Intro to path analysis Richard Williams, University of Notre Dame, Last revised April 6, 2015

Transcription

1 Intro to path analysis Richard Williams, Uniersity of Notre Dame, Last reised April 6, 05 Sorces. This discssion dras heaily from Otis Ddley Dncan s Introdction to Strctral Eqation Models. Oerie. Or theories often lead s to be interested in ho a series of ariables are interrelated. It is therefore often desirable to deelop a system of eqations, i.e. a model, hich specifies all the casal linkages beteen ariables.for example, stats attainment research asks ho family backgrond, edcational attainment and other ariables prodce socio-economic stats in later life. Here is one of the early stats attainment models (see Haser, Tsai, Seell 983 for a discssion): Among the many implications of this model are that Parents Socio-Economic Stats (7) indirectly affects the Edcational Attainment () and Occpational Aspirations () of children. These, in trn, directly affect children s Occpational Attainment (). In other ords, higher parental SES helps children to become better edcated and gies them higher occpational aspirations, hich in trn leads to greater occpational achieement. Or earlier discssion of the Logic of Casal Order, combined ith the crrent discssion of Path Analysis, can help s better nderstand ho models sch as the aboe ork. Reie of key lessons from the logic of casal order. In the logic of casal order, e learned that the correlation beteen to ariables says little abot the casal relationship beteen them. This is becase the correlation beteen to ariables can be de to Intro to path analysis Page

2 the direct effect of one ariable on another indirect effects; one ariable affects another ariable hich in trn affects a third common cases, e.g. affects both Y and Z. This is sprios association correlated cases, e.g. is a case of Z and is correlated ith Y reciprocal casation; each ariable is a case of the other Hence, a correlation can reflect many non-casal inflences. Frther, a correlation can t tell yo anything abot the direction of casality. At the same time, only looking at the direct effect of one ariable on another may also not be optimal. Direct effects tell yo ho a nit change in ill affect Y, holding all other ariables constant. Hoeer, it may be that other ariables are not likely to remain constant if changes, e.g. a change in can prodce a change in Z hich in trn prodces a change in Y. Pt another ay, both the direct and indirect effects of on Y mst be considered if e ant to kno hat effect a change in ill hae on Y, i.e. e ant to kno the total effects (direct + indirect). We hae done all this conceptally. No, e ill see ho, sing path analysis, this is done mathematically and statistically. We ill sho ho the correlation beteen to ariables can be decomposed into its component parts, i.e. e ill sho ho mch of a correlation is de to direct effects, indirect effects, common cases and correlated cases. We ill frther sho ho each of the strctral effects in a model affects the correlations in the model. Path analysis terminology. Consider the folloing diagram: In this diagram, is an exogenos ariable. Exogenos ariables are those ariables hose cases are not explicitly represented in the model. Exogenos ariables are casally prior to all dependent ariables in the model. There is no casal ordering of the exogenos ariables. There can be more than one exogenos ariable in a model. For example, if there as a -headed arro linking and instead of a -headed arro, then and old both be exogenos. Conersely,,, and are endogenos ariables. The cases of endogenos ariables are specified in the model. Exogenos ariables mst alays be independent ariables. Hoeer, endogenos ariables can be either dependent or independent. For example, is a case of, bt is itself a case of and. Intro to path analysis Page

3 ,, and are distrbances, or, if yo prefer, the residal terms. Many notations are sed for distrbances; indeed, sometimes no notation is sed at all, there is jst an arro coming in from ot of nohere. ε, ε3, and ε4 old also be a good notation, gien or past practices. The one ay arros represent the direct casal effects in the model, also knon as the strctral effects. Sometimes, the names for these effects are specifically labeled, bt other times they are left implicit. The strctral eqations in the aboe diagram can be ritten as = β + = β + β = β + β + β Note that e se sbscripts for each strctral effect. The first sbscript stands for the DV, the second stands for the IV. When there are mltiple eqations, this kind of notation is necessary to keep things straight. Note, too, that intercepts are not inclded. Discssions of path analysis are simplified by assming that all ariables are centered, i.e. the mean of the ariable has been sbtracted from each case. Finally, note that the paths linking the distrbances to their respectie ariables are set eqal to. In the aboe example, each DV as affected by all the other predetermined ariables, i.e. those ariables hich are casally prior to it. We refer to sch a model as being flly recrsie, for reasons e ill explain later. There is no reqirement that each DV be affected by all the predetermined ariables, of corse. For example, β cold eqal zero, in hich case that path old be deleted from the model. Indeed, it is fairly easy to inclde paths in a model; the theoretically difficlt part is deciding hich paths to leae ot. Determining correlations and coefficients in a path model sing standardized ariables. We ill no start to examine the mathematics behind a path model. For conenience, WE WILL ASSUME THAT ALL VARIABLES HAVE A MEAN OF 0 AND A VARIANCE OF, i.e. are standardized. This makes the math easier, and it is easy enogh later on to go back to nstandardized ariables. Recall that, hen ariables are standardized, E( ) = V() =, E() = COV(,) = ρ (here ρ is the poplation conterpart to the sample estimate r) Also, e assme (at least for no) that the distrbance in an eqation is ncorrelated ith any of the IVs in the eqation. (Note, hoeer, that the distrbance in each eqation has a nonzero correlation ith the dependent ariable in that eqation and (in general) ith the dependent ariable in each later eqation.) Keeping the aboe in mind, if e kno the strctral parameters, it is fairly easy to compte the nderlying correlations. Perhaps more importantly, it is possible to decompose the correlation beteen to ariables into the sorces of association noted aboe, e.g. correlation de to direct effects, correlation de to indirect effects, etc. And, of corse, if e kno the correlations, e can compte the strctral parameters, althogh this is somehat harder to do by hand. There are a cople of ays of doing this. The normal eqations approach is more mathematical; hile perhaps less intitie, it is less prone to mistakes. second, Seell Wright s rle, is ery Intro to path analysis Page 3

4 diagram-oriented and is perhaps more intitie to most people once yo nderstand it. I find that sing both together is often helpfl. (Both approaches are probably best learned ia examples, so in class I ill probably jst skip to the examples and then let yo re-read the folloing explanations on yor on). Normal eqations. To get the normal eqations, each strctral eqation is mltiplied by its predetermined ariables, and then expectations are taken. If the strctral parameters are knon, simple algebra then yields the correlations. We ll sho ho to se normal eqations in the more complicated example. Seell Wright s mltiplication rle: To find the correlation beteen h and j, here j appears later in the model, begin at j and read back to h along each distinct direct and indirect (compond) path, forming the prodct of the coefficients along that path. (This ill gie yo the correlation beteen j and h that is de to the direct and indirect effects of h on j) After reading back, read forard (if necessary), bt only one reersal from back to forard is permitted. (This ill gie yo correlation that is de to common cases.) A doble-headed arro may be read either forard or backard, bt yo can only pass throgh doble-headed arro on each transit. (This ill gie yo correlation de to correlated cases) If yo pass throgh a ariable, yo may not retrn to it on that transit. Sm the prodcts obtained for all the linkages beteen j and h. (The main trick to sing Wright s rle is to make sre yo don t miss any linkages, cont linkages tice, or make illegal doble reersals.) This ill gie yo the total correlation beteen the ariables. To illstrate path analysis principles, e ll first go oer a generic and complicated example. We ll then present a fairly simple sbstantie (albeit hypothetical) example similar to hat e e discssed before. Generic, Complicated Example (pretty mch stolen from Dncan). We ill illstrate both the Wright rle and the se of normal eqations for each of the 3 strctral eqations in the model presented earlier: Intro to path analysis Page 4

5 (). For, the strctral eqation is = β + The only predetermined ariable is. Hence, if e mltiply both sides of the aboe eqation by and then take expectations, e get the normal eqation E( ) ( ) ( ) ρ = β = β E + E NOTE: Ho did e get from the strctral eqation to the normal eqation? First, e mltiplied both sides of the strctral eqation by, and then e took the expectations of both sides, i.e. E( = β = β ρ = β ) = β E( + => + => = ) + E( ) = Again, remember that hen ariables are standardized, E( ) = and E( ) = ρ (here ρ is the poplation conterpart to the sample estimate r ). Also remember that e are assming that the distrbance in an eqation is ncorrelated ith any of the IVs in the eqation, ergo E( ) = 0. Hence, as e hae seen before, in a biariate regression, the correlation is the same as the standardized regression coefficient. Also, all of the correlation beteen and is casal. SW Rle: Go back from to. (). For, the strctral eqation is = β + β There are to predetermined ariables, and. Taking each in trn, the normal eqations are E( 3) = β3e( ) + β3e( ) + E( ) ρ3 = β3 + β3ρ = β + β β 3 3 (Remember that β = ρ). As the aboe makes clear, there are to sorces of correlation beteen and : = Intro to path analysis Page 5

6 (a) There is a direct effect of on (represented in β3) SW Rle: Go back from to. (b) An indirect effect of operating throgh (reflected by β3β). All of the association beteen and is casal. SW Rle: Go back from to, and then back from to. NOTE: Recall that the sm of a ariable s direct effect and its indirect effects is knon as its total effect. So, in this case, the total effect of on is β 3 + β3β. Doing the same thing for and, e get E( ) = β E( + β E + E 3 3 ) 3 ( ) ( ) ρ = β ρ + β = β β + β Again, as the aboe makes clear, there are to sorces of correlation beteen and : (a) There is a direct effect of on (represented in β3). = SW Rle: Go back from to. Intro to path analysis Page 6

7 (b) Bt, there is also correlation de to a common case, (reflected by β3β). Hence, part of the correlation beteen and is sprios. SW Rle: Go back from to, go forard from to. (3). For, the predetermined ariables are,, and. The strctral eqation is The normal eqations are, first, for, E( 4 4 = β + β + β ) = β E( ) + β E( ) + β E( ) + E( ) = = β + β β + β β + β β 4 ρ = β + β ρ + β ρ = β + β β + β ( β + β β ) This shos there are 4 sorces of association beteen and : (a) Association de to the direct effect of on (β4) β 3 SW Rle: Go back from to. Intro to path analysis Page 7

8 (b) Association de to an indirect effect: affects hich then affects (β4β) SW Rle: Go back from to, go back from to. (c) Association de to another indirect effect: affects hich then affects (ββ3) SW Rle: Go back from to, go back from to. (d) Association de to yet another indirect effect: affects, hich then affects, hich then affects (ββ3β) SW Rle: Go back from to, back from to, back from to. Note that yo sm (b), (c) and (d) to get the total indirect effect of on. Note too that all of the correlation beteen and is casal. Intro to path analysis Page 8

9 The normal eqations for and are E( ) = β E( ) + β E( ) + β E( ) + E( ) = ρ 4 4 = β ρ + β + β ρ = β β + β + β ( β + β β ) = β β + β + β β + β β β This shos there are 4 sorces of association beteen and : (a) Association de to being a common case of and (β4β) SW Rle: GO back from to, go forard from to. (b) Association de to the direct effect of on (β4) SW Rle: Go back from to. (c) Association de to the indirect effect of affecting hich in trn affects (ββ3) SW Rle: Go back from to, go back from to. Intro to path analysis Page 9

10 (d) Association de to being a common case of and : directly affects and indirectly affects throgh (ββ3β). SW Rle: Go back from to, back from to, forard from to. Note that yo sm (a) and (d) to get the correlation de to common cases. This represents sprios association, hile (b) + (c) represents casal association. The normal eqations for and are, E( 3 ) = β E( ) + β E( ) + β E( ) + E( ) = ρ 4 = β = β ( β + β ρ + β ρ + β 4 3 β ) + β ( β = β β + β β β + β β + β β β + β β β ) + β This shos there are 5 sorces of association beteen and : (a) Association de to being a common case of and (β4β3) 3 SW Rle: Go back from to, go forard from to. Intro to path analysis Page 0

11 (b) Association de to being a common case of (by first affecting, hich in trn affects ) and (β4β β3) SW rle: Go back from to, forard from to, forard from to. (c) Association de to being a common case of and (β4β3) SW Rle: Back from to, go forard from to. (d) Association de to being a common case of and : directly affects and indirectly affects throgh (β4ββ3). SW Rle: Go back from to, back from to, forard from to. Intro to path analysis Page

12 (e) Association de to being a direct case of (β) SW Rle: Go back from to. Note that yo sm (a), (b) (c) and (d) to get the correlation de to common cases. This is the sprios association. There are no indirect effects of on. In reieing the aboe, note that, if there are no doble-headed arros in the model If yo go back once and then stop, it is a direct effect If yo go back or more times and neer come forard, it is an indirect effect If yo go back and later come forard, it is correlation de to a common case Correlated cases. Sppose that, in the aboe model, and ere both exogenos, i.e. there as a doble-headed arro beteen them instead of a -ay arro. This old not hae any significant effect on the math, bt it old affect or interpretation of the sorces of correlation. Anything inoling ρ old then hae to be interpreted as correlation de to correlated cases. Frther, e cold not alays say hat effect changes in old hae on other ariables, since e oldn t kno hether changes in old also prodce changes in (nless e hae good reasons for belieing that that coldn t be the case, e.g. gender and race might both be exogeneos ariables in a model, bt e are pretty confident that changes in one are not going to prodce changes in the other.). That is, ith to-headed arros e often can t be sre hat the indirect effects are, hich also means that e can t be sre hat the total effects are. Ergo, the feer -headed arros in a model, the more poerfl the model is in terms of the statements it makes. For example: Instead of and being correlated becase of the indirect effect of affecting hich in trn affects (hich is a casal relationship) and are correlated becase of the Intro to path analysis Page

13 correlated cases of and (hich e do not assme to be casal), i.e. is correlated ith a case of. Or, Instead of and being correlated becase they share a common case, they are correlated becase of a correlated case, i.e. is a case of and is correlated ith. SUBSTANTIVE HYPOTHETICAL EAMPLE (Adapted From the 995 Soc 593 Exam ): A demographer beliees that the folloing model describes the relationship beteen Income, Health of the Mother, Use of Infant formla, and Infant deaths. All ariables are in standardized form. The hypothesized ale of each path is inclded in the diagram. Mother's Health -.8 Income Infant Deaths Infant Formla Usage a. Write ot the strctral eqation for each endogenos ariable. MH = β IF = β ID = β MH If, MH ID, MH, IncIncome + =.7* Income + MH + =.8* MH + * MH + β ID, IF * IF + =.8* MH.5* IF + b. Determine the complete correlation matrix. (Remember, ariables are standardized. Yo can se either normal eqations or Seell Wright, bt yo might ant to se both as a doble-check.) Intro to path analysis Page 3

14 Correlation rmh,inc =.7 Seell-Wright Approach Go back from Mother s health to Income. (Direct effect of Income on MH) rif,mh = -.8 Go back from IF to MH. (Direct effect of MH on IF) rif,inc = -.8 *.7 = -.56 rid,if = *-.8 =.4 Go back from IF to MH, then back from MH to income. (Indirect effect of Income Income affects mother s health hich in trn affects Infant formla sage) Go back from ID to IF. (Direct effect of Infant formla on infant deaths) Then, go back from ID to MH, then go forard from MH to IF. (Mother s health is a common case of both Infant formla sage and infant deaths) Note that, een thogh the direct effect of infant formla sage on infant deaths is negatie (hich means that sing formla redces infant deaths) the correlation beteen infant formla sage and infant deaths is positie (hich means that those ho se formla are more likely to experience infant deaths). We discss this frther belo. rid,mh = *-.5 = -.4 Go back from ID to MH. (Direct effect of Mother s Health on Infant deaths) Then, go back from ID to IF to MH. (Indirect effect of Mother s health on infant deaths Mother s health affects infant formla sage hich in trn affects infant deaths) rid,inc = -.8* *-.8*.7 = -.8 Go back from Infant Death to Mother s Health, then back to Income. (Income is an indirect case of Infant deaths Income affects mother s health hich in trn affects infant deaths.) Then go back from Infant deaths, then back to Mother s Health, then back to Income. (Income is yet again an indirect case Income affects Mother s Health, hich affects Infant Formla Usage, hich affects Infant Deaths.) Intro to path analysis Page 4

15 into c. Decompose the correlation beteen Infant deaths and Usage of Infant formla Correlation de to direct effects -.5 (see path from IF to ID) Correlation de to common cases -.8 * -.8 =.64 (Mother s health is a case of both IF and ID) d. Sppose the aboe model is correct, bt instead the researcher belieed in and estimated the folloing model: Infant Formla Usage Infant Deaths What conclsions old the researcher likely dra? Why old he make these mistakes? Discss the conseqences of this mis-specification. The correlation beteen IF and ID is positie, hence, if the aboe model as estimated, the expected ale of the coefficient old be.4. This old imply that infant formla sage increases infant deaths, hen in reality the correct model shos that it decreases them. The correlation is positie becase of the common case of Mother s health: less healthy mothers are more likely to se infant formla, and they are also more likely to hae higher infant death rates. Belief in the aboe model cold lead to a redction in infant formla sage, hich old hae exactly the opposite effect of hat as intended. Intro to path analysis Page 5

16 Appendix: Basic Path Analysis ith Stata We hae been doing things a bit backards here. We hae been starting ith the coefficients, and then figred ot hat the correlations mst be. Normally, of corse, e start ith the data/correlations and then estimate the coefficients. Nonetheless, e can se Stata to erify e hae calclated the correlations correctly. Jst gie Stata the correlations e compted by hand and then se one of the methods belo to estimate the arios regressions. If e e done eerything right, the regression parameters shold come ot the same as in the path diagram. Remember, this is easier if yo se the inpt matrix by hand sbmen. (Click Data/ Matrices / Inpt matrix by hand.). matrix inpt Corr = (,.7,-.56,-.8\.7,,-.80,-.40\-.56,-.80,,.4\-.8,-.40,.4,). matrix inpt SDs = (,,,). matrix inpt Means = (0,0,0,0). corrdata income mhealth formla death, corr(corr) mean(means) sd(sds) n(00) (obs 00) There are no at least three ays to estimate the path models (or at least, the simple models e are estimating here; approach, the sem commands, is probably best for more complicated models.) I. Estimate separate regressions for each dependent ariable.. reg mhealth income Sorce SS df MS Nmber of obs = F(, 98) = 94.6 Model Prob > F = Residal R-sqared = Adj R-sqared = Total Root MSE =.7778 mhealth Coef. Std. Err. t P> t [95% Conf. Interal] income _cons 6.4e reg formla income mhealth Sorce SS df MS Nmber of obs = F(, 97) = 86. Model Prob > F = Residal R-sqared = Adj R-sqared = Total Root MSE =.6065 formla Coef. Std. Err. t P> t [95% Conf. Interal] income 4.93e mhealth _cons -.3e Intro to path analysis Page 6

17 . reg death income mhealth formla Sorce SS df MS Nmber of obs = F( 3, 96) = 0.67 Model Prob > F = Residal R-sqared = Adj R-sqared = 0.66 Total Root MSE = death Coef. Std. Err. t P> t [95% Conf. Interal] income.63e mhealth formla _cons -6.54e * The mis-specified model. reg death formla Sorce SS df MS Nmber of obs = F(, 98) =.96 Model Prob > F = Residal R-sqared = Adj R-sqared = Total Root MSE =.9959 death Coef. Std. Err. t P> t [95% Conf. Interal] formla _cons -5.3e II. The sem commands. We can also se the sem (Strctral Eqation Modeling) commands that ere introdced in Stata. This example is pretty simple so it isn t too hard to do. Among the nice featres of sem is that yo can specify all the eqations at once, and yo can get estimates of the direct, indirect and total effects. Time permitting, e ill talk abot sem more later in the semester.. sem (mhealth <- income) (formla <- income mhealth) (death <- income mhealth formla) Endogenos ariables Obsered: mhealth formla death Exogenos ariables Obsered: income Fitting target model: Iteration 0: log likelihood = Iteration : log likelihood = Strctral eqation model Nmber of obs = 00 Estimation method = ml Log likelihood = Intro to path analysis Page 7

18 OIM Coef. Std. Err. z P> z [95% Conf. Interal] Strctral mhealth <- income _cons 6.4e formla <- mhealth income 4.93e _cons -.3e death <- mhealth formla income.63e _cons -6.54e Variance e.mhealth e.formla e.death LR test of model s. satrated: chi(0) = 0.00, Prob > chi =.. * Estimate the direct, indirect, and total effects of each ariable. estat teffects Direct effects OIM Coef. Std. Err. z P> z [95% Conf. Interal] Strctral mhealth <- income formla <- mhealth income 4.93e death <- mhealth formla income.63e Intro to path analysis Page 8

19 Indirect effects OIM Coef. Std. Err. z P> z [95% Conf. Interal] Strctral mhealth <- income 0 (no path) formla <- mhealth 0 (no path) income death <- mhealth formla 0 (no path) income Total effects OIM Coef. Std. Err. z P> z [95% Conf. Interal] Strctral mhealth <- income formla <- mhealth income death <- mhealth formla income * Incorrect model. sem death <- formla Endogenos ariables Obsered: death Exogenos ariables Obsered: formla Fitting target model: Iteration 0: log likelihood = Iteration : log likelihood = Strctral eqation model Nmber of obs = 00 Estimation method = ml Log likelihood = Intro to path analysis Page 9

20 OIM Coef. Std. Err. z P> z [95% Conf. Interal] Strctral death <- formla _cons -5.3e Variance e.death LR test of model s. satrated: chi(0) = 0.00, Prob > chi =. III. UCLA s pathreg command. Yo can get this ith the findit command. Again, it lets yo specify all the eqations at once, bt doesn t offer the many additional featres that sem does. Also, pathreg does not spport factor ariables as of March 03.. pathreg (mhealth income) (formla income mhealth) (death income mhealth formla) mhealth Coef. Std. Err. t P> t Beta income _cons 6.4e n = 00 R = sqrt( - R) = 0.74 formla Coef. Std. Err. t P> t Beta income 4.93e e-09 mhealth _cons -.3e n = 00 R = sqrt( - R) = death Coef. Std. Err. t P> t Beta income.63e e-09 mhealth formla _cons -6.54e n = 00 R = sqrt( - R) = Intro to path analysis Page 0