Université d’Ottawa / University of Ottawa 2001 Bio 8100s Allied Multivariate Biostatistics L6.1...

2001

Bio 8100s Allied Multivariate Biostatistics L6.1

Université d’Ottawa / University of Ottawa

Lecture 6: Single-classification Lecture 6: Single-classification multivariate ANOVA (multivariate ANOVA (kk-group -group

MANOVA)MANOVA)

Lecture 6: Single-classification Lecture 6: Single-classification multivariate ANOVA (multivariate ANOVA (kk-group -group

MANOVA)MANOVA) Rationale and

underlying principles Univariate ANOVA Multivariate ANOVA

(MANOVA): principles and procedures

Rationale and underlying principles

Univariate ANOVA Multivariate ANOVA

(MANOVA): principles and procedures

MANOVA test statistics MANOVA assumptions Planned and unplanned

comparisons

MANOVA test statistics MANOVA assumptions Planned and unplanned

comparisons

2001



When to use ANOVAWhen to use ANOVAWhen to use ANOVAWhen to use ANOVA Tests for effect of “discrete”

independent variables. Each independent variable is

called a factor, and each factor may have two or more levels or treatments (e.g. crop yields with nitrogen (N) or nitrogen and phosphorous (N + P) added).

ANOVA tests whether all group means are the same.

Use when number of levels (groups) is greater than two.

Tests for effect of “discrete” independent variables.

Each independent variable is called a factor, and each factor may have two or more levels or treatments (e.g. crop yields with nitrogen (N) or nitrogen and phosphorous (N + P) added).

ANOVA tests whether all group means are the same.

Use when number of levels (groups) is greater than two.

ControlExperimental (N)Experimental (N+P)

Fre

qu

en

cy

Yield

C N N+P

2001



Why not use multiple 2-sample Why not use multiple 2-sample tests?tests?

Why not use multiple 2-sample Why not use multiple 2-sample tests?tests?

For k comparisons, the probability of accepting a true H0 for all k is (1 - )k.

For 4 means, (1 - )k = (0.95)6 = .735.

So (for all comparisons) = 0.265. So, when comparing the means of

four samples from the same population, we would expect to detect significant differences among at least one pair 27% of the time.

For k comparisons, the probability of accepting a true H0 for all k is (1 - )k.

For 4 means, (1 - )k = (0.95)6 = .735.

So (for all comparisons) = 0.265. So, when comparing the means of

four samples from the same population, we would expect to detect significant differences among at least one pair 27% of the time. Yield

C N N+P


c:N N:N+P

C: N+P

Fre

qu

en

cy

2001



What ANOVA What ANOVA does/doesn’t dodoes/doesn’t do

What ANOVA What ANOVA does/doesn’t dodoes/doesn’t do

Tells us whether all group means are equal (at a specified level)...

...but if we reject H0, the ANOVA does not tell us which pairs of means are different from one another.

Tells us whether all group means are equal (at a specified level)...

...but if we reject H0, the ANOVA does not tell us which pairs of means are different from one another.

ControlExperimental (N)Experimental (N+ P) Yield

Fre

qu

en

cyC N N+P

Fre

qu

en

cy

C N

N+P

2001



Model I ANOVA: effects of Model I ANOVA: effects of temperature on trout growthtemperature on trout growth

Model I ANOVA: effects of Model I ANOVA: effects of temperature on trout growthtemperature on trout growth

3 treatments determined (set) by investigator.

Dependent variable is growth rate (), factor (X) is temperature.

Since X is controlled, we can estimate the effect of a unit increase in X (temperature) on theeffect size...

… and can predictat other temperatures.

3 treatments determined (set) by investigator.

Dependent variable is growth rate (), factor (X) is temperature.

Since X is controlled, we can estimate the effect of a unit increase in X (temperature) on theeffect size...

… and can predictat other temperatures.

Water temperature (°C)

16 20 24 28

0.00

0.04

0.08

0.12

0.16

0.20

Gro

wth

ra

te

(c

m/d

ay)

2001



Model II ANOVA: geographical Model II ANOVA: geographical variation in body size of black bearsvariation in body size of black bears

Model II ANOVA: geographical Model II ANOVA: geographical variation in body size of black bearsvariation in body size of black bears

3 locations (groups) sampled from set of possible locations.

Dependent variable is body size, factor (X) is location.

Even if locations differ, we have no idea what factors are controlling this variability...

…so we cannot predictbody sizeat other locations.

3 locations (groups) sampled from set of possible locations.

Dependent variable is body size, factor (X) is location.

Even if locations differ, we have no idea what factors are controlling this variability...

…so we cannot predictbody sizeat other locations.

Bo

dy

siz

e (

kg

)

120

160

200

240

280

RidingMountain

Kluane Algonquin

2001



Model differencesModel differences

In Model I, the putative causal factor(s) can be manipulated by the experimenter, whereas in Model II they cannot.

In Model I, we can estimate the magnitude of treatment effects and make predictions, whereas in Model II we can do neither.

In one-way (single classification) ANOVA, calculations are identical for both models…

…but this is NOT so for multiple classification ANOVA!

2001



How is it done? And why call it How is it done? And why call it ANOVA?ANOVA?

In ANOVA, the total variance in the dependent variable is partitioned into two components:

among-groups: variance of means of different groups (treatments)

within-groups (error): variance of individual observations within groups around the mean of the group

2001



The general The general ANOVA modelANOVA model

The general The general ANOVA modelANOVA model The general model is:

ANOVA algorithms fit the above model (by least squares) to estimate the i’s.

H0: all i’s = 0

The general model is:

ANOVA algorithms fit the above model (by least squares) to estimate the i’s.

H0: all i’s = 0

ij i ijY

Group 1Group 2Group 3

Group

Y

2

2

42

Y

2001



Partitioning the total sums of Partitioning the total sums of squaressquares

Partitioning the total sums of Partitioning the total sums of squaressquares

Group 1Group 2Group 3

Y

Total SS Model (Groups) SS Error SS

2001



The ANOVA tableThe ANOVA table

Source of Variation

Sum ofSquares

MeanSquare

Degrees offreedom (df)

F

Total

Error

n - 1

n - k

SS/df

SS/df

Groups k - 1 SS/dfMSgroups

MSerror

i 1

k

ijj 1

n2(Y Y)

i

i ii

k

n Y Y( )

1

2

i 1

k

ij 1

n2(Y Yi)

i

j

2001



Use of single-classification Use of single-classification MANOVAMANOVA

Use of single-classification Use of single-classification MANOVAMANOVA

Data set consists of k groups (“treatments”), with ni observations per group, and p variables per observation.

Question: do the groups differ with respect to their multivariate means?

Data set consists of k groups (“treatments”), with ni observations per group, and p variables per observation.

Question: do the groups differ with respect to their multivariate means?

In single-classification ANOVA, we assume that a single factor is variable among groups, i.e., that all other factors which may possible affect the variables in question are randomized among groups.

In single-classification ANOVA, we assume that a single factor is variable among groups, i.e., that all other factors which may possible affect the variables in question are randomized among groups.

2001



ExamplesExamplesExamplesExamples

4 different concentrations of some suspected contaminant; 10 young fish randomly assigned to each treatment; at age 2 months, a number of measurements taken on each surviving fish.

4 different concentrations of some suspected contaminant; 10 young fish randomly assigned to each treatment; at age 2 months, a number of measurements taken on each surviving fish.

10 young fish reared in 4 different “treatments”, each treatment consisting of water samples taken at different stages of treatment in a water treatment plant.

10 young fish reared in 4 different “treatments”, each treatment consisting of water samples taken at different stages of treatment in a water treatment plant.

Good(ish) Bad(ish)

2001



Multivariate variance: a geometric Multivariate variance: a geometric interpretationinterpretationMultivariate variance: a geometric Multivariate variance: a geometric interpretationinterpretation

Univariate variance is a measure of the “volume” occupied by sample points in one dimension.

Multivariate variance involving m variables is the volume occupied by sample points in an m -dimensional space.

Univariate variance is a measure of the “volume” occupied by sample points in one dimension.

Multivariate variance involving m variables is the volume occupied by sample points in an m -dimensional space.

X X

Largervariance

Smallervariance

X1

X2Occupiedvolume

2001



Multivariate variance: Multivariate variance: effects of correlations effects of correlations among variablesamong variables

Multivariate variance: Multivariate variance: effects of correlations effects of correlations among variablesamong variables

Correlations between pairs of variables reduce the volume occupied by sample points…

…and hence, reduce the multivariate variance.

Correlations between pairs of variables reduce the volume occupied by sample points…

…and hence, reduce the multivariate variance.

No correlation

X1

X2

X2

X1

Positivecorrelation

Negativecorrelation

Occupiedvolume

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C and the generalized C and the generalized multivariate variancemultivariate varianceC and the generalized C and the generalized multivariate variancemultivariate variance

The determinant of the sample covariance matrix C is a generalized multivariate variance…

… because area2 of a parallelogram with sides given by the individual standard deviations and angle determined by the correlation between variables equals the determinant of C.

The determinant of the sample covariance matrix C is a generalized multivariate variance…

… because area2 of a parallelogram with sides given by the individual standard deviations and angle determined by the correlation between variables equals the determinant of C.

rc

s s1212

1 2

05 60 . cos , o

C CLNMOQP

1 1

1 43

2s

1s

h

2

sin 60 ; 3.2

3,

opposite hh

hypotenuse

Area Base Height Area

C

2001



ANOVA vs MANOVA: procedureANOVA vs MANOVA: procedureANOVA vs MANOVA: procedureANOVA vs MANOVA: procedure

In ANOVA, the total sums of squares is partitioned into a within-groups (SSw) and between-group SSb sums of squares:

In ANOVA, the total sums of squares is partitioned into a within-groups (SSw) and between-group SSb sums of squares:

In MANOVA, the total sums of squares and cross-products (SSCP) matrix is partitioned into a within groups SSCP (W) and a between-groups SSCP (B)

In MANOVA, the total sums of squares and cross-products (SSCP) matrix is partitioned into a within groups SSCP (W) and a between-groups SSCP (B)

T b wSS SS SS T B W

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In ANOVA, the null hypothesis is:

This is tested by means of the F statistic:

In ANOVA, the null hypothesis is:

This is tested by means of the F statistic:

ANOVA vs MANOVA: hypothesis ANOVA vs MANOVA: hypothesis testingtesting

ANOVA vs MANOVA: hypothesis ANOVA vs MANOVA: hypothesis testingtesting

In MANOVA, the null hypothesis is

This is tested by (among other things) Wilk’s lambda:

In MANOVA, the null hypothesis is

This is tested by (among other things) Wilk’s lambda:

0 1 2: kH

b b

w e

MS MSF

MS MS

0 1 2: kH μ μ μ

, 0 1

W W

T B W

2001



SSCP matrices: SSCP matrices: within, between, and within, between, and

totaltotal

SSCP matrices: SSCP matrices: within, between, and within, between, and

totaltotal The total (T) SSCP matrix

(based on p variables X1, X2,…, Xp ) in a sample of objects belonging to m groups G1, G2,…, Gm with sizes n1, n2,…, nm can be partitioned into within-groups (W) and between-groups (B) SSCP matrices:

The total (T) SSCP matrix (based on p variables X1, X2,…, Xp ) in a sample of objects belonging to m groups G1, G2,…, Gm with sizes n1, n2,…, nm can be partitioned into within-groups (W) and between-groups (B) SSCP matrices:

ijkx

jkx

kx

1 1

1 1

( )( )

( )( )

j

j

nm

rc ijr r ijc cj i

nm

rc ijr jr ijc jcj i

t x x x x

w x x x x

Value of variable Xk forith observation in group j

Mean of variable Xk forgroup j

Overall mean of variable Xk

T B W

,rc rct w Element in row r andcolumn c of total (T, t) and within (W, w) SSCP

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The distribution of The distribution of The distribution of The distribution of

Unlike F, has a very complicated distribution…

…but, given certain assumptions it can be approximated b as Bartlett’s 2 (for moderate to large samples) or Rao’s F (for small samples)

Unlike F, has a very complicated distribution…

…but, given certain assumptions it can be approximated b as Bartlett’s 2 (for moderate to large samples) or Rao’s F (for small samples)

2 [( 1) 0.5( )]ln

( 1)

N p k

df p k

1/

1/

2 2

2 2

1 ( 1) / 2 1

( 1)

1 ( ) / 2

( 1) 4

( 1) 5

( 1), ( 1) / 2 1

s

s

ms p kF

p k

m N p k

p ks

p k

df p k ms p k

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AssumptionsAssumptionsAssumptionsAssumptions

All observations are independent (residuals are uncorrelated)

Within each sample (group), variables (residuals) are multivariate normally distributed

Each sample (group) has the same covariance matrix (compound symmetry)

All observations are independent (residuals are uncorrelated)

Within each sample (group), variables (residuals) are multivariate normally distributed

Each sample (group) has the same covariance matrix (compound symmetry)

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Effect of violation of assumptionsEffect of violation of assumptionsEffect of violation of assumptionsEffect of violation of assumptions

Assumption Effect on Effect on power

Independence of observations

Very large, actual much larger than nominal

Large, power much reduced

Normality Small to negligible

Reduced power for platykurtotic distributions, skewness has little effect

Equality of covariance matrices

Small to negligible if group Ns similar, if Ns very unequal, actual larger than nominal

Power reduced, reduction greater for unequal Ns.

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Checking assumptions in MANOVAChecking assumptions in MANOVAChecking assumptions in MANOVAChecking assumptions in MANOVA

Independence(intraclass correlation,

ACF)

Use groupmeans as unit

of analysis

AssessMV normality

Check groupsizes

MVNgraph test

Check Univariatenormality

No

Yes Ni > 20

Ni < 20

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Checking assumptions in MANOVA (cont’d)Checking assumptions in MANOVA (cont’d)Checking assumptions in MANOVA (cont’d)Checking assumptions in MANOVA (cont’d)

MV normal?Check homogeneity

ofcovariance

matrices

Most variablesnormal?

Transformoffendingvariables

Group sizesmore or

less equal(R < 1.5)?

Groupsreasonably

large(> 15)?

Yes

No

Yes

Yes

No

END

Yes

Yes

No Transformvariables, or

adjust

2001



Then what?Then what?Then what?Then what?

Question Procedure

What variables are responsible for detected differences among groups?

Check univariate F tests as a guide; use another multivariate procedure (e.g. discriminant function analysis)

Do certain groups (determined beforehand) differ from one another?

Planned multiple comparisons

Which pairs of groups differ from one another (groups not specified beforehand)?

Unplanned multiple comparisons

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What are multiple comparisons?What are multiple comparisons?What are multiple comparisons?What are multiple comparisons?

Pair-wise comparisons of different treatments

These comparisons may involve group means, medians, variances, etc.

for means, done after ANOVA

In all cases, H0 is that the groups in question do not differ.

Pair-wise comparisons of different treatments

These comparisons may involve group means, medians, variances, etc.

for means, done after ANOVA

In all cases, H0 is that the groups in question do not differ. Yield

C N N+P


c:N N:N+P

C: N+P

Fre

qu

en

cy

2001



Types of Types of comparisonscomparisons

Types of Types of comparisonscomparisons

planned (a priori): independent of ANOVA results; theory predicts which treatments should be different.

unplanned (a posteriori): depend on ANOVA results; unclear which treatments should be different.

Test of significance are very different between the two!

planned (a priori): independent of ANOVA results; theory predicts which treatments should be different.

unplanned (a posteriori): depend on ANOVA results; unclear which treatments should be different.

Test of significance are very different between the two!

Y

Y

X1 X2 X3 X4 X5

X1 X2 X3 X4 X5

Planned

unplanned

2001



Planned comparisons (Planned comparisons (a prioria priori contrasts): contrasts): catecholamine levels in stressed fishcatecholamine levels in stressed fish

Planned comparisons (Planned comparisons (a prioria priori contrasts): contrasts): catecholamine levels in stressed fishcatecholamine levels in stressed fish

Comparisons of interest are determined by experimenter beforehand based on theory and do not depend on ANOVA results.

Prediction from theory: catecholamine levels increase above basal levels only after threshold PAO2 = 30 torr is reached.

So, compare only treatments above and below 30 torr (NT = 12).

Comparisons of interest are determined by experimenter beforehand based on theory and do not depend on ANOVA results.

Prediction from theory: catecholamine levels increase above basal levels only after threshold PAO2 = 30 torr is reached.

So, compare only treatments above and below 30 torr (NT = 12). Predicted threshold

PAO2 (torr)

10 20 30 40 50

[Ca

tech

ola

min

e]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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Unplanned comparisons (Unplanned comparisons (a posterioria posteriori contrasts): catecholamine levels in contrasts): catecholamine levels in

stressed fishstressed fish

Unplanned comparisons (Unplanned comparisons (a posterioria posteriori contrasts): catecholamine levels in contrasts): catecholamine levels in

stressed fishstressed fish Comparisons are

determined by ANOVA results.

Prediction from theory: catecholamine levels increase with increasing PAO2 .

So, comparisons between any pairs of treatments may be warranted (NT = 21).

Comparisons are determined by ANOVA results.

Prediction from theory: catecholamine levels increase with increasing PAO2 .

So, comparisons between any pairs of treatments may be warranted (NT = 21).

Predicted relationship

PAO2 (torr)

10 20 30 40 50[C

ate

cho

lam

ine]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2001



The problem: controlling The problem: controlling experiment-wise experiment-wise error errorThe problem: controlling The problem: controlling experiment-wise experiment-wise error error

For k comparisons, the probability of accepting H0 (no difference) is (1 - )k.

For 4 treatments, (1 - )k = (0.95)6 = .735, so experiment-wise (e) = 0.265.

Thus we would expect to reject H0 for at least one paired comparison about 27% of the time, even if all four treatments are identical.

For k comparisons, the probability of accepting H0 (no difference) is (1 - )k.

For 4 treatments, (1 - )k = (0.95)6 = .735, so experiment-wise (e) = 0.265.

Thus we would expect to reject H0 for at least one paired comparison about 27% of the time, even if all four treatments are identical.

Nominal = .05

Number of treatments

0 2 4 6 8 10

Ex

pe

rim

ent-

wis

e

(

e)

0.0

0.2

0.4

0.6

0.8

1.0

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Unplanned comparisons: Hotelling Unplanned comparisons: Hotelling TT22 and univariate F tests and univariate F tests

Unplanned comparisons: Hotelling Unplanned comparisons: Hotelling TT22 and univariate F tests and univariate F tests

Follow rejection of null in original MANOVA by all pairwise multivariate tests using Hotelling T2 to determine which groups are different

…but test at modified to maintain overall nominal type I error rate (e.g. Bonferroni correction)

Follow rejection of null in original MANOVA by all pairwise multivariate tests using Hotelling T2 to determine which groups are different

…but test at modified to maintain overall nominal type I error rate (e.g. Bonferroni correction)

Then use univariate t-tests to determine which variables are contributing to the detected pairwise differences…

…opinion is divided as to whether these should be done at a modified .

Then use univariate t-tests to determine which variables are contributing to the detected pairwise differences…

…opinion is divided as to whether these should be done at a modified .

2001



How many different variables for a How many different variables for a MANOVA?MANOVA?

How many different variables for a How many different variables for a MANOVA?MANOVA?

In general, try to use a small number of variables because:

In MANOVA, power generally declines with increasing number of variables.

If a number of variables are included that do not differ among groups, this will obscure differences on a few variables

In general, try to use a small number of variables because:

In MANOVA, power generally declines with increasing number of variables.

If a number of variables are included that do not differ among groups, this will obscure differences on a few variables

Measurement error is multiplicative among variables: the larger the number of variables, the larger the measurement noise

Interpretation is easier with a smaller number of variables

Measurement error is multiplicative among variables: the larger the number of variables, the larger the measurement noise

Interpretation is easier with a smaller number of variables

2001



How many different variables for a How many different variables for a MANOVA : recommendationMANOVA : recommendation

How many different variables for a How many different variables for a MANOVA : recommendationMANOVA : recommendation

Choose variables carefully, attempting to keep them to a minimum

Try to reduce the number of variables by using multivariate procedures (e.g. PCA) to generate composite, uncorrelated variables which can then be used as input.

Use multivariate procedures (such as discriminant function analysis) to “optimize” set of variables.

Choose variables carefully, attempting to keep them to a minimum

Try to reduce the number of variables by using multivariate procedures (e.g. PCA) to generate composite, uncorrelated variables which can then be used as input.

Use multivariate procedures (such as discriminant function analysis) to “optimize” set of variables.

Université d’Ottawa / University of Ottawa 2001 Bio 8100s Allied Multivariate Biostatistics L6.1...

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