How does Prism compute the normalized covariance matrix? How does it relate to the actual covariance matrix?
How to ask Prism to report the normalized covariance matrix
The normalized covariance is reported for each pair of parameters, and quantifies the degree to which those two parameters are intertwined. Check an option on the diagnostics tab of the nonlinear regression dialog to see these values.
Interpreting the normalized covariance matrix
By definition, the normalized covariance matrix takes on unitless values between -1.0 and 1.0. This makes them easy to compare and interpret (like a correlation coefficient).
Each value in the normalized covariance matrix ranges from -1.0 to 1.0. A value equal to -1.0 or 1.0 means the two parameters are redundant. A value of 0.0 means the parameters are completely independent or orthogonal -- if you change the value of one parameter you will make the fit worse and changing the value of the other parameter can't make it better. You can interpret a normalized covariance much as you interpret a correlation coefficient. Note the difference between covariance and dependency. Each value in the covariance matrix tells you how much two specified parameters are intertwined. In contrast, each dependency value tells you how much a specified parameter is intertwined with all other parameters.
Why does Prism report the normalized covariance matrix rather than the covariance matrix itself?
The normalized covariance matrix is easy to interpret. In contrast, each value in the actual covariance between two parameters is expressed in strange units that are hard to interpret: the units of one of the parameters times the units of the other parameter.
How to convert to the nonnormalized matrix
Some other programs report the actual (not normalized) variance-covariance matrix. Compute the actual covariance -- Cov(i,j) -- of any two parameters from the normalized matrix Prism reports -- NormCov(i,j) -- by multiplying by the standard error of the two parameters:
Cov(i, j) = NormCov(i, j) * SE(i) * SE(j)
Prism does not report the normalized covariance matrix for a parameter with itself, because the normalized covariance of any parameter with itself equals, by definition, 1.0. The "covariance" of a parameter with itself is better called the variance of that parameter. You can calculate the variance of any parameter (a diagonal value in the variance-covariance matrix) as the square of its standard error:
Cov(i, i) = SE(i)^2
Note that this equation matches the previous one, when you set j equal to i, and set NormCov (i,i) equal to 1.0, since by definition, the normalized covariance of a parameter with itself is 1.0.
For a mathematical explanation of nonlinear regression, including the covariance matrix, see this page.
This faq was rewritten in April 2013. The prior version gave invalid equations for computing the nonnormalized covariance matrix. Prism has always computed the normalized covariance matrix correctly.