Uni/bivariate Probleme
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Transcript of Uni/bivariate Probleme
26/06/2008-1-
Praktische Statistik für Umwelt- und Geowissenschaftler
Uni/bivariate Probleme
Parametrische Verfahren Nicht parametrische Verfahren
Unabhängigkeit Normalverteilung Ausreißer (Phasen-Iterationstest) (KS-Test / Chi-Quadrat Test) (Dixon / Chebyshev's Theorem)
Verteilungstest
Vergleich von Mittelwerten mit dem Parameter der GG
Zusammenhangsanalyse
Voraussetzungen erfüllt ?
Vergleich von 2 unabhängigen Stichproben
Vergleich von k unabhängigen Stichproben
KS-Test / Chi-quadrat Test
H-Test
Rangkorrelation nachSpearmann
Einstichproben T-testChi-quadrat Test
Zweistichproben T-testF-Test / Levene Test
Varianzanalyse (ANOVA)
Pearson‘s Korrelationsanalyse/Regressionsanalyse
Vergleich von 2 verbundenen Stichproben Wilcoxon-Test
U-Test
T-Test für verbundene Stichproben
NeinJa
Mehrfachvergleiche: Post hoc tests
Mehrfachvergleiche: Bonferroni Korrektur,
Šidàk-Bonferonni correction
26/06/2008-2-
Praktische Statistik für Umwelt- und Geowissenschaftler
Data analysis Data mining
Reduction Classification Data Relationships
Principal Component
Analysis
FactorAnalysis
CorrespondenceAnalysis
HomogeneityAnalysis
Non-linearPCA
ProcrustesAnalysis
FactorAnalysis
DiscriminantAnalysis
HierarchicalCluster Analysis
MultidimensionalScaling
K-MeansArtificialNeural
Networks
MultipleRegression
PrincipalComponentRegression
LinearMixtureAnalysis
PartialLeast
Squares - 2
PartialLeast
Squares -1
Canonical Analysis
SupportVector
Machines
ANNSVM
ANNSVM
Categorization of multivariate methods
26/06/2008-3-
Praktische Statistik für Umwelt- und Geowissenschaftler
Vorgehen beim statistischen testen:
a) Aufstellen der H0/H1-Hypothese
b) Ein- oder zweiseitige Fragestellung
c) Auswahl des Testverfahrens
d) Festlegen des Signikanzniveaus (Fehler 1. und 2. Art)
e) Testen
f) Interpretation
26/06/2008-4-
Praktische Statistik für Umwelt- und Geowissenschaftler
In Population giltE
ntsc
heid
ung
aufg
rund
der
Stic
hpro
be
H0 H1H
1H
0 richtig, mit 1-α β-Fehler P(H0¦H1)= β
α-Fehler P(H1¦H0)= α richtig, mit 1- β
Fehler 1. und 2. Art
26/06/2008-5-
Praktische Statistik für Umwelt- und Geowissenschaftler
Bestimmen von Irrtumswahrscheinlichkeiten
42 / K x mg g sei eine normalverteilte Stichprobe (nach 1. Grenzwertsatz) unbekannter Herkunft, mit 8, 100n
5%
40 / Hunsrück mg g K
43 / Eifel mg g K
.1: HunsH x .0 : HunsH x Probe stammt aus dem Hunsrück
Probe stammt aus der Eifel
26/06/2008-6-
Praktische Statistik für Umwelt- und Geowissenschaftler
0
x
xz
Test: Einstichproben Gauss Test
mit 0.8x n
42 40 2.50.8
z Wert schneidet 0.62% von NV ab
(P-Wert = Irrtumswahrscheinlichkeit)
H0 muss verworfen werden!
P-Wert wird gleinermit > Diff.mit <mit > n
0x
x
α=5%, ~Z=1.65
26/06/2008-7-
Praktische Statistik für Umwelt- und Geowissenschaftler
Frage: Welches muss überschritten werden, um H0 mit gerade verwerfen zu können?
(1 )critx
5%
schneided von der rechten Seite der SNV genau 5% ab1.65z
( ) 0
= 40 + 1.65 0.8 = 41.32
crit xx z
26/06/2008-8-
Praktische Statistik für Umwelt- und Geowissenschaftler
Zweiseitiger Test:
1.96z
Hunsx
schneidet auf jeder Seite der SNV genau 2.5% ab
( /2) 40 - 1.96 0.8 = 38.43 40 + 1.96 0.8 = 41.57
critx H0 wird knapper abgelehnt!
Entscheidung ein-/zweiseitiger Test muss im Vorfeld erfolgen!
26/06/2008-9-
Praktische Statistik für Umwelt- und Geowissenschaftler
Der β-Fehler
Kann nur bei spezifischer H1 bestimmt werden!
Wir testen, ob sich die Stichprobe mit dem Parameter der Eifelproben verträgt
0
x
xz
42 43 1.25
0.8
Wert schneidet auf der linken Seite der SNV 10.6% ab.
Entscheidet man sich aufgrund des Ereignisses für die H0, so wird man mit einer p von 10.6% einen β-Fehler begehen, d.h. H1 (« Probe stammt aus der Eifel ») verwerfen, obwohl sie richtig ist.
42 / K x mg g
26/06/2008-10-
Praktische Statistik für Umwelt- und Geowissenschaftler
Die Teststärke
Die β-Fehlerwahrscheinlichkeit gibt an, mit welcher p die H1 verworfen wird, obwohl ein Unterschied besteht
1- β gibt die p an zugunsten von H1 zu entscheiden, wenn H1 gilt.
Bestimmen der Teststärke
Wir habe herausgefunden, dass ab einem Wert der Test gerade signifikant wird (« Probe stammt aus der Eifel »)
41.32 / K x mg g
26/06/2008-11-
Praktische Statistik für Umwelt- und Geowissenschaftler
41.32 43 2.10.8
Bestimmen der Teststärke
β-Wahrscheinlichkeit: 0.0179
Teststärke: 1-β =1-0.0179 = 0.9821
Die p, dass wir uns aufgrund des gewählten Signifikanzniveaus (α=5%) zu Recht zugunsten der H1 entscheiden, beträgt 98.21%
Determinanten der Teststärke:
Mit kleiner werdener Diff. µ0-µ1 verringert sich 1- βMit wachsendem n vergrössert sich 1- βMit wachsender Merkmalsstreuung sinkt 1- β
26/06/2008-12-
Praktische Statistik für Umwelt- und Geowissenschaftler
Why multivariate statistics?
Fancy statistics do not make up for poor planning
Design is more important than analysis
Remember
26/06/2008-13-
Praktische Statistik für Umwelt- und Geowissenschaftler
• Prediction Methods– Use some variables to predict unknown or future values of
other variables.
• Description Methods– Find human-interpretable patterns that describe the data.
From [Fayyad, et.al.] Advances in Knowledge Discovery and Data Mining, 1996
Categorization of multivariate methods
26/06/2008-14-
Praktische Statistik für Umwelt- und Geowissenschaftler
Multiple Linear Regression Analysis
The General Linear Model
A general linear model can be:straight-line modelquadratic model (second-order model) more than one independent variables. E.g.
222110
222110
)(
)(
xxyE
orxxyE iii
x1
x2
y
0=10
0
(xi1, xi2)
E(yi)yi
i
Response Surface
26/06/2008-15-
Praktische Statistik für Umwelt- und Geowissenschaftler
y=x1 + x2 – x1 + 2 x12 + 2 x2
2
Multiple Linear Regression Analysis
26/06/2008-16-
Praktische Statistik für Umwelt- und Geowissenschaftler
The goal of an estimator is to provide an estimate of a particular statistic based on the data. There are several ways to characterize estimators:
Bias: an unbiased estimator converges to the true value with large enough sample size. Each parameter is neither consistently over or under estimated
Likelihood: the maximum likelihood (ML) estimator is the one that makes the observed data most likely ML estimators are not always unbiased for small N
Efficient: an estimator with lower variance is more efficient, in the sense that it is likely to be closer to the true value over samples the “best” estimator is the one with minimum variance of all estimators
Parameter Estimation
Multiple Linear Regression Analysis
26/06/2008-17-
Praktische Statistik für Umwelt- und Geowissenschaftler
A linear model can be written as XY
TNyyY ,...,1Where: is an N-dimensional column vector of observations
Tk ,...,0 is a (k+1)-dimensional column vector of unknown parameters
TN ,...1 is an N-dimensional random column vector of unobserved errors
Matrix X is written as
NkN
k
k
XX
XXXX
X
1
221
111
1
11
TN 1,...,11 0The first column of X is the vector , so that the first coefficient is the intercept.
N
it
Tt XxRSS
1
2)(
The unknown coefficient vector is estimated by minimizing the residual sum of squares
Multiple Linear Regression Analysis
26/06/2008-18-
Praktische Statistik für Umwelt- und Geowissenschaftler
Mean of errors is zero: Errors have a constant variance: Errors from different observations are independent of each other: forErrors follow a Normal Distribution.Errors are not uncorrelated with explanatory variable:
Model assumptionsThe OLS estimator can be considered as the best linear unbiased estimator (BLUE) of provided some basic assumptions regarding the error term are satisfied :
te
0)( tE 22)(
tE
0)( ktE kt
0)(: ktt XEX
Multiple Linear Regression Analysis
26/06/2008-19-
Praktische Statistik für Umwelt- und Geowissenschaftler
For a multiple regression model :
1 should be interpreted as change in y when a unit change is observed in x1 and x2 is kept constant. This statement is not very clear when x1 and x2 are not independent.
Misunderstanding: i always measures the effect of xi on E(y), independent of other x variables.
Misunderstanding: a statistically significant value establishes a cause and effect relationship between x and y.
iiii exxy 22110
Interpreting Multiple Regression Model X2
X1
Y
Multiple Linear Regression Analysis
26/06/2008-20-
Praktische Statistik für Umwelt- und Geowissenschaftler
If the model is useful…At least one estimated must 0
But wait …What is the chance of having one estimated significant if I have 2 random x?
For each , prob(b 0) = 0.05At least one happen to be b 0, the chance is:
Prob(b1 0 or b2 0) = 1 – prob(b1=0 and b2=0) = 1-(0.95)2 = 0.0975 Implication?
Explanation Power by
Multiple Linear Regression Analysis
26/06/2008-21-
Praktische Statistik für Umwelt- und Geowissenschaftler
RR22 (multiple correlation squared) – variation in (multiple correlation squared) – variation in YY accounted for by the set of accounted for by the set of predictorspredictorsAdjusted RAdjusted R22. . The adjustment takes into account the size of the sample and The adjustment takes into account the size of the sample and number of predictors to adjust the value to be a better estimate of the number of predictors to adjust the value to be a better estimate of the population value.population value.
Adjusted RAdjusted R22 = R = R22 - ( - (kk - 1) / ( - 1) / (n - kn - k) * (1 - R) * (1 - R22))Where: Where:
nn = # of observations, = # of observations,kk = # of independent variables, = # of independent variables,
Accordingly: smaller Accordingly: smaller nn decreases R decreases R22 value; larger value; larger nn increases R increases R22 value; value; smaller smaller kk, increases R, increases R22 value; larger value; larger kk, decreases R, decreases R22 value. value. The The F-F-test in the ANOVA table to judge whether the explanatory variables test in the ANOVA table to judge whether the explanatory variables in the model adequately describe the outcome variable.in the model adequately describe the outcome variable.The The t-t-test of each partial regression coefficient. Significanttest of each partial regression coefficient. Significant t t indicates that indicates that the variable in question influences the the variable in question influences the YY response while controlling for other response while controlling for other explanatory variables.explanatory variables.
Analysis
Multiple Linear Regression Analysis
26/06/2008-22-
Praktische Statistik für Umwelt- und Geowissenschaftler
Source of Variance SS df MS
Regression p-1 MSR=SSR/(p-1)
Error n-p MSE=SSE/(n-p)
Total n-1
JYY'YX''β
)yy()'yy(
)1(ˆ
ˆˆ
n
SSR
)y(y)'y(y SST
yX'βyy'
)y(y)'y(yˆ
ˆˆ
SSE
ANOVA
where J is an nn matrix of 1s
Multiple Linear Regression Analysis
26/06/2008-23-
Praktische Statistik für Umwelt- und Geowissenschaftler
The R2 statistic measures the overall contribution of Xs.
Then test hypothesis:H0: 1=… k=0H1: at least one parameter is nonzero
Since there is no probability distribution form for R2, F statistic is used instead.
2 1 SSE SSRRSST SST
Multiple Linear Regression Analysis
26/06/2008-24-
Praktische Statistik für Umwelt- und Geowissenschaftler
dof 1)(k-n vdof, p v where,FF :regionRejection )1/()1(
/
)1/(/
21
2
2
knRpRF
knSSEkSSR
MSEMSRF
F-statistics
Multiple Linear Regression Analysis
26/06/2008-25-
Praktische Statistik für Umwelt- und Geowissenschaftler
How many variables should be included in the model?
Basic strategies:Sequential forwardSequential backwardForce entire
1
1,1 kk
kNRSS
RSSRSSF kNkk
The first two strategies determine a suitable number of explanatory variables using the semi-partial correlation as criterion and a partial F-statistics which is calculated from the error terms from the restricted (RSS1) and unrestricted (RSS) models:
where k, k1 denotes the number of lags of the unrestricted and restricted model, and N is the number of observations.
Multiple Linear Regression Analysis
26/06/2008-26-
Praktische Statistik für Umwelt- und Geowissenschaftler
Measures the relationship between a predictor and the outcome, controlling for the relationship between that predictor and any others already in the model.
It measures the unique contribution of a predictor to explaining the variance of the outcome.
The semi-partial correlation Z
X
Y
Multiple Linear Regression Analysis
26/06/2008-27-
Praktische Statistik für Umwelt- und Geowissenschaftler
2An unbiased estimator for the variance is
kNRSSs
2
The regression coefficients are tested for significance under the Null-Hypothesis using a standard t-test
0:0 iH
iiikN cst /^
Where denotes the ith diagonal element of the matrix . is also referred to as standard error of a regression coefficient .
iic 1 XXC T
iics
i
Testing the regression coefficients
Multiple Linear Regression Analysis
26/06/2008-28-
Praktische Statistik für Umwelt- und Geowissenschaftler
Which X is contributing the most to the prediction of Y?
Cannot interpret relative size of bs because each are relative to the variables scalebut s (Betas; standardized Bs) can be interpreted.
a is the mean on Y which is zero when Y is standardized
1 1 2 2 3 3
1 1 2 2 3 3
'' ( ) ( ) ( )
y a b x b x b xZy Zx Zx Zx
Multiple Linear Regression Analysis
26/06/2008-29-
Praktische Statistik für Umwelt- und Geowissenschaftler
Can the regression equation be generalized to other data?
Can be evaluated by randomly separating a data set into two halves. Estimate regression equation with one half and apply it to the other half and see if it predicts Cross-validation
Multiple Linear Regression Analysis
26/06/2008-30-
Praktische Statistik für Umwelt- und Geowissenschaftler
MSEe
n
)(
:sample from estimate weunknonw, is population Since
ˆˆ
: termsresidual theand
ˆˆ
:is ,ˆ denoted , of valuefitted The
2
2
1
s
βXyyye
βXy
yy
1n
Residual analysis
n
ippii xxySSEMin
1
211110 )(
Multiple Linear Regression Analysis
26/06/2008-31-
Praktische Statistik für Umwelt- und Geowissenschaftler
Divide the residuals into two (or more) groups based the level of x, The variances and the means of the two groups are supposed to be equal. A
standard t-test can be used to test the difference in mean. A large t indicates nonconsistancy.
e
x/E(y)
0
The Revised Levene’s test
Multiple Linear Regression Analysis
26/06/2008-32-
Praktische Statistik für Umwelt- und Geowissenschaftler
Influential points are those whose exclusion will cause major change in fitted line.
“Leave-one-out” crossvalidation. If ei > 4s, it is considered as outlier. True outlier should not be
removed, but should be explained.
Detecting Outliers and Influential Observations
0.10.0-0.1-0.2
0.4
0.3
0.2
0.1
0.0
-0.1
Fitted Value
Re
sid
ual
Residuals Versus the Fitted Values(response is m1)
Multiple Linear Regression Analysis
26/06/2008-33-
Praktische Statistik für Umwelt- und Geowissenschaftler
Example for a Generalized Least-Square model which can be used instead of OLS-regression in the case of autocorrelated error terms (e.g. in Distributed Lag-Models)
Generalized Least-Squares
Multiple Linear Regression Analysis
26/06/2008-34-
Praktische Statistik für Umwelt- und Geowissenschaftler
SPSS-Example
Multiple Linear Regression Analysis
26/06/2008-35-
Praktische Statistik für Umwelt- und Geowissenschaftler
SPSS-Example
Multiple Linear Regression Analysis
26/06/2008-36-
Praktische Statistik für Umwelt- und Geowissenschaftler
SPSS-ExampleModel evaluation
Multiple Linear Regression Analysis
26/06/2008-37-
Praktische Statistik für Umwelt- und Geowissenschaftler
Studying residual helps to detect if:Model is nonlinear in functionMissing xOne or more assumptions of is violated.Outliers
SPSS-ExampleModel evaluation
Multiple Linear Regression Analysis
26/06/2008-38-
Praktische Statistik für Umwelt- und Geowissenschaftler
ANalysis Of VAriance
ANOVA (ONE-WAY)ANOVA (TWO-WAY)
MANOVA
ANOVA
26/06/2008-39-
Praktische Statistik für Umwelt- und Geowissenschaftler
Comparing more than two groups
• ANOVA deals with situations with one observation per object, and three or more groups of objects
• The most important question is as usual: Do the numbers in the groups come from the same population, or from different populations?
ANOVA
26/06/2008-40-
Praktische Statistik für Umwelt- und Geowissenschaftler
One-way ANOVA: Example• Assume ”treatment results” from 13 soil
plots from three different regions: – Region A: 24,26,31,27– Region B: 29,31,30,36,33– Region C: 29,27,34,26
• H0: The treatment results are from the same population of results
• H1: They are from different populations
ANOVA
26/06/2008-41-
Praktische Statistik für Umwelt- und Geowissenschaftler
Comparing the groups• Averages within groups:
– Region A: 27– Region B: 31.8– Region C: 29
• Total average: • Variance around the mean matters for comparison. • We must compare the variance within the groups
to the variance between the group means.
4 27 5 31.8 4 29 29.464 5 4
ANOVA
26/06/2008-42-
Praktische Statistik für Umwelt- und Geowissenschaftler
Variance within and between groups• Sum of squares within groups:
• Sum of squares between groups:
• The number of observations and sizes of groups has to be taken into account!
2 2 2(24 27) (26 27) ... (29 31.8) .... 94.8SSW
2 2 2
2 2 2
(27 29.46) (27 29.46) ... (31.8 29.46) ....
4(27 29.46) 5(31.8 29.46) 4(29 29.46) 52.43
SSG
ANOVA
26/06/2008-43-
Praktische Statistik für Umwelt- und Geowissenschaftler
Adjusting for group sizesSSWMSWn K
1SSGMSGK
Both are estimates of population variance of error under H0
n: number of observationsK: number of groups
• If populations are normal, with the same variance, then we can show that under the null hypothesis,
• Reject at confidence level if
1,~ K n KMSG FMSW
1, ,K n KMSG FMSW
ANOVA
26/06/2008-44-
Praktische Statistik für Umwelt- und Geowissenschaftler
Continuing example
• -> H0 can not be rejected
94.8 9.4813 3
SSWMSWn K
52.43 26.21 3 1
SSGMSGK
26.2 2.769.48
MSGMSW
3 1,13 3,0.05 4.10F
ANOVA
26/06/2008-45-
Praktische Statistik für Umwelt- und Geowissenschaftler
ANOVA table
Source of variation
Sum of squares
Deg. of freedom
Mean squares
F ratio
Between groups
SSG K-1 MSG
Within groups
SSW n-K MSW
Total SST n-1
MSGMSW
2 2 2(24 29.46) (26 29.46) ... (26 29.46)SST SSG SSW SST NOTE:
26/06/2008-46-
Praktische Statistik für Umwelt- und Geowissenschaftler
When to use which method• In situations where we have one observation per
object, and want to compare two or more groups: – Use non-parametric tests if you have enough data
• For two groups: Mann-Whitney U-test (Wilcoxon rank sum)• For three or more groups use Kruskal-Wallis
– If data analysis indicate assumption of normally distributed independent errors is OK
• For two groups use t-test (equal or unequal variances assumed)• For three or more groups use ANOVA
ANOVA
26/06/2008-47-
Praktische Statistik für Umwelt- und Geowissenschaftler
Two-way ANOVA (without interaction)• In two-way ANOVA, data fall into categories in
two different ways: Each observation can be placed in a table.
• Example: Both type of fertilization and crop type should influence soil properties.
• Sometimes we are interested in studying both categories, sometimes the second category is used only to reduce unexplained variance. Then it is called a blocking variable
ANOVA
26/06/2008-48-
Praktische Statistik für Umwelt- und Geowissenschaftler
Sums of squares for two-way ANOVA• Assume K categories, H blocks, and assume
one observation xij for each category i and each block j block, so we have n=KH observations. – Mean for category i: – Mean for block j: – Overall mean:
ix
jx
x
ANOVA
26/06/2008-49-
Praktische Statistik für Umwelt- und Geowissenschaftler
Sums of squares for two-way ANOVA
2
1
( )K
ii
SSG H x x
2
1
( )H
jj
SSB K x x
2
1 1
( )K H
ij i ji j
SSE x x x x
2
1 1
( )K H
iji j
SST x x
SSG SSB SSE SST
ANOVA
26/06/2008-50-
Praktische Statistik für Umwelt- und Geowissenschaftler
ANOVA table for two-way data
Source of variation
Sums of squares
Deg. of freedom
Mean squares F ratio
Between groups SSG K-1 MSG= SSG/(K-1) MSG/MSE
Between blocks SSB H-1 MSB= SSB/(H-1) MSB/MSE
Error SSE (K-1)(H-1) MSE= SSE/(K-1)(H-1)
Total SST n-1
Test for between groups effect: compare to
Test for between blocks effect: compare to
MSGMSEMSBMSE
1,( 1)( 1)K K HF
1,( 1)( 1)H K HF
26/06/2008-51-
Praktische Statistik für Umwelt- und Geowissenschaftler
Two-way ANOVA (with interaction)• The setup above assumes that the blocking
variable influences outcomes in the same way in all categories (and vice versa)
• Checking interaction between the blocking variable and the categories by extending the model with an interaction term
ANOVA
26/06/2008-52-
Praktische Statistik für Umwelt- und Geowissenschaftler
Sums of squares for two-way ANOVA (with interaction)
• Assume K categories, H blocks, and assume L observations xij1, xij2, …,xijL for each category i and each block j block, so we have n=KHL observations. – Mean for category i: – Mean for block j:– Mean for cell ij: – Overall mean:
ix
jx
xijx
ANOVA
26/06/2008-53-
Praktische Statistik für Umwelt- und Geowissenschaftler
Sums of squares for two-way ANOVA (with interaction)
2
1
( )K
ii
SSG HL x x
2
1
( )H
jj
SSB KL x x
2
1 1
( )K H
ij i ji j
SSI L x x x x
2
1 1 1
( )K H L
ijli j l
SST x x
SSG SSB SSI SSE SST
2
1 1 1
( )K H L
ijl iji j l
SSE x x
ANOVA
26/06/2008-54-
Praktische Statistik für Umwelt- und Geowissenschaftler
ANOVA table for two-way data (with interaction)
Source of variation Sums of squares
Deg. of freedom
Mean squares F ratio
Between groups SSG K-1 MSG= SSG/(K-1) MSG/MSE
Between blocks SSB H-1 MSB= SSB/(H-1) MSB/MSE
Interaction SSI (K-1)(H-1) MSI= SSI/(K-1)(H-1)
MSI/MSE
Error SSE KH(L-1) MSE= SSE/KH(L-1)
Total SST n-1
Test for interaction: compare MSI/MSE with Test for block effect: compare MSB/MSE with Test for group effect: compare MSG/MSE with 1, ( 1)K KH LF
1, ( 1)H KH LF
( 1)( 1), ( 1)K H KH LF
26/06/2008-55-
Praktische Statistik für Umwelt- und Geowissenschaftler
Notes on ANOVA• All analysis of variance (ANOVA) methods
are based on the assumptions of normally distributed and independent errors
• The same problems can be described using the regression framework. We get exactly the same tests and results!
• There are many extensions beyond those mentioned
ANOVA
26/06/2008-56-
Praktische Statistik für Umwelt- und Geowissenschaftler
MANOVA Uses Multiple DVs
• Various measures of soil properties– Corg, Cmik, N, pH,…
• Various outcome measures following different types of categories– Fertilization, point in time, crop type,…
Predictors (IVs) Criterion (DV(s))ANOVA Multiple, discrete Single, continuousMANOVA Multiple, discrete Multiple, continuous
MANOVA
26/06/2008-57-
Praktische Statistik für Umwelt- und Geowissenschaftler
• Multiple DVs could be analysed using multiple ANOVAs, but:– The FW increases with each ANOVA– Scores on the DVs are likely correlated
• Non-independent, and taken from the same subjects• Hard to interpret results if multiple ANOVAs are
significant
• MANOVA solves this by conducting only one overall test– Creates a ‘composite’ DV– Tests for significance of the composite DV
MANOVA
26/06/2008-58-
Praktische Statistik für Umwelt- und Geowissenschaftler
• The Composite DV is a linear combination of the DVs– i.e., a discriminant function, or root– The weights maximally separate the groups on the
composite DV
C = W1Y1 + W2Y2 + W3Y3 + …+ WnYn
where, C is a subject’s score on the composite DVYi are scores on each of the DVsWi are the weights, one for each DV
A composite DV is required for each main effect and interaction
MANOVA
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• Considering the DVs together can enhance power
a. Frequency distributions show considerable overlap between groups on the individual DVs
b. The elipses, that reflect the DVs in combination, show less overlap
c. Small differences on each DV combine to make a larger multivariate difference
MANOVA
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• In ANOVA, the sums of squared deviations are partitioned: SST = SSA + SSB + SSAxB + SSS/AB
• In MANOVA, the sum of squares cross-products are partitioned: ST = SD + STr + SDxTr + SS(DTr)
• The SSCP matrices (S) are analogous to the SS– SSCP matrix is a squared deviation that also
reflects correlations among the DVs
2TTSS Y Y
MANOVA
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Scores and Means in MANOVA are Vectors
• Y: Scores for each subject• T and D: Row and column marginals• GM: the grand mean • DTr: the average scores of subjects within cells
MANOVA
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MANOVA
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• The deviation score for the first subject is:
• The squared deviation is obtained by multiplying by the transpose:
SS are on the diagonal: (25.89)2 = 670, and (20.78)2 = 431 Cross-products are on the off-diagonals: (25.89)(20.78)=538
• And:
111
115 89 26108 87 21
Y GM
111 111
26 670 53826 21
21 538 431Y GM Y GM
T iii iiiS Y GM Y GM
MANOVA
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• The squaring of a matrix is carried out by multiplying it by its transpose
• The transpose is obtained by flipping the matrix about its diagonal:
• To multiply, the ijth element in the resulting matrix is obtained by the sum of products of the ith row in A and the jth column in A'
• For a vector, the transpose is a row vector, and:
a b cA d e f
g h i
a d gA b e h
c f i
( )( ) ( )( )( )( ) ( )( )
a a a a bAxA a b
b b a b b
MANOVA
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• Main Effects in ANOVA vs. MANOVA:
D i iS n t D GM D GM Tr i iS n d Tr GM Tr GM
2A TASS n b Y Y
DxTr cells D TrS S S S
2
/ ABS ABSS Y Y ( )S DTrS Y DTr Y DTr
• The Interaction:
• The Error Term:
SS n Y YC ells A B T 2
CellsS n DTr GM DTr GM
AxB Cells A BSS SS SS SS
MANOVA
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• In ANOVA, variance estimates (MS) are obtained from the SS for significance testing using the F-statistic
• In MANOVA, variance estimates (determinants) are obtained from the SSCP matrices for significance testing e.g. using Wilk’s Lambda ()
ANOVA MANOVA SS ~ SSCP MS ~ |SSCP|
~
Note that F and are inverse to one another
Effect
Error
MSF
MS Error
Effect Error
SS S
MANOVA
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• The determinant of a 2x2 matrix is given by:
( ) ( )
544 31, det 544 539 31 31 292434
31 529S DT S DTS S
( ) ( )
546 36, det 546 529 36 36 322040
36 529DT S DT DT S DTS S S S
, deta b
if A then A A a d b cc d
• The determinants required to test the interaction are:
( )
( )
292434 0.908322040
S DT
DT S DT
S
S S
Error
Effect Error
SS S
• Wilk’s Lambda for the Interaction is obtained by:
MANOVA
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• If the effect is small, then approaches 1.0
– Here SDT was small, and was 0.91
Error
Effect Error
SS S
• Eta Squared for MANOVA is:• 2 = 1 - Effect
• = 1 – 0.91 • = 0.09
• The interaction accounts for only 9% of the variance in the group means on the composite DV
MANOVA
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MANOVA SPSS ExampleMANOVA
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MANOVA SPSS Example
MANOVA
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MANOVA
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MANOVA
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MANOVA
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Discriminant analysis is used to predict group memberships from a set of continuous predictors
Analogy to MANOVA: in MANOVA linearly combined DVs
are created to answer the question if groups can be separated.
The same “DVs” can be used to predict group membership!!
Discriminant Analysis
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What is the goal of Discriminant Analysis?
− Perform dimensionality reduction “while preserving as much of the class discriminatory information as possible”.
− Seeks to find directions along which the classes are best separated.
− Takes into consideration the scatter within-classes but also the scatter between-classes.
Discriminant Analysis
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MANOVA and Disriminant Analysis (DA) are mathematically identical but are different in terms of emphasis:
– DA is usually concerned with grouping of objects (classification) and testing how well objects were classified (one grouping variable, one or more predictor variables)
– Discriminant functions are identical to canonical correlations between the groups on one side and the predictors on the other side.
– MANOVA is applied to test if groups significantly differ from each other (one or more grouping variables, one or more predictor variables)
Discriminant Analysis
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Discriminant Analysis
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Assumptions – small number of samples might lead to overfitting.– If there are more DVs than objects in any cell the cell will
become singular and cannot be inverted. – If only a few cases more than DVs equality of covariance
matrices is likely to be rejected.– With a small objects/DV ratio power is likely to be very small– Multivariate normality: the means of the various DVs in each
cell and all linear combinations of them are normally distributed
– Absence of outliers – significance assessment is very sensitive to outlying cases
– Homogeneity of Covariance Matrices. DA is relatively robust to violations of this assumption if interference is the focus of the analysis, but not in classification.
Discriminant Analysis
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Assumptions
— For classification purposes DA is highly influenced by violations for the last assumption, since subjects will tend to be classified into groups with the largest variance
— Homogeneity of class variances can be assessed by plotting pairwise the discriminant function scores for the first discriminant functions.
— LDA assumes linear relationships between all predictors within each group. Violations tend to reduce power and not increase alpha.
— Absence of Multicollinearity/Singularity in each cell of the design: Avoid redundant predictors
Discriminant Analysis
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Interpreting a Two-Group Discriminant Function
In the two-group case, discriminant function analysis is analogous to multiple regression; the two-group discriminant analysis is also called Fisher linear discriminant analysis.
In general, in the two-group case we fit a linear equation of the type:
c = a + d1*x1 + d2*x2 + ... + dm*xm
where a is a constant and d1 through dm are regression coefficients and c is the predicted class. The interpretation of the results of a two-group problem is straightforward and closely follows the logic of multiple regression: Those variables with the largest (standardized) regression coefficients are the ones that contribute most to the prediction of group membership.
Discriminant Analysis
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Discriminant Functions for Multiple Groups
When there are more than two groups, then we can estimate more than one discriminant function. For instance, when there are three groups, there exist a function for discriminating between group 1 and groups 2 and 3 combined, and another function for discriminating between group 2 and group 3.
Canonical analysis. In a multiple group discriminant analysis, the first function is defined such that it provides the most overall discrimination between groups, the second provides second most, and so on.
All functions are independent or orthogonal. Computationally, a canonical correlation analysis is performed that determines the successive functions and canonical roots.
The number of function that can be calculated is:
Min [number of groups-1;number of variables]
Discriminant Analysis
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Eigenvalues
Eigenvalus can be interpreted as the proportion of variance accounted for by the correlation between the respective canonical variates.
Successive eigenvalues will be of smaller and smaller size. First, compute the weights that maximize the correlation of the two sum scores. After this first root has been extracted, you will find the weights that produce the second largest correlation between sum scores, subject to the constraint that the next set of sum scores does not correlate with the previous one, and so on.
Canonical correlations. If the square root of the eigenvalues is taken, then the resulting numbers can be interpreted as correlation coefficients. Because the correlations pertain to the canonical variates, they are called canonical correlations.
Discriminant Analysis
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Let be the total number of samples. And
1 1
( )( )iMC
Tw j i j i
i j
S x x
μ μ
1
1/C
ii
C
Suppose there are C classesLet µi be the mean vector of class i, i = 1,2,…, C
Within-class scatter matrix:
1
( )( )C
Tb i i
i
S
μ μ μ μ
1
1/C
ii
C
Between-class scatter matrix:
Where = mean of the entire data set
and t B WS S S
Discriminant Analysis
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• Methodology
– LDA computes a transformation that maximizes the between-class scatter while minimizing the within-class scatter:
| | | |max max| | | |
Tb b
Tw w
U S U SU S U S
TUy x
products of eigenvalues !
projection matrix
,b wS S : scatter matrices of the projected data y
Discriminant Analysis
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Linear transformation implied by LDA
– The linear transformation is given by a matrix U whose columns are the eigenvectors of the above problem.
– The LDA solution is given by the eigenvectors of the generalized eigenvector problem:
– Important: Since Sb has at most rank C-1, the max number of eigenvectors with non-zero eigenvalues is C-1 (i.e., max dimensionality of sub-space is C-1)
B k k W kS u S u
1 1
2 2
... ...
T
TT
Tk K
b ub u
x U x
b u
Discriminant Analysis
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1W B k k kS S u u
• Does Sw-1 always exist?
– If Sw is non-singular, we can obtain a conventional eigenvalue problem by writing:
– In practice, Sw is often singular when more variables than cases are involved in the analysis (M << N )
Discriminant Analysis
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