[R] Doubt about Student t distribution simulation
Dear R list,
I would like to illustrate the origin of the Student t distribution using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean) has t
distribution with (sample.size - 1) degree free, what is wrong with the
simulation below? I think that the theoretical curve should agree with
the relative frequencies of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n))
}
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@stat.math.ethz.ch mailing list
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] Doubt about Student t distribution simulation
Dear Jose,
The problem is that you're using the population standard deviation (sigma)
rather than the sample SD of each sample [i.e., t[i] = (mean(amo.i) - mu) /
(sd(amo.i) / sqrt(n)) ], so your values should be normally distributed, as
they appear to be.
A couple of smaller points: (1) Even after this correction, you're sampling
from a discrete population (albeit with replacement) and so the values won't
be exactly t-distributed. You could draw the samples directly from N(mu,
sigma) instead. (2) It would be preferable to make a quantile-comparison
plot against the t-distribution, since you'd get a better picture of what's
going on in the tails.
I hope this helps,
John
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox
-Original Message-
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of Jose
Claudio Faria
Sent: Friday, August 04, 2006 3:09 PM
To: R-help@stat.math.ethz.ch
Subject: [R] Doubt about Student t distribution simulation
Dear R list,
I would like to illustrate the origin of the Student t
distribution using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean)
has t distribution with (sample.size - 1) degree free, what
is wrong with the simulation below? I think that the
theoretical curve should agree with the relative frequencies
of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n)) }
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@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.
__
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] Doubt about Student t distribution simulation
Jose Claudio Faria [EMAIL PROTECTED] writes:
Dear R list,
I would like to illustrate the origin of the Student t distribution using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean) has t
distribution with (sample.size - 1) degree free, what is wrong with the
simulation below? I think that the theoretical curve should agree with
the relative frequencies of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n))
At the very least, you need a sample-based standard error: sd(amo.i),
not sigma. Also, resampling from pop is not really what the
t-distribution is based on, but I don't think that matters much.
}
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@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.
--
O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907
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R-help@stat.math.ethz.ch mailing list
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] Doubt about Student t distribution simulation
Hi, Jose/John,
Here's an example to help Jose and highlights John's advice. Also
includes set.seed which should be included in all simulations posted to
R-help.
set.seed(42)
mu - 10
sigma - 5
n - 3
nsim - 1
m - matrix(rnorm(n * nsim, mu, sigma), nsim, n)
t - apply(m, 1, function(x) (mean(x) - mu)/(sd(x)/sqrt(n)))
library(lattice)
qqmath(t, distribution = function(x) qt(x, n - 1),
panel = function(x, ...) {
panel.qqmath(x, col = darkblue, ...)
panel.qqmathline(x, col = darkred, ...)
})
With n = 3, expect a few outliers.
--sundar
John Fox wrote:
Dear Jose,
The problem is that you're using the population standard deviation (sigma)
rather than the sample SD of each sample [i.e., t[i] = (mean(amo.i) - mu) /
(sd(amo.i) / sqrt(n)) ], so your values should be normally distributed, as
they appear to be.
A couple of smaller points: (1) Even after this correction, you're sampling
from a discrete population (albeit with replacement) and so the values won't
be exactly t-distributed. You could draw the samples directly from N(mu,
sigma) instead. (2) It would be preferable to make a quantile-comparison
plot against the t-distribution, since you'd get a better picture of what's
going on in the tails.
I hope this helps,
John
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox
-Original Message-
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of Jose
Claudio Faria
Sent: Friday, August 04, 2006 3:09 PM
To: R-help@stat.math.ethz.ch
Subject: [R] Doubt about Student t distribution simulation
Dear R list,
I would like to illustrate the origin of the Student t
distribution using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean)
has t distribution with (sample.size - 1) degree free, what
is wrong with the simulation below? I think that the
theoretical curve should agree with the relative frequencies
of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n)) }
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@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.
__
R-help@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.
__
R-help@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.
Re: [R] Doubt about Student t distribution simulation
Dears John, Peter and Sundar,
Many thanks for the quick answers!!!
.. and sorry for all..
[]s
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
John Fox escreveu:
Dear Jose,
The problem is that you're using the population standard deviation (sigma)
rather than the sample SD of each sample [i.e., t[i] = (mean(amo.i) - mu) /
(sd(amo.i) / sqrt(n)) ], so your values should be normally distributed, as
they appear to be.
A couple of smaller points: (1) Even after this correction, you're sampling
from a discrete population (albeit with replacement) and so the values won't
be exactly t-distributed. You could draw the samples directly from N(mu,
sigma) instead. (2) It would be preferable to make a quantile-comparison
plot against the t-distribution, since you'd get a better picture of what's
going on in the tails.
I hope this helps,
John
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox
-Original Message-
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of Jose
Claudio Faria
Sent: Friday, August 04, 2006 3:09 PM
To: R-help@stat.math.ethz.ch
Subject: [R] Doubt about Student t distribution simulation
Dear R list,
I would like to illustrate the origin of the Student t
distribution using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean)
has t distribution with (sample.size - 1) degree free, what
is wrong with the simulation below? I think that the
theoretical curve should agree with the relative frequencies
of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n)) }
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@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.
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] Doubt about Student t distribution simulation
Dear Sundar,
Try qq.plot(t, dist=t, df=n-1) from the car package, which include a
95-percent point-wise confidence envelope that helps you judge how extreme
the outliers are relative to expectations.
Regards,
John
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox
-Original Message-
From: Sundar Dorai-Raj [mailto:[EMAIL PROTECTED]
Sent: Friday, August 04, 2006 3:27 PM
To: John Fox
Cc: [EMAIL PROTECTED]; R-help@stat.math.ethz.ch
Subject: Re: [R] Doubt about Student t distribution simulation
Hi, Jose/John,
Here's an example to help Jose and highlights John's advice.
Also includes set.seed which should be included in all
simulations posted to R-help.
set.seed(42)
mu - 10
sigma - 5
n - 3
nsim - 1
m - matrix(rnorm(n * nsim, mu, sigma), nsim, n) t -
apply(m, 1, function(x) (mean(x) - mu)/(sd(x)/sqrt(n)))
library(lattice)
qqmath(t, distribution = function(x) qt(x, n - 1),
panel = function(x, ...) {
panel.qqmath(x, col = darkblue, ...)
panel.qqmathline(x, col = darkred, ...)
})
With n = 3, expect a few outliers.
--sundar
John Fox wrote:
Dear Jose,
The problem is that you're using the population standard deviation
(sigma) rather than the sample SD of each sample [i.e., t[i] =
(mean(amo.i) - mu) /
(sd(amo.i) / sqrt(n)) ], so your values should be normally
distributed, as they appear to be.
A couple of smaller points: (1) Even after this correction, you're
sampling from a discrete population (albeit with
replacement) and so
the values won't be exactly t-distributed. You could draw
the samples
directly from N(mu,
sigma) instead. (2) It would be preferable to make a
quantile-comparison plot against the t-distribution, since
you'd get a
better picture of what's going on in the tails.
I hope this helps,
John
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario
Canada L8S 4M4
905-525-9140x23604
http://socserv.mcmaster.ca/jfox
-Original Message-
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of Jose Claudio
Faria
Sent: Friday, August 04, 2006 3:09 PM
To: R-help@stat.math.ethz.ch
Subject: [R] Doubt about Student t distribution simulation
Dear R list,
I would like to illustrate the origin of the Student t distribution
using R.
So, if (sample.mean - pop.mean) / standard.error(sample.mean) has t
distribution with (sample.size - 1) degree free, what is wrong with
the simulation below? I think that the theoretical curve
should agree
with the relative frequencies of the t values calculated:
#== begin options=
# parameters
mu= 10
sigma = 5
# size of sample
n = 3
# repetitions
nsim = 1
# histogram parameter
nchist = 150
#== end options===
t = numeric()
pop = rnorm(1, mean = mu, sd = sigma)
for (i in 1:nsim) {
amo.i = sample(pop, n, replace = TRUE)
t[i] = (mean(amo.i) - mu) / (sigma / sqrt(n)) }
win.graph(w = 5, h = 7)
split.screen(c(2,1))
screen(1)
hist(t,
main = histogram,
breaks = nchist,
col = lightgray,
xlab = '', ylab = Fi,
font.lab = 2, font = 2)
screen(2)
hist(t,
probability = T,
main= 'f.d.p and histogram',
breaks = nchist,
col = 'lightgray',
xlab= 't', ylab = 'f(t)',
font.lab= 2, font = 2)
x = t
curve(dt(x, df = n-1), add = T, col = red, lwd = 2)
Many thanks for any help,
___
Jose Claudio Faria
Brasil/Bahia/Ilheus/UESC/DCET
Estatística Experimental/Prof. Adjunto
mails: [EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
__
R-help@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.
__
R-help@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.
__
R-help@stat.math.ethz.ch mailing list
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
