Dependency and covariance matrix

When your model has two or more parameters, as is almost always the case, the parameters can be intertwined.

What does it mean for parameters to be intertwined? After fitting a model, change the value of one parameter but leave the others alone. The curve moves away from the points. Now, try to bring the curve back so it is close to the points by changing the other parameter(s). If you can bring the curve closer to the points, the parameters are intertwined. If you can bring the curve back to its original position, then the parameters are redundant.

Prism can quantify the relationships between parameters in two ways. If you are in doubt, we suggest that you focus on the dependency values and not bother with the covariance matrix.

Dependency is reported for each parameter, and quantifies the degree to which that parameter is intertwined with others. Its value ranges from 0.0 to 1.0. A high value means that the value of a parameter is intertwined with the values of other parameters. Changing the value of the parameter will make the curve fit the data much worse. But changing other parameters can bring the curve back.

A dependency of 0.0 is an ideal case when the parameters are entirely independent (mathematicians would say orthogonal). In this case the increase in sum-of-squares caused by changing the value of one parameter cannot be reduced at all by also changing the values of other parameters. This is a very rare case.

A dependency of 1.0 means the parameters are redundant. After changing the value of one parameter, you can change the values of other parameters to reconstruct exactly the same curve. If any dependency is greater than 0.9999, GraphPad calls the fit 'ambiguous'.

With experimental data, of course, the value lies between 0.0 and 1.0. How high is too high? Obviously, any rule-of-thumb is arbitrary. But dependency values up to 0.90 and even 0.95 are not uncommon, and are not really a sign that anything is wrong.

A dependency greater than 0.99 is really high, and suggests that something is wrong. This means that you can create essentially the same curve, over the range of X values for which you collected data, with multiple sets of parameter values. Your data simply do not define all the parameters in your model. If your dependency is really high, ask yourself these questions:

•	Would it help to collect data over a wider range of X, or at closer spaced X values? It depends on the meaning of the parameters.

•	Can you collect data from two kinds of experiments, and fit the two data sets together using global fitting?

•	Can you constrain one of the parameters to have a constant value based on results from another experiment?

If the dependency is high, and you are not sure why, look at the covariance matrix (see below). While the dependency is a single value for each parameter, the covariance matrix reports the normalized covariance for each pair of parameters. If the dependency is high, then the covariance with at least one other parameter will also be high. Figuring out which parameter that is may help you figure out where to collect more data, or how to set a constraint.

If you want to see the covariance matrix, check an option on the Diagnostics tab. Prism will then report the normalized covariance between every pair of parameters. The value 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.

Each value in the covariance matrix tell you how much two parameters are intertwined. In contrast, each dependency value tells you how much that parameter is intertwined with all other parameters.

Tip. Rather than interpreting the covariance matrix, we suggest you focus on the dependency instead.

SPAN & K	0.5390
SPAN & PLATEAU	0.1048
K & PLATEAU	0.8109

With only three parameters, there are three values for dependency (one per parameter) and three values in the covariance matrix (one for each pair of parameters). With more parameters, the covariance matrix grows faster than the list of dependencies, which is why we suggest that the dependency is usually more useful.