Path analysis from the

Path Analysis Program Page
is an extension of
the multiple correlation analysis described in the

Multiple Regression Explained Page
.

It is a method of describing complex sequential relationship between measurements. It was first used in
population genetics to describe the contributions from multiple influences
on attributes of a target population, such as the influences of the parents
genetic characteristics and the environment on some attribute of the child.
More recently the method has been found to be useful in sociological and
epidemiological studies.

Conceptually, the variables (measurements) used in the model are assumed to
be all the measurements that matters. These are assigned to different levels
in a sequence or cascade of influences, where the earlier levels affect the
subsequent ones, but the reverse does not happen. In this sequence, all
measurements in prior levels affect all subsequent levels, and the scale of
the influence is described by the **path coefficient**, and **partial correlation**
between measurements in the same level describes the size of the as yet
unexplained common preceding influences.

Mathematically, path analysis consists of a repeated sequence of multiple
correlation calculations from a correlation matrix, following the cascade of
influences. This is carried out one level at a time, using all the measurements
in preceding levels as independent variables, and the Standardised Partial
Regression Coefficients (Path Coefficients) represents the size of each
influence. This is followed by calculating Partial Correlation Coefficients
between all the variables in the same level, corrected for all preceding
variables as well. These Partial Correlation Coefficients represents common
influences that has not as yet been explained by the model.

| MGF | MGM | PGF | PGM | Dad | Mum | Child |

MGF | 1 | 0.5 | 0.2 | 0.2 | 0.3 | 0.6 | 0.4 |

MGM | 0.5 | 1 | 0.1 | 0.1 | 0.3 | 0.7 | 0.5 |

PGF | 0.2 | 0.1 | 1 | 0.4 | 0.5 | 0.1 | 0.5 |

PGM | 0.2 | 0.1 | 0.4 | 1 | 0.6 | 0.1 | 0.4 |

Dad | 0.3 | 0.3 | 0.5 | 0.6 | 1 | 0.3 | 0.7 |

Mum | 0.6 | 0.7 | 0.1 | 0.1 | 0.3 | 1 | 0.8 |

Child | 0.4 | 0.5 | 0.5 | 0.4 | 0.7 | 0.8 | 1 |

We will use the default example data from the

Path Analysis Program Page
to demonstrate the method. The data
was made up to demonstrate the procedure, and does not reflect any reality.

The input data in the correlation matrix text box is a correlation matrix of IQs measurements between members of the families,
as shown to the right.

We then divide the members of the family into 3 layers, along generations.

- The first layer consists of the grandparents, 1=Maternal Grandfathers(MGF), 2=Maternal Grandmothers(MGM), 3=Paternal
Grandfathers(PGF), and 4=Paternal Grandmothers(PGM)
- The second layer consists of the parents, 5=Dads and 6=Mums
- The third layer is the dependent variable, 7=Children.

The variables are separated by spaces or tabs, and each layer is in a separate
line, and these are entered into the path variables text box. The following results are then produced.

Layer 1 : Partial correlation coefficient
PCor 1.MGF - 2.MGM **0.50**
PCor 1.MGF - 3.PGF 0.12
PCor 1.MGF - 4.PGM 0.12
PCor 2.MGM - 3.PGF 0.00
PCor 2.MGM - 4.PGM 0.00
PCor 3.PGF - 4.PGM **0.38** |

It can be seen that, at the grand parent level, there are strong (>0.2) correlations between husbands and wives
from the same family, but virtually none between the families.

The next step is to examine the influences of the grandparents on the parents,
and the correlation between Dads and Mums after correcting for the influence of
the grandparents.

layer 2 : Path Coefficients
Path: 1.MGF - 5.Dad 0.05
Path: 2.MGM - 5.Dad 0.20
Path: 3.PGF - 5.Dad **0.29**
Path: 4.PGM - 5.Dad **0.45**
Path: 1.MGF - 6.Mum **0.34**
Path: 2.MGM - 6.Mum **0.53**
Path: 3.PGF - 6.Mum -0.02
Path: 4.PGM - 6.Mum -0.05
Patial Correlation coefficients
PCor: 5.Dad - 6.Mum 0.1179 |

From this we can see that the IQs of mums and Dads are influenced by their
respective parents and, after correcting for those influence, there was no
correlation between the IQs of mums and Dads.

Finally, the influence of everyone in the family on the child (corrected for all inter-correlations)
is shown.

layer 3 : Path Coefficients
Path: 1.MGF - 7.Child -0.24
Path: 2.MGM - 7.Child -0.17
Path: 3.PGF - 7.Child 0.26
Path: 4.PGM - 7.Child 0.03
Path: 5.Dad - 7.Child **0.40**
Path: 6.Mum - 7.Child **0.91** |

The analysis provided by the program ends here. However, the results cannot
be only presented as a list of coefficients, and they are too cumbersome to
understand. The results are therefore presented as a path diagram, which has
to be constructed from these coefficients.

The convention is for the variables of different layers to line up vertically, and
the layers display horizontally from left to right. All variables in the
same layers must be connected to each other using curve lines, and labelled with
the values of the Partial Correlation Coefficient. All variables are also
connected to those in subsequent layers using straight lines, and labelled with
the values of the path coefficients. An example (from a different study) may be as follows.

However, if the number of variables involved in the path analysis exceeds 4-5,
connecting each variable with every other one becomes unmanageable,
and the path diagram becomes cluttered and confused. One way to deal with it is
to show only major paths, using statistical significance or an arbitrarily determined
size to decide. If only those paths with coefficient values exceeding 0.25, then the path diagram from this
example will looks as follows.

Pedhazur E. Multiple Regression in Behavioral Research.
Explanation and Prediction.( 3rd. ed. 1997) Harcourt Brace College Publishers,
Fort Worth, USA. ISBN 0-03-072831-2 . Chapter 18 Structural Equation Models with
Observed Variables: Path Analysis. p765-840