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This is likely a basic problem but as a non-statistician I don't know how to formulate it or where to start with the analysis.

I have (paired) observations for individuals in two conditions. Each observation is 10 data points. The independent variable $x_i$ ranges from 0 to 10 for each individual and the dependent variable $y_i \in \mathbb{R_+}$. So for each individual I have 20 data points, 10 in each condition. I expect that $y_i$ varies as a quadratic function in $x_i$ for each individual, however: (a) the responses are noisy, and (b) I expect the coefficients of the quadratic to be different for each individual.

I would like to:

  • Independently fit a quadratic model to each individual in each condition and analyze the goodness of fit ($R^2$) per individual. I would like to report some kind of summary statistic as evidence that a quadratic model is appropriate for our setting; however, I don't know what to report. The average $R^2$ value doesn't seem right. How can one report this type of statistic when I expect the true coefficients to differ between individuals (so fitting a single, aggregate model is not a good option)?
  • Compare the model fits between conditions to understand if there is a significant change per individual between conditions. For this point, my thought was to fit a quadratic to the difference in observations between the conditions and then perform a significance test (Wald test?) to see if the coefficients are greater than 0. However, I'm not sure if I need to take into account that some individuals may fit the quadratic model better than others. What is an appropriate way to test this difference?

I would appreciate any suggestions on how to address these two questions.

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Perhaps a mixed linear model on subject ID with random intercept, linear and quadratic effects for $x_i$ would best fit your data. If one does not wish to measure the variation through variance parameters, but wishes to encode the variation in the mean function, you can create a factor variable for subject ID, call it $z_i$, and create interaction variables with $x_i$ that will force each subject to have a different set of quadratic coefficients. This full model will give you 1 measure of $R^2$ which you can interpret normally. You should also make sure that the individual quadratic models have significant coefficients.

For your second question, you could reformulate it as: Is there a statistically significant difference in the mean value of the response variable when the condition $x_i = c$ between subject $i$ and subject $j$? In this case one makes assumptions about independence between subjects and homogeneity of variances, computes a point estimate of the mean value for the the two subjects at condition $c$, and computes a $t$ statistic to test that the difference is significantly different than 0. You may extend this by constructing a confidence band through Scheffe's method and have this interpretation at all conditions.

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  • $\begingroup$ Thanks, this helps a lot! Regarding the first paragraph "You should also make sure that the individual quadratic models have significant coefficients." what are you suggesting is the hypothesis being tested here for significance? I thought the hypothesis test only comes into play in the second part but maybe I'm wrong. $\endgroup$
    – user219923
    Nov 20, 2021 at 23:54

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