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I'm building regression models on four the different but related data set and at the end, I want to test the significance of models. Since my models are built in a different data set, it's not comparable. But there are some hierarchy in my dataset.

1) The output in my main dataset is the sum of the outputs of data set A, B and C

2) The value of feature is the same in all data sets, but all features are NOT present in all data set and some data sets only having a subset of the features of the main data set.

3) The union of the features of the dataset A, B, C are the features of the main data set.

So, I want to build a regression model on these data sets seprately and compare the performance and significance of the models togethers.Becuase the dataset are not the exactly the same for all four models, I can not use standard statstics to test the sigificance of models like AIC, CP-statstics,..... Is there anyway to compare the performance of the model ?

Here is my dataset(I just included first two data points) for more clarification:

enter image description here

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I'm not sure how to interpret the diagram, but for asymmetric/messy problems the bootstrap is often your friend. Suppose that you want to get a confidence interval on the difference between $R^2$ from two different models on two different or overlapping datasets, and that the number of independent experimental units for the two is $n_{1}, n_{2}$ with there being $n_{u}$ unique experimental units from the union of the two samples. You could sample with replacement from the $n_{u}$ units 1000 times, each time recreating dataset 1 on the basis of which and how often the $n_{1}$ units were selected, and likewise for dataset 2. For each resample estimate two sets of model parameters and two $R^2$ and their difference. Get a bootstrap confidence interval for the difference using the 1000 estimated differences.

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  • $\begingroup$ I edited my post. the point is that, I want to compare the models from each datasets together, and because there the datasets are not the exactly the same, I can not use standard test statistics to compare them. I'm looking way to use the advantage of relationships between my dataset to make them comparabel. $\endgroup$
    – user51661
    Jul 14 '14 at 15:32
  • $\begingroup$ My suggestion answered that question. $\endgroup$ Jul 14 '14 at 16:56
  • $\begingroup$ Thanks Frank, but the problem is that, we can not compare the R^2 of the models together, because we learned them on different data set. but if we had just only one data set, then comparing R^2 of different model would make sense. $\endgroup$
    – user51661
    Jul 14 '14 at 21:26
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    $\begingroup$ First you need to settle on a measure of model performance (e.g., predictive discrimination) that is not sample size dependent (this rules out AIC,for example). Two possibilities are $R^2$ (or generalized $R^2$) or rank correlation measures ($c$-index or generalized ROC area; Spearmans $\rho$). Then you can use the procedure I outlined. You have to be sure to repeat all feature selection and other $Y$-data-driven steps for each bootstrap resample. $\endgroup$ Jul 15 '14 at 12:05

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