Transcript The t test - chezshaw.demon.co.uk

The t test
Peter Shaw
RIl
"Small samples are slippery customers
whose word is not to be taken as
gospel" (Moroney).
Introduction
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Last week we met the MannWhitney U test, a non-parametric
test to examine how likely it is that
2 samples could have come from
the same population.
This week we explore other
approaches to this and related
situations.
Student’s t test
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This test was invented by a
statistician working for the brewer
Guinness. He was called WS
Gosset (1867-1937), but preferred
to keep anonymous so wrote under
the name “Student”.
Hence we have Student’s t test,
the Studentised range, etc - in
memory of Mr Gosset (!).
T vs U?
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These 2 tests are identical in hypothesis
formulation.
They require 2 samples which may be from
the same population. These samples need
not be of equal #, nor are they paired.
 H0:
The 2 samples are from the same
population - any differences are due to chance
 H1: The 2 samples come from different
populations.
1 big difference (+ a few
small ones):
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The t test is a parametric test - it
assumes the data are normally
distributed.
There are several different
versions of the t test, depending on
exactly what assumptions you
make about the data. I’ll stick to
the simplest.
The basic idea
Remember Z scores? These apply to the
idealised normal distribution
How many s.d.s is this data
point from the mean? Zi = (Xiμ)/σ
μ σ
We can look up Z in tables, but
these assume that the values of
μ and σ are known perfectly.
Gosset’s discovery:
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Was the formulae appropriate to Z when the
sample is small, so that μ and S are based on
inadequate data.
To distinguish this distribution from the idealised
normal distribution, Gosset named the function
the “t statistic”, and the value of (Xi- μ)/S when μ
and S are estimates was renamed from Z to t.
Hence t is really just a special, unreliable Z
score. To identify a t score you must also
specify how many data points it comes from: a
value based on 6 observations is FAR less
reliable than one based on 6000.
The theory...
You have 2 samples which may be from 1
distribution or 2. To assess the likelihood,
find how many s.d.s the means of the 2
populations are apart:
How many S.D.’s?
Calculate t = (μ1 - μ2) / pooled sd
μ1
μ2
The details are slightly
more messy..
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Because of the question “How do we calculate the
pooled sd?”
There are several ways of doing this which make
different assumptions, and give slightly different
answers.
The simplest model assumes that the 2 samples
have a common variance, and gives t as follows:
Given data X1, X2 which have N1, N2 datapoints
each, and sums of squares SSx1, SSx2
t
=
(μ1 - μ2)
with N1+n2-1 df
__________
sq.root[(SSx1 + SSx2)*(1/Nx+1/Ny) / (n1+n2-2)]
Beware!
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I spent an afternoon in the library once checking
ways to calculate t.
I found 3 different formulae, plus several
confusing ways to express the relationship I just
showed you.
Another one widely used differs in assuming that
the 2 samples have unequal variance. This gives
a messier formula, plus another even messier
formula for the df.
The third approach assumes that samples are
accurately paired - the paired samples t test.
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x1
57.8
56.2
61.9
54.4
53.6
56.4
53.2
n
Sum x
sumx*2
mean
ss
x2
64.2
58.7
63.1
62.5
59.8
59.2
7
393.5
22174.41
56.21%
54.089
6
367.5
22535.87
61.25%
26.495
So you know what to do
to compare 2 groups!
X1 X2 X3
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You have the choice of M-W U, or
Student’s t test.
But what if there are 3 groups, or 4, or 5?
You may work out the following routine:
Test group 1 vs group 2, then 2 vs 3, etc.
 Clever, but WRONG! (The danger with
multiple tests is that you will get a
“p=0.05” significant result more often
than 1:20).
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Multiple groups can be
compared..
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With a suitable multiple test.
There are 2 options here, both of which
are usually run on PCs.
Parametric data: Analysis of variance
ANOVA
Non-Parametric data: Kruskal-Wallis
ANOVA.
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I make M.Sc. students run ANOVA
calculations by hand, but K-W ANOVA is
PC only.
Type of data
Number
of
groups:
2
Parametric
NonParametric
T test
Mann-Whitney
U test
>=2
Analysis of
variance
(ANOVA)
Kruskal_Wallis
ANOVA
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n
Sum x
sumx*2
mean
SS
7
393.5
22174.41
56.21%
54.089
6
367.5
22535.87
61.25%
26.495
┌─
─┐
Sediffere = sqrt│(SSxx + SSyy)*(1/Nx+1/Ny)│
│──────────────── │
│ Nx+Ny-2
│
└─
─┘
Example:
SEdiff= sqrt[(26.495+54.089)*(1/6+1/7)/(6+7-2)]
= sqrt[2.2675] = 1.506
Hence t = (61.25 - 56.21)/1.506 = 3.35 with 12df
This is significant at p<0.05