Hi, everybody!
In my regression analyses (performed through LM and GLM models) I found R-squared values from 8% to 15%, but high P values. The predictor variable was incisors' procumbency angle and the response variables were the mechanical advantages of jaw adductor muscles in a subterranean rodent species.
Can I include such low R-squared values in my research paper? Or R-squared values always have to be 70% or more. If anyone can refer me any books or journal articles about validity of low R-squared values, it would be highly appreciated.
Hi Alejandra,
"High" or "low" R squared really does depend on the context. In some context, R-squared of 70% is considered to be appropriate, but in some other context, an R-squared of about 20% may satisfy the scientists already.
In your model, you have only one predictor, and it is significantly related to the dependent variable, but you have R-squared of 8% to 15%. You can, consider, to include other possibly important covariates.
It is shown that R-squared always increases if you add more covariates into your model. So what we can consider, is to use adjusted R-squared instead of R-squared (in case you have many covariates). The adjusted R-squared can even decrease when we add more covariates in the model.
Below you can see a clear discussion on this problem:
http://blog.minitab.com/blog/adventures-in-statistics/how-high-should-r-squared-be-in-regression-analysis
I hope it help.
Hi Alejandra,
"High" or "low" R squared really does depend on the context. In some context, R-squared of 70% is considered to be appropriate, but in some other context, an R-squared of about 20% may satisfy the scientists already.
In your model, you have only one predictor, and it is significantly related to the dependent variable, but you have R-squared of 8% to 15%. You can, consider, to include other possibly important covariates.
It is shown that R-squared always increases if you add more covariates into your model. So what we can consider, is to use adjusted R-squared instead of R-squared (in case you have many covariates). The adjusted R-squared can even decrease when we add more covariates in the model.
Below you can see a clear discussion on this problem:
http://blog.minitab.com/blog/adventures-in-statistics/how-high-should-r-squared-be-in-regression-analysis
I hope it help.
This really depends what you're interested in testing. If you're just interested in testing whether one predictor has a significant effect (i.e., if you predict the outcome better with that predictor than without it) then the overall magnitude of R2 doesn't matter. It's not unusual at all to observe small but significant effects (and thus to have a low R2 but high p). "R2 values always have to be 70% or more" is nonsense. That might be the case for some research questions, but not for many others. A statistical analysis means nothing except in the context of what research question you are trying to answer with it.
Hello Alejandra,
When you do linear regression, test these hypothesis :
Based on this model : y=ax+b
So you test a=0 and b=0. So you get two p-values, one for the intercept (b) the other for the slope (a).
The coefficient of determination (R-squared) give you (somehow) "how your model explain well your data distribution".
Please find below an example (in R code + attached figures) of very low p-values obtained but higher R-square in "well defined model" :
> x y plot(x,y)
> abline(lm(y~x))
> summary(lm(y~x))
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-22.865 -14.492 -4.803 13.247 48.871
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -41.476 7.782 -5.33 4.40e-06 ***
x 29.862 2.413 12.38 4.42e-15 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 18.28 on 39 degrees of freedom
Multiple R-squared: 0.797, Adjusted R-squared: 0.7918
F-statistic: 153.2 on 1 and 39 DF, p-value: 4.416e-15
> summary(lm(log(y)~x))
Call:
lm(formula = log(y) ~ x)
Residuals:
Min 1Q Median 3Q Max
-0.52382 -0.12886 0.00583 0.12591 0.46825
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.65034 0.08615 19.16 abline(lm(log(y)~x))
So, I would recommend you to pay more attention on your model if you get low R-squared. Because, if you have many values and somehow a increase (or decrease) of y with x, you'll find for sure low p-values with linear model but that doesn't mean that the model is "valid".
Moreover, I would also recommend you to have a look to linear model assumptions (if you didn't check it already).
Finally, the lower R-squared I remembered is of 0.5 and you'll find it in the paper on the link below.
Regards
http://www.nature.com/nature/journal/v444/n7122/abs/4441022a.html
Another link:
http://blog.minitab.com/blog/adventures-in-statistics/multiple-regession-analysis-use-adjusted-r-squared-and-predicted-r-squared-to-include-the-correct-number-of-variables
I think if you've found a dependency with R-squared>10%, it's a good result already. So you can publish it in your research paper. If reviewers an explain that it is not a valuable result, you can always remove it.
Look at this page:
https://books.google.ru/books?id=wG06_c4yzC4C&pg=PA191&dq=r+squared+value+interpretation&hl=en&sa=X&ved=0ahUKEwjZifzPi8HMAhVFXCwKHdGbAw8Q6AEILTAD#v=onepage&q=r%20squared%20value%20interpretation&f=false
perhaps you may think about global and local solution or false optimums !!
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