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Hi SPMers, Happy New Year! I was hoping someone might be able to advise on the following question concerning the specification of a design matrix for an fMRI paradigm (balloon analogue risk task, BART) that includes non-linear parametric modulators. My design matrix contains (amongst other regressors) a response onset vector (pumping in experimental balloons) with a parametric modulator (balloon size), as well as a control response onset vector (pumping on control balloons), also with a parametric modulator (balloon size). I am interested in linear as well as quadratic trends to see if balloon size gets tracked linearly or potentially (and perhaps more realistically given the trial dynamics on this task) following a U-shaped trajectory. To examine this question, I have specified a 2nd order polynomial expansion for the parametric modulator (balloon size). My question concerns the quadratic (second order) function: If I wanted to examine whether the increase in the parametric modulator (size) is tracked linearly, I would set my contrast to be 1 for the column containing the 1st order parametric modulator for the experimental balloons and -1 for the 1st order parametric modulator for control balloons, and 0 on every other column. If I now wanted to know whether the tracking can be thought of as non-linear, is it sufficient to set the regressor containing the 2nd order parametric modulator for experimental balloons to 1 and the regressor containing the 2nd order parametric modulator for control balloons to -1, and 0 for everything else? As far as I am aware, the 2nd order expansion is orthogonalised w.r.t. the onset vector and the 1st order vector, hence I anticipated the contrast as described to yield neural areas in which the association between activation and the parametric modulator (balloon size) follows a U-shaped trajectory? Also, quadratic functions can result in a parabola that opens upwards (U-shape, vertex at minimum) or downwards (inverted U-shape, vertex at maximum), depending on the sign of the constant. How exactly does SPM handle this distinction? In other words, when I specify a 2nd order expansion, what would be the resulting curve? Does the constant get estimated somehow, and if so, how can I then check its value to interpret my results? Any advice on these related issues would be massively appreciated. Many thanks in advance! Best wishes, Loreen