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I'm in the process of conducting a multiple regression model and have created the following test to see the variance of the residuals in my model. That is the graph for the original data and has had no transformations done to it.

The only transformation which made the homogeneity and r-squared better was a square-root of the response variable. However, I understand its not ideal to transform the response variable? As well as this, if I was to use this transformation I can't conduct a anova test between the original and transformed model as they differ in their response variables...

1 - Would that residuals graph suggest a good enough homogeneity to satisfy that assumption of linear regression?

2 - Would it be okay/better to use the sqrt transformation of the response variable instead?

  • The first one is the original
  • The second is the transformed

No transformation

TRANSFORMED

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2 Answers 2

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Usually I would just post a comment, but this may be too long a response for this. First implied question. You can transform the response variable however you wish to, nothing wrong with that. Second, you can perform ANOVA on your transformed variables and then detransform the results back into the original measurement scheme. Alternatively, you can change what you are minimizing to be non-linear to match the original data. In other words, instead of fitting $\sqrt{y}=m x +b$ you could fit $y=(m x+b)^2$, which is the same thing. Third, how do you know that square rooting is best? Did you, for example, try to fit $y=(m x+b)^c$? That might give you something better yet. Fourth, your residuals are curvilinear, so there is still some other transformation that potentially needs to be done, and potentially lots more you could try.

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I agree with the answer of @Carl, but I'll add a couple of thoughts.

a) You are right that you can't conduct an anova test (or likelihood ratio test, etc.) between the models because the transformed and un-transformed models are not nested. As you suggest, you could look at other goodness-of-fit statistics like R-squared or one one of various accuracy measures. But more importantly than eking out a few more points in the R-squared value or improving the accuracy of the model, is making sure the model meets model assumptions if you are relying on p-values.

1/2) I wouldn't be bothered by the heteroscedasticity shown in either plot. Remember that there aren't truly normal or truly homoscedastic data in the real world. I am somewhat bothered by the distribution in the vertical axis in the first plot. The residuals appear to be right skewed. That is the points above the 0 line extend further vertically than do the points below the 0 line. Because of this, I like the the second plot better, but would definitely use a histogram or q-q plot to look at the normality of the residuals.

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