# adjacency.polyReg: Adjacency matrix based on polynomial regression In WGCNA: Weighted Correlation Network Analysis

## Adjacency matrix based on polynomial regression

### Description

adjacency.polyReg calculates a network adjacency matrix by fitting polynomial regression models to pairs of variables (i.e. pairs of columns from datExpr). Each polynomial fit results in a model fitting index R.squared. Thus, the n columns of datExpr result in an n x n dimensional matrix whose entries contain R.squared measures. This matrix is typically non-symmetric. To arrive at a (symmetric) adjacency matrix, one can specify different symmetrization methods with symmetrizationMethod.

### Arguments

 datExpr data frame containing numeric variables. Example: Columns may correspond to genes and rows to observations (samples). degree the degree of the polynomial. Must be less than the number of unique points. symmetrizationMethod character string (eg "none", "min","max","mean") that specifies the method used to symmetrize the pairwise model fitting index matrix (see details).

### Details

A network adjacency matrix is a symmetric matrix whose entries lie between 0 and 1. It is a special case of a similarity matrix. Each variable (column of datExpr) is regressed on every other variable, with each model fitting index recorded in a square matrix. Note that the model fitting index of regressing variable x and variable y is usually different from that of regressing y on x. From the polynomial regression model glm(y ~ poly(x,degree)) one can calculate the model fitting index R.squared(y,x). R.squared(y,x) is a number between 0 and 1. The closer it is to 1, the better the polynomial describes the relationship between x and y and the more significant is the pairwise relationship between the 2 variables. One can also reverse the roles of x and y to arrive at a model fitting index R.squared(x,y). If degree>1 then R.squared(x,y) is typically different from R.squared(y,x). Assume a set of n variables x1,...,xn (corresponding to the columns of datExpr then one can define R.squared(xi,xj). The model fitting indices for the elements of an n x n dimensional matrix (R.squared(ij)). symmetrizationMethod implements the following symmetrization methods: A.min(ij)=min(R.squared(ij),R.squared(ji)), A.ave(ij)=(R.squared(ij)+R.squared(ji))/2, A.max(ij)=max(R.squared(ij),R.squared(ji)).

### Value

An adjacency matrix of dimensions ncol(datExpr) times ncol(datExpr).

### Author(s)

Lin Song, Steve Horvath

### References

Song L, Langfelder P, Horvath S Avoiding mutual information based co-expression measures (to appear).

Horvath S (2011) Weighted Network Analysis. Applications in Genomics and Systems Biology. Springer Book. ISBN: 978-1-4419-8818-8

### Examples

#Simulate a data frame datE which contains 5 columns and 50 observations
m=50
x1=rnorm(m)
r=.5; x2=r*x1+sqrt(1-r^2)*rnorm(m)
r=.3; x3=r*(x1-.5)^2+sqrt(1-r^2)*rnorm(m)
x4=rnorm(m)
r=.3; x5=r*x4+sqrt(1-r^2)*rnorm(m)
datE=data.frame(x1,x2,x3,x4,x5)
#calculate adjacency by symmetrizing using max