I was asked by Topher Bill at Mary Washington College if I had an example of a dataset where a trend analysis would show a cubic relationship. A cubic relationship is one in which a line relating the means to the level of the independent variable has two inflection points--in other words it changes direction twice. Obviously we can only ask such a question if there is a ordered relationship on the independent variable.
I didn't have such a dataset, but I did have an example based on a study by Drake Bradley at Bates that had a quadratic relationship. A lab based on this study can be seen at Multcomp.html I took that data set, where the independent variable referred to the speed of rotation of a stimulus, and moved one of the groups to the end so that the line came back down again. (Group 5 used to fall between Groups 1 and 2.)
The scatterplot of the points follows. (I know that we don't usually draw scatterplots of group data, but it works here.) Note that there are two inflection points, which is characteristic of a cubic trend.
The SPSS analysis of variance, requesting cubic trend, follows.
Note that there is a significant cubic trend, but not a significant linear or quadratic trend.
The dependent variable in this case is the perceived diameter of the circle inscribed by the rotating object. In Bradley's actual study, as the speed of rotation increased, the circle first appeared to decrease in diameter. With a further increase in speed the diameter appeared to increase. For the faked data that I have here, the appropriate conclusion would be that at the highest speed the circle perceptually decreased again. I don't believe that you could set up an actual study like this.
Although there are situations in psychology where data do take on a cubic trend, they are rare. It is not hard to imagine a process that would reverse itself at some point, becoming quadratic. (As an aging runner, I can well attest to reversal effects.) But it is difficult to imagine that reversal itself reversing at even higher levels of the independent variable, although Justin Joffe still has the naive notion that he is going to make a comeback after years of decline.
Last revised: 05/22/00