[R] Difference betweeen cor.test() and formula everyone says to use

[Jeremy Miles]

I'm trying to understand how cor.test() is calculating the p-value of
a correlation. It gives a p-value based on t, but every text I've ever
seen gives the calculation based on z.

data(cars)
with(cars[1:10, ], cor.test(speed, dist))

Pearson's product-moment correlation

data: speed and dist
t = 2.3893, df = 8, p-value = 0.04391
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.02641348 0.90658582
sample estimates:
cor
0.6453079

r <- cor(cars[1:10, ])[1, 2]
r.z <- fisherz(r)
se <- se <- 1/sqrt(10 - 3)
z <- r.z / se
(1 - pnorm(z))*2

[1] 0.04237039

My p-value is different. The help file for cor.test doesn't (seem to)
have any reference to this, and I can see in the source code that it
is doing something different. I'm just not sure what.

Thanks,

Jeremy

[Joshua Wiley]

Hi Jeremy,

I don't know about references, but this around. See for example:
http://afni.nimh.nih.gov/sscc/gangc/tr.html

the relevant line in cor.test is:

STATISTIC <- c(t = sqrt(df) * r/sqrt(1 - r^2))

You can convert *t*s to *r*s and vice versa.

Best,

Josh



On Fri, Oct 17, 2014 at 10:32 AM, Jeremy Miles <jeremy.miles at gmail.com>

Post by Jeremy Miles
I'm trying to understand how cor.test() is calculating the p-value of
a correlation. It gives a p-value based on t, but every text I've ever
seen gives the calculation based on z.

data(cars)
with(cars[1:10, ], cor.test(speed, dist))

Pearson's product-moment correlation
data: speed and dist
t = 2.3893, df = 8, p-value = 0.04391
alternative hypothesis: true correlation is not equal to 0
0.02641348 0.90658582
cor
0.6453079

r <- cor(cars[1:10, ])[1, 2]
r.z <- fisherz(r)
se <- se <- 1/sqrt(10 - 3)
z <- r.z / se
(1 - pnorm(z))*2

[1] 0.04237039
My p-value is different. The help file for cor.test doesn't (seem to)
have any reference to this, and I can see in the source code that it
is doing something different. I'm just not sure what.
Thanks,
Jeremy
______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

[peter dalgaard]

This is pretty much standard. I'm quite sure that other stats packages do likewise and I wouldn't know who "everyone" is. It is not unheard of that textbook authors give suboptimal formulas in order not to confuse students, though.

The basic point is that the t transformation gives the exact distribution under the null. Fisher's Z is only approximately normally distributed.

The t transformation works because if beta is the regression coefficient of y on x, beta==0 iff rho==0, and we have exact theory for testing beta==0 by a t-test.

Off-null, the t-approach does not readily transfer, so confidence intervals tend to be based on the Z-transformation.

-Peter D.

Post by Joshua Wiley
Hi Jeremy,
http://afni.nimh.nih.gov/sscc/gangc/tr.html
STATISTIC <- c(t = sqrt(df) * r/sqrt(1 - r^2))
You can convert *t*s to *r*s and vice versa.
Best,
Josh
On Fri, Oct 17, 2014 at 10:32 AM, Jeremy Miles <jeremy.miles at gmail.com>

Post by Jeremy Miles
I'm trying to understand how cor.test() is calculating the p-value of
a correlation. It gives a p-value based on t, but every text I've ever
seen gives the calculation based on z.

data(cars)
with(cars[1:10, ], cor.test(speed, dist))

Pearson's product-moment correlation
data: speed and dist
t = 2.3893, df = 8, p-value = 0.04391
alternative hypothesis: true correlation is not equal to 0
0.02641348 0.90658582
cor
0.6453079

r <- cor(cars[1:10, ])[1, 2]
r.z <- fisherz(r)
se <- se <- 1/sqrt(10 - 3)
z <- r.z / se
(1 - pnorm(z))*2

[1] 0.04237039
My p-value is different. The help file for cor.test doesn't (seem to)
have any reference to this, and I can see in the source code that it
is doing something different. I'm just not sure what.
Thanks,
Jeremy
______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

--
Joshua F. Wiley
Ph.D. Student, UCLA Department of Psychology
http://joshuawiley.com/
Senior Analyst, Elkhart Group Ltd.
http://elkhartgroup.com
Office: 260.673.5518
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______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

[JLucke]

The distribution of the statistic $ndf * r^2 / (1-r^2)$ with the true
value $\rho = zero$ follows an $F(1,ndf)$ distribution.
So the t-test is the correct test for $\rho=0$.
Fisher's z is an asymptotically normal transformation for any value of
$\rho$.
Thus Fisher's z is better for testing $\rho= \rho_0 $ or $\rho_1 =
\rho_2$.
The two statistics will not be equivalent at $\rho=0$ because the
statistics are based on different assumptions.




Jeremy Miles <jeremy.miles at gmail.com>
Sent by: r-help-bounces at r-project.org
10/16/2014 07:32 PM

To
r-help <r-help at r-project.org>,
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Subject
[R] Difference betweeen cor.test() and formula everyone says to use






I'm trying to understand how cor.test() is calculating the p-value of
a correlation. It gives a p-value based on t, but every text I've ever
seen gives the calculation based on z.

data(cars)
with(cars[1:10, ], cor.test(speed, dist))

Pearson's product-moment correlation

data: speed and dist
t = 2.3893, df = 8, p-value = 0.04391
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.02641348 0.90658582
sample estimates:
cor
0.6453079

r <- cor(cars[1:10, ])[1, 2]
r.z <- fisherz(r)
se <- se <- 1/sqrt(10 - 3)
z <- r.z / se
(1 - pnorm(z))*2

[1] 0.04237039

My p-value is different. The help file for cor.test doesn't (seem to)
have any reference to this, and I can see in the source code that it
is doing something different. I'm just not sure what.

Thanks,

Jeremy

______________________________________________
R-help at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.


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