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## Structural Equations Path Diagram Path Analysis

Bryan College, US has reference to this Academic Journal, Path Analysis Path Diagram Single headed arrow runs from cause so that effect Double 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+e1 1stOc=.440*Ed+.224*FaOc+e2 Ed=.279*FaOc+.310*FaEd+e3 a. Exogeneous vs. Endogeneous Variables Exogenous variables are never dependent variables: FaOcc, FaEd Endogenous variables are dependent variables at least once: Occ, 1stOc, Ed b. Dependent vs, Independent Variables While the exogenous vs. endogenous distinction is alongside respect so that the model as a whole, D vs. I variables are defined alongside respect so that individual equations c. Recursive vs. Non-Recursive Models A recursive model is one where the flow of causation is one-way: you start from any variable in addition to if you follow the one headed arrows, you cannot encounter the same variable twice Structural Equations EQ1: ÿ PREMARSX=a+b1*RELITEN+b2*EDUC+e ÿEQ2: ÿ ABSINGLE=a?+b1?*RELITEN+b2?EDUC+b3?*PREMARSX+e? ÿÿ Standardized Structural Equations ÿEQ1: ÿZpremarsx=ppr*Zreliten+ppe*Zeduc+e1 ÿEQ2: ÿZabsingle=par*Zreliten+pae*Zeduc+pap*Zpremarsx+e1 pyx= the path (standardized regression) coefficient of X in a regression where X is one independent in addition to Y is the dependent variable Observed Variables UnobservedVariables

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Normal Equations in consideration of each equation the number of normal equations is K where K= number of independent variables rvariable1, variable2= rvariable2,variable1 bivariate Pearson?s correlation coefficient measuring the linear relationship between x in addition to y EQ1: 1) rreliten,premarsx=ppr*rreliten,reliten+ppe*rreliten,educ+ rreliten,e1 as rreliten,reliten=1 in addition to rreliten,e1=0 rreliten,premarsx=ppr +ppe*rreliten,educ 2) reduc,premarsx=ppr*rreliten,educ +ppe ÿ EQ2: 3) rreliten,absingle =par +pae*rreliten,educ + pap*r reliten,premarsx, ÿ 4) reduc,absingle =par* reduc, reliten +pae + pap*reduc, premarsx ÿ 5) rpremarsx,absingle=par* rpremarsx,reliten +pae*rpremarsx,educ+ pap 5 normal equations, 5 path coefficients: 5 equations, 5 unknowns: JUST-IDENTIFIED model Effects of Religion (RELITEN) on Support in consideration of Abortion in consideration of Single Women (ABSINGLE) Take the normal equation which has the correlation of RELITEN in addition to ABSINGLE on the right-hand side (Normal Equation #1) in consideration of EQ2). ÿ rreliten,absingle =par +pae*rreliten,educ + pap*r reliten,premarsx, ÿÿ Notice that ÿrreliten,premarsx=ppr +ppe*rreliten,educ (Normal Equation #1 in consideration of EQ1). ÿ So ÿrreliten,absingle =par + pae*rreliten,educ+ pap*( ppr +ppe*rreliten,educ)= ÿrreliten,absingle =par + pae*rreliten,educ+ pap* ppr + pap*ppe*rreliten,educ ÿÿ Total direct unanalyzed 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 in addition to support in consideration of abortion in consideration of single women Direct effect = -.074 Indirect effect through Premarsx = -.291*.335= -.097485 Unanalyzed effect due so that Educ = -.019*.230= -.00437 Unanalyzed effect due so that Educ in addition to Premarsx= -.019*.171*.335= -.001088415 Total association = -.074 + -.097485 + -.00437 + -.001088415 = -.176943415 Compare so that rAbsingle,Reliten = -.177 Reliten Educ Premarsx Absingle e1 e2 -.074 -.291 .335 .230 -.019 .171

Effects of Education (EDUC) on Support in consideration of Abortion in consideration of Single Women (ABSINGLE) Take the normal equation which has the correlation of RELITEN in addition to ABSINGLE on the right-hand side (Normal Equation #2) in consideration of EQ2). ÿÿ reduc,absingle =par* reduc, reliten +pae + pap*reduc, premarsx ÿ Notice that reduc,premarsx=ppr*rreliten,educ +ppe (Normal Equation #2 in consideration of EQ1) ÿ So reduc,absingle =par* reduc, reliten +pae + pap* (ppr*rreliten,educ +ppe)= reduc,absingle =par* reduc, reliten +pae + pap* ppr*rreliten,educ + pap* ppe ÿ Total unanalyzed direct unanalyzed indirect association= effect through + effect + effect through + effect through religion religion in addition to premarital sex premarital sex Decomposing the relationship between education in addition to support in consideration of abortion in consideration of single women Direct effect = .230 Indirect effect through Premarsx = .171*.335= .057285 Unanalyzed effect due so that Reliten = -.019*-.074 = .001406 Unanalyzed effect due so that Reliten in addition to Premarsx= -.019*-.291*.335=.001852215 Total association = .230+ .057285 + .001406 +.001852215= 0.290543215 Compare so that rAbsingle,Educ= .291 Reliten Educ Premarsx Absingle e1 e2 -.074 -.291 .335 .230 -.019 .171 Effects of support in consideration of pre-marital sex on Support in consideration of Abortion in consideration of Single Women Take the normal equation which has the correlation of PREMARSX in addition to ABSINGLE on the right-hand side (Normal Equation #3 in consideration of EQ2). ÿ rpremarsx,absingle=par* rpremarsx,reliten +pae*rpremarsx,educ+ pap ÿ Notice that ÿrreliten,premarsx =ppr +ppe*rreliten,educ (Normal Equation #1) in consideration of EQ1). ÿ (keep in mind that rreliten,premarsx=rpremarsx, reliten) in addition to ÿreduc,premarsx=ppr*rreliten,educ +ppe (Normal Equation #2) in consideration of EQ1) So rpremarsx,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 + pap ÿ Total association unanalyzed unanalyzed association direct association= due so that + effect through + effect through + due so that + effect common cause education in addition to religion in addition to common cause religion religion education education

Chapter 11 Corporations: Organization, Stock Transactions, Dividends, in addition to Retained Earnings Objectives of The Chapter I. Corporations: An Overview Procedures of Forming a Corporation Procedures of Forming a Corporation (contd.) Organization of a Corporation Advantages of a Corporation Disadvantages of a Corporation The Stockholders? Equity Section of a Balance Sheet The Stockholders? Equity Section of a Balance Sheet Terminology Related so that Stockholders? Equity of a Corporation Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) Terminology Related so that Stockholders? Equity of a Corporation (contd.) 2. Accounting in consideration of Issuance of Stock Stock Issued in consideration of Cash ?common stock alongside par value Stock Issued in consideration of Cash (contd.) Stock Issued in consideration of Cash (contd.) Stock Issued in consideration of Cash (contd.) Stock Issued in consideration of services or noncash assets Stock Issued in consideration of Noncash Proposition (contd.) Preferred Stock Characteristics Preferred Stock Characteristics (contd.) Cumulative Preferred Stock Dividends Allocation: Examples Dividends Allocation: (contd.) Dividends Allocation (contd.) Dividends Allocation (Contd.) Convertible Preferred Stock (skip) Accounting in consideration of Conversion of Preferred so that Common Stock (skip) 3. Treasury Stock Treasury Stock (contd.) Accounting in consideration of Treasury Stock (T.S.) ?the Cost Method Accounting Methods in consideration of Treasury Stock (T.S.) (contd.) Accounting Methods in consideration of Treasury Stock (T.S.) (contd.) Accounting Methods in consideration of Treasury Stock (T.S.) (contd.) Balance Sheet Presentation of Treasury Stock Balance Sheet Presentation of Treasury Stock (contd) Retirement of Stock 4. Retained Earnings in addition to Dividends Dividends Dividends (Contd.) Dividends (Contd.) Dividends (contd.) Cash Dividends Cash Dividends Example Example (contd.) Stock Dividends Stock Dividends (contd.) Stock Dividends (contd.) Example 1: Small Stock Dividend Example 1 (contd.) Example 2: Large Stock Dividend Example 2 (contd.) Stock Splits Accounting in consideration of Stock Splits (proportionate stock splits) Similarities in addition to Differences Between Stock Dividends in addition to Stock Splits (Skip) Stock Dividends in addition to Stock Splits (contd.) (Skip) Accounting in consideration of Prior Years? Errors Prior Period Adjustment- An Example Prior Period Adjustment- An Example (contd.) Different Values of Stock Different Values of Stock (contd.) Different Values of Stock (contd.) Different Values of Stock (contd.) Decision Making Ratio

Decomposing the relationship between attitude towards pre-marital sex in addition to support in consideration of abortion in consideration of single women Direct effect = .335 Spurious effect due so that common cause Reliten = .-.291*-.074= .021534 Spurious effect due so that common cause Educ = .171*.230= .03933. Unanalyzed effect due so that correlation of common cause Reliten alongside Educ= -.291* -.019*.230= .00127167 Unanalyzed effect due so that correlation of common cause Educ alongside Reliten= .171*-.019*-.074= .00074727 Total association = .335 + .021534 + .03933 + .00127167+ .00074727 =.039788294 Compare so that rAbsingle,Premarsx= .398 Reliten Educ Premarsx Absingle e1 e2 -.074 -.291 .335 .230 -.019 .171 Rules of calculating the various effects of (X) on (Y) a. Direct effect path coefficient b. Indirect effects Start from the variable (Y) later in the causal chain so that your right. Trace backwards (right so that left) against arrows passing intervening variables until you get so that variable (X) Each combination of intervening variables is a separate indirect effect. c. Spurious effects (due so that common causes) Start from variable (Y). Trace backwards so that a variable (Z) that has a direct or indirect effect on both (X) in addition to (Y). Move from (Z) so that (X). There are as many spurious effects of (X) on Y due so that (Z) as many ways you can get from Y so that (X) through (Z) following the rule above. d. Correlated (unanalyzed) effects If (X) is one of several exogenous variables find (Z) that is both exogenous in addition to has a direct or indirect effect on Y. Start from variable (Y). Trace back so that (Z). Make the last step through the double headed arrow so that (X) If (X) is an endogenous variable, find an exogenous variable (Z) that has a direct or indirect effect on (Y) in addition to is correlated so that another exogenous variable (W) that has a direct or indirect effect on (X). Start from variable (Y). Trace back so that (Z). Travel through the double headed arrow so that (W). Move from (W) so that (X). Comment: A Correlated (unanalyzed) effect is like an indirect effect or a spurious effect due so that 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 rules No loops Within one path you cannot go through the same variable twice. No going forward then backward Only common causes matter, common consequences (effects) don’t. Maximum of one curved arrow per path 2. Calculate compound paths (indirect, spurious, correlated) by multiplying coefficients encountered on the way 3. Add up all direct in addition to 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 Models As many coefficients as normal equations (a necessary but not sufficient condition) With K variables this means k*(k-1)/2 single headed in addition to double headed arrows Just-identified models can be estimated in SPSS as separate regression equations. Underidentified Models More coefficients than normal equations Underindentified models cannot be estimated A model can be locally underidentified even when you have the same or more normal equations than coefficient so that estimate. Overidentified Models Fewer coefficients than normal equations (a necessary but not sufficient condition) Degrees of freedom: = #normal equations-#coefficients

## Bogert, John Host

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