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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).

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Introduction n 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. n This week we explore other approaches to this and related situations.

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Student’s t test n 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”. n Hence we have Student’s t test, the Studentised range, etc - in memory of Mr Gosset (!).

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T vs U? n These 2 tests are identical in hypothesis formulation. n They require 2 samples which may be from the same population. These samples need not be of equal #, nor are they paired. u H0: The 2 samples are from the same population - any differences are due to chance u H1: The 2 samples come from different populations.

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1 big difference (+ a few small ones): n The t test is a parametric test - it assumes the data are normally distributed. n 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.

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The basic idea σ μ How many s.d.s is this data point from the mean? Z i = (X i - μ)/σ We can look up Z in tables, but these assume that the values of μ and σ are known perfectly. Remember Z scores? These apply to the idealised normal distribution

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Gosset’s discovery: n Was the formulae appropriate to Z when the sample is small, so that μ and S are based on inadequate data. n To distinguish this distribution from the idealised normal distribution, Gosset named the function the “t statistic”, and the value of (X i - μ)/S when μ and S are estimates was renamed from Z to t. n 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.

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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

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The details are slightly more messy.. n Because of the question “How do we calculate the pooled sd?” n There are several ways of doing this which make different assumptions, and give slightly different answers. n The simplest model assumes that the 2 samples have a common variance, and gives t as follows: n Given data X1, X2 which have N1, N2 datapoints each, and sums of squares SSx1, SSx2 n t = (μ1 - μ2) with N1+n2-1 df n __________ sq.root [ (SSx1 + SSx2)*(1/Nx+1/Ny) / (n1+n2-2)]

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Beware! n I spent an afternoon in the library once checking ways to calculate t. n I found 3 different formulae, plus several confusing ways to express the relationship I just showed you. n 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. n The third approach assumes that samples are accurately paired - the paired samples t test.

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n x1x2 n 57.864.2 n 56.258.7 n 61.963.1 n 54.462.5 n 53.659.8 n 56.459.2 n 53.2 n n76 n Sum x393.5367.5 n sumx*222174.4122535.87 n mean56.21%61.25% n ss54.08926.495

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So you know what to do to compare 2 groups! n You have the choice of M-W U, or Student’s t test. n But what if there are 3 groups, or 4, or 5? n You may work out the following routine: u Test group 1 vs group 2, then 2 vs 3, etc. u Clever, but WRONG! (The danger with multiple tests is that you will get a “p=0.05” significant result more often than 1:20). X1X2X3

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Multiple groups can be compared.. n With a suitable multiple test. n There are 2 options here, both of which are usually run on PCs. n Parametric data: Analysis of variance ANOVA n Non-Parametric data: Kruskal-Wallis ANOVA. u I make M.Sc. students run ANOVA calculations by hand, but K-W ANOVA is PC only.

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Kruskal_Wallis ANOVA Analysis of variance (ANOVA) >=2 Mann-Whitney U test T test2 Non- Parametric ParametricNumber of groups: Type of data

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Example: n n76 n Sum x393.5367.5 n sumx*222174.4122535.87 n mean56.21%61.25% n SS54.089 26.495 ┌─ ─┐ n Se differe = sqrt│(SSxx + SSyy)*(1/Nx+1/Ny)│ n │──────────────── │ n │ Nx+Ny-2 │ n └─ ─┘ SE diff = 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

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