Since three points are not sufficient to perform a quadratic regression, are there other methods for fitting a quadratic model which is capable of testing statistical significance?
If you have several groups/individuals (as you speak of repeated measures), see in mixed model approaches, so you fit an average quadratic polynomial for the whole population and variances on each coefficients of this polynomial to account for inter-individual variabilities.
But with only 3 points/individual, I'm not sure how far the results can be interpretated...
Any regression requires at least one more point than the number of parameters to be tested statistically. As the quadratic regression has three parameters, it requires at least four points. Any approach to a statistical solution, either by resampling or other, cannot be recommended. At best, line fitting techniques can be used to make the relationship more pleasing to the eye but even this is somewhat artificial.
I don't see why you can't use a quadratic term in your mixed model. Your time-effcect at say baseline, after 6 weeks and 18 weeks can be modelled as y=i+b1Time+b2Time-squared+e. You could be overfitting but that is a general issue: Babyak MA. (2004) What You See May Not Be What You Get: A Brief, Nontechnical Introduction to Overfitting in Regression-Type Models. Psychosomatic Medicine 66: 411-421.
Dear Emmanuel,
I have 15 individuals measured in three test conditions. I used repeated measure for this three conditions.
you mean that I should used mixed model repeated measures to be able to fit a quadratic model?
best,
"I have 15 individuals measured in three test conditions. I used repeated measure for this three conditions."
This is different than what you explained above. Can you be more explicit about the experimental design. Are the 15 individuals subsamples or repeated measurements on a single experimental unit per condition or else?
Thank you all for your replies.
As Gaetan kindly requested, I clarify the experimental design.
15 participants ran at three different speeds (low to high) on a treadmill. I am trying to investigate the effect of speed on their stability (using LyE as a measure of stability). if the effect of speed is significant I want to know the trend of this effect (i.e. linear or quadratic).
This is clearer now. You don't have 3 points but 3 levels and many data points. This is a basic mixed model in which you'll have to define which observations belong to the same participant.
Dear Gaetan
could this be done with a one-way repeated measure too?since I have only within-subject factors (three speeds), then whiy a mixed model should be used?
I suggested mixed model because I prefer the way these models deal with random factors, in this case the participants. A one-way repeated measures ANOVA will do fine if you are not used to mixed models. By the way, the 3 levels give you 2 df for the term "speed", which permits a decomposition in 2 term, the linear and the quadratic level that you are looking for. You just have to include the term speed (as a continuous variable) and its quadratic value (speed*speed) in your model. Good luck!
Dear Sina,
I suspect what you had in mind in a bit of shorthand often used in psychology. Since an ANOVA on your data would have 2df, often people find it convenient to consider the "linear" effect (1st vs 3rd data point) and the "Quadratic" effect (Does pt 2 fall on the line drawn between the 1st and 3rd). This has little to do with actually establishing that the effects in question are specifically linear or quadratic apart from the fact that one would expect such trends to load onto the two degrees of freedom.
In SPSS you would ask for polynomial contrasts inside the repeated measures ANOVA dialogue. The default situation is the one which corresponds to your question.
Thank you Gaetan, Bruce and all who participated in this discussion.
You can try fitting both linear and quadratic models, one of which may represent the trend. in such sampling for regression lack of fit is the most important test of model. besides, if your speed intervals are not same, it gets a little complex. I tried the program in SAS for such data.
By the way, if the lack of fit was significant you should test the subject*speed interactions. if the interaction was showed significant too, then you need more than 3 levels of speed, otherwise you have a relatively reliable model.
I wish it could help
Good luck
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