Intro to basic statistics

Lilla ZölleiFeb 5, 2009Why’N’How

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Outline

Correlations

t-tests

ANOVA

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CorrelationCovariance: measures strength of linear link between two (numerical) random variables

large: strong linear link between RVssmall: weak OR nonlinear link between RVsdepends on measurement unit!!!

Correlation coefficient: standardized version on covariance

( ) ( )( )[ ]22121,Cov μμ −−= 1 XXEXX

( )( ) ( )21

21

VarVar,Cov

XXXX

=ρ

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andsymmetrical: if both have unit variances:

distributions known through a sample (Pearson):

Correlation coefficient facts11 +≤≤− ρ

21 XX = ( ) ( ) ( )2121 VarVar,Cov XXXX ==

( ) ( )1221 ,, XXXX ρρ =( ) 1, 11 +=XXρ

( ) ( )2121 ,,Cov XXXX ρ=

( )( )( ) ( )∑

∑−−

−−=

22 yyxx

yyxx

ii

iiρ

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: perfect linear functional relationship between X1 and X2

iff the relationship between X1 and X2 is linear, their relationship is weak

uncorrelated variables

≠ causality≠ independence!!!independence is much stronger than lack of correlation

Interpretation( ) 1, 21 ±=XXρ

( ) 0, 21 ≈XXρ

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Anscombe’s quartets

Image: courtesy of http://wikipedia.org

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Non-parametric corr. coeff.’s

When the sample distribution is not normal the following tests are more useful:

Chi-square, Point biserial correlation, Spearman’s, Kendall’s, Goodman’s, Kruskal’s, …

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t-testTwo strong assumptions

samples are drawn from normal populationsfor the two samples the variances (either known or unknown) are identical

“How much can we trust the sample mean as a guess of the mean of the normal distribution from which the sample was drawn ?”

Examples: Single sampleTwo samples (paired an unpaired)

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One sample t-test with reference

μ0: reference valuem : sample mean

QUESTION: "Is m significantly different from μ0?"

Null hypothesis: H0 : μ = μ0.

Image: courtesy of http://www.aiaccess.net

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Assumptions about variance

variance of the sample generating normal distribution

known distribution of the standardized sample mean is the standard normal distribution N(0, 1)unknown variance has to be estimated from the sample; but distribution of the standardized sample mean is known: (Student's) t-distribution

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t-distribution wrt N(0,1)DOF = 1, 2, 3, 5, 10, 30

Image: courtesy of http://wikipedia.org

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t-test: one- or two-sided

Two-sided:“Is there is a significant difference between m and μ0?” (in absolute!) alternative hypothesis: H1: μ ≠ μ0

One-sided:“Is the mean of the population larger (smaller) than μ0?”alternative hypothesis H1: μ ≥ μ0 (μ ≤ μ0)

H0 : μ = μ0

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Two paired samples t-test

QUESTION: “Is the average shift of the observations due to the treatment significantly different from 0?”

not : “Are the means of the two samples significantly different?”

Image: courtesy of http://www.aiaccess.net

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Two independent samples t-test

QUESTION: “Are the means m1 and m2 significantly different?”

large difference between them rejects the null hypothesis H0: μ1 = μ2

Image: courtesy of http://www.aiaccess.net

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Variations

variance of the two populations known difference between the two standardized sample means is N(0, 1)unknown difference between the two standardized sample means is t distributed

can also be one- or two-sided

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A more general scenario…

What if there are more than two groups of observations?

series of t-tests on every pair of groups; find at least one pair of groups where hypothesis is rejected

ANOVA (global test)

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REMINDER: t-testtwo strong assumptions

samples are drawn from normal populationsfor the two samples the variances (either known or unknown) are identical

test for normality: Kolmogorov-Smirnov test, Shapiro-Wilks test, …test for equality of variances: F-test, Levene’s test, Bartlett’s test…

in case the above assumptions failMann-Whitney test (non-parametric)

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To proceedSelect significance level α = 0.05 or 0.01t-test p-value

if (p < α): reject H0

else: the data is not incompatible (at this significance level) with H0 (does notmean that H0 is true!!!)

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nSXXt

XX 221

21 −=2

2221

21

XXXX

SSS

+=

21

21

1121 nn

S

XXt

XX +

−= ( ) ( )

111

21

22

21 21

21 −+

−+−=

nnSnSn

S XXXX

21

21

XXsXXt

−

−=

2

22

1

21

21 ns

nss XX +=−

NsXt

D

D 0μ−=

differences are computed!dependent (paired)

two sample

WSindep. two samples: unequal sample

size, unequal variance (Welsch)

(n1+n2-1)indep. two samples:

unequal sample size, equal variance

(2n-2)indep. two samples: equal sample size,

equal variance

(n-1)s: std; n: sample sizeindep. one sample with reference

DOFSTATISTICSCENARIO

nsxt 0μ−=

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Some history…

1908: first t-distribution derivation publishedWilliam Sealy Gosset (Student); Guinness Brewery, Dublin

t-test and theory: through work of R.A. Fisher --“Student's distribution”

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ANOVA

“ANalysis Of VAriance”a type of parametric statistical test~ generalization of t-test for > 2 groupsuses the F-testto show that there is a difference between distribution means

also: Fisher’s ANOVA (Fisher’s F-distribution)

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Example

Null hypothesis: H0 : μ1 = μ2 = μ3

Assumptions about the sampling distributions: normal, identical variances (homoscedasticity), independent

Image: courtesy of http://www.aiaccess.net

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To proceedSelect significance level α = 0.05 or 0.01ANOVA F-value

if (F < α): reject H0

else: the data is not incompatible (at this significance level) with H0 (does notmean that H0 is true!!!)

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After ANOVA rejects H0 …

Follow-up tests to analyze the reasons why the hypothesis was rejected: planned (a priori) vs. post hoc (a posteriori)post hoc examples: Tukey’s test, Dunnett'stest

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Power analysis

To determine sufficient sample size to possibly reject null hypothesis

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REMINDER: ANOVAStrong assumptions

normal sampling distributionshomogeneous variance of the sampling distributions

To test for them:normality: Kolmogorov-Smirnov test, Shapiro-Wilk test, …homoscedasticity: Bartlett test, Levene test, …

In case the above assumptions failKruskal-Wallace test (non-parametric)

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References / useful citesWIKIPEDIAhttp://www.aiaccess.net/English/home.htm(images in the talk were obtained from above)

http://www.physics.csbsju.edu/stats/anova.htmlhttp://udel.edu/~mcdonald/statanovaintro.html