An iterative method for the computation of the correlation matrix implied by a recursive path model
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An iterative method for the computation of the correlation matrix implied by a recursive path model M’barek Iaousse1 · Zouhair El Hadri2 · Amal Hmimou1 · Yousfi El Kettani1
© Springer Nature B.V. 2020
Abstract In Path Analysis, especially in social sciences studies, many researchers usually assume that errors in the model are uncorrelated with all exogenous variables as well as with each other. These assumptions, in most cases, are not valid in reality and were introduced to facilitate the model estimation. This article establishes a new algorithm for the computation of the correlation matrix implied by a recursive path model that overcomes these drawbacks. We compare our algorithm to two other methods used in the literature. The comparison was made mathematically through an illustrated example and numerically with a simulation study. The findings show that, unlike the classical methods, the proposed method gives more accurate results. Keywords Path analysis · Finite iterative method · Implied correlation matrix · Recursive model · Correlated errors
1 Introduction Path Analysis is a statistical technique that measures the magnitude of a hypothesized causal relationships between a set of observed variables (Wright 1921, 1923, 1934, 1960; Elhadri and Hanafi 2015). It is a widely used method in many areas such as biology (Shipley 2016; Pugesek 2003; Griesemer 1991), ecology (Scheiner and Gurevich 2001; Shipley 1997), and sociology (Blalock 1961; Duncan 1966).
* M’barek Iaousse [email protected]; [email protected]
Zouhair El Hadri [email protected]; [email protected]
Amal Hmimou [email protected] Yousfi El Kettani [email protected] 1
Department of mathematics, Ibn Tofail University, Kenitra, Morocco
2
Faculty of Sciences, Center of Mathematical Researches and Applications, Mohammed V University, Rabat, Morocco
13
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M. Iaousse et al. γ 12 ξ1 ρ12
η1
γ 11 ξ2
β31 β21
γ 22
η3
δ3
ζ3
β32
ρ32
η2
γ 23 ξ3
Fig. 1 Example of a path diagram with three exogenous variables and three endogenous variables
A Path Analysis model for a set of (standardized) random variables {𝜉1 , … , 𝜉q , 𝜂1 , … , 𝜂p } is defined as the combination of two parts: (a) A set of equations of the form:
𝜂i =
p ∑
𝛽ij 𝜂j +
j=1
q ∑
𝛾ik 𝜉k + 𝛿i 𝜁i
k=1
(1)
(b) A path diagram. A path diagram is Directed Graph (example: Fig. 1). Each vertex corresponds to a variable (including error terms). It illustrates the sense of a hypothesized cause between each pair of variables (vertices) by a one-headed arrow from the cause to the effect. The non-existence of an arrow between two variables means that there is no direct causation between them. A two-headed arrow between two variables means that the relationship between them is not analyzed by the model (unanalyzed association). By setting:
𝜼t = (𝜂1 , … , 𝜂p ), 𝝃 t = (𝜉1 , … , 𝜉p ), 𝜻 t = (𝜁1 , … , 𝜁p ) and
B = (𝛽ij ) 1 ⩽ i ⩽ p , 𝜞 = (𝛾ik ) 1 ⩽ i ⩽ p , 𝜟 = diag(𝛿i )1⩽i⩽p 1⩽j⩽p
1⩽k⩽q
The set of the above equations can be written in the followin
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