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Canonical Correlation

Canonical correlation analyses attempts to

simultaneously solve the goals of multiple

correlation and principal components analysis. It

finds the linear combinations of variables in two

sets that are maximally correlated across sets

but orthogonal within sets.

X1 X2 X3 X4 . . . Xq

Y1 Y2 Y3 Y4 . . . Yp

What is the best way to understand how the

variables in these two sets are related?

- Bivariate correlations across sets
- Multiple correlations across sets
- Principal components within sets correlations

between principal components across sets

X1 X2 X3 X4 . . . Xq

Y1 Y2 Y3 Y4 . . . Yp

What linear combinations of the X variables (u)

and the Y variables (t) will maximize their

correlation?

b1X1 b2X2 b3X3 b4X4 . bpXp u

a1Y1 a2Y2 a3Y3 a4Y4 . aqYq t

What linear combinations of the X variables (u)

and the Y variables (t) will maximize their

correlation?

If X and Y are in standard score form, and u

Xb t Ya then find a and b to maximize rt,u

while

If X and Y are in standard score form, and u

Xb t Ya then find a and b to maximize rt,u

while

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The correlation between the two sets is called

the canonical correlation and is the largest

possible correlation that can be found between

linear combinations. The weights (a and b) that

are used to create the linear combinations are

called the standardized canonical coefficients.

The linear combinations created are called the

canonical variates.

Additional canonical variates and their

correlations can be found provided they satisfy

Additional canonical variates and their

correlations can be found provided they satisfy

The extraction of canonical variates can continue

up to a maximum defined by the number of measures

in the smaller of the two sets.

The standardized canonical coefficients (a and b)

are interpreted in the same way as standardized

regression coefficients in multiple

regressionthey indicate the unique contribution

of a variable to the linear combination. It is

also possible to derive the correlations between

each variable and the linear combination. These

are called canonical loadings and are interpreted

the same way as loadings in principal components.

These loadings can be calculated as

As in principal components analysis, the loadings

can assist in understanding the nature of the

linear combinations in each set.

Fader and Lodish (1990) collected data for 331

different grocery products. They sought

relations between what they called structural

variables and promotional variables. The

structural variables were characteristics not

likely to be changed by short-term promotional

activities. The promotional variables represented

promotional activities. The major goal was to

determine if different promotional activities

were associated with different types of grocery

products.

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market

share for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

Promotional variables (Y) FEAT Percent of

volume sold on feature (advertised in local

newspaper) DISP Percent of volume sold on

display (e.g., end of aisle) PCUT Percent of

volume sold at a temporary reduced

price SCOUP Percent of volume purchased using a

retailers store coupon MCOUP Percent of

volume purchased using a manufacturers coupon

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the

category per purchase occasion PVTSH Combined

market share for all private-label and generic

products PURHH Average number of

purchase occasions per household during the year

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using

a manufacturers coupon

PENET PURHH PCYCLE PRICE PVTSH FEAT DISP PCUT

SCOUP MCOUP BEER 62.3 11.1 46 5.16 .4 19 32

27 1 1 WINE 42.9 5.8 59 4.58 1.0 14 26 8 0 1

FRESH BREAD 98.6 26.6 21 1.30 39.4 12 4 15 1 2 CU

PCAKES 27.4 2.5 60 1.11 3.5 4 10 10 1 4

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Canonical correlation analysis must be obtained

using syntax statements in SPSS

MANOVA penet purhh pcycle price pvtsh with feat

disp pcut scoup mcoup /print signif(multiv dimenr

eigen stepdown univ hypoth) error(cor) /discrim

raw stan cor alpha(1).

Test Name Value Approx. F Hypoth. DF

Error DF Sig. of F Pillais .73057

11.12256 25.00 1625.00 .000

Hotellings 1.09732 14.01931 25.00

1597.00 .000 Wilks .41262

12.85124 25.00 1193.96 .000 Roys

.41271

These tests indicate whether there is any

significant relationship between the two sets of

variables. They do not indicate how many of those

sets of linear combinations are significant. With

5 variables in each set, there are up to 5 sets

of linear combinations that could be derived.

This test tells us that at least the first one is

significant.

Test Name Value Approx. F Hypoth. DF

Error DF Sig. of F Pillais .73057

11.12256 25.00 1625.00 .000

Hotellings 1.09732 14.01931 25.00

1597.00 .000 Wilks .41262

12.85124 25.00 1193.96 .000 Roys

.41271

Ra has an approximate F distribution with pq and

(1ts-.5pq) degrees of freedom.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 .703 64.040

64.040 .642 .413 2

.305 27.790 91.830 .483

.234 3 .075 6.877

98.708 .265 .070 4

.013 1.198 99.906 .114

.013 5 .001 .094

100.000 .032 .001

The canonical correlations are extracted in

decreasing size. At each step they represent the

largest correlation possible between linear

combinations in the two sets, provided the linear

combinations are independent of any previously

derived linear combinations.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 .703 64.040

64.040 .642 .413 2

.305 27.790 91.830 .483

.234 3 .075 6.877

98.708 .265 .070 4

.013 1.198 99.906 .114

.013 5 .001 .094

100.000 .032 .001

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F 1

TO 5 .41262 12.85124 25.00

1193.96 .000 2 TO 5 .70257

7.53593 16.00 984.36 .000 3 TO 5

.91682 3.17374 9.00 786.25

.001 4 TO 5 .98600 1.14582

4.00 648.00 .334 5 TO 5 .99897

.33534 1.00 325.00 .563

Procedures for testing the significance of the

canonical correlations can be applied

sequentially. At each step, the test indicates

whether there is any remaining significant

relationships between the two sets. In this case,

three sets of linear combinations can be formed.

As in principal components, identifying the

number of significant sets of linear combinations

is just the beginning. The nature of those linear

combinations must also be determined. This

requires interpreting the canonical weights and

loadings.

The linear combinations can be formed using the

variables in their original metrics. Sometimes

this makes it easier to understand the role a

particular variable plays because the metric is

well understood.

Raw canonical coefficients for DEPENDENT

variables Function No. Variable

1 2 3 4

5 PENET .036 -.018 .016

.016 .011 PURHH -.073

-.013 -.175 .072 -.329 PCYCLE

-.012 -.031 -.019 .049

-.020 PRICE .198 -.838

-.417 -.299 .305 PVTSH

.000 .024 -.061 .002 .039

More typically the linear combinations are formed

after the variables have been standardized. The

weights are then interpreted as standardized

regression coefficients and the resulting linear

combinations are in standard score form.

Standardized canonical coefficients for

DEPENDENT variables Function No.

Variable 1 2 3

4 5 PENET 1.066 -.527

.484 .483 .326 PURHH

-.307 -.055 -.737 .304

-1.382 PCYCLE -.262 -.695

-.417 1.104 -.455 PRICE

.208 -.883 -.439 -.315 .321

PVTSH .000 .359 -.898

.024 .576

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market share

for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

The loadings provide information about the

bivariate relationship between each variable and

each linear combination.

Correlations between DEPENDENT and canonical

variables Function No. Variable

1 2 3 4

5 PENET .956 .114 -.042

.223 -.145 PURHH .555

.148 -.389 -.207 -.690 PCYCLE

-.582 -.320 .060 .697

.263 PRICE -.011 -.769 -.285

-.569 .059 PVTSH .336

.465 -.705 .245 .337

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market share

for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

The same coefficients exist for the other set of

variables.

Raw canonical coefficients for COVARIATES

Function No. COVARIATE 1

2 3 4 5 FEAT

.083 -.151 -.058 -.232

.215 DISP .044 .011 .108

.091 .074 PCUT .021

.199 .037 .079 -.247 SCOUP

-.015 -.385 -.788 1.124

-.268 MCOUP .022 -.079

.043 -.003 -.057

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 4 5

FEAT .637 -1.160 -.448

-1.780 1.649 DISP .318

.077 .770 .653 .532 PCUT

.164 1.530 .281 .611

-1.898 SCOUP -.014 -.362

-.740 1.056 -.252 MCOUP

.202 -.728 .400 -.029 -.523

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using a

manufacturers coupon

Correlations between COVARIATES and canonical

variables CAN. VAR. Covariate

1 2 3 4 5

FEAT .939 .073 -.293

-.157 .046 DISP .730

.136 .384 .412 .362 PCUT

.896 .321 -.184 -.063

-.238 SCOUP .617 -.167

-.614 .462 -.024 MCOUP

.156 -.717 .427 -.069 -.523

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using a

manufacturers coupon

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What if the linear combinations had been formed

within each set separately and then correlated

across sets? What might justify this approach?

penet purhh pcycle price pvtsh

feat disp pcut scoup mcoup

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- Crosby, Evans, and Cowles (1990) examined the

impact of relationship quality on the outcome of

insurance sales. They examined relationship

characteristics and outcomes for 151

transactions. - Relationship Characteristics
- Appearance similarity
- Lifestyle similarity
- Status similarity
- Interaction intensity
- Mutual disclosure
- Cooperative intentions

- Crosby, Evans, and Cowles (1990) examined the

impact of relationship quality on the outcome of

insurance sales. They examined relationship

characteristics and outcomes for 151

transactions. - Outcomes
- Trust in the salesperson
- Satisfaction with the salesperson
- Cross-sell
- Total insurance sales

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Matrix data Variables rowtype_ trust satis

cross total appear life status interact mutual

coop . Begin data N 151 151 151 151 151 151 151

151 151 151 Mean 0 0 0 0 0 0 0 0 0 0 STDDEV 1 1 1

1 1 1 1 1 1 1 Corr 1.00 corr .63 1.00 corr .28

.22 1.00 corr .23 .24 .51 1.00 corr .38

.33 .29 .20 1.00 corr .42 .28 .36 .39

.57 1.00 corr .37 .30 .39 .29 .48 .59

1.00 corr .30 .36 .21 .18 .15 .29

.30 1.00 corr .45 .37 .31 .39 .29 .41

.35 .44 1.00 corr .56 .56 .24 .29 .18

.33 .30 .46 .63 1.00 end data.

Variable labels trust ' Trust in the

salesperson' Satis 'Satisfaction with the

salesperson' cross 'Cross-sell' total 'Total

insurance sales' appear 'Appearance

similarity' life 'Lifestyle similarity' status

'Status similarity' interact 'Interaction

intensity' mutual 'Mutual disclosure' coop

'Cooperative intentions' . MANOVA trust satis

cross total with appear life status interact

mutual coop /matrixIN() /print signif(multiv

dimenr eigen stepdown univ hypoth)

error(cor) /discrim raw stan cor alpha(1).

Multivariate Tests of Significance (S 4, M

1/2, N 69 1/2) Test Name Value

Approx. F Hypoth. DF Error DF Sig. of F

Pillais .73301 5.38481 24.00

576.00 .000 Hotellings 1.35153

7.85574 24.00 558.00 .000 Wilks

.37940 6.57954 24.00 493.10

.000 Roys .52771

There is at least one significant relationship

between the two sets of measures. With 6 and 4

measures in the two sets, there are a maximum of

4 possible sets of linear combinations that can

be formed.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 1.117 82.672

82.672 .726 .528 2

.176 13.050 95.722 .387

.150 3 .050 3.706

99.428 .218 .048 4

.008 .572 100.000 .088

.008

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F 1

TO 4 .37940 6.57954 24.00

493.10 .000 2 TO 4 .80331

2.15996 15.00 392.40 .007 3 TO 4

.94500 1.02566 8.00 286.00

.417 4 TO 4 .99233 .37087

3.00 144.00 .774

Two of the four possible sets of linear

combinations are significant.

Standardized canonical coefficients for

DEPENDENT variables Function No.

Variable 1 2 3

4 TRUST -.543 .317 -.390

1.082 SATIS -.364 -.936

.103 -.816 CROSS -.186

.148 1.160 .057 TOTAL -.239

.721 -.672 -.597

Outcomes Trust in the salesperson Satisfaction

with the salesperson Cross-sell Total insurance

sales

Correlations between DEPENDENT and canonical

variables Function No. Variable

1 2 3 4 TRUST

-.879 -.065 -.155 .447

SATIS -.804 -.530 -.048

-.265 CROSS -.540 .399

.731 -.124 TOTAL -.546 .645

-.145 -.515

Outcomes Trust in the salesperson Satisfaction

with the salesperson Cross-sell Total insurance

sales

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 4 APPEAR

-.268 -.561 .342 .552 LIFE

-.164 .833 -.467

.138 STATUS -.156 .128 .906

-.007 INTERACT -.049 -.379

.361 -.853 MUTUAL -.128

.749 -.209 -.441 COOP -.603

-.773 -.566 .408

Relationship Characteristics Appearance

similarity Lifestyle similarity Status

similarity Interaction intensity Mutual

disclosure Cooperative intentions

Correlations between COVARIATES and canonical

variables CAN. VAR. Covariate

1 2 3 4 APPEAR

-.589 -.003 .402 .445 LIFE

-.674 .531 .095 .155

STATUS -.622 .267 .660

.052 INTERACT -.517 -.209 .196

-.739 MUTUAL -.729 .319

-.182 -.345 COOP -.855

-.263 -.353 -.120

Relationship Characteristics Appearance

similarity Lifestyle similarity Status

similarity Interaction intensity Mutual

disclosure Cooperative intentions

- Remaining issues
- How much variance is really accounted for?
- How easily does the procedure capitalize on

chance? - How are canonical correlations cross-validated?
- Can the results be rotated?

How much variance isreally accounted

for? Reliance on the canonical correlations for

evidence of variance accounted for across sets of

variables can be misleading. Each linear

combination only captures a portion of the

variance in its own set. That needs to be taken

into account when judging the variance accounted

for across sets.

The squared canonical correlation indicates the

shared variance between linear combinations from

the two sets.

Each linear combination accounts for only a

portion of the variance in the variables in its

set.

Redundancy coefficients indicate the proportion

of variance in the variables of the opposite set

that is accounted for by the linear combination.

Canonical Loadings

Adequacy Coefficients

Canonical communality coefficients

Redundancy coefficients are defined as the

product of adequacy coefficients and the square

of canonical correlations.

Fader and Lodish (1990) collected data for 331

different grocery products. They sought

relations between what they called structural

variables and promotional variables. The

structural variables were characteristics not

likely to be changed by short-term promotional

activities. The promotional variables represented

promotional activities. The major goal was to

determine if different promotional activities

were associated with different types of grocery

products.

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market

share for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

Promotional variables (Y) FEAT Percent of

volume sold on feature (advertised in local

newspaper) DISP Percent of volume sold on

display (e.g., end of aisle) PCUT Percent of

volume sold at a temporary reduced

price SCOUP Percent of volume purchased using a

retailers store coupon MCOUP Percent of

volume purchased using a manufacturers coupon

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the

category per purchase occasion PVTSH Combined

market share for all private-label and generic

products PURHH Average number of

purchase occasions per household during the year

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using

a manufacturers coupon

PENET PURHH PCYCLE PRICE PVTSH FEAT DISP PCUT

SCOUP MCOUP BEER 62.3 11.1 46 5.16 .4 19 32

27 1 1 WINE 42.9 5.8 59 4.58 1.0 14 26 8 0 1

FRESH BREAD 98.6 26.6 21 1.30 39.4 12 4 15 1 2 CU

PCAKES 27.4 2.5 60 1.11 3.5 4 10 10 1 4

Test Name Value Approx. F Hypoth. DF

Error DF Sig. of F Pillais .73057

11.12256 25.00 1625.00 .000

Hotellings 1.09732 14.01931 25.00

1597.00 .000 Wilks .41262

12.85124 25.00 1193.96 .000 Roys

.41271

These tests indicate whether there is any

significant relationship between the two sets of

variables. They do not indicate how many of those

sets of linear combinations are significant. With

5 variables in each set, there are up to 5 sets

of linear combinations that could be derived.

This test tells us that at least the first one is

significant.

Test Name Value Approx. F Hypoth. DF

Error DF Sig. of F Pillais .73057

11.12256 25.00 1625.00 .000

Hotellings 1.09732 14.01931 25.00

1597.00 .000 Wilks .41262

12.85124 25.00 1193.96 .000 Roys

.41271

Ra has an approximate F distribution with pq and

(1ts-.5pq) degrees of freedom.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 .703 64.040

64.040 .642 .413 2

.305 27.790 91.830 .483

.234 3 .075 6.877

98.708 .265 .070 4

.013 1.198 99.906 .114

.013 5 .001 .094

100.000 .032 .001

The canonical correlations are extracted in

decreasing size. At each step they represent the

largest correlation possible between linear

combinations in the two sets, provided the linear

combinations are independent of any previously

derived linear combinations.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 .703 64.040

64.040 .642 .413 2

.305 27.790 91.830 .483

.234 3 .075 6.877

98.708 .265 .070 4

.013 1.198 99.906 .114

.013 5 .001 .094

100.000 .032 .001

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F 1

TO 5 .41262 12.85124 25.00

1193.96 .000 2 TO 5 .70257

7.53593 16.00 984.36 .000 3 TO 5

.91682 3.17374 9.00 786.25

.001 4 TO 5 .98600 1.14582

4.00 648.00 .334 5 TO 5 .99897

.33534 1.00 325.00 .563

Procedures for testing the significance of the

canonical correlations can be applied

sequentially. At each step, the test indicates

whether there is any remaining significant

relationships between the two sets. In this case,

three sets of linear combinations can be formed.

The standardized canonical coefficients are the

weights applied to standardized variables to

create the new linear combinations.

Standardized canonical coefficients for

DEPENDENT variables Function No.

Variable 1 2 3

4 5 PENET 1.066 -.527

.484 .483 .326 PURHH

-.307 -.055 -.737 .304

-1.382 PCYCLE -.262 -.695

-.417 1.104 -.455 PRICE

.208 -.883 -.439 -.315 .321

PVTSH .000 .359 -.898

.024 .576

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market share

for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

The loadings provide information about the

bivariate relationship between each variable and

each linear combination.

Correlations between DEPENDENT and canonical

variables Function No. Variable

1 2 3 4

5 PENET .956 .114 -.042

.223 -.145 PURHH .555

.148 -.389 -.207 -.690 PCYCLE

-.582 -.320 .060 .697

.263 PRICE -.011 -.769 -.285

-.569 .059 PVTSH .336

.465 -.705 .245 .337

Structural variables (X) PENET Percentage of

households making at least one category

purchase PCYCLE Average interpurchase

time PRICE Average dollars spent in the category

per purchase occasion PVTSH Combined market share

for all private-label and generic

products PURHH Average number of purchase

occasions per household during the year

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 4 5

FEAT .637 -1.160 -.448

-1.780 1.649 DISP .318

.077 .770 .653 .532 PCUT

.164 1.530 .281 .611

-1.898 SCOUP -.014 -.362

-.740 1.056 -.252 MCOUP

.202 -.728 .400 -.029 -.523

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using a

manufacturers coupon

Correlations between COVARIATES and canonical

variables CAN. VAR. Covariate

1 2 3 4 5

FEAT .939 .073 -.293

-.157 .046 DISP .730

.136 .384 .412 .362 PCUT

.896 .321 -.184 -.063

-.238 SCOUP .617 -.167

-.614 .462 -.024 MCOUP

.156 -.717 .427 -.069 -.523

Promotional variables (Y) FEAT Percent of volume

sold on feature (advertised in local

newspaper) DISP Percent of volume sold on display

(e.g., end of aisle) PCUT Percent of volume sold

at a temporary reduced price SCOUP Percent of

volume purchased using a retailers store

coupon MCOUP Percent of volume purchased using a

manufacturers coupon

Variance in dependent variables explained by

canonical variables CAN. VAR. Pct Var DE Cum

Pct DE Pct Var CO Cum Pct CO 1

33.462 33.462 13.810 13.810 2

18.895 52.357 4.415 18.226

3 14.708 67.065 1.032

19.258 4 19.263 86.328

.250 19.508 5 13.672 100.000

.014 19.522

Correlations between DEPENDENT and canonical

variables Function No. Variable

1 PENET .956 PURHH

.555 PCYCLE -.582 PRICE

-.011 PVTSH .336

Variance in covariates explained by canonical

variables CAN. VAR. Pct Var DE Cum Pct DE Pct

Var CO Cum Pct CO 1 21.654

21.654 52.467 52.467 2

3.127 24.781 13.382 65.849 3

1.159 25.940 16.521 82.371

4 .108 26.048 8.337

90.708 5 .010 26.058

9.292 100.000

Adequacy (33.462) times the squared canonical

correlation (.413)

(SL2i,1)/i

Adequacy Coefficients

RedundancyCoefficients

Variance in dependent variables explained by

canonical variables CAN. VAR. Pct Var DE Cum

Pct DE Pct Var CO Cum Pct CO 1

33.462 33.462 13.810 13.810 2

18.895 52.357 4.415 18.226

3 14.708 67.065 1.032

19.258 4 19.263 86.328

.250 19.508 5 13.672 100.000

.014 19.522

Variance in covariates explained by canonical

variables CAN. VAR. Pct Var DE Cum Pct DE Pct

Var CO Cum Pct CO 1 21.654

21.654 52.467 52.467 2

3.127 24.781 13.382 65.849 3

1.159 25.940 16.521 82.371

4 .108 26.048 8.337

90.708 5 .010 26.058

9.292 100.000

Any given loading can be squared to indicate the

proportion of the variance in that variable that

is accounted for by that canonical variate. The

sum of the squared loadings for a given variable

indicates the total proportion of variance

accounted for by the collection of canonical

variates. The average of the squared loadings for

a canonical variate is the adequacy coefficient

and indicates the proportion of variance in the

collection of variables that is accounted for by

the canonical variate. The redundancy coefficient

is the proportion of variance in a set of

variables that is accounted for by a linear

combination from the other set. The sum of the

redundancy coefficients gives the total

proportion of variance in one set that is

accounted for by the other set. These will

usually be different values for each set.

How easily does the procedure capitalize on

chance? Canonical correlation analysis has

elements of two proceduresprincipal components

analysis and multiple regression analysisthat

can capitalize on chance. It is important to

gauge how susceptible canonical correlation

analysis is to this problem.

A sample of 500 cases was generated, each with 10

variables from random normal distributions (m

100, s 10). The first 5 variables are

considered one set the remaining 5 variables are

considered the other set.

A few correlations are significant by chance

alone . . .

MANOVA x1 x2 x3 x4 x5 with y1 y2 y3 y4 y5 /print

signif(multiv dimenr eigen univ)

error(cor) /discrim raw stan cor alpha(1).

Multivariate Tests of Significance (S 5, M

-1/2, N 244 ) Test Name Value

Approx. F Hypoth. DF Error DF Sig. of F

Pillais .05241 1.04651 25.00

2470.00 .400 Hotellings .05346

1.04436 25.00 2442.00 .403 Wilks

.94845 1.04568 25.00 1821.77

.401 Roys .02742

The overall test of significance indicates that

there are no linear combinations that can be

formed between the two sets that would provide a

significant association.

Eigenvalues and Canonical Correlations Root No.

Eigenvalue Pct. Cum. Pct. Canon Cor.

Sq. Cor 1 .028 52.738

52.738 .166 .027 2

.015 28.362 81.100 .122

.015 3 .006 11.294

92.394 .077 .006 4

.004 7.561 99.955 .063

.004 5 .000 .045

100.000 .005 .000

The largest canonical correlation represents the

best attempt to make sense out of the random

associations. That this value is so small

indicates that the procedure will not generate

strong associations where none are known to exist.

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F

1 TO 5 .94845 1.04568 25.00

1821.77 .401 2 TO 5 .97519

.77454 16.00 1500.67 .716 3 TO 5

.98997 .55211 9.00 1197.55

.837 4 TO 5 .99595 .50066

4.00 986.00 .735 5 TO 5 .99998

.01187 1.00 494.00 .913

Given the overall nonsignificant test of

association, the dimension reduction analysis is

unnecessary. There cannot be any sets of linear

combinations that provide nonrandom associations

between sets.

Standardized canonical coefficients for DEPENDENT

variables Function No. Variable

1 2 3 4

5 X1 -.912 -.216 .111

.061 -.333 X2 -.174

.980 -.013 .114 -.132 X3

-.368 .025 .046 -.305

.884 X4 .139 -.117 .935

.306 .130 X5 .042

.052 .343 -.857 -.394

Correlations between DEPENDENT and canonical

variables Function No. Variable

1 2 3 4

5 X1 -.902 -.219 .108

.137 -.328 X2 -.182

.966 .116 .126 -.065 X3

-.350 .097 .069 -.374

.850 X4 .097 .005 .933

.330 .105 X5 .070

.081 .332 -.889 -.297

Although there are no significant canonical

correlations, individual weights and correlations

can be sizeable. Why?

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 4

5 Y1 -.729 .348 .173

-.175 .547 Y2 -.412

-.621 -.648 .170 -.091 Y3

.042 -.591 .612 .389

.375 Y4 -.427 -.128

.411 -.301 -.742 Y5 .195

-.317 -.065 -.899 .244

Correlations between COVARIATES and canonical

variables CAN. VAR. Covariate

1 2 3 4

5 Y1 -.784 .293 .109

-.140 .518 Y2 -.484

-.597 -.623 .145 -.022 Y3

.025 -.645 .629 .278

.333 Y4 -.434 -.167

.467 -.261 -.705 Y5 .217

-.393 -.024 -.850 .275

Variance in dependent variables explained by

canonical variables CAN. VAR. Pct Var DE Cum

Pct DE Pct Var CO Cum Pct CO 1

19.686 19.686 .540 .540 2

19.943 39.629 .298 .838

3 20.215 59.844 .121

.959 4 21.474 81.318 .086

1.045 5 18.682 100.000

.000 1.046

Variance in covariates explained by canonical

variables CAN. VAR. Pct Var DE Cum Pct DE Pct

Var CO Cum Pct CO 1 .595

.595 21.717 21.717 2 .311

.906 20.807 42.524 3

.122 1.028 20.268 62.792 4

.073 1.101 18.165 80.957

5 .000 1.101 19.043 100.000

The linear combinations will faithfully reproduce

the variance of the variables within sets. But,

because the data are random, each set accounts

for trivial variance in the other set.

Provided the significance tests are used to guide

decisions about the presence of canonical

correlations, the procedure will not unfairly

capitalize on chance. Nonetheless, like other

statistical procedures, our faith in the

conclusions is likely to be bolstered

considerably with cross-validation.

How are canonical correlations cross-validated?

The most convincing approach to cross-validation

requires a calibration sample and a hold-out

sample. The calibration sample is analyzed and

the canonical coefficients derived. Those

coefficients are then applied to the hold-out

sample. A standard canonical correlation analysis

is also conducted on the hold-out sample and the

correlations among the actual and estimated

canonical variates are computed.

- The data from Crosby, Evans, and Cowles (1990)

was used as the basis for the cross-validation.

In that study, characteristics of insurance sales

transactions were measured. - Relationship Characteristics
- Appearance similarity
- Lifestyle similarity
- Status similarity
- Interaction intensity
- Mutual disclosure
- Cooperative intentions

- Outcomes
- Trust in the salesperson
- Satisfaction with the salesperson
- Cross-sell
- Total insurance sales

In the original study, the variables had the

following correlation matrix

For the cross-validation, two separate sample of

250 cases were generated, each having this

correlation matrix as the population correlation

matrix. One sample was then used as the

calibration sample and the other was used as the

hold-out sample.

Multivariate Tests of Significance (S 4, M

1/2, N 119 ) Test Name Value Approx.

F Hypoth. DF Error DF Sig. of F Pillais

.88474 11.50214 24.00 972.00

.000 Hotellings 1.63071 16.20515

24.00 954.00 .000 Wilks

.31038 13.92044 24.00 838.47

.000 Roys .54378 - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - - Eigenvalues and Canonical Correlations

Root No. Eigenvalue Pct. Cum. Pct.

Canon Cor. Sq. Cor 1 1.192

73.093 73.093 .737 .544

2 .353 21.648 94.741

.511 .261 3 .078

4.784 99.525 .269 .072

4 .008 .475 100.000

.088 .008

Sample 1 Calibration Sample

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F

1 TO 4 .31038 13.92044 24.00

838.47 .000 2 TO 4 .68034

6.64488 15.00 665.70 .000 3 TO 4

.92050 2.55832 8.00 484.00

.010 4 TO 4 .99231 .62784

3.00 243.00 .598

Sample 1 Calibration Sample

Standardized canonical coefficients for

DEPENDENT variables Function No.

Variable 1 2 3

TRUST -.458 .649 .722

SATIS -.499 -.991 -.148

CROSS -.167 -.152 -.955

TOTAL -.156 .821 -.034 - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - Correlations between

DEPENDENT and canonical variables

Function No. Variable 1 2

3 TRUST -.874 .151

.331 SATIS -.885 -.392

.065 CROSS -.506 .151

-.795 TOTAL -.471 .653 -.330

Sample 1 Calibration Sample

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 APPEAR

-.378 -.399 .364 LIFESTYL

-.019 .764 -.258 STATUS

-.192 -.161 -.719 INTENSIT

-.150 -.507 -.531 MUTUAL

-.070 .865 .087 COOP

-.587 -.481 .767 - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - Correlations between COVARIATES and

canonical variables CAN. VAR.

Covariate 1 2 3

APPEAR -.634 .008 -.014

LIFESTYL -.626 .504 -.325

STATUS -.632 .060 -.623

INTENSIT -.594 -.281 -.416

MUTUAL -.670 .542 .098 COOP

-.837 -.034 .318

Sample 1 Calibration Sample

Multivariate Tests of Significance (S 4, M

1/2, N 119 ) Test Name Value Approx.

F Hypoth. DF Error DF Sig. of F Pillais

.78926 9.95559 24.00 972.00

.000 Hotellings 1.53353 15.23948

24.00 954.00 .000 Wilks

.34466 12.47535 24.00 838.47

.000 Roys .56014 - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - - Eigenvalues and Canonical Correlations

Root No. Eigenvalue Pct. Cum. Pct.

Canon Cor. Sq. Cor 1 1.273

83.042 83.042 .748 .560

2 .159 10.400 93.442

.371 .138 3 .100

6.490 99.932 .301 .091

4 .001 .068 100.000

.032 .001

Sample 2 Hold-out Sample

Dimension Reduction Analysis Roots Wilks

L. F Hypoth. DF Error DF Sig. of F

1 TO 4 .34466 12.47535 24.00

838.47 .000 2 TO 4 .78356

4.09951 15.00 665.70 .000 3 TO 4

.90854 2.97232 8.00 484.00

.003 4 TO 4 .99896 .08406

3.00 243.00 .969

Sample 2 Hold-out Sample

Standardized canonical coefficients for DEPENDENT

variables Function No. Variable

1 2 3 TRUST

-.556 -.066 .299 SATIS

-.329 .722 .010 CROSS

-.034 -.060 -1.193 TOTAL

-.401 -.822 .558 - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - Correlations between DEPENDENT and canonical

variables Function No. Variable

1 2 3 TRUST

-.878 .248 .074 SATIS

-.790 .527 -.052 CROSS

-.501 -.285 -.816 TOTAL

-.586 -.752 .008

Sample 2 Hold-out Sample

Standardized canonical coefficients for

COVARIATES CAN. VAR. COVARIATE

1 2 3 APPEAR

-.247 .717 -.467 LIFESTYL

-.184 -.967 .726 STATUS

-.040 -.208 -1.040 INTENSIT

-.105 .240 -.143 MUTUAL

-.061 -.682 .037 COOP

-.688 .756 .457 - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - Correlations between COVARIATES and

canonical variables CAN. VAR.

Covariate 1 2 3

APPEAR -.560 .019 -.472

LIFESTYL -.648 -.579 -.100

STATUS -.550 -.350 -.701

INTENSIT -.526 .058 -.073

MUTUAL -.692 -.293 .099 COOP

-.906 .185 .239

Sample 2 Hold-out Sample

The patterns of canonical correlations, weights,

and loadings are similar across samples, but the

best evidence for cross-validation comes from a

close correspondence between canonical variate

scores calculated in the hold-out sample using

that samples weights and using the weights from

the calibration sample.

Four sets of canonical scores need to be

calculated Actual Dependent Canonical Variate

Scores Z DV Sample 2 W DV Sample 2 Actual

Covariate Canonical Variate Scores Z CV Sample 2

W CV Sample 2 Estimated Dependent Canonical

Variate Scores Z DV Sample 2 W DV Sample

1 Estimated Covariate Canonical Variate Scores Z

CV Sample 2 W CV Sample 1

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Can the results be rotated? Given the resemblance

to principal components analysis, it might seem

sensible to rotate the variates to make

interpretation easier. This can be done, with

some restrictions and knowledge of how it alters

the nature of the analysis. It must be kept in

mind that any rotation will destroy one key

feature of the original analysisthat successive

pairs of linear combinations have maximum

correlations.

An orthogonal rotation to simple structure is

usually done on the structure matrix from one of

the sets. This resembles rotation to simple

structure in principal components analysis.

Varimax is the usual method used. Then the same

transformation is done on the other structure

matrix. This may not produce simple structure in

the other matrix but it preserves one important

feature of the original analysisthe total amount

of variance in one set accounted for by the other

set is preserved (i.e., the redundancies are the

same).

The transformation matrix can be applied to the

original weights to get the new weights for the

rotated canonical variates. Because the

transformation is orthogonal, the new variates

will be independent as well within sets. If the

same transformation is done for both sets, then

the correlations among the new canonical variates

will preserve the information contained in the

original canonical variates, but distribute it

differently.

MANOVA trust,satis,cross,total with

appear,lifestyl,status,intensit,mutual,coop /print

signif(multiv dimenr eigen) /discrim raw stan

cor rotate.

VARIMAX rotated correlations between canonical

variables and COVARIATES Can. Var.

DEP. VAR. 1 2 3

TRUST .398 .285 .432

SATIS .382 .768 .112

CROSS .875 .345 .189

TOTAL .409 .130 .465 - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - Transformation Matrix

1 2 3 1

-.278 -.601 -.749 2

.011 .778 -.628 3

-.961 .183 .209

This transformation matrix is applied to the

original matrices of canonical weights to get the

new weights. Those weights can be applied to the

correlation matrix relating the sets of variables

to produce the new canonical correlations.

VARIMAX rotated correlations between canonical

variables and COVARIATES Can. Var.

DEP. VAR. 1 2 3

APPEAR .846 .039 .360

LIFESTYL .946 .185 -.087

STATUS .155 .958 .212

INTENSIT .110 .356 .816

MUTUAL -.026 -.001 .077 COOP

.100 -.046 -.018 - - - -

- - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - Transformation Matrix

1 2 3 1

-.855 -.310 -.417 2

-.460 .080 .884 3

.241 -.947 .211

If maximum interpretability is desired within

sets, then an alternative approach would be to

conduct principal components analyses within sets

followed by multiple regression analyses between

sets.

- Assumptions and other odds and ends
- Interval level data
- Linear relations
- Homoskedasticity
- Low measurement error
- Unrestricted variances
- Low multicollinearity

- Assumptions and other odds and ends
- Similar distributions for all measures
- Multivariate normality for significance tests
- Sufficient sample size (20 times as many cases as

variables to interpret the first canonical

correlation 40-60 times as many cases as

variables for more than one canonical

correlation) - No outliers

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