Modeling and Multivariate Methods - SAS

Chapter 17 Correlations and Multivariate Techniques 455
Computations and Statistical Details
where K is a constant equal to the 0.75 quantile of a chi-square distribution with the degrees of freedom
equal to the number of columns in the data table, and
Q = ( y i
– μ) T ( S 2 ) – 1
( y i
– μ)
where y i = the response for the i th observation, μ = the current estimate of the mean vector, S 2 = current
estimate of the covariance matrix, and T = the transpose matrix operation. The final step is a bias
reduction of the variance matrix.
The tradeoff of this method is that you can have higher variance estimates when the data do not have many
outliers, but can have a much more precise estimate of the variances when the data do have outliers.
Pearson Product-Moment Correlation
The Pearson product-moment correlation coefficient measures the strength of the linear relationship
between two variables. For response variables X and Y, it is denoted as r and computed as
r =
( x–
x) ( y – y)
---------------------------------------------------------
( x–
x) 2
( y–
y )2
If there is an exact linear relationship between two variables, the correlation is 1 or –1, depending on
whether the variables are positively or negatively related. If there is no linear relationship, the correlation
tends toward zero.
Nonparametric Measures of Association
For the Spearman, Kendall, or Hoeffding correlations, the data are first ranked. Computations are then
performed on the ranks of the data values. Average ranks are used in case of ties.
Spearman’s ρ (rho) Coefficients
Spearman’s ρ correlation coefficient is computed on the ranks of the data using the formula for the Pearson’s
correlation previously described.
Kendall’s τ b Coefficients
Kendall’s τ b coefficients are based on the number of concordant and discordant pairs. A pair of rows for two
variables is concordant if they agree in which variable is greater. Otherwise they are discordant, or tied.
The formula
sgn( x i
– x j
) sgn( y i
– y j
)
i<
j
τ b
= ---------------------------------------------------------------
( T 0
– T 1
)( T 0
– T 2
)