Path Analysis

Path Diagram

Single headed arrowruns from cause to effectDouble headed bent arrow: correlation

The model above assumes that all 5 variables are turned into z-scores (standardized)It can be represented by a set of standardized STRUCTURAL (regression) equations:Occ=.281*1stOc+.394*Ed+.115*FaOc+e11stOc=.440*Ed+.224*FaOc+e2Ed=.279*FaOc+.310*FaEd+e3a. Exogeneous vs. Endogeneous VariablesExogenous variables are never dependent variables: FaOcc, FaEdEndogenous variables are dependent variables at least once: Occ, 1stOc, Edb. Dependent vs, Independent VariablesWhile the exogenous vs. endogenous distinction is with respect to the model as a whole, D vs. I variables are defined with respect to individual equationsc. Recursive vs. Non-Recursive ModelsA recursive model is one where the flow of causation is one-way: you start from any variable and if you follow the one headed arrows, you cannot encounter the same variable twice

Structural Equations

EQ1:PREMARSX=a+b1*RELITEN+b2*EDUC+eEQ2:ABSINGLE=a+b1*RELITEN+b2EDUC+b3*PREMARSX+eStandardized Structural EquationsEQ1: Zpremarsx=ppr*Zreliten+ppe*Zeduc+e1EQ2: Zabsingle=par*Zreliten+pae*Zeduc+pap*Zpremarsx+e1pyx= the path (standardized regression) coefficient of X in a regression where X is one independent and Y is the dependent variableObserved VariablesUnobservedVariables

Normal Equationsfor each equation the number of normal equations is K where K= number of independent variables rvariable1, variable2= rvariable2,variable1 bivariate Pearsons correlation coefficient measuring the linear relationship between x and yEQ1:1)rreliten,premarsx=ppr*rreliten,reliten+ppe*rreliten,educ+ rreliten,e1 as rreliten,reliten=1 and rreliten,e1=0rreliten,premarsx=ppr +ppe*rreliten,educ

2)reduc,premarsx=ppr*rreliten,educ +ppeEQ2:3)rreliten,absingle =par +pae*rreliten,educ + pap*r reliten,premarsx,4)reduc,absingle =par* reduc, reliten +pae + pap*reduc, premarsx5)rpremarsx,absingle=par* rpremarsx,reliten +pae*rpremarsx,educ+ pap5 normal equations, 5 path coefficients:5 equations, 5 unknowns:JUST-IDENTIFIED model

Effects of Religion (RELITEN) on Support for Abortion for Single Women (ABSINGLE)Take the normal equation which has the correlation of RELITEN and ABSINGLE on the right-hand side (Normal Equation #1) for EQ2).rreliten,absingle =par +pae*rreliten,educ + pap*r reliten,premarsx,Notice that rreliten,premarsx=ppr +ppe*rreliten,educ(Normal Equation #1 for EQ1).Sorreliten,absingle =par + pae*rreliten,educ+ pap*( ppr +ppe*rreliten,educ)=rreliten,absingle =par + pae*rreliten,educ+ pap* ppr + pap*ppe*rreliten,educTotal directunanalyzed indirect unanalyzed association= effect + effect through +effect through + effect through education premarital sex education AND premarital sex

Decomposing the relationship between intensity of religious beliefs and support for abortion for single women

Direct effect = -.074Indirect effect through Premarsx = -.291*.335= -.097485Unanalyzed effect due to Educ = -.019*.230= -.00437Unanalyzed effect due to Educ and Premarsx= -.019*.171*.335= -.001088415Total association = -.074 + -.097485 + -.00437 + -.001088415 = -.176943415Compare to rAbsingle,Reliten = -.177

RelitenEducPremarsxAbsinglee1e2-.074-.291.335.230-.019.171

Effects of Education (EDUC) on Support for Abortion for Single Women (ABSINGLE)Take the normal equation which has the correlation of RELITEN and ABSINGLE on the right-hand side (Normal Equation #2) for EQ2).reduc,absingle =par* reduc, reliten +pae + pap*reduc, premarsxNotice thatreduc,premarsx=ppr*rreliten,educ +ppe(Normal Equation #2 for EQ1)Soreduc,absingle =par* reduc, reliten +pae + pap* (ppr*rreliten,educ +ppe)=reduc,absingle =par* reduc, reliten +pae + pap* ppr*rreliten,educ + pap* ppeTotal unanalyzed direct unanalyzed indirect association= effect through + effect + effect through + effect through religion religion and premarital sex premarital sex

Decomposing the relationship between education and support for abortion for single women

Direct effect = .230Indirect effect through Premarsx = .171*.335= .057285Unanalyzed effect due to Reliten = -.019*-.074 = .001406Unanalyzed effect due to Reliten and Premarsx= -.019*-.291*.335=.001852215Total association = .230+ .057285 + .001406 +.001852215= 0.290543215Compare to rAbsingle,Educ= .291RelitenEducPremarsxAbsinglee1e2-.074-.291.335.230-.019.171

Effects of support for pre-marital sex on Support for Abortion for Single WomenTake the normal equation which has the correlation of PREMARSX and ABSINGLE on the right-hand side (Normal Equation #3 for EQ2).rpremarsx,absingle=par* rpremarsx,reliten +pae*rpremarsx,educ+ papNotice that rreliten,premarsx =ppr +ppe*rreliten,educ(Normal Equation #1) for EQ1).(keep in mind that rreliten,premarsx=rpremarsx, reliten)andreduc,premarsx=ppr*rreliten,educ +ppe(Normal Equation #2) for EQ1)

Sorpremarsx,absingle=par*( ppr +ppe*rreliten,educ )+pae*( ppr*rreliten,educ +ppe )+ pap=rpremarsx,absingle =par* ppr + par* ppe*rreliten,educ +pae* ppr *rreliten,educ + pae *ppe + papTotal association unanalyzed unanalyzed association directassociation= due to + effect through + effect through + due to + effect common cause education and religion and common causereligion religion education education

Decomposing the relationship between attitude towards pre-marital sex and support for abortion for single women

Direct effect = .335Spurious effect due to common cause Reliten = .-.291*-.074= .021534Spurious effect due to common cause Educ = .171*.230= .03933. Unanalyzed effect due to correlation of common cause Reliten with Educ= -.291* -.019*.230= .00127167Unanalyzed effect due to correlation of common cause Educ with Reliten= .171*-.019*-.074= .00074727Total association = .335 + .021534 + .03933 + .00127167+ .00074727 =.039788294Compare to rAbsingle,Premarsx= .398RelitenEducPremarsxAbsinglee1e2-.074-.291.335.230-.019.171

Rules of calculating the various effects of (X) on (Y)a. Direct effect path coefficientb. Indirect effectsStart from the variable (Y) later in the causal chain to your right. Trace backwards (right to left) against arrows passing intervening variables until you get to variable (X)Each combination of intervening variables is a separate indirect effect.c. Spurious effects (due to common causes)Start from variable (Y). Trace backwards to a variable (Z) that has a direct or indirect effect on both (X) and (Y). Move from (Z) to (X).There are as many spurious effects of (X) on Y due to (Z) as many ways you can get from Y to (X) through (Z) following the rule above.d. Correlated (unanalyzed) effectsIf (X) is one of several exogenous variables find (Z) that is both exogenous and has a direct or indirect effect on Y. Start from variable (Y). Trace back to (Z). Make the last step through the double headed arrow to (X)If (X) is an endogenous variable, find an exogenous variable (Z) that has a direct or indirect effect on (Y) and is correlated to another exogenous variable (W) that has a direct or indirect effect on (X). Start from variable (Y). Trace back to (Z). Travel through the double headed arrow to (W). Move from (W) to (X).Comment: A Correlated (unanalyzed) effect is like an indirect effect or a spurious effect due to common causes, except it includes one (and only one) double headed arrow.

Rules of calculating the total association 1. Find all paths Sewall Wright's rulesNo loopsWithin one path you cannot go through the same variable twice.No going forward then backwardOnly common causes matter, common consequences (effects) don't.Maximum of one curved arrow per path2. Calculate compound paths (indirect, spurious, correlated) by multiplying coefficients encountered on the way3. Add up all direct and compound effects

Identification of fully recursive models Rules of thumb (The actual rules of identification are bit more complicated but the following rules will work most of the time)Just-Identified ModelsAs many coefficients as normal equations (a necessary but not sufficient condition)With K variables this means k*(k-1)/2 single headed and double headed arrowsJust-identified models can be estimated in SPSS as separate regression equations.Underidentified ModelsMore coefficients than normal equationsUnderindentified models cannot be estimatedA model can be locally underidentified even when you have the same or more normal equations than coefficient to estimate.Overidentified ModelsFewer coefficients than normal equations (a necessary but not sufficient condition) Degrees of freedom: = #normal equations-#coefficients