# Conversion of ANOVA to Pearson r

#### DrK89

##### New Member
Hello all,

I am working on a meta-analysis that assesses the relationship between amount of substance A in the diet and the concentration of substance A in the blood. In the identified studies, this relationship is shown as a Pearson r in the majority and as beta coefficients in a large minority. However, a few studies report P values for ANOVA. Instead of looking at the relationship between the two continuous variables, they categorized the amount of substance A in the diet into tertiles or quartiles, then provided the average concentration of substance A in the blood for those tertiles or quartiles.

For example, this leads to results like:
Lowest tertile of amount of substance A in the diet: concentration of A in the blood of 1g/L
Middle tertile: concentration 1.5g/L
Highest tertile: concentration 2.0g/L

They then provide n = X and a P-value for the ANOVA. Let’s say for this study: n = 350 and P<0.001. This study also provided an F-value, which is not that common in medicine, which in this case was 158.

I am looking for a way to convert the F-value for the ANOVA into a Pearson r – if this is possible of course.

Based on this online lecture, it is possible:
(for example stop the video at 7.50). However, apparently this is specific for 2 groups. If I use the values above (F = 158, df = 347), I obtain: r = 0.56, which is a large correlation, in agreement with the aforementioned results.

However, I cannot find the formula for an ANOVA with 3 groups, nor do I know if my reasoning above is correct. I have looked at tables to convert F-values into P-values, but I only found “Critical value tables”. I found online converters, but they don’t seem to match what I want (besides, I’d like to understand). Can anyone help me with the formula to convert an ANOVA with 3 groups into a Pearson r, and can anyone confirm whether I can use it this way – or explain to me if I cannot?

Best,
Martin

#### katxt

##### Well-Known Member
I am looking for a way to convert the F-value for the ANOVA into a Pearson r – if this is possible of course
This works for regression with 1 predictor, which sounds like your case.
You need n, the number of points and the F value for the regression. J, K and L are holders.
Find d = n-2 d is also commonly quoted as part of the F value F(1,37)=25 the 37 is d
Find J = F/d
Find K = 1/J
Find L = K + 1
Find R2 = 1/L
R = sqrt(R2)
This could all be made into one formula, but this is just as easy.

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#### DrK89

##### New Member

If I use n-2, I get almost the same answer as the formula shown in the video. If I use n-3, I get exactly the same answer. I guess I should use n-2 because I have three groups, so df = n-2? I am trying to understand the formula.

Is there a source for the formula you posted? I would like to refer to the source in my meta-analysis.

Thanks again.

Best,
Martin

#### katxt

##### Well-Known Member
I must admit that I only lasted about 30s into your video, and I'm not sure what you mean by three groups.
I will show you what I did for the regression and probably you can adapt the method.
For a simple regression I got this anova table with n = 8. F=MSReg/MSRes = (SSReg/dfReg)/(SSRes/dfRes) by defintion. Unjumble this we get
J = F/d=SSReg/SSRes
K = 1/J = SSRes/SSReg Now add 1
L = 1+ SSRes/SSReg = (SSReg+SSRes)/SSReg = SSTot/SSReg
R2 = 1/K =SSReg/SSTot by definition.
As far as the n-2 or n-3 is concerned, presumably there would have been another row in the anova table (group?) using up 1 df making the dfRes n-3. If n-3 works for you, go ahead. No reference, just say "from first principles".
Did the quoted F value include the df? if so use that.