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I am working on an analysis of a simple linear regression and I don't know what to do. This is my graph: enter image description here

the p-value is <0.0001 but the data is clearly not linear and $R^2$ value is really small. What do you do in this situation? Transform it? Leave it as it is?

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    $\begingroup$ What does the plot show? What is on the $y$-axis? What exactly is your model? $\endgroup$
    – Tim
    Commented Jun 20, 2018 at 21:15
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    $\begingroup$ Nothing to transform. Use splines or local polynomial or fractional polynomial to match the turning point. This looks like two effects, but they are not symmetric: temperature rising means more work in air conditioning; temperature falling means more work in heating. Good scientific manners to state units of measurement (temperature in Fahrenheit; no idea about vertical axis). $\endgroup$
    – Nick Cox
    Commented Jun 20, 2018 at 21:15
  • $\begingroup$ You should use piecewise linear function to do this. Piecewise linear regression in numpy $\endgroup$
    – user1876887
    Commented Jun 20, 2018 at 21:33
  • $\begingroup$ The plot is strongly reminiscent of data about energy usage vs. temperature I analyzed at stats.stackexchange.com/a/148166/919. Regardless, the same principles I advocated there will serve you well. $\endgroup$
    – whuber ♦
    Commented Jun 21, 2018 at 1:06

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The p-value only tells you that the linear model is significantly better than a constant model.

The plot looks like a second-degree polynomial could fit well, i.e.

$$y=\beta_1 x^2 + \beta_2 x+\beta_3$$

This is linear - all you need to do is add a polynomial term to your predictors (i.e. create a new predictor $\mathrm{Temperature}^2$)

In R you can automagically do this with the formula interface: y ~ poly(x, 2)

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