Path Analysis

Overview Path analysis is an extension of the regression model, used to test the fit of the correlation matrix against two or more causal models which are being compared by the researcher. The model is usually depicted in a circle-and-arrow figure in which single-headed arrows indicate causation. A regression is done for each variable in the model as a dependent on others which the model indicates are causes. The regression weights predicted by the model are compared with the observed correlation matrix for the variables, and a goodness-of-fit statistic is calculated. The best-fitting of two or more models is selected by the researcher as the best model for advancement of theory. Path analysis requires the usual assumptions of regression. It is particularly sensitive to model specification because failure to include relevant causal variables or inclusion of extraneous variables often substantially affects the path coefficients, which are used to assess the relative importance of various direct and indirect causal paths to the dependent variable. Such interpretations should be undertaken in the context of comparing alternative models, after assessing their goodness of fit discussed in the section on structural equation modeling (SEM packages are commonly used today for path analysis in lieu of stand-alone path analysis programs). When the variables in the model are latent variables measured by multiple observed indicators, path analysis is termed structural equation modeling, treated separately. We follow the conventional terminology by which path analysis refers to modeling single-indicator variables. .

Contents Key concepts and terms Path coefficients Path multiplication rule Effect decomposition Path analysis with structural equation modeling Assumptions Frequently asked questions Bibliography

Key Concepts and Terms

• Estimation. Note that path estimates may be calculated by OLS regression or by MLE maximum likelihood estimation, depending on the computer package. Two-Stage Least Squares (2SLS), discussed separately, is another path estimation procedure designed to extend the OLS regression model to situations where non-recursivity is introduced because the researcher must assume the covariances of some disturbance terms are not 0 (this assumption is discussed below). Click here for a separate discussion of 2SLS.

• Path model. A path model is a diagram relating independent, intermediary, and dependent variables. Single arrows indicate causation between exogenous or intermediary variables and the dependent(s). Arrows also connect the error terms with their respective endogenous variables. Double arrows indicate correlation between pairs of exogenous variables. Sometimes the width of the arrows in the path model are drawn in a width which is proportional to the absolute magnitude of the corresponding path coefficients (see below).

• Causal paths to a given variable include (1) the direct paths from arrows leading to it, and (2) correlated paths from endogenous variables correlated with others which have arrows leading to the given variable. Consider this model:
  

  This model has correlated exogenous variables A, B, and C, and endogenous variables D and E. Error terms are not shown. The causal paths relevant to variable D are the paths from A to D, from B to D, and the paths reflecting common anteceding causes -- the paths from B to A to D, from C to A to D, and from C to B to D. Paths involving two correlations (C to B to A to D) are not relevant. Likewise, paths that go backward (E to B to D, or E to B to A to D) reflect common effects and are not relevant.

• Exogenous and endogenous variables. Exogenous variables in a path model are those with no explicit causes (no arrows going to them, other than the measurement error term). If exogenous variables are correlated, this is indicated by a double-headed arrow connecting them. Endogenous variables, then, are those which do have incoming arrows. Endogenous variables include intervening causal variables and dependents. Intervening endogenous variables have both incoming and outgoing causal arrows in the path diagram. The dependent variable(s) have only incoming arrows.

• Path coefficient/path weight. A path coefficient is a standardized regression coefficient (beta) showing the direct effect of an independent variable on a dependent variable in the path model. Thus when the model has two or more causal variables, path coefficients are partial regression coefficients which measure the extent of effect of one variable on another in the path model controlling for other prior variables, using standardized data or a correlation matrix as input. Recall that for bivariate regression, the beta weight (the b coefficient for standardized data) is the same as the correlation coefficient, so for the case of a path model with a variable as a dependent of a single exogenous variable (and an error residual term), the path coefficient in this special case is a zero-order correlation coefficient.

  Consider this model, based on Bryman, A. and D. Cramer (1990). Quantitative data analysis for social scientists, pp. 246-251.

  

  This model is specified by the following path equations:

  Equation 1. satisfaction = b₁₁age + b₁₂autonomy + b₁₃ income + e₁
  Equation 2. income = b₂₁age + b₂₂autonomy + e₂
  Equation 3. autonomy = b₃₁age + e₃

  where the b's are the regression coefficients and their subscripts are the equation number and variable number (thus b₂₁ is the coefficient in Equation 2 for variable 1, which is age.

  Note: In each equation, only (and all of) the direct priors of the endogenous variable being used as the dependent are considered. The path coefficients, which are the betas in these equations, are thus the standardized partial regression coefficients of each endogenous variable on its priors. That is, the beta for any path (that is, the path coefficient) is a partial weight controlling for other priors for the given dependent variable.

  Formerly called p coefficients, now path coefficients are called simply beta weights, based on usage in multiple regression models. Bryman and Cramer computed the path coefficients = standardized regression coefficients = beta weights, to be:

  
  Correlated Exogenous Variables. If exogenous variables are correlated, it is common to label the corresponding double-headed arrow between them with its correlation coefficient.

  Disturbance terms.The residual error terms, also called disturbance terms, reflect unexplained variance (the effect of unmeasured variables) plus measurement error. Note that the dependent in each equation is an endogenous variable (in this case, all variables except age, which is exogenous). Note also that the independents in each equation are all the variables with arrows to the dependent.

  The effect size of the disturbance term for a given endogenous variable, which reflects unmeasured variables, is (1 - R²), and its variance is (1 - R²) times the variance of that endogenous variable, where R² is based on the regression in which it is the dependent and those variables with arrows to it are independents. The path coefficient is SQRT(1 - R²).

  The correlation between two disturbance terms is the partial correlation of the two endogenous variables, using as controls all their common causes (all variables with arrows to both). The covariance estimate is the partial covariance: the partial correlation times the product of the standard deviations of the two endogenous variables.

• Path multiplication rule: The value of any compound path is the product of its path coefficients. Imagine a simple three-variable compound path where education causes income causes conservatism. Let the regression coefficient of income on education be 1000: for each year of education, income goes up $1,000. Let the regression coefficient of conservatism on income be .0002: for every dollar income goes up, conservativism goes up .0002 points on a 5-point scale. Thus if education goes up 1 year, income goes up $1,000, which means conservatism goes up .2 points. This is the same as multiplying the coefficients: 1000*.0002 = .2. The same principle would apply if there were more links in the path. If standardized path coefficients (beta weights) were used, the path multiplication rule would still apply, but the the interpretation is in standardized terms. Either way, the product of the coefficients along the path reflects the weight of that path.

• Effect decomposition. Path coefficients may be used to decompose correlations in the model into direct and indirect effects, corresponding, of course, to direct and indirect paths reflected in the arrows in the model. This is based on the rule that in a linear system, the total causal effect of variable i on variable j is the sum of the values of all the paths from i to j. Considering "satisfaction" as the dependent in the model above, and considering "age" as the independent, the indirect effects are calculated by multiplying the path coefficients for each path from age to satisfaction:

  age -> income -> satisfaction is .57*.47 = .26
  age -> autonomy -> satisfaction is .28*.58 = .16
  age -> autonomy -> income -> satisfaction is .28*.22 x .47 = .03
  total indirect effect = .45

  That is, the total indirect effect of age on satisfaction is plus .45. In comparison, the direct effect is only minus .08. The total causal effect of age on satisfaction is (-.08 + .45) = .37.

  Effect decomposition is equivalent to effects analysis in regression with one dependent variable. Path analysis, however, can also handle effect decomposition for the case of two or more dependent variables.

  In general, any bivariate correlation may be decomposed into spurious and total causal effects, and the total causal effect can be decomposed into a direct and an indirect effect. The total causal effect is the coefficient in a regression with all of the model's prior but not intervening variables for x and y controlled (the beta coefficient for the usual standardized solution, the partial b coefficient for the unstandardized or raw solution). The spurious effect is the total effect minus the total causal effect. The direct effect is the partial coefficient (beta for standardized, b for unstandardized) for y on x controlling for all prior variables and all intervening variables in the model. The indirect effect is the total causal effect minus the direct effect, and measures the effect of the intervening variables. Where effects analysis in regression may use a variety of coefficients (partial correlation or regression, for instance), effect decomposition in path analysis is restricted to use of regression.

  For instance, imagine a five-variable model in which the exogenous variable Education is correlated with the exogenous variable Skill Level, and both Education and Skill Level are correlated with the exogenous variable Job Status. Further imagine that Education and each of the other two exogenous variables are modeled to be direct causes of Income and also of Median House Value, which are the two dependent variables. We might then decompose the correlation of Education and Income:

  1. Direct effect of Education on Income, indicated by the path coefficient of the single-headed arrow from Education to Income.
  2. Indirect effect due to Education's correlation with Skill Level, and Skill Level's direct effect on Income, indicated by multiplying the correlation of Education and Skill Level by the path coefficient from Skill Level to Income.
  3. Indirect effect due to Education's correlation with Job Status, and Job Status's direct effect on Income, indicated by multiplying the correlation of Education and Job Status by the path coefficient from Job Status to Income.

  As a second example decomposition for the same five-variable model is a bit more complex if we wish to break down the correlation of the two dependent variables, Income and Median House Value. Since here somewhat implausibly the two dependents are modeled not to have a direct effect from Income to House Value, the true correlation is hypothesized to be zero and all correlations are spurious.

  1. The spurious direct effect of Education as a common anteceding variable directly causing both dependents, indicated by multiplying the path coefficient from Education to Income by the path coefficient of Education to House Value.
  2. The spurious direct effect of Skill Level as a common anteceding variable directly causing both dependents, indicated by multiplying the path coefficient from Skill Level to Income by the path coefficient of Skill Level to House Value.
  3. The spurious direct effect of Job Status as a common anteceding variable directly causing both dependents, indicated by multiplying the path coefficient from Job Status to Income by the path coefficient of Job Status to House Value.
  4. The spurious indirect effect of Education and Skill Level as a common antecedings variable directly causing both dependents, indicated by multiplying the path coefficient from Education to Income by the correlation of Education and Skill Level by the path from Skill Level to House Value and adding the product of the path from Skill Level to Income by the correlation of Education and Skill Level by the path from Education to Median House Value.
  5. The spurious indirect effect of Education and Job Status as a common anteceding variables directly causing both dependents, indicated by multiplying the path coefficient from Education to Income by the correlation of Education and Job Status by the path from Job Status to House Value and adding the product of the path from Job Status to Income by the correlation of Education and Job Status by the path from Education to Median House Value..
  6. The spurious indirect effect of Skill Level and Job Status as a common anteceding variables directly causing both dependents, indicated by multiplying the path coefficient from Skill Level to Income by the correlation of Skill Level and Job Status by the path from Job Status to House Value and adding the product of the path from Job Status to Income by the correlation of Skill Level and Job Status by the path from Skill Level to Median House Value..
  7. The residual effect is the difference between the correlation of Income and Median House Value and the sum of the spurious direct and indirect effects.

  Correlated exogenous variables. The path weights connecting correlated exogenous variables are equal to the Pearson correlations. When calculating indirect paths, not only direct arrows but also the double-headed arrows connecting correlated exogenous variables, are used in tracing possible indirect paths, except:

  Tracing rule: An indirect path cannot enter and exit on an arrowhead. This means that you cannot have a direct path composed of the paths of two correlated exogenous variables.

  
  

• Significance and Goodness of Fit in SEM Path Models
  • OLS vs. SEM While a series of OLS regressions may be used to implement path analysis, testing individual path coefficients using the standard t or F test from regression output, today it is far more common to use structural equation modeling (SEM) software. This section uses AMOS with a model based on Ingram et al. (2000), used with the kind permission of Karl Wuensch. Use of AMOS is described more fully in the section on structural equation modeling. A structural equation model with simple rather than latent variables is a path model.

  • The SEM path model. The path model is drawn as usual in SEM. Illustrated below is the model for the Ingram data, which deals with application to graduate schools. In this model, Attitude, SubNorm, and PBC all predict Intent, while the ultimate dependent variable, Behavior, is predicted by Intent and also directly by PBC. As customary, the straight arrows represent regression paths for presumed causal relationships, which the curved double-headed arrows represent assumed covariances among the exogenous variables. The endogenous variables are depicted with associated error terms.
    

    In this model, Attitude is the individual's attitude toward graduate school; SubNorm is subjective norms, reflected by attitudes toward graduate school of others around the individual; PBC is planned behavioral control, which is the individual's level of control over behaviors related to graduate school. Intent is the individual's intent to go to graduate school. Behavior is applying to graduate school.

  • Select outputs. Statistical tests and other outputs are selected under View, Analysis Properties, in the AMOS menu system, yielding the dialog shown below:
    

  • Path coefficients in standardized and unstandardized form are generated by AMOS by selecting Analyze, Compute Estimates. Un like the OLS regression method, all parameters are calculated simultaneously. These coefficients may be caused to be displayed on the path diagram, and also appear in the output, which is obtained in the menu system by selecting View, Text Output. For this example, note that the paths from SubNorm and PBC to Intent are not significant. That is, Intent is primarily a function of Attitude.
    

  • Overall test of the model. The likehood ratio chi-square test, also called the model chi-square test or deviance test, assesses the overall fit of the model. A finding of nonsignificance corresponds to an adequate model - one whose model-implied covariance matrix does not differ from the observed covariance matrix. For this example, there is adequate fit:

    Result (Default model)
    Minimum was achieved
    Chi-square = .847
    Degrees of freedom = 2
    Probability level = .655
    

    However, the likelihood ratio chi-square test cannot be relied upon alone, particularly for large samples, because a finding of significance (rejecting the model) can occur even with very small differences of the model-implied and observed covariance matrices (note below that AMOS labels the likelihood ratio chi-square CMIN). Therefore a large variety of other goodness of fit measures have been devised. Their use and relative merits are described in the section on structural equation modeling. In the output below, the "default model" is the researcher's model. The "saturated model" is the perfectly explanatory but trivial model with all possible arrows. he "independence model" is the model with no regression arrows (straight arrows). Suffice it to say, these goodness of fit measures support the adequacy of the model in the example (for example, RMSEA should be <.05 for a good model and here is .000). Note, however, that in spite of statistically adequate goodness of fit, normally the researcher would drop non-significant structural arrows indicated above in the path coefficients section.
    

  • Correlations. Also included in AMOS output are the correlations among the exogenous variables (correlations in the upper output) and between the exogenous variables and the endogenous variables (squared multiple correlation in lower output) as illustrated below. The model explains 34.3% of the variance in the dependent variable, Behavior.

    Correlations: (Group number 1 - Default model)
       Estimate 
    SubNorm <--> PBC .505 
    Attitude <--> SubNorm .472 
    Attitude <--> PBC .665 
    
    Squared Multiple Correlations: (Group number 1 - Default model)
       Estimate 
    Intent   .600 
    Behavior   .343 
    

  • Direct and indirect effects. AMOS will use the muliplication rule automatically to partition overall effects into direct and indirect effects for the endogenous variables (for Intent and Behavior in this example).
    

  • Modification indexes. Modificiation indexes (MI) may be used to add arrows to the model. The larger the MI, the more adding the model will improve model fit. As discussed in the section on structural equation modeling, MIs should be interpreted in relation to critical ratios (CR), which are a measure of the change in likelihood ratio chi square. And as noted above, nonsignificance of path coefficients is used to drop arrows in a model-building and model-trimming process discussed in the section on structural equation modeling. The conservative approach calls for adding or dropping one arrow at a time as each change will affect the coefficients. For this example, the MIs are so small that no addition of arrows is warranted. In fact, all MIs are well below the usual lower threshold of 4.0.
    

Assumptions

• Linearity: relationships among variables are linear (though, of course, variables may be nonlinear transforms).

• Additivity: there are no interaction effects (though, of course, variables may be interaction crossproduct terms)

• Interval level data for all variables, if regression is being used to estimate path parameters. As in other forms of regression modeling, it is common to use dichotomies and ordinal data in practice. If dummy variables are used to code a categorical variable, one must be careful that they are represented as a block in the path diagram (ex., if an arrow is drawn to one dummy it must be drawn to all others in the set). If an arrow were to be drawn from one dummy variable to another dummy variable in the same set, this would violate the recursivity assumption discussed below.

• Residual (unmeasured) variables are uncorrelated with any of the variables in the model other than the one they cause.

• Disturbance terms are uncorrelated with endogenous variables. As a corollary of the previous assumption, path analysis assumes that for any endogenous variable, its distubance term is uncorrelated with any other endogenous variable in the model. This is a critical assumption, violation of which may make regression inappropriate as a method of estimating path parameters. This assumption may be violated due to measurement error in measuring an endogenous variable; when an endogenous variable is actually a direct or indirect cause of a variable which the model states is the cause of that endogenous variable (reverse causation); or when a variable not in the model is a cause of an endogenous variable and a variable the model specifies as a cause of that endogenous variable (spurious causation).

• Low multicollinearity (otherwise one will have large standard errors of the b coefficients used in removing the common variance in partial correlation analysis).

• No underidentification or underdetermination of the model is required. For underidentified models there are too few structural equations to solve for the unknowns. Overidentification usually provides better estimates of the underlying true values than does just identification.
  • Recursivity: all arrows flow one way, with no feedback looping. Also, it is assumed that disturbance (residual error) terms for the endogenous variables are uncorrelated. Recursive models are never underidentified.

• Proper specification of the model is required for interpretation of path coefficients. Specification error occurs when a significant causal variable is left out of the model. The path coefficients will reflect the shared covariance with such unmeasured variables and will not be accurately interpretable in terms of direct and indirect effects. In particular, if a variable specified as prior to a given variable is really consequent to it, "we can do ourselves considerable damage" (Davis, 1985: 64) because if a variable is consequent it would be estimated to have no path effect, whereas when it is included as a prior variable in the model, this erroneously changes the coefficients for other variables in the model. Note, however, that while interpretation of path coefficients is inaccurate under specification error, it is still possible to compare the relative fit of two models, perhaps both with specification error.

• Appropriate correlation input. When using a correlation matrix as input, it is appropriate to use Pearsonian correlation for two interval variables, polychoric correlation for two ordinals, tetrachoric for two dichotomies, polyserial for an interval and an ordinal, and biserial for an interval and a dichotomy.

• Adequate sample size is needed to assess significance. Kline (1998) recommends 10 times as many cases as parameters (or ideally 20 times). He states that 5 times or less is insufficient for significance testing of model effects.

• The same sample is required for all regressions used to calculate the path model. This may require reducing the data set down so that there are no missing values for any of the variables included in the model. This might be achieved by listwise dropping of cases or by data imputation.

Frequently Asked Questions

• Does path analysis confirm causation in a model?
  No, although this is sometimes said. Everitt and Dunn (1991) note, "However convincing, respectable and reasonable a path diagram... may appear, any causal inferences extracted are rarely more than a form of statistical fantasy". The authors are referring to the fact that ultimately path analysis deals with correlation, not causation of variables. The arrows in path models do indeed reflect hypotheses about causation. However, many models may be consistent with a given dataset. Path analysis merely illuminates which of two or more competing models, derived from theory, is most consistent with the pattern of correlations found in the data. The competing theories may be represented in separate path models with separate path analyses, or may be combined in a single path diagram, in which case the researcher is concerned with comparing the relative importance of different paths within the diagram.

• Can path analysis be used for exploratory rather than confirmatory purposes?
  Methodologists favor a priori formulation of hypotheses about the results to be obtained from path analysis, not post factum conclusions based on the results. That is, the researcher should be seeking to confirm hypotheses made beforehand. At a minimum, the researcher should posit the sign of each relationship (arrow) in the model, and ideally should go further to posit the arrangement by magnitude of the importance of the independents, or even better yet, the intervals within which the path coefficients will be expected to lie. Since data can support multiple models, path analysis should focus on determining which of two or more theoretically-derived models most conform to the underlying data.

  If used in an exploratory way, note that the more models you test, the more likely you will confirm one just by chance. Thus the actual confidence level is equal to the individual test confidence level to the power of the number of models tested, here .95 cubed = .86. Thus testing 3 models at .95 would mean we were actually operating at about .86 confidence. At the .99 level, however, actual confidence would still be .95 with 5 models.

• How do I know if my model is "underidentified" and what difference does it make? How does this relate to "recursivity?
  A unique path solution cannot be calculated if a model is underidentified. Identification is defined and steps the researcher can take to deal with underidentification are dealt with in a section on structural equation modeling. How to determine beforehand if a model is underidentified, other than by running a path analysis program for sample or fictional data and seeing if there are error flags, is discussed more fully under a second section on structural equation modeling.

• How does the significance of a path coefficient compare with the significance of the corresponding regression coefficient?
  They are identical. The path coefficient is the beta for the regression of the dependent or other endogenous variable on the other variables with arrows to it. The significance of the beta and b coefficients will be the same, and is displayed on the same line in SPSS regression output. Naturally, all paths in the model should be significant!

• How do you assess the significance of the total (direct and indirect) effect of exogenous variable x on endogenous variable y?
  Run a regression with y as dependent and all others as independents, leaving out any variable which mediates between x and y. The significance of the b or beta for x in this equation is a test of the significance of the total effect.

• How are path coefficients related to the correlation matrix for purposes of testing a model?
  First, recognize that computation of the model-estimated correlations and their comparison with observed correlations is best done by relying on a model-estimating program such as LISREL or AMOS. The model path coefficients can be compared to the predicted path coefficients as computed from the correlation matrix, following which the model coefficients can be tested for goodness-of-fit with the predicted coefficients.

  The tracing rule is a rule for identifying all the paths, the sum of effects of which is the estimated correlation between two variables in the model. This model-estimated correlation can be compared to the observed correlation to assess the fit of the model to the data. The tracing rule is simply that the model-implied correlation between two variables in a model is the sum of all valid paths (tracings) between the two variables. These include the total effect (which is the sum of direct and indirect effects) plus any associational effects due to correlated exogenous variables. These associational effects are calculated by multiplying the correlation between the exogenous variable under consideration with a second exogenous variable, by this second exogenous variable"s total effect on the target variable under consideration.

  For simplicity, consider this simple model:

  

  The actual (observed) correlation matrix might look like this:

         A          B         D   
  A     1         .379      -.652 
  B                  1      -.451
  C                             1   
  
  

  When running AMOS or another path analysis program, the path coefficients (standardized regression coefficients) would be:

         A          B         D   
  A     1         .379      -.562 
  B                  1      -.238
  C                             1   
  
  

  If one had only the path output and wanted to estimate back to the correlation matrix, one would use these equations, one for each path:

  r(AB) = n = .379 (the correlation is the standardized path coefficient, in the bivariate case
                          where the independent is exogenous and the dependent has no other inputs)
  r(BD) = q + np = -.238 + .379(-.562) = -.451
  r(AD) = p + nq = -.562 + .379(-.238) = -.652
  

  In testing a model, somewhat similar reasoning is followed to compare model-implied covariance matrices with observed covariance matrices, with smaller differences indicating better goodness of fit.

• How, exactly, can I compute path coefficients in SPSS?
  The recommended method is to enter a path model into AMOS, which is the SPSS program for structural equation modeling, discussed in that section. However, in the SPSS regression module, for a recursive model, let VARA cause VARB and VARC, and let VARB cause VARC. A series of regressions is conducted with each non-exogenous variable considered as a dependent in turn. For the foregoing model, the path coefficient from VARA to VARB is given by this SPSS code:

  REGRESSION
  /MISSING LISTWISE
  /STATISTCS COEFF OUTS R ANOVA
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT VARB
  /METHOD=ENTER VARA.
  
  

  The path coefficients from VARA to VARC and from VARB to VARC are given by this second regression command:

  REGRESSION
  /MISSING LISTWISE
  /STATISTCS COEFF OUTS R ANOVA
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT VARC
  /METHOD=ENTER VARA VARB.
  
  

• How do I compute the value of the path from an error term to an endogenous variable?
  The path is the square root of (1 - R-squared), where R-squared is from the regression equation for the corresponding dependent variable. Do not use adjusted R-squared.

• How can multiple group path analysis determine if the path model differs across groups in my sample?
  Multiple group path analysis may be accomplished simply by running separate path analysis for each group in the sample, then comparing the path estimates. A more sophisticated approach supported by some path analysis and SEM packages involves a second step: to impose a cross-group equality constraint on the path estimates, then run the analysis separately for each group, then see if the goodness-of-fit for the constrained models is as good as for the unconstrained models. If the fit of the constrained model is worse than that for the corresponding unconstrained model, then the researcher concludes that model direct effects differ by group.

• Could I substitute logistic regression in doing effect decomposition?
  No. Forward paths cannot be decomposed accurately with log linear techniques. See Davis (1985): 48, 59.

  Are regression and SEM the only approaches to path analysis?

  • Partial least squares path analysis is also available, through custom software (not SPSS or SAS, which only support PLS regression). PLS can support small sample models, even where there are more variables than observations, but it is lower in power than SEM approaches.

Bibliography

• Alwin, Duane F. and Robert M. Hauser (1975). The decomposition of effects in path analysis. American Sociological Review 40(Feb.): 37-47.
• Cohen, Jacob, et al (2003). Applied multiple regression/correlation analysis for the behavioral sciences. Mahwah, NJ. Chapter 12 is a good generalized introduction to path analysis.
• Davis, James A. (1985). The logic of causal order. Quantitative applications in the social sciences series, no. 55. Thousand Oaks, CA: Sage Publications. Pp. 44-48 provide a non-technical introduction to effects analysis. Pp. 48-68 discuss path analysis generally.
• Everitt, B. S., and G. Dunn, G. (1991). Applied multivariate data analysis. London: Edward Arnold.
• Heise, David R. (1975). Causal analysis. NY: Wiley.
• Ingram, K. L., Cope, J. G., Harju, B. L., & Wuensch, K. L. (2000). Applying to graduate school: A test of the theory of planned behavior. Journal of Social Behavior and Personality, 15, 215-226.
• Kerlinger, F. N. and E. J. Pedhazur (1973). Multiple regression in behavioral research. BY: Holt, Rinehart, and Winston.
• Kline, Rex B. (1998). Principles and practice of structural equation modeling. NY: Guilford Press. A very readable introduction to the subject, with good coverage of assumptions and SEM's relation to underlying regression, factor, and other techniques.
• Loehlin, J. (1991). Latent variable models: An introduction to factor, path and structural analysis. Hillsdale, NJ: Lawrence Erlbaum. Widely used textbook.
• Pedhazur, Elazer J. (1982). Multiple regression in behavioral research, 2nd edition. NY: Holt. Chapter 15 (pp. 577-635) covers path analysis. Widely used textbook.
• Schumacker, Randall E. and Richard G. Lomax (2004). A beginner's guide to structural equation modeling, Second edition. Mahwah, NJ: Lawrence Erlbaum Associates. Chapters 7 discusses path models in the context of SEM packages.
• Wright, S. (1921). Correlation and causation. Journal of Agricultural Research 20: 557-585.
• Wright, S. (1934). The method of path coefficients. Annals of Mathematical Statistics, Vol. 5: 161-215. The seminal article.