From the RF Toolbox documentation:
The delay factor is a scaling factor that controls the amount of delay to fit to the data, specified as the comma-separated pair consisting of 'DelayFactor' and a scalar between 0 and 1 inclusive. The Delay parameter, τ, of the rational function object is equal to the specified value of 'DelayFactor' times an estimate of the group delay of the data. If the original data has delay, increasing this value might allow rationalfit to fit the data with a lower-order object.
In simple words - in the attempt to give an intuitive explanation - with this parameter you can "extract" a constant delay from the data (equal to a percentage of the average group delay), so that the rational fitting function fits only the residual of the frequency dependent group delay. This allows reducing the overall number of poles required for the fitting (the delay needs to be separately taken into account when you simulate the rational function). This is particularly relevant in data that is affected by relatively large delays, such as backplanes, cables, transmission lines.
You can estimate the group delay of the data using this simple MATLAB expression: groupdelay = squeeze(-diff(unwrap(angle(S_Parameters(2,1,:))))) ./ diff(Freq*2*pi);
Where Freq represents the vector of the frequency values over which the S_Parameters matrix is measured. Alternatively, you can build an rfckt object and use the analyze method on it.
To play with your S-parameters data and this parameter, you can use the Rational Fitting GUI available on MATLAB Central File Exchange - it will help in assessing if the quality of the fitting improves:
With best regards, Giorgia