Path AnalysisMethod of Path AnalysisConventions of Path Analysis IConventions of Path Analysis IIConventions of Path Analysis IIIConventions of Path Analysis IVTracing Rules of Path AnalysisExampleExercisesCovariance between A and BExpectationsMZ Twins Reared ApartMZ Twins Reared TogetherDZ Twins Reared TogetherDZ Twins Reared ApartTwins and ParentsTwins & Parents IITwins & Parents IIIPath AnalysisHGEN619 class 2005Method of Path Analysisallows us to represent linear models for the relationship between variables in diagrammatic form, e.g. a genetic model; a factor model; a regression modelmakes it easy to derive expectations for the variances and covariances of variables in terms of the parameters of the proposed linear modelpermits easy translation into matrix formulation as used by programs such as MxConventions of Path Analysis ISquares or rectangles denote observed variables.Circles or ellipses denote latent (unmeasured) variables.Upper-case letters are used to denote variables.Lower-case letters (or numeric values) are used to denote covariances or path coefficients.Conventions of Path Analysis IISingle-headed arrows or paths (–>) are used to represent causal relationships between variables under a particular model - where the variable at the tail is hypothesized to have a direct influence on the variable at the head.Double-headed arrows (<–>) are used to represent a covariance between two variables, which may arise through common causes not represented in the model. They may also be used to represent the variance of a variable.Conventions of Path Analysis IIIDouble-headed arrows may not be used for any variable which has one or more single-headed arrows pointing to it - these variables are called endogenous variables. Other variables are exogenous variables.Single-headed arrows may be drawn from exogenous to endogenous variables or from endogenous variables to other endogenous variables.Conventions of Path Analysis IVOmission of a two-headed arrow between two exogenous variables implies the assumption that the covariance of those variables is zero (e.g., no genotype-environment correlation).Omission of a direct path from an exogenous (or endogenous) variable to an endogenous variable implies that there is no direct causal effect of the former on the latter variable.Tracing Rules of Path AnalysisTrace backwards, change direction at a double-headed arrow, then trace forwards. This implies that we can never trace through double-headed arrows in the same chain.The expected covariance between two variables, or the expected variance of a variable, is computed by multiplying together all the coefficients in a chain, and then summing over all possible chains.qBonAFDlk11Em1CpExamplepExercisesCov AB = Cov BC = Cov AC = Cov AE =Var A = Var B = Var C = Var ECovariance between A and BqBpnAFDlk11Em1CpqBpnAFDlk11Em1CpCov AB = kl + mqn + mplqBpnAFDlk11Em1CpExpectationsCov AB = kl + mqn + mplCov BC = npCov AC = mqpCov AE = m + pkVar A = k2 + m2 + 2 kpmVar B = l2 + n2Var C = p2Var E = 1MZ Twins Reared ApartT2AEae11Cc11T1EAea11Cc1MZ Twins Reared TogetherT2AEae11Cc111T1EAea11Cc1DZ Twins Reared TogetherT2AEae11Cc1.51T1EAea11Cc1DZ Twins Reared ApartT2AEae11Cc1.5T1EAea11Cc1Twins and ParentsMAEae11Cc1FaEAea11Cc1T2AEae.51Cc1T1EAea1.5Cc1.5.5.5.5Twins & Parents IIMAEae11Cc11FaEAea11Cc1T2AEae.51Cc1T1EAea1.5Cc1.5 .5.5.5Twins & Parents IIIMAEae11Cc1FaEAea11Cc1T2AEae.51cT1EAea1.5Ccr.5.5.5.5p
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