How Collinearity Affects Mixture Regression Results: Online Supplement
Jan-Michael Becker1, Christian M. Ringle2,4, Marko Sarstedt3,4, and Franziska Völckner5
Marketing Letters
1Department of Marketing and Brand Management, University of Cologne, Albertus Magnus Platz 1, 50923 Cologne, Germany, E-mail:
2Institute of Human Resource Management and Organizations (HRMO), Hamburg University of Technology (TUHH), Schwarzenbergstraße 95, 21073 Hamburg, Germany, E-mail:
3Institute of Marketing, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, E-mail: .
4School of Business and Law, University of Newcastle, University Drive, Callaghan NSW 2308, Australia.
5Department of Marketing and Brand Management, University of Cologne, Albertus Magnus Platz 1, 50923 Cologne, Germany, E-mail:
This document is the Online Supplement for the publication: How collinearity affects mixture regression results, Marketing Letters(doi:10.1007/s11002-014-9299-9)
ABSTRACT
Mixture regression models are an important methodforuncovering unobserved heterogeneity. A fundamental challenge in their application relates to the identification of the appropriate number of segments to retain from the data. Prior research has provided several simulation studies that compare the performance of different segment retention criteria. Although collinearity between the predictor variables is a common phenomenon inregression models, itseffect on theperformance of these criteriahas not been analyzed thus far.We address this gap in research by examining the performance of segment retention criteria in mixture regression models characterized by systematically increased collinearity levels. The results have fundamental implications and provide guidance for using mixture regression models in empirical (marketing) studies.
Keywords:Market segmentation, Segment retention, Mixture regression, Collinearity
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ONLINE SupPlement
Criterion / Description / ReferenceInformation Criteria
Akaike Information Criterion / AIC = -2 ∙ lnL + 2 ∙ Ds / Akaike(1973)
Modified AIC 3 / AIC3= -2 ∙ lnL + 3 ∙ Ds / Bozdogan(1994)
Modified AIC 4 / AIC4 = -2 ∙ lnL + 4 ∙ Ds / Bozdogan(1994)
Bayes Information Criterion / BIC = -2 ∙ lnL + ln(N) ∙ Ds / Schwarz (1978)
Consistent AIC / CAIC = -2 ∙ lnL + [ln(N) + 1] ∙ Ds / Bozdogan(1987)
Informational Complexity Criterion / ICOMP = -2∙ lnL + Ns ∙ ln(tr(F-1)/ Ds ) – ln(det(F-1)) / Bozdogan(1988)
Hannan-Quinn Criterion / HQ = -2 ∙ lnL + 2 ln[ln(N)] ∙ Ds / Hannanand Quinn (1979)
Minimum Description Length 2 / MDL2 = -2 ∙ lnL + 2 ∙ ln(N) ∙ Ds / Liang et al. (1992)
Minimum Description Length 5 / MDL2 = -2 ∙ lnL + 5 ∙ ln(N) ∙ Ds / Liang et al. (1992)
Classification Criteria
Entropy Criterion / / Ramaswamy et al. (1993)
Non-Fuzzy Index / / Roubens(1978)
Normalized Entropy Criterion / NEC = E(S) / [ln(S) – ln(1)] / Celeuxand Soromenho(1996)
Integrated Completed Likelihood-BIC / ICL-BIC = -2 ∙ lnL + [ln(N)] ∙ Ds + 2 ∙ E(S) / Biernacki et al. (2000)
Partition Entropy / / Bezdek(1981)
Classification Likelihood Criterion / CLC = -2∙lnL + 2 ∙ E(S) / BiernackiandGovaert(1997)
Approximate Weight of Evidence / AWE = -2 ∙ lnLc + 2 ∙ Ns ∙ [1.5 + ln(N)] / BanfieldandRaftery(1993)
Partition Coefficient / / Bezdek(1981)
Notes: lnL is the log-likelihood of the model, Ds is the number of parameters required to estimate the model with S segments, N is the sample size, Ns is the sample size in segment s, pis is the a-posteriori probability of observation i belonging to segment s, F-1 denotes the estimate of the inverse Fisher information matrix, and E(S) the estimated entropy for a model with S segments defined as .
Table A1. Information and Classification Criteria
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X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4X1 / 1 / 0.00 / 0.00 / 0 / X1 / 1 / 0.50 / 0.20 / 0 / X1 / 1 / 0.65 / 0.40 / 0
X2 / 0.00 / 1 / 0.00 / 0 / X2 / 0.50 / 1 / 0.20 / 0 / X2 / 0.65 / 1 / 0.40 / 0
X3 / 0.00 / 0.00 / 1 / 0 / X3 / 0.20 / 0.20 / 1 / 0 / X3 / 0.40 / 0.40 / 1 / 0
X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1
Highest VIF / 1.00 / Highest VIF / 1.35 / Highest VIF / 1.80
Trace(X'X)-1 / 1.00 / Trace(X'X)-1 / 4.76 / Trace(X'X)-1 / 5.85
Cond. No. / 1.00 / Cond. No. / 1.80 / Cond. No. / 2.38
X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
X1 / 1 / 0.75 / 0.50 / 0 / X1 / 1 / 0.80 / 0.60 / 0 / X1 / 1 / 0.85 / 0.65 / 0
X2 / 0.75 / 1 / 0.50 / 0 / X2 / 0.80 / 1 / 0.60 / 0 / X2 / 0.85 / 1 / 0.65 / 0
X3 / 0.50 / 0.50 / 1 / 0 / X3 / 0.60 / 0.60 / 1 / 0 / X3 / 0.65 / 0.65 / 1 / 0
X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1
Highest VIF / 2.40 / Highest VIF / 3.00 / Highest VIF / 3.80
Trace(X'X)-1 / 7.20 / Trace(X'X)-1 / 8.59 / Trace(X'X)-1 / 10.50
Cond. No. / 2.95 / Cond. No. / 3.42 / Cond. No. / 4.03
X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
X1 / 1 / 0.925 / 0.75 / 0 / X1 / 1 / 0.90 / 0.70 / 0 / X1 / 1 / 0.95 / 0.80 / 0
X2 / 0.925 / 1 / 0.75 / 0 / X2 / 0.90 / 1 / 0.70 / 0 / X2 / 0.95 / 1 / 0.80 / 0
X3 / 0.75 / 0.75 / 1 / 0 / X3 / 0.70 / 0.70 / 1 / 0 / X3 / 0.80 / 0.80 / 1 / 0
X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1 / X4 / 0 / 0 / 0 / 1
Highest VIF / 5.50 / Highest VIF / 7.30 / Highest VIF / 10.70
Trace(X'X)-1 / 14.15 / Trace(X'X)-1 / 17.99 / Trace(X'X)-1 / 25.40
Cond. No. / 5.04 / Cond. No. / 5.91 / Cond. No. / 7.35
Table A2. Correlation Matrices for the Different Collinearity Levels
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Table A3 illustrates the difference between consistent and inconsistent correlation matrices between segments regarding the factor-level combination of two groups of data, a .40 mean difference between the standardized group-specific regression coefficients and a R2of 40%. The correlation matrices exhibit relatively low values, which translate into a maximum VIF value of 1.80 for each segment. Note thatin contrast to the consistent condition, the VIF of the aggregated dataset in the inconsistent condition is not 1.80, but lower due to the averaging of the correlations.
Group 1 / Group 2X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
Group-specific regression coefficients / .565 / .165 / .165 / .165 / .165 / .165 / .165 / .565
Group-specific correlation between the independent variables consistent) / 1.00 / 1.00
.65 / 1.00 / .65 / 1.00
.40 / .40 / 1.00 / .40 / .40 / 1.00
.00 / .00 / .00 / 1.00 / .00 / .00 / .00 / 1.00
Group-specific correlation between the independent variables (inconsistent) / 1.00 / 1.00
.65 / 1.00 / .00 / 1.00
.40 / .40 / 1.00 / .00 / .40 / 1.00
.00 / .00 / .00 / 1.00 / .00 / .40 / .65 / 1.00
Mean separation = .40; R2 = 40%; Highest VIF = 1.80;
Table A3. Example of Consistent and Inconsistent Correlation Matrices between Two Groups of Data
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AIC3 / AIC4 / ICOMP / HQB / Sig. / B / Sig. / B / Sig. / B / Sig.
U / Constant Term / -3.70 / 0.00 / -3.09 / 0.00 / -3.53 / 0.00 / -3.46 / 0.00
[Sample Size =3000] / -4.29 / 0.00 / -5.68 / 0.00 / -2.47 / 0.00 / -5.07 / 0.00
[Sample Size =1,500] / -1.97 / 0.00 / -2.66 / 0.00 / -0.95 / 0.00 / -2.25 / 0.00
[Sample Size =1,000] / -2.23 / 0.00 / -3.06 / 0.00 / -1.36 / 0.00 / -2.74 / 0.00
[Sample Size =500] / * / * / * / * / * / * / * / *
[R²=.80] / -3.65 / 0.00 / -4.67 / 0.00 / -2.49 / 0.00 / -4.55 / 0.00
[R²=.60] / -1.39 / 0.00 / -1.86 / 0.00 / -0.90 / 0.00 / -1.82 / 0.00
[R²=.40] / * / * / * / * / * / * / * / *
[VIF=10.70] / 6.08 / 0.00 / 7.40 / 0.00 / 5.12 / 0.00 / 7.31 / 0.00
[VIF=7.30] / 5.40 / 0.00 / 6.56 / 0.00 / 4.62 / 0.00 / 6.48 / 0.00
[VIF=5.50] / 4.86 / 0.00 / 5.90 / 0.00 / 4.20 / 0.00 / 5.82 / 0.00
[VIF=3.80] / 4.14 / 0.00 / 5.01 / 0.00 / 3.61 / 0.00 / 4.96 / 0.00
[VIF=3.00] / 3.58 / 0.00 / 4.30 / 0.00 / 3.09 / 0.00 / 4.26 / 0.00
[VIF=2.40] / 2.99 / 0.00 / 3.55 / 0.00 / 2.59 / 0.00 / 3.54 / 0.00
[VIF=1.80] / 2.19 / 0.00 / 2.62 / 0.00 / 1.91 / 0.00 / 2.62 / 0.00
[VIF=1.35] / 1.24 / 0.00 / 1.43 / 0.00 / 1.13 / 0.00 / 1.45 / 0.00
[VIF=1.00] / * / * / * / * / * / * / * / *
[Mean Separation =.40] / -4.35 / 0.00 / -5.42 / 0.00 / -3.72 / 0.00 / -5.26 / 0.00
[Mean Separation =.30] / -2.65 / 0.00 / -3.42 / 0.00 / -2.26 / 0.00 / -3.30 / 0.00
[Mean Separation =.20] / * / * / * / * / * / * / * / *
[Segments=4] / 6.09 / 0.00 / 7.65 / 0.00 / 3.92 / 0.00 / 7.46 / 0.00
[Segments=3] / 4.15 / 0.00 / 5.24 / 0.00 / 2.62 / 0.00 / 5.09 / 0.00
[Segments=2] / * / * / * / * / * / * / * / *
[Inconsistent Correlation Matrix] / 1.79 / 0.00 / 2.15 / 0.00 / 1.75 / 0.00 / 2.10 / 0.00
[Consistent Correlation Matrix] / * / * / * / * / * / * / * / *
[Very Unbalanced] / -0.22 / 0.00 / -0.29 / 0.00 / -0.15 / 0.00 / -0.26 / 0.00
[Unbalanced] / 0.54 / 0.00 / 0.71 / 0.00 / 0.44 / 0.00 / 0.71 / 0.00
[Balanced] / * / * / * / * / * / * / * / *
O / Constant Term / -2.34 / 0.00 / -4.89 / 0.00 / -1.22 / 0.00 / -4.14 / 0.00
[Sample Size =3000] / -0.44 / 0.00 / 0.27 / 0.00 / -1.73 / 0.00 / -0.35 / 0.00
[Sample Size =1,500] / -0.12 / 0.00 / 0.26 / 0.00 / -0.70 / 0.00 / -0.21 / 0.00
[Sample Size =1,000] / -1.04 / 0.00 / -1.20 / 0.00 / -1.34 / 0.00 / -1.46 / 0.00
[Sample Size =500] / * / * / * / * / * / * / * / *
[R²=.80] / 0.22 / 0.00 / 0.30 / 0.00 / -1.34 / 0.00 / 0.30 / 0.00
[R²=.60] / 0.12 / 0.00 / 0.20 / 0.00 / -0.44 / 0.00 / 0.22 / 0.00
[R²=.40] / * / * / * / * / * / * / * / *
[VIF=10.70] / 0.56 / 0.00 / 1.33 / 0.00 / 1.57 / 0.00 / 1.18 / 0.00
[VIF=7.30] / 0.60 / 0.00 / 1.43 / 0.00 / 1.59 / 0.00 / 1.26 / 0.00
[VIF=5.50] / 0.62 / 0.00 / 1.43 / 0.00 / 1.47 / 0.00 / 1.26 / 0.00
[VIF=3.80] / 0.57 / 0.00 / 1.25 / 0.00 / 1.32 / 0.00 / 1.03 / 0.00
[VIF=3.00] / 0.46 / 0.00 / 0.98 / 0.00 / 1.15 / 0.00 / 0.78 / 0.00
[VIF=2.40] / 0.35 / 0.00 / 0.56 / 0.00 / 0.90 / 0.00 / 0.38 / 0.00
[VIF=1.80] / 0.08 / 0.04 / 0.05 / 0.50 / 0.62 / 0.00 / -0.02 / 0.74
[VIF=1.35] / 0.00 / 0.99 / -0.02 / 0.84 / 0.34 / 0.00 / -0.07 / 0.34
[VIF=1.00] / * / * / * / * / * / * / * / *
[Mean Separation =.40] / 0.31 / 0.00 / 0.59 / 0.00 / -0.52 / 0.00 / 0.54 / 0.00
[Mean Separation =.30] / 0.22 / 0.00 / 0.48 / 0.00 / -0.30 / 0.00 / 0.41 / 0.00
[Mean Separation =.20] / * / * / * / * / * / * / * / *
[Segments=4] / -0.12 / 0.00 / -0.83 / 0.00 / 1.03 / 0.00 / -0.59 / 0.00
[Segments=3] / -0.41 / 0.00 / -1.29 / 0.00 / 0.42 / 0.00 / -1.06 / 0.00
[Segments=2] / * / * / * / * / * / * / * / *
[Inconsistent Correlation Matrix] / 0.84 / 0.00 / 1.62 / 0.00 / 0.83 / 0.00 / 1.43 / 0.00
[Consistent Correlation Matrix] / * / * / * / * / * / * / * / *
[Very Unbalanced] / 0.08 / 0.00 / 0.11 / 0.00 / 0.01 / 0.73 / 0.15 / 0.00
[Unbalanced] / -0.40 / 0.00 / -0.60 / 0.00 / -0.10 / 0.00 / -0.55 / 0.00
[Balanced] / * / * / * / * / * / * / * / *
- 2 LnL / 54635.99 / 44976.90 / 65549.43 / 46387.70
% Correct / 80.7% / 88.0% / 72.5% / 87.5%
Notes: * = reference category; U = underestimation; O = overestimation
Table A4. Multinomial Logistic Regression Results of the Four Best-Performing Criteria’s Segment Retentionand the Design Factors
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Pre-specified / AIC / AIC3 / AIC4 / BIC / CAIC / ICOMP / HQ / MDL2 / MDL5Groups / 2 / .631 / .561 / .602 / .608 / .597 / .589 / .596 / .609 / .534 / .363
3 / .460 / .434 / .453 / .448 / .419 / .408 / .446 / .450 / .343 / .186
4 / .366 / .350 / .359 / .351 / .316 / .303 / .354 / .353 / .240 / .112
Sample
Size / 5x100 / .360 / .317 / .339 / .330 / .290 / .271 / .331 / .334 / .185 / .050
10x100 / .583 / .551 / .572 / .567 / .530 / .515 / .564 / .569 / .435 / .214
5x300 / .392 / .355 / .379 / .380 / .364 / .357 / .373 / .380 / .314 / .179
10x300 / .609 / .572 / .595 / .598 / .593 / .590 / .594 / .598 / .556 / .439
Relative
Segment
Size / balanced / .476 / .437 / .456 / .455 / .434 / .425 / .452 / .456 / .370 / .228
unbalanced / .482 / .445 / .466 / .464 / .441 / .431 / .461 / .466 / .373 / .225
veryunbalanced / .500 / .464 / .491 / .487 / .457 / .444 / .483 / .489 / .374 / .208
Mean
Separation / .20 / .288 / .265 / .270 / .259 / .219 / .205 / .264 / .262 / .141 / .046
.30 / .510 / .470 / .496 / .494 / .470 / .458 / .491 / .496 / .389 / .212
.40 / .660 / .610 / .647 / .653 / .644 / .637 / .641 / .653 / .588 / .404
R² / .40 / .346 / .319 / .330 / .323 / .290 / .277 / .321 / .325 / .214 / .091
.60 / .461 / .427 / .447 / .443 / .416 / .404 / .439 / .445 / .336 / .180
.80 / .650 / .600 / .636 / .640 / .627 / .619 / .636 / .641 / .567 / .391
VIF / 1.00 / .669 / .626 / .657 / .655 / .625 / .612 / .654 / .656 / .536 / .333
1.35 / .608 / .565 / .594 / .590 / .558 / .544 / .590 / .592 / .469 / .281
1.80 / .558 / .516 / .543 / .540 / .507 / .494 / .537 / .542 / .421 / .244
2.40 / .513 / .471 / .497 / .495 / .467 / .455 / .491 / .497 / .388 / .224
3.00 / .478 / .436 / .460 / .458 / .433 / .421 / .453 / .460 / .359 / .209
3.80 / .443 / .405 / .425 / .423 / .401 / .391 / .418 / .424 / .332 / .191
5.50 / .402 / .368 / .386 / .385 / .366 / .357 / .379 / .386 / .306 / .177
7.30 / .369 / .340 / .355 / .353 / .336 / .328 / .348 / .354 / .283 / .170
10.70 / .333 / .310 / .322 / .320 / .304 / .298 / .317 / .321 / .258 / .155
Correlation
Matrix / consistent / .568 / .535 / .559 / .555 / .525 / .513 / .554 / .556 / .448 / .280
inconsistent / .403 / .362 / .383 / .383 / .364 / .354 / .376 / .385 / .297 / .161
Overall / .486 / .449 / .471 / .469 / .444 / .433 / .465 / .470 / .372 / .220
Table A5. Adjusted Rand Index
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Pre-specified / NEC / PE / EN / NFI / PC / AWE / CLC / ICL-BICGroups / 2 / .631 / .617 / .628 / .597 / .613 / .629 / .433 / .540 / .541
3 / .460 / .395 / .369 / .392 / .387 / .369 / .241 / .389 / .353
4 / .366 / .294 / .250 / .291 / .278 / .250 / .155 / .306 / .255
Sample
Size / 5x100 / .360 / .323 / .318 / .312 / .319 / .318 / .103 / .273 / .229
10x100 / .583 / .522 / .499 / .513 / .512 / .499 / .326 / .494 / .490
5x300 / .392 / .356 / .335 / .345 / .346 / .336 / .203 / .313 / .275
10x300 / .609 / .541 / .510 / .536 / .529 / .510 / .473 / .567 / .538
Relative
Segment
Size / balanced / .476 / .415 / .382 / .403 / .396 / .382 / .280 / .399 / .378
unbalanced / .482 / .427 / .407 / .418 / .419 / .408 / .279 / .408 / .381
very unbalanced / .500 / .465 / .457 / .458 / .464 / .458 / .270 / .429 / .390
Mean
Separation / .20 / .288 / .259 / .252 / .246 / .252 / .252 / .068 / .207 / .143
.30 / .510 / .456 / .432 / .446 / .445 / .433 / .276 / .438 / .403
.40 / .660 / .591 / .563 / .587 / .581 / .563 / .484 / .591 / .604
R² / .40 / .346 / .315 / .308 / .305 / .309 / .308 / .125 / .273 / .215
.60 / .461 / .414 / .394 / .404 / .404 / .394 / .235 / .386 / .351
.80 / .650 / .577 / .545 / .571 / .565 / .546 / .469 / .577 / .583
VIF / 1.00 / .669 / .643 / .559 / .627 / .619 / .561 / .414 / .578 / .556
1.35 / .608 / .566 / .505 / .552 / .540 / .506 / .352 / .516 / .489
1.80 / .558 / .489 / .457 / .480 / .474 / .457 / .313 / .471 / .438
2.40 / .513 / .440 / .426 / .434 / .432 / .426 / .281 / .434 / .399
3.00 / .478 / .407 / .403 / .402 / .403 / .403 / .261 / .401 / .368
3.80 / .443 / .381 / .383 / .374 / .379 / .382 / .243 / .373 / .340
5.50 / .402 / .354 / .359 / .347 / .353 / .359 / .222 / .339 / .309
7.30 / .369 / .331 / .336 / .324 / .330 / .336 / .207 / .311 / .286
10.70 / .333 / .307 / .312 / .299 / .306 / .312 / .191 / .283 / .261
Correlation Matrix / consistent / .568 / .489 / .471 / .479 / .478 / .471 / .346 / .490 / .465
inconsistent / .403 / .382 / .361 / .374 / .374 / .361 / .206 / .334 / .301
Overall / .486 / .435 / .416 / .426 / .426 / .416 / .276 / .412 / .383
Table A5. Adjusted Rand Index (continued)
Correct / AIC3 / AIC4 / ICOMP / HQDf / Partial Eta-Square / Sig. / Partial Eta-Square / Sig. / Partial Eta-Square / Sig. / Partial Eta-Square / Sig. / Partial Eta-Square / Sig.
Constant Term / 1 / 0.950 / 0.000 / 0.946 / 0.000 / 0.942 / 0.000 / 0.942 / 0.000 / 0.944 / 0.000
Mean Separation / 2 / 0.651 / 0.000 / 0.654 / 0.000 / 0.660 / 0.000 / 0.645 / 0.000 / 0.661 / 0.000
R² / 2 / 0.557 / 0.000 / 0.556 / 0.000 / 0.559 / 0.000 / 0.560 / 0.000 / 0.561 / 0.000
VIF / 8 / 0.472 / 0.000 / 0.469 / 0.000 / 0.452 / 0.000 / 0.463 / 0.000 / 0.458 / 0.000
Segments / 2 / 0.490 / 0.000 / 0.441 / 0.000 / 0.455 / 0.000 / 0.429 / 0.000 / 0.458 / 0.000
Sample Size / 3 / 0.496 / 0.000 / 0.502 / 0.000 / 0.500 / 0.000 / 0.500 / 0.000 / 0.499 / 0.000
Relative Segment Size / 2 / 0.009 / 0.000 / 0.017 / 0.000 / 0.014 / 0.000 / 0.013 / 0.000 / 0.014 / 0.000
Correlation Matrix / 1 / 0.353 / 0.000 / 0.379 / 0.000 / 0.354 / 0.000 / 0.374 / 0.000 / 0.358 / 0.000
VIF * Segments / 16 / 0.043 / 0.000 / 0.029 / 0.000 / 0.031 / 0.000 / 0.027 / 0.000 / 0.032 / 0.000
R² * Segments / 4 / 0.001 / 0.000 / 0.001 / 0.000 / 0.001 / 0.000 / 0.002 / 0.000 / 0.001 / 0.000
Segments * Correlation Matrix / 2 / 0.047 / 0.000 / 0.069 / 0.000 / 0.062 / 0.000 / 0.065 / 0.000 / 0.063 / 0.000
Segments * Sample Size / 6 / 0.002 / 0.000 / 0.005 / 0.000 / 0.006 / 0.000 / 0.005 / 0.000 / 0.006 / 0.000
Mean Separation * Segments / 4 / 0.012 / 0.000 / 0.007 / 0.000 / 0.009 / 0.000 / 0.008 / 0.000 / 0.008 / 0.000
Segments * Relative Segment Size / 4 / 0.004 / 0.000 / 0.003 / 0.000 / 0.004 / 0.000 / 0.003 / 0.000 / 0.004 / 0.000
R² * VIF / 16 / 0.015 / 0.000 / 0.018 / 0.000 / 0.019 / 0.000 / 0.021 / 0.000 / 0.019 / 0.000
VIF * Correlation Matrix / 8 / 0.169 / 0.000 / 0.183 / 0.000 / 0.170 / 0.000 / 0.175 / 0.000 / 0.173 / 0.000
VIF * Sample Size / 24 / 0.014 / 0.000 / 0.020 / 0.000 / 0.021 / 0.000 / 0.020 / 0.000 / 0.020 / 0.000
Mean Separation * VIF / 16 / 0.027 / 0.000 / 0.030 / 0.000 / 0.032 / 0.000 / 0.030 / 0.000 / 0.032 / 0.000
VIF * Relative Segment Size / 16 / 0.003 / 0.000 / 0.001 / 0.000 / 0.001 / 0.000 / 0.001 / 0.000 / 0.001 / 0.000
R² * Correlation Matrix / 2 / 0.004 / 0.000 / 0.003 / 0.000 / 0.003 / 0.000 / 0.004 / 0.000 / 0.003 / 0.000
R² * Sample Size / 6 / 0.002 / 0.000 / 0.003 / 0.000 / 0.004 / 0.000 / 0.003 / 0.000 / 0.004 / 0.000
Mean Separation * R² / 4 / 0.012 / 0.000 / 0.009 / 0.000 / 0.009 / 0.000 / 0.009 / 0.000 / 0.008 / 0.000
R² * Relative Segment Size / 4 / 0.000 / 0.236 / 0.000 / 0.386 / 0.000 / 0.007 / 0.000 / 0.011 / 0.000 / 0.005
Sample Size * Correlation Matrix / 3 / 0.000 / 0.000 / 0.003 / 0.000 / 0.002 / 0.000 / 0.003 / 0.000 / 0.002 / 0.000
Mean Separation * Correlation Matrix / 2 / 0.010 / 0.000 / 0.010 / 0.000 / 0.010 / 0.000 / 0.010 / 0.000 / 0.011 / 0.000
Relative Segment Size * Correlation Matrix / 2 / 0.000 / 0.000 / 0.003 / 0.000 / 0.003 / 0.000 / 0.003 / 0.000 / 0.002 / 0.000
Mean Separation * Sample Size / 6 / 0.011 / 0.000 / 0.010 / 0.000 / 0.012 / 0.000 / 0.011 / 0.000 / 0.011 / 0.000
Sample Size * Relative Segment Size / 6 / 0.001 / 0.000 / 0.001 / 0.000 / 0.001 / 0.000 / 0.002 / 0.000 / 0.001 / 0.000
Mean Separation * Relative Segment Size / 4 / 0.000 / 0.000 / 0.000 / 0.000 / 0.001 / 0.000 / 0.000 / 0.000 / 0.001 / 0.000
Error / 174784
Overall / 174960
R² = .874 / R² = .873 / R² = .873 / R2 = .871 / R² = .874
Table A6. ANCOVA Results of the ARI Value for the Four Best-Performing Criteria and the Design Factors
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