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I am building a linear regression model for 5000 gene expressions, and each gene has its own independent variables. So in total, there are 5000 linear models and for each model, I have 360 samples and I used 5 fold cross-validation for those samples. so using 288 samples to model the linear regression and use 72 samples to test it.

After that, I calculated the variance of the dependent variable(each gene) and error MSE for each model, and calculate the correlation between them, which yields really high positive value(0.99). And this means that the higher error residual's MSE, the higher is the dependent variable's variance. Does this make any sense? What is the proper relationship between error MSE and dependent variable's variance?

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I'll go for an intuitive explanation. If I'm understanding, this makes sense. Think of your independent variables as one variable, and imagine two different scatter plots of your 1D independent variable on the X-axis and dependent variable on the Y. Holding constant the scatter in the X-direction, the scatter plot with more variation in the Y-direction (taller) will likely have larger error for a linear fit than the scatter plot with less variation in the Y-direction (shorter).

Someone may jump in with a more rigorous argument.

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