R devel - Oct 2004 - same test statistic for t-test with and without equal variance assumption

Could some kindly tell me if I am supposed to be getting the same test
statistic value with var.equal=TRUE and var.equal=FALSE in t.test ?

set.seed(1066)
x1 <- rnorm(50)
x2 <- rnorm(50)

t.test(x1, x2, var.equal=FALSE)$statistic # 0.5989774
t.test(x1, x2, var.equal=TRUE)$statistic  # 0.5989774 ???


Here are my own calculations that shows that perhaps the result when
var.equal=TRUE is wrong.

n1 <- length(x1); n2 <- length(x2)
m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
v1 <- var(x1)   ; v2 <- var(x2)

# t-test with UNequal variance
denom1 <- sqrt( v1/n1 + v2/n2 )
num / denom1   # gives 0.5989774

# t-test with equal variance 
sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
denom2 <- sp * sqrt( 1/n1 + 1/n2 )
num / denom2   # gives 0.5913777


I tested this using R-1.9.1 (21/06/2004) on Redhat Fedora Core 2 and
Windows 2000 Professional with the same results.

Any suggestions would be kindly appreciated.

Regards,
-- 
Adaikalavan Ramasamy                    ramasamy@cancer.org.uk
Centre for Statistics in Medicine       http://www.ihs.ox.ac.uk/csm/
Cancer Research UK                      Tel : 01865 226 677
Old Road Campus, Headington, Oxford     Fax : 01865 226 962

Adaikalavan Ramasamy <ramasamy@cancer.org.uk> writes:
> Could some kindly tell me if I am supposed to be getting the same test
> statistic value with var.equal=TRUE and var.equal=FALSE in t.test ?
> 
> set.seed(1066)
> x1 <- rnorm(50)
> x2 <- rnorm(50)
> 
> t.test(x1, x2, var.equal=FALSE)$statistic # 0.5989774
> t.test(x1, x2, var.equal=TRUE)$statistic  # 0.5989774 ???
> 
> 
> Here are my own calculations that shows that perhaps the result when
> var.equal=TRUE is wrong.
> 
> n1 <- length(x1); n2 <- length(x2)
> m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
> v1 <- var(x1)   ; v2 <- var(x2)
> 
> # t-test with UNequal variance
> denom1 <- sqrt( v1/n1 + v2/n2 )
> num / denom1   # gives 0.5989774
> 
> # t-test with equal variance 
> sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> num / denom2   # gives 0.5913777
> 
> 
> I tested this using R-1.9.1 (21/06/2004) on Redhat Fedora Core 2 and
> Windows 2000 Professional with the same results.
> 
> Any suggestions would be kindly appreciated.
Your calculation is wrong. Try increasing the variance of x1 and x2...
> set.seed(1066)
> x1 <- rnorm(50,,100)
> x2 <- rnorm(50,,100)
> m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
>  v1 <- var(x1)   ; v2 <- var(x2)
> denom1 <- sqrt( v1/n1 + v2/n2 )
> num / denom1   # gives 0.5989774
[1] 0.5989774
>  sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> num / denom2   # gives 0.5913777
[1] 0.005913777


-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk)             FAX: (+45) 35327907

Peter, thank you! I forgot the to square root in calculating sp.

sp <- sqrt( ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2) )

For several simulation runs, the test statistics from both tests are
remarkably similar (difference is less than 10e-16). I naively assumed
that the statistics value should be slightly but visibly different too. 

It appear that the effects of equal and unequal variance assumptions are
only felt through the degrees of freedom calculation. 

Regards, Adai


On Fri, 2004-10-01 at 12:15, Peter Dalgaard wrote:
> Adaikalavan Ramasamy <ramasamy@cancer.org.uk> writes:
> 
> > Could some kindly tell me if I am supposed to be getting the same test
> > statistic value with var.equal=TRUE and var.equal=FALSE in t.test ?
> > 
> > set.seed(1066)
> > x1 <- rnorm(50)
> > x2 <- rnorm(50)
> > 
> > t.test(x1, x2, var.equal=FALSE)$statistic # 0.5989774
> > t.test(x1, x2, var.equal=TRUE)$statistic  # 0.5989774 ???
> > 
> > 
> > Here are my own calculations that shows that perhaps the result when
> > var.equal=TRUE is wrong.
> > 
> > n1 <- length(x1); n2 <- length(x2)
> > m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
> > v1 <- var(x1)   ; v2 <- var(x2)
> > 
> > # t-test with UNequal variance
> > denom1 <- sqrt( v1/n1 + v2/n2 )
> > num / denom1   # gives 0.5989774
> > 
> > # t-test with equal variance 
> > sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> > denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> > num / denom2   # gives 0.5913777
> > 
> > 
> > I tested this using R-1.9.1 (21/06/2004) on Redhat Fedora Core 2 and
> > Windows 2000 Professional with the same results.
> > 
> > Any suggestions would be kindly appreciated.
> 
> Your calculation is wrong. Try increasing the variance of x1 and x2...
> 
> > set.seed(1066)
> > x1 <- rnorm(50,,100)
> > x2 <- rnorm(50,,100)
> > m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
> >  v1 <- var(x1)   ; v2 <- var(x2)
> > denom1 <- sqrt( v1/n1 + v2/n2 )
> > num / denom1   # gives 0.5989774
> [1] 0.5989774
> >  sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> > denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> > num / denom2   # gives 0.5913777
> [1] 0.005913777
>

Apologies, I made an incorrect statement. The statistics from t-test
with difference variance assumptions are the same if both groups have
the same length. Sorry for troubling everyone again.


On Fri, 2004-10-01 at 13:56, Adaikalavan Ramasamy wrote:
> Peter, thank you! I forgot the to square root in calculating sp.
> 
> sp <- sqrt( ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2) )
> 
> For several simulation runs, the test statistics from both tests are
> remarkably similar (difference is less than 10e-16). I naively assumed
> that the statistics value should be slightly but visibly different too. 
> 
> It appear that the effects of equal and unequal variance assumptions are
> only felt through the degrees of freedom calculation. 
> 
> Regards, Adai
> 
> 
> On Fri, 2004-10-01 at 12:15, Peter Dalgaard wrote:
> > Adaikalavan Ramasamy <ramasamy@cancer.org.uk> writes:
> > 
> > > Could some kindly tell me if I am supposed to be getting the same
test
> > > statistic value with var.equal=TRUE and var.equal=FALSE in t.test
?
> > > 
> > > set.seed(1066)
> > > x1 <- rnorm(50)
> > > x2 <- rnorm(50)
> > > 
> > > t.test(x1, x2, var.equal=FALSE)$statistic # 0.5989774
> > > t.test(x1, x2, var.equal=TRUE)$statistic  # 0.5989774 ???
> > > 
> > > 
> > > Here are my own calculations that shows that perhaps the result
when
> > > var.equal=TRUE is wrong.
> > > 
> > > n1 <- length(x1); n2 <- length(x2)
> > > m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
> > > v1 <- var(x1)   ; v2 <- var(x2)
> > > 
> > > # t-test with UNequal variance
> > > denom1 <- sqrt( v1/n1 + v2/n2 )
> > > num / denom1   # gives 0.5989774
> > > 
> > > # t-test with equal variance 
> > > sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> > > denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> > > num / denom2   # gives 0.5913777
> > > 
> > > 
> > > I tested this using R-1.9.1 (21/06/2004) on Redhat Fedora Core 2
and
> > > Windows 2000 Professional with the same results.
> > > 
> > > Any suggestions would be kindly appreciated.
> > 
> > Your calculation is wrong. Try increasing the variance of x1 and x2...
> > 
> > > set.seed(1066)
> > > x1 <- rnorm(50,,100)
> > > x2 <- rnorm(50,,100)
> > > m1 <- mean(x1)  ; m2 <- mean(x2)  ; num   <- (m1 - m2)
> > >  v1 <- var(x1)   ; v2 <- var(x2)
> > > denom1 <- sqrt( v1/n1 + v2/n2 )
> > > num / denom1   # gives 0.5989774
> > [1] 0.5989774
> > >  sp <- ( (n1-1)*v1 + (n2-1)*v2 )/(n1 + n2 - 2)
> > > denom2 <- sp * sqrt( 1/n1 + 1/n2 )
> > > num / denom2   # gives 0.5913777
> > [1] 0.005913777
> >
> 
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