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I am studying behaviour of particulate matters(pm10) concentration in respose to change in rain and tempretaure. my data was not normaly distributed so i have to transform data I did log transformation and inverse transformation.The Adjusted R-squared for log transformation is :0.07918 and Adjusted R-squared for inverse transform is :0.1002.Now according to rule i must select model with high value of Adjusted R-square but on the other hand if we look to the QQ-plot and histogram of log(x) and 1/X it seemes that model with log transformation best fits the data. Now i am confused in selecting the best model. your help will be highly appreciated . all the necessary graphs are attached here Y is pm10 (response) and x represents rain (dependent varaible) . Histogram of both models

QQ-plots of Models

Model 1

Model 2

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    $\begingroup$ You're fixating on whether the response is normally distributed but that's not a requirement for regression. At most it is good if residuals are approximately normally distributed. The graphs don't allow a good recommendation of which model is better because that should depend on physical reasoning; it is far from clear that rainfall should be used as it comes (perhaps it should be cube rooted); and there is no information here on temperature. You want scales on which behaviour is approximately linear; that's immensely more important than whether marginal distributions are normal. $\endgroup$
    – Nick Cox
    Jan 20, 2017 at 9:29
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    $\begingroup$ It difficult to conceive of any situation where any statistic related to R-squared would be valid or useful for comparing fits that differ by some nonlinear re-expression of the response. If you "have to transform," that suggests this might be some kind of textbook exercise. If so, find a better textbook! $\endgroup$
    – whuber
    Dec 31, 2023 at 16:12

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You write

My data was not normally distributed so i have to transform data

This is not correct. OLS regression makes no assumptions about the distribution of the data, other than the DV having to be continuous. It makes assumptions about the errors, which we examine by looking at residuals.

I did log transformation and inverse transformation.

Do either of these make sense in your context? The inverse of particulate matter per parts of air is parts of air per particulate matter. Does that make sense to use? The log of a ratio is the log of the numerator minus the log of the denominator. That doesn't seem sensible to me, but I am no expert in this field.

Instead of changing your data to fit the model, change your model to fit the data. You could use robust regression, quantile regression, maybe a regression tree, or something else.

The Adjusted R-squared for log transformation is :0.07918 and Adjusted R-squared for inverse transform is :0.1002.Now according to rule i must select model with high value of Adjusted R-square

No, you don't. Data analysis is rarely a matter of rules. You need to think about your problem. As David Cox put it:

There are no routine statistical questions, only questionable statistical routines

As for your graphs, they are very nice and they show that neither of your models is really great. You do have some extreme values of rain. I'd first check if those were errors. It might make sense to take the log of rain. This essentially changes additivity (what happens when there is 1 more cm of rain?) to multiplicativity (what happens when there is twice as much rain?) and that might be good. It would emphasize small changes at low levels of rain, which might make sense, substantively.

Or, maybe a spline or rainfall would be good.

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