9.1 Fitting the cusp catastrophe in SPSS

In the file Cusp Attitude.sav, you can find data from a (simulated) experiment. Assume the experiment tried to measure the effects of explicit predisposition and affective conditioning on attitudes towards Complexity Science measured in a sample of psychology students using a specially designed Implicit Attitude Test (IAT).

  1. Look at a Histogram of the difference score (post-pre) \(dZY\). This should look quite normal (pun intended).
  • Perform a regular linear regression analysis predicting $dZY$ (Change in Attitude) from $\alpha$ (Predisposition). 
  • Are you happy with the \(R^2\)?
  1. Look for Catastrophe Flags: Bimodality. Examine what the data look like if we split them across the conditions.
  • Use \(\beta\) (Conditioning) as a Split File Variable (Data >> Split File). And again, make a histogram of \(dZY\).
  • Try to describe what you see in terms of the experiment.
  • Turn Split File off. Make a Scatterplot of \(dZY\) (x-axis) and \(\alpha\) (y-axis). Here you see the bimodality flag in a more dramatic fashion.
  • Can you see another flag?
  1. Perhaps we should look at a cusp Catastrophe model:
  • Go to Analyse >> Regression >> Nonlinear (also see Basic Time Series Analysis). First we need to tell SPSS which parameters we want to use, press Parameters.
    • Now you can fill in the following:
      • Intercept (Starting value \(0.01\))
      • B1 through B4 (Starting value \(0.01\))
      • Press Continue and use \(dZY\) as the dependent.
    • Now we build the model in Model Expression, it should say this:

      Intercept + B1 * Beta * ZY1 + B2 * Alpha + B3 * ZY1 ** 2 + B4 * ZY1 ** 3
    • Run! And have a look at \(R^2\).
  1. The model can also be fitted with linear regression in SPSS, but we need to make some extra (nonlinear) variables using COMPUTE:
BetaZY1 = Beta*ZY1  *(Bifurcation, splitting parameter).
ZY1_2 = ZY1 ** 2    *(ZY1 Squared).
ZY1_3 = ZY1 ** 3    *(ZY1 Cubed).
  • Create a linear regression model with \(dZY\) as dependent and \(Alpha\), \(BetaZY1\) and \(ZY1_2\) en \(ZY1_3\) as predictors.
  • Run! The parameter estimates should be the same.
  1. Finally try to can make a 3D-scatterplot with a smoother surface to have look at the response surface.
  • HINT: this is a lot easier in R or Matlab perhaps you can export your SPSS solution.
  1. How to evaluate a fit?
  • Check the last slides of lecture 8 in which the technique is summarised.
  • The cusp has to outperform the pre-post model.