# [R] R's Spearman

Frank E Harrell Jr f.harrell at vanderbilt.edu
Thu May 31 23:39:03 CEST 2007

```Mendiburu, Felipe (CIP) wrote:
> Dear Ray,
>
> The R's Spearman calculated by R is correct for ties or nonties, which is not correct is the probability for the case of ties. I send to you formulates it for the correlation with ties, that is equal to R.
>
> Regards,
>
> Felipe de Mendiburu
> Statistician

Just use midranks for ties (as with rank()) and compute the ordinary
correlation on those (see also the spearman2 and rcorr functions in
Hmisc package).  No need to use complex formulas.  And the t
approximation for p-values works pretty well.

Frank Harrell

>
>
> # Spearman correlation "rs" with ties or no ties
> rs<-function(x,y) {
> d<-rank(x)-rank(y)
> tx<-as.numeric(table(x))
> ty<-as.numeric(table(y))
> Lx<-sum((tx^3-tx)/12)
> Ly<-sum((ty^3-ty)/12)
> N<-length(x)
> SX2<- (N^3-N)/12 - Lx
> SY2<- (N^3-N)/12 - Ly
> rs<- (SX2+SY2-sum(d^2))/(2*sqrt(SX2*SY2))
> return(rs)
> }
>
> # Aplicacion
>> cor(y[,1],y[,2],method="spearman")
> [1] 0.2319084
>> rs(y[,1],y[,2])
> [1] 0.2319084
>
>
>
> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch
> [mailto:r-help-bounces at stat.math.ethz.ch]On Behalf Of Raymond Wan
> Sent: Monday, May 28, 2007 10:29 PM
> To: r-help at stat.math.ethz.ch
> Subject: [R] R's Spearman
>
>
>
> Hi all,
>
> I am trying to figure out the formula used by R's Spearman rho (using
> cor(method="spearman")) because I can't seem to get the same value as by
> calculating "by hand".  Perhaps I'm using "cor" wrong, but I don't know
> where.  Basically, I am running these commands:
>
>  > y
>    IQ Hours
> 1 106     7
> 2  86     0
> 3  97    20
> 4 113    12
> 5 120    12
> 6 110    17
>  > cor(y[1],y[2],method="spearman")
>        Hours
> IQ 0.2319084
>
> [it's an abbreviated example of one I took from Wikipedia].  I
> calculated by hand (apologies if the table looks strange when pasted
> into e-mail):
>
>       IQ    Hours    rank(IQ)  rank(hours)    diff    diff^2
> 1    106    7        3         2                 1    1
> 2     86    0        1         1                 0    0
> 3     97    20       2         6                -4    16
> 4    113    12       5         3.5             1.5    2.25
> 5    120    12       6         3.5             2.5    6.25
> 6    110    17       4         5                -1    1
>                                                       26.5
>
>                                               rho=    0.242857
>
> where rho = (1 - ((6 * 26.5) / 6 * (6^2 - 1))).  I kept modifying the
> table and realized that the difference in result comes from ties.  i.e.,
> if I remove the tie in rows 4 and 5, I get the same result from both cor
> and calculating by hand.  Perhaps I'm handling ties wrong...does anyone
> know how R does it or perhaps I need to change how I'm using it?
>
> Thank you!
>
> Ray
>
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>
> ______________________________________________
> R-help at stat.math.ethz.ch 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.
>

--
Frank E Harrell Jr   Professor and Chair           School of Medicine
Department of Biostatistics   Vanderbilt University

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