## 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.