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Is this simple linear regression a good fit? Are there any transformations that would improve it?

The data is discrete interval count vs discrete interval count (the count of steps walked per time)

  • Variables: time spend walking ~ steps walked
  • Dependent: time spend walking
  • Explanatory: steps walked

without log transform enter image description here

with log transform on explanatory

enter image description here

with log transform on dependent enter image description here

with log transform on dependent and explanatory

enter image description here

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    $\begingroup$ What are your data? What are your variables? How many explanatory variables are there? What are you hoping to accomplish with this model? Etc. $\endgroup$ Sep 14 at 0:55
  • $\begingroup$ QQ plot shows normality assumption may be violated (a bit heavy in the tails), and Residual vs Fitted plot gives some evidence that linearity assumption may be violated. A log transformation on the explanatory may improve your linearity, and a log transformation on the dependent may improve normality. My advice is to start with the linearity as that may solve the normality. $\endgroup$
    – jros
    Sep 14 at 15:00
  • $\begingroup$ @jros added those log transforms. It appears linearity was fixed by a log transform on both. Log transform on both does seem to fatten the tails, I am not sure of the implications of this. $\endgroup$ Sep 15 at 6:27
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    $\begingroup$ You've missed out the most basic graph of all: a scatter plot of time versus steps with fitted line superimposed In this example, I'd be concerned with whether a fitted function should respect the origin 0 steps, 0 time. $\endgroup$
    – Nick Cox
    Sep 15 at 9:10
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    $\begingroup$ This should be about the substance as well as the statistics. On the face of it time and steps should be proportional as a good first approximation. But people get tired etc. and so I would expect some curvature. And there is nothing here about whether your data are heterogeneous because of mixing different people and/or different conditions. $\endgroup$
    – Nick Cox
    Sep 15 at 9:34

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