Multivariate analysis

STEPWISE DISCRIMINANT ANALYSIS

The STEPDISC Procedure

The Method for Selecting Variables is STEPWISE

Total Sample Size 569 Variable(s) in the Analysis 14

Class Levels 2 Variable(s) Will Be Included 0

Significance Level to Enter 0.15

Significance Level to Stay 0.15

Number of Observations Read 716

Number of Observations Used 569

DV

(2 levels)

Class Level Information

Variable

ADOPT Name Frequency Weight Proportion

0 _0 468 468.0000 0.822496

p-values: Ho tested is no difference between means of IV for 2 Adopt status levels, using all other IVs as covariates

1 _1 101 101.0000 0.177504

IVs selected by stepwise routine

The STEPDISC Procedure

Stepwise Selection Summary

Number Partial

Step In Entered Removed Label R-Square F Value Pr > F

1 1 NUMIT 0.0747 45.77 <.0001

2 2 REVDUM 0.0306 17.84 <.0001

3 3 CHLEADER My company is obligated to do as 0.0324 18.91 <.0001

4 4 P2HDUM 0.0310 18.04 <.0001

5 5 WMLDUM 0.0162 9.26 0.0025

6 6 P2LDUM 0.0129 7.34 0.0069

7 7 QUALITY Product quality 0.0076 4.28 0.0390

8 8 SERVICE My firm feels my channel or supply 0.0055 3.08 0.0796

9 9 FIRMDUM 0.0050 2.81 0.0942

10 10 WMHDUM 0.0054 3.04 0.0820

11 9 WMLDUM 0.0033 1.83 0.1763

12 10 BRHDUM 0.0056 3.16 0.0759

13 11 WMLDUM 0.0041 2.32 0.1285

Note: With 14 potential IVs, Step 1 conducts 14 ANCOVAs, Step 2 conducts 13 ANCOVAs, Step 3 conducts 12 ANCOVAs, etc. At each step, IV with smallest p-value is selected

DA function includes the 1st 7 IVs selected by Stepwise DA

SAS output for Discriminant Analysis (assuming MV normal distribution)

NORMAL DISCRIMINANT ANALYSIS

The DISCRIM Procedure

Total Sample Size 626 DF Total 625

Variables 7 DF Within Classes 624

Classes 2 DF Between Classes 1

Prior probabilities used

Number of Observations Read 716

DV

(2 levels)

Number of Observations Used 626

Class Level Information

Variable Prior

ADOPT Name Frequency Weight Proportion Probability

0 _0 512 512.0000 0.817891 0.817891

1 _1 114 114.0000 0.182109 0.182109

NORMAL DISCRIMINANT ANALYSIS

The DISCRIM Procedure

Test of Homogeneity of Within Covariance Matrices

Notation: K = Number of Groups

P = Number of Variables

N = Total Number of Observations - Number of Groups

N(i) = Number of Observations in the i'th Group - 1

__ N(i)/2

|| |Within SS Matrix(i)|

V = -----------------------------------

N/2

|Pooled SS Matrix|

_ _ 2

| 1 1 | 2P + 3P - 1

RHO = 1.0 - | SUM ----- - --- | -------------

|_ N(i) N _| 6(P+1)(K-1)

DF = .5(K-1)P(P+1)

_ _

| PN/2 |

| N V |

p-value for testing Ho: equal Var-Cov matrix

Under the null hypothesis: -2 RHO ln | ------------------ |

| __ PN(i)/2 |

|_ || N(i) _|

SAS uses Linear DA if equal Var-Cov matrix; uses Quadratic DA if unequal (i.e., if reject Ho at alpha=.10)

is distributed approximately as Chi-Square(DF).

Chi-Square DF Pr > ChiSq

234.291358 28 <.0001

Since the Chi-Square value is significant at the 0.1 level, the within

covariance matrices will be used in the discriminant function.

Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252.

p-values for MANOVA test of Ho: no mean vector differences between the 2 ADOPT groups

NORMAL DISCRIMINANT ANALYSIS

The DISCRIM Procedure

Multivariate Statistics and Exact F Statistics

S=1 M=2.5 N=308

Statistic Value F Value Num DF Den DF Pr > F

Wilks' Lambda 0.80289363 21.67 7 618 <.0001

Pillai's Trace 0.19710637 21.67 7 618 <.0001

Hotelling-Lawley Trace 0.24549500 21.67 7 618 <.0001

Roy's Greatest Root 0.24549500 21.67 7 618 <.0001

NORMAL DISCRIMINANT ANALYSIS

The DISCRIM Procedure

Classification Summary for Calibration Data: WORK.RFID

Resubstitution Summary using Quadratic Discriminant Function

Number of Observations and Percent Classified into ADOPT

From ADOPT 0 1 Total

0 400 112 512

Summary table of Hit Rates when using all data to estimate DA function

78.13 21.88 100.00

1 37 77 114

32.46 67.54 100.00

Total 437 189 626

69.81 30.19 100.00

Priors 0.81789 0.18211

Error Count Estimates for ADOPT

0 1 Total

Rate 0.2188 0.3246 0.2380

Priors 0.8179 0.1821

NORMAL DISCRIMINANT ANALYSIS

The DISCRIM Procedure

Classification Summary for Calibration Data: WORK.RFID

Cross-validation Summary using Quadratic Discriminant Function

Hit Rate for ADOPT=0

Number of Observations and Percent Classified into ADOPT

From ADOPT 0 1 Total

0 398 114 512

Summary table of Hit Rates when using jackknife method to estimate DA function

77.73 22.27 100.00

1 50 64 114

43.86 56.14 100.00

Total 448 178 626

71.57 28.43 100.00

Priors 0.81789 0.18211

Hit Rate for ADOPT=1

Error Count Estimates for ADOPT

0 1 Total

Rate 0.2227 0.4386 0.2620

Priors 0.8179 0.1821

Overall hit rate = 1 - .262 = .738

SAS output Stepwise Logistic Regression

STEPWISE LOGISTIC REGRESSION

The LOGISTIC Procedure

Model Information

Data Set WORK.RFIDSUB

Response Variable ADOPT

Number of Response Levels 2

Model binary logit

Number of Observations Read 716

Number of Observations Used 569

DV (2 levels):

1=Adopt RFID

0=No adoption

Response Profile

Ordered Total

Value ADOPT Frequency

1 0 468

2 1 101

Probability modeled is ADOPT=1.

NOTE: 147 observations were deleted due to missing values.

Stepwise Selection Procedure

SAS creates dummy variables for QL IVs

Class Level Information

Design

Class Value Variables

WLAN HI-USE 1 0

LO-USE 0 1

NO-USE -1 -1

WMS HI-USE 1 0

LO-USE 0 1

NO-USE -1 -1

BAR HI-USE 1 0

LO-USE 0 1

P2LS HI-USE 1 0

LO-USE 0 1

NO-USE -1 -1

REVENUE HIGH 1

LOW -1

FIRMTYPE DOM 1

INT -1

p-values for Ho: beta associated with IV = 0

STEPWISE LOGISTIC REGRESSION

IVs selected by the stepwise routine

Summary of Stepwise Selection

Effect Number Score Wald Variable

Step Entered Removed DF In Chi-Square Chi-Square Pr > ChiSq Label

1 NUMIT 1 1 42.4998 <.0001

2 REVENUE 1 2 20.5906 <.0001

3 CHLEADER 1 3 17.1828 <.0001

4 P2LS 2 4 18.1073 0.0001

5 WMS 2 5 12.3397 0.0021

Type 3 Analysis of Effects

Note: With 14 potential IVs, Step 1 conducts 14 chi-sq tests (one for each IV), Step 2 conducts 13 chi-sq tests, Step 3 conducts 12 chi-sq tests, etc. At each step, IV with smallest p-value is selected

Wald

Effect DF Chi-Square Pr > ChiSq

CHLEADER 1 16.5384 <.0001

NUMIT 1 7.2375 0.0071

WMS 2 11.3212 0.0035

P2LS 2 19.0501 <.0001

REVENUE 1 23.4229 <.0001

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -5.6041 0.7805 51.5490 <.0001

CHLEADER 1 0.3095 0.0761 16.5384 <.0001

NUMIT 1 0.3620 0.1346 7.2375 0.0071

WMS HI-USE 1 0.8305 0.2512 10.9344 0.0009

WMS LO-USE 1 -0.6511 0.3810 2.9205 0.0875

P2LS HI-USE 1 -1.2804 0.3046 17.6678 <.0001

P2LS LO-USE 1 0.9976 0.2706 13.5920 0.0002

REVENUE HIGH 1 0.8824 0.1823 23.4229 <.0001

SAS output for fit of Logistic Regression model with only main effects of IVs selected by stepwise

MAIN EFFECTS LOGISTIC REGRESSION

The LOGISTIC Procedure

Model Information

Data Set WORK.RFID

Response Variable ADOPT

Number of Response Levels 2

Model binary logit

Number of Observations Read 716

Number of Observations Used 626

Response Profile

Ordered Total

π = P(Adopt RFID) is probability modeled in logistic regression equation

Value ADOPT Frequency

DV (2 levels):

1=Adopt RFID

0=No adoption

1 0 512

2 1 114

Probability modeled is ADOPT=1.

NOTE: 90 observations were deleted due to missing values.

Rsq statistic used to assess fit:

Values near 1 are excellent fit, values near 0 are poor fit

(Subjective decision)

Model Convergence Status

Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Intercept

Intercept and

Criterion Only Covariates

AIC 596.169 465.647

SC 600.608 501.162

-2 Log L 594.169 449.647

p-value for overall model chi-sq test of Ho: all betas in model = 0

(Reject Ho implies a statistically useful model)

R-Square 0.2062 Max-rescaled R-Square 0.3363

Number of IVs in model is 7 -- the IVs selected by the stepwise routine

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq

Likelihood Ratio 144.5219 7 <.0001

Score 123.3886 7 <.0001

Wald 86.0149 7 <.0001

MAIN EFFECTS LOGISTIC REGRESSION

p-values for testing each IV in model, Ho: beta associated with IV = 0

(Reject Ho implies IV is a statistically useful predictor)

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -6.5899 0.7008 88.4291 <.0001

Independent variables (IV) in model

CHLEADER 1 0.3148 0.0728 18.6977 <.0001

NUMIT 1 0.4359 0.1296 11.3081 0.0008

REVDUM 1 1.7115 0.3389 25.5101 <.0001

P2HDUM 1 -1.6445 0.4420 13.8436 0.0002

P2LDUM 1 0.6271 0.3881 2.6102 0.1062

WMHDUM 1 0.9445 0.5089 3.4439 0.0635

WMLDUM 1 -0.3953 0.6878 0.3303 0.5655

95% Confidence Interval for odds ratio (OR) estimate of each IV in model

Odds Ratio Estimates

Odds ratio (OR) estimates for each IV in model

Point 95% Wald

Effect Estimate Confidence Limits

CHLEADER 1.370 1.188 1.580

NUMIT 1.546 1.199 1.994

REVDUM 5.537 2.850 10.758

OR values > 1 imply odds increase as IV increases;

OR values < 1 imply odds decrease as IV increases

P2HDUM 0.193 0.081 0.459

Odds of Adopt RFID increase 1.55 times (i.e., by 55%) for each unit increase in NUMIT

P2LDUM 1.872 0.875 4.006

WMHDUM 2.572 0.948 6.973

WMLDUM 0.673 0.175 2.593

Association of Predicted Probabilities and Observed Responses

Percent Concordant 82.6 Somers' D 0.667

Percent Discordant 16.0 Gamma 0.676

Summary table of predictions using jackknife method

Percent Tied 1.4 Tau-a 0.199

Pairs 58368 c 0.833

Level used for making predictions (e.g., .5). If predicted prob. of adopting RFID (π) > .5 then predict company will adopt. If predicted π < .5, then predict company will not adopt.

Classification Table

Correct Incorrect Percentages

Prob Non- Non- Sensi- Speci- False False

Level Event Event Event Event Correct tivity ficity POS NEG

For a given prob. level:

Sensitivity is Hit Rate for Adopters;

Specificity is Hit Rate for Non-adopters

Select prob. level based on maximizing these hit rates

0.100 111 285 227 3 63.3 97.4 55.7 67.2 1.0

0.200 81 370 142 33 72.0 71.1 72.3 63.7 8.2

0.300 60 437 75 54 79.4 52.6 85.4 55.6 11.0

0.400 39 464 48 75 80.4 34.2 90.6 55.2 13.9

0.500 23 487 25 91 81.5 20.2 95.1 52.1 15.7

0.600 16 496 16 98 81.8 14.0 96.9 50.0 16.5

0.700 8 507 5 106 82.3 7.0 99.0 38.5 17.3

0.800 4 512 0 110 82.4 3.5 100.0 0.0 17.7

0.900 0 512 0 114 81.8 0.0 100.0 . 18.2

SAS output for testing interactions of channel leader IV with other IVs in Logistic Regression model

TEST CHANNEL LEADER INTERACTIONS

The LOGISTIC Procedure

Model Information

Data Set WORK.RFID

Response Variable ADOPT

Number of Response Levels 2

Model binary logit

Number of Observations Read 716

Number of Observations Used 626

Response Profile

Ordered Total

Value ADOPT Frequency

1 0 512

2 1 114

Probability modeled is ADOPT=1.

NOTE: 90 observations were deleted due to missing valuess.

Model Convergence Status

Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Intercept

Intercept and

Criterion Only Covariates

AIC 596.169 470.688

SC 600.608 532.839

-2 Log L 594.169 442.688

R-Square 0.2149 Max-rescaled R-Square 0.3507

Number of IVs in model is 13 -- the 7 IVs selected by the stepwise routine + 6 interactions with channel leader

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq

Likelihood Ratio 151.4814 13 <.0001

Score 127.2428 13 <.0001

Wald 76.3402 13 <.0001

TEST CHANNEL LEADER INTERACTIONS

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -11.8112 2.6631 19.6704 <.0001

CHLEADER 1 1.3518 0.4700 8.2711 0.0040

NUMIT 1 0.6703 0.3899 2.9559 0.0856

REVDUM 1 2.6683 1.1137 5.7407 0.0166

P2HDUM 1 -2.4134 1.2934 3.4819 0.0620

P2LDUM 1 0.3411 1.0803 0.0997 0.7522

WMHDUM 1 4.4171 2.5652 2.9651 0.0851

WMLDUM 1 3.7808 2.9408 1.6528 0.1986

CHL_NUM 1 -0.0469 0.0812 0.3335 0.5636

CHL_REV 1 -0.2023 0.2204 0.8426 0.3587

CHL_WMH 1 -0.6879 0.4753 2.0947 0.1478

CHL_WML 1 -0.8488 0.5723 2.1997 0.1380

CHL_P2H 1 0.1676 0.2725 0.3782 0.5385

CHL_P2L 1 0.0599 0.2414 0.0617 0.8039

Odds Ratio Estimates

Point 95% Wald

Effect Estimate Confidence Limits

These are the 7 IVs selected by stepwise

CHLEADER 3.864 1.538 9.709

NUMIT 1.955 0.910 4.197

REVDUM 14.416 1.625 127.881

P2HDUM 0.090 0.007 1.129

P2LDUM 1.407 0.169 11.687

WMHDUM 82.854 0.543 >999.999

p-value for testing the interactions (nested model test); Ho: all interaction betas =0

(Fail to reject Ho implies interactions are not significant)

WMLDUM 43.850 0.138 >999.999

CHL_NUM 0.954 0.814 1.119

These are the 6 channel leader interactions

CHL_REV 0.817 0.530 1.258

CHL_WMH 0.503 0.198 1.276

CHL_WML 0.428 0.139 1.314

CHL_P2H 1.182 0.693 2.017

CHL_P2L 1.062 0.662 1.704

Linear Hypotheses Testing Results

Wald

Label Chi-Square DF Pr > ChiSq

CHLEADER_INTERACTION 5.7630 6 0.4503

Number of terms (interactions) tested

TEST NUMBER IT INTERACTIONS

SAS output for testing interactions of NUMIT IV with other IVs in Logistic Regression model

The LOGISTIC Procedure

Model Information

Data Set WORK.RFID

Response Variable ADOPT

Number of Response Levels 2

Model binary logit

Optimization Technique Fisher's scoring

Number of Observations Read 716

Number of Observations Used 626

Response Profile

Ordered Total

Value ADOPT Frequency

1 0 512

2 1 114

Probability modeled is ADOPT=1.

NOTE: 90 observations were deleted due to missing values.

Model Convergence Status

Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Intercept

Intercept and

Criterion Only Covariates

AIC 596.169 460.609

SC 600.608 522.760

-2 Log L 594.169 432.609

Number of IVs in model is 13 -- the 7 IVs selected by the stepwise routine + 6 interactions with NUMIT

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq

Likelihood Ratio 161.5598 13 <.0001

Score 130.3523 13 <.0001

Wald 69.9183 13 <.0001

TEST NUMBER IT INTERACTIONS

The LOGISTIC Procedure

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -11.4784 2.5018 21.0504 <.0001

CHLEADER 1 0.7476 0.3019 6.1310 0.0133

NUMIT 1 1.3536 0.5507 6.0409 0.0140

REVDUM 1 5.2846 1.6370 10.4219 0.0012

P2HDUM 1 -4.4241 3.1177 2.0136 0.1559

P2LDUM 1 4.4983 2.2370 4.0436 0.0443

WMHDUM 1 1.0095 1.3328 0.5737 0.4488

WMLDUM 1 -7.4836 3.7309 4.0234 0.0449

CHL_NUM 1 -0.0858 0.0598 2.0609 0.1511

NUM_REV 1 -0.7198 0.3025 5.6619 0.0173

NUM_WMH 1 0.0565 0.3817 0.0220 0.8822

NUM_WML 1 1.3533 0.7071 3.6632 0.0556

NUM_P2H 1 0.4742 0.5318 0.7952 0.3725

NUM_P2L 1 -0.7049 0.3867 3.3223 0.0683

Odds Ratio Estimates

Point 95% Wald

Effect Estimate Confidence Limits

CHLEADER 2.112 1.169 3.817

NUMIT 3.871 1.315 11.393

REVDUM 197.280 7.974 >999.999

P2HDUM 0.012 <0.001 5.401

P2LDUM 89.865 1.121 >999.999

WMHDUM 2.744 0.201 37.399

WMLDUM <0.001 <0.001 0.843

These are the 6 NUMIT interactions

CHL_NUM 0.918 0.816 1.032

NUM_REV 0.487 0.269 0.881

NUM_WMH 1.058 0.501 2.236

NUM_P2H 1.607 0.567 4.556

NUM_P2L 0.494 0.232 1.055

Linear Hypotheses Testing Results

p-value for testing the interactions (nested model test); Ho: all interaction betas =0

(Fail to reject Ho implies interactions are not significant)

Wald

Label Chi-Square DF Pr > ChiSq

NUMIT_INTERACTION 14.3850 6 0.0256

Note: When conducting multiple tests better to test each at a small α (e.g., .01)

Number of terms (interactions) tested