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I am carrying out linear regressions with quite different independent variables and I am unsure how to treat each appropriately without compromising my ability to compare the final $R^2$ values.

I have 150 individuals in the analysis, each with 6 proxy measures for body size, as well as the actual body size (my DV). All individuals will later be analysed altogether, but they are made up of 3 species (n=50 each) and the focus of my analysis is to look at them separately (to see if there is one proxy for body size that suits them all, or if each species has its own that's most suitable).

Within each species, some of my variables show a normal distribution and some of them don't, and some of them show clear heteroscedasticity while others are homoscedastic.

I understand that to carry out a linear regression I need normally distributed residuals, but $\frac{5}{18}$ of my independent variables are not normal (or even approximately normal) and there is no one transformation that is suitable for them all. I have been told by a colleague that each variable should be considered independently and transformed to normality accordingly.

Ultimately I am concerned that having a combination of both transformed (with various transformations) and non-transformed data as well as using a traditional regression and a weighted regression, will make it difficult to compare my final $R^2$ values.

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    $\begingroup$ Why do you want to compare the final R2 values? R2 do not only depend on the quality of fit, but also on the distribution of the predictors, so is not a measure of quality of fit. It is probably more natural to compare residual variance. Transforming predictors should not be necessary, maybe transforming the response .... Comparison of different models will be easier if the response have a common scale, so maybe search for a compromise transformation ... $\endgroup$ Feb 27, 2017 at 14:18

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There are no distributional assumptions on your independent variables in linear regression. Only the dependent variable is assumed to be normally distributed (with constant variance, homoscedastic), conditional on the values of the covariates in the model.

Based on the information you give I would fit separate models for every species and compare them to see if there is a common best proxy.

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  • $\begingroup$ Thank you for your help. Are the R2 values generated through OLS and WLS then comparable? $\endgroup$
    – Joanne
    Feb 27, 2017 at 14:52
  • $\begingroup$ @Joanne Which weights would you use? The $R^2$ are comparable as long as you use the same dependent variable, but better use an adjusted version to account for different numbers of independent variables. $\endgroup$
    – Knarpie
    Feb 27, 2017 at 14:57
  • $\begingroup$ I used my unstandardised predictor values to calculate my weights. I am using the same dependent variable throughout. $\endgroup$
    – Joanne
    Mar 3, 2017 at 13:45
  • $\begingroup$ @Joanne I need more explanation with respect to these weights, they are not self-evident to me. $\endgroup$
    – Knarpie
    Mar 3, 2017 at 13:49

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