent variables, and it is relatively simple to carry outsince the Hauser-Goldberger solution can be derived from theoutput of computer programs for canonical analysis.

NOTES ’

1. When the unmeasured construct is a nominal variable, as in this example,gamma exists because of errors of classification. The gamma corrections would becorrelated with one or more of the indicator variables, though this is not necessarilytrue in the case of model for induced or block variables discussed below.

2. If gamma were correlated with the indicators, as is necessarily true in nominalvariable problems, then these correlations also would be unknowns to be estimated.

3. Goldberger (1964: 177, 185) shows that the F statistic in formula 4 can beobtained without actually conducting two regression analyses. His suggestedprocedure is more efficient, but it is not part of standard computer programs, soordinarily it is more practical to conduct the regressions with and without the subsetof indicator variables.

4. The errors of measurement in the indicator variables have been assumed

independent. This could not be true in a nominal variable problem, so the examplehere is strictly representative of only some types of induced variable and blockvariable problems.



[169]

APPENDIX

The following derivation is for the general case where a dependentvariable is a function of a set of independent variables, each of which is anunmeasured construct assessed perfectly in terms of its determinants. Theformulas for estimating the effect parameters in this general situation canbe applied to directly measured independent variables by treating these asunmeasured variables assessed perfectly with a single indicator.

For simplicity, it is assumed that all variables are measured as deviationsfrom their means; however, it is not further assumed that the deviations are

expressed in standard units.The basic equation expressing the relation between the independent

variables and the dependent variable is:

z=piyi+p2y2+...p~-~z [A1]

where z is the dependent variable, the y’s are the unmeasured independentvariables, the p’s are the effect parameters, m is the total number of

independent variables in the analysis, and Uz is a random disturbance termthat is assumed uncorrelated with any of the y’s.

Each independent variable can be expressed as a function of its own setof determinants, as indicated by the following equations.

yl al lxll + a12x12 + ... alixli i

y2 a21x21 + a22x22 + ... &dquo; ~2j~2j [A2]

ym aml xml + am2xm2 + ... amkxmk

The x’s in these equations are the determinants (and indicators) of theunmeasured y’s; the a’s are the effect parameters linking the x’s to the y’s;and i, j, and k represent the number of indicator variables used to measure

YI y~, and ym, respectively. No error term is included in these equationsbecause it is assumed that the y’s are completely specified in terms of theirindicators.

The expressions in A2 can be substituted back into equation Al toobtain the reduced form of A 1.



[170]

plalixll + PI a12xl2 + ... PI alixli + P2a21 X2I +

P2’22&dquo;22 ~ - P2a2jX2j + ... pmamlxml + . [A3]

Pmam2Xm2 + pmamkxmk + Uz ’

Let the products of the original parameters be represented by a singlesymbol.

°

snh - pnanh [A4]

where n specifies the subscript of one particular y and h specifies one ofthe indicators used to measure that y variable. Then A3 can be rewritten asfollows:

z sl1 Xu + s12x12 + ... slixli + s2I Xu + s 22x22 +[A5]

- ’2~2j + Smlxml + sm2xm2 + ... smkxmk + uz z

A procedure for estimating the s’s in equation A5 is to carry out a

multiple linear regression analysis of z on the x variables (Goldberger,1968: 13-22). The b’s or partial regression coefficients that result areestimates of the s’s, and when the regression is carried out over the entirepopulation so that the b’s are not subject to sampling error, the followingis true: .

snh = b~ [A6]

where it is understood that bnh is the partial regression coefficient forvariable xnh. Equations A4 and A6 can be combined to get:

anh &dquo;~b nh/Pn _

[A7]

This can be used to rewrite the first of equations A2 as follows.

yi - _ bil 1 x11

~12 xi2 +

... b1i xli [A8]yi - p, JI-11 ’ p, X12 + ... - X ii z

PI PI PI



[171]

Squaring both sides of this equation, one obtains the following.

Y1 = llP2 blixil + b12xi2 + ... b2ix2i 1 1 + 2(bilbi2x11x12 +...b~b~x~x~+...b~x~x~+...)j 1 [A9]

Taking expectations on both sides and rearranging terms, this becomes:

p2 I = I/a Yi 2 bi1QX11 + bi2Qxi2 + ... biiUXl’ +2(bllbl2uxiixl2 + ... biiblioxiixli + [A10]

bl2blioxl2xli ... _

where o with a single subscript represents a variance and a with a doublex subscript represents a covariance. Clearly, this procedure can be carriedout for each equation in the set A2, so:

pn = 1/QYn bniUXnl + bn2QXn2 + ... b~QXnh +~(bnl bn2oxnl xn2 + ... bni bnhaxni xnh + [All]

bn2b&dquo;haxn2xnh ... )The variances of the unmeasured variables (a 2 must be defined to

2 ’ncompute pn from All, but these variances can be set at any value byadjusting the a’s in A2 since the unmeasured variances have no empiricalscale of their own. Two alternative procedures are available for settingthese variances so that they have a meaningful interpretation within agiven problem. (a) Let the variance of Yn be a function only of thevariances of its indicator variables, each weighted by its regressioncoefficient bnh- In this case, the variance of a Yn would reflect its

predictive power, while the sheaf coefficients (Pn) would be 1.0 in all

cases. (b) Let the variance of each unmeasured variable equal the varianceof the dependent variable, a Z, 2 In this case, the variances of differentunmeasured variables in the same problem would be equal, and the sheaf



[172]

coefficients would vary according to the predictive power of each

unmeasured variable. Taking the latter approach, and recalling the

relationship between regression coefficients and standardized regressioncoefficients [(jnh = bnh(aXnh/aZ)] and also the relationship betweencovariances and correlations (e.g., ax n 1 x nh = n 1 x nh ~x n 1 ~ ~x nh ), wen n n n n nget: ,

pn gnl 2 + an2 + ... Onh 2 fJnlfJn2Px x’.,n n2

[A12]

Ønl {jnhPx nl x nh + ... (jn2{jnh{jx n2 x nh + ... )

where the #’s are the standardized partial regression coefficients and thep’s are the true correlations among indicators (the latter are estimatedfrom the observed correlations). Once the p’s have been estimated fromformula A12, the a’s can be estimated from formula A7.

REFERENCES &dquo;

BLALOCK, H. M., Jr. (1971) "Causal models involving unmeasured variables instimulus-response situations," pp. 335-347 in H. M. Blalock, Jr. (ed.) CausalModels in the Social Sciences. New York: Aldine-Atherton.

--- (1960) Social Statistics. New York: McGraw-Hill.BOYLE, R. P. (1970) "Path analysis and ordinal data." Amer. J. of Sociology 75:

461-480.

COHEN, J. (1968) "Multiple regression as a general data-analytic system." Psych.Bull. 70: 426-443.

DUNCAN, O. D. (1970) "Partials, partitions, and paths," pp. 38-47 in E. F. Borgatta(ed.) Sociological Methodology: 1970. San Francisco: Jossey-Bass.

GOLDBERGER, A. S. (1968) Topics in Regression Analysis. London: Collier-Macmillan.

--- (1964) Econometric Theory. New York: John Wiley.HAUSER, R. M. and A. S. GOLDBERGER (1971) "The treatment of unobservable

variables in path analysis," pp. 81-117 in H. L. Costner (ed.) SociologicalMethodology: 1971. San Francisco: Jossey-Bass.

HAYS, W. L. (1968) Statistics for Psychologists. New York: Holt, Rinehart &Winston.

HEISE, D. R. (1969) "Problems in path analysis and causal inference," pp. 38-73 inE. F. Borgatta (ed.) Sociological Methodology: 1969. San Francisco: Jossey-Bass.

JOHNSTON, J. (1963) Econometric Methods. New York: McGraw-Hill.KLEIN, L. R. (1953) A Textbook of Econometrics. Evanston, Ill.: Row, Peterson.LAND, K. C. (1970) "On the estimation of path coefficients for unmeasured

variables from correlations among observed variables." Social Forces 48: 506-511.



[173]

LINN, R. L. and C. E. WERTS (1969) "Assumptions in making causal inferencesfrom part correlations, partial correlations, and partial regression coefficients."Psych. Bull. 72: 307-310.

LYONS, M. and T. M. CARTER (1971) "Further comments on Boyle." Amer. J. ofSociology 76: 112-132.

MORGAN, J. N., M. H. DAVID, W. J. COHEN, and H. E. BRAZER (1962) Incomeand Welfare in the United States. New York: McGraw-Hill.

SULLIVAN, J. L. (1971) "Multiple indicators and complex causal models," pp.327-334 in H. M. Blalock, Jr. (ed.) Causal Models in the Social Sciences. NewYork: Aldine-Atherton.

WERTS, C. E. and R. L. LINN (1971) "Comments on Boyle’s ’path analysis andordinal data.’ " Amer. J. of Sociology 76: 1109-1 112.



sociological methods & research...

Documents