Basically in a research project I am looking at the linear regression between my independent variable: Government Stringency Index, and dependent real GDP growth.

One area I investigate assumes if real GDP growth is more precisely measured, switching my variables in linear regression; to x on y linear regression; would associate the error with my independent variable (which is presumably less accurate).

My question is: After doing this, what statistical tests could I use to determine which is a better fit for my model?

enter image description here

r,r^2 are of course the same. Visually the best fit line in x on y appears to be worse.

  • $\begingroup$ There is no need for a test for goodness of fit. If all the data are plotted, then it is clear the blue line is a better fit than the red line. I don't need a p value for that. $\endgroup$ Jan 18 at 17:11
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    $\begingroup$ Since the models are incompatible--each one assumes one variable is measured without error and the other is not--it doesn't seem to make sense to determine which may be a "better fit." You need to decide based on an understanding of the data. $\endgroup$
    – whuber
    Jan 18 at 18:09

For simple linear regression using Ordinary Least Squares, you are minimising the sum of squares of the vertical residuals, and the outlying point has much more influence on the red line than on the blue line. Flipping your chart round (below) so horizontal becomes vertical and vice versa makes this obvious, especially when you remember that regression lines pass through the mean of the data; the square of the vertical residual of the outlying point from the blue line would be huge in this second chart.

Since the outlying point is presumably Covid-19 related, it is fairly clear that in such extreme circumstances you do not in fact have a linear relationship between your two variables.

You might want to ask yourself whether (a) you want a linear model, (b) the outlying point is actually part of the relationship you are trying to model (removing it will affect both lines but will mean that your model will not extend to current circumstances) and (c) which variable is really the dependent variable (if you are trying to predict one from the other, that might give you a clue).

flipped chart


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