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# The t test - PowerPoint PPT Presentation

The t test. Peter Shaw RIl. "Small samples are slippery customers whose word is not to be taken as gospel" (Moroney). Introduction. Last week we met the Mann-Whitney U test, a non-parametric test to examine how likely it is that 2 samples could have come from the same population.

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### The t test

Peter Shaw

RIl

"Small samples are slippery customers whose word is not to be taken as gospel" (Moroney).

• Last week we met the Mann-Whitney 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.

• 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 (!).

• 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.

• 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.

Remember Z scores? These apply to the idealised normal distribution

How many s.d.s is this data point from the mean? Zi = (Xi- μ)/σ

We can look up Z in tables, but these assume that the values of μ and σ are known perfectly.

σ

μ

• 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.

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

• 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)]

• 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.

• x1 x2

• 57.8 64.2

• 56.2 58.7

• 61.9 63.1

• 54.4 62.5

• 53.6 59.8

• 56.4 59.2

• 53.2

• n 7 6

• Sum x 393.5 367.5

• sumx*2 22174.41 22535.87

• mean 56.21% 61.25%

• ss 54.089 26.495

X1

X2

X3

• 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).

• 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.

• I make M.Sc. students run ANOVA calculations by hand, but K-W ANOVA is PC only.

Number of groups:

Parametric

Non-Parametric

2

T test

Mann-Whitney U test

>=2

Analysis of variance (ANOVA)

Kruskal_Wallis ANOVA

• n 7 6

• Sum x 393.5 367.5

• sumx*2 22174.41 22535.87

• mean 56.21% 61.25%

• SS 54.089 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