# TABLES by yaoyufang

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```									             TESTING LATENT VARIABLE MODELS WITH SURVEY DATA

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TABLES

Table 1- The Number of Cases per Unique Covariance Matrix Element, Based on
the Number of Variables in the Covariance Matrix and the Number of Cases per
Variable in a Data Set

Table 2a- The Number of Cases Required for 4 Indicators per Latent Variable (LV),
Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator
Variable in a Data Set

Table 2b- The Number of Cases Required for 5 Indicators per Latent Variable (LV),
Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator
Variable in a Data Set

Table 2c- The Number of Cases Required for 6 Indicators per Latent Variable (LV),
Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator
Variable in a Data Set

APPENDICES

Appendix A- Interaction Specification Using a Single Product-of-Sums Indicator and
Structural Equation Analysis
Figure A- An Abbreviated Structural Model

Appendix B- OLS Regression and Structural Equation Analysis
Table B - Structural Equation Analysis and OLS Regression Coefficient Estimates

Appendix C- Interaction Interpretation
Table C1- Appendix A Structural Model Estimation Results
Table C2- Table C1 UxT Interaction Statistical Significance

Appendix D- Indirect and Total Effects

 2002 Robert A. Ping, Jr. 9/20/02 i
Table D1- Figure A Model Standardized Indirect Effects
Table D2- Figure A Model Standardized Total Effects
Appendix E- Consistency Improvement using Summed First Derivatives
Table E1- First Derivatives for the Eight Item Measure
Table E2- First Derivatives with x4 Deleted
Table E3- First Derivatives with x3 and x4 Deleted
Table E4- First Derivatives with x1, x3 and x4 Deleted

Appendix F- Ordered Similarity Coefficients and Consistency
Table F- Ordered Similarity Coefficients for the Appendix E Items

Appendix G- Error Adjusted Regression Estimates for the Figure A Model
Table G1-- Unadjusted Covariances for A, B, C, D, and E: with Reliabilities,
Table G2-- Adjusted Covariances for A, B, C, D, and E, with Coefficient
Estimates
Appendix H- Scenario Example
Exhibit H- A Scenario

Appendix I- Structural Equation Analysis with Summed Indicators
Table I- Coefficient Estimates

Appendix J- Second-Order Construct Example
Figure J- Second-Order Constructs

Appendix K- Scenario Analysis Results Comparison
Table K1- Comparison of Scenario and Survey Data from a Common
Questionnaire Using Factor Analysis
Table K2- Comparison of Scenario and Field Survey Data from a Common
Questionnaire Using Regression

Appendix L- Average Variance Extracted
Table L- AVE and Reliability Estimates for the Appendix A Variables T, U, V,
W, and UxT., and the Appendix G Variables A, B, C, D, and E

Appendix M- Nonrecursive Analysis Example
Figure M- A Nonrecursive Model
Appendix N- Second-Order Interactions
Table N1- Results for T First-Order and UxT as Single Indicator with Ping
(1995) Specification (χ2/df = 189/112, GFI = .91, AGFI = .88, CFI = .96,
RMSEA = .05)
Table N2- Results for T First-Order and UxT as Single Indicator with Single
Indicator structural equation analysis Specification (χ2/df = 189/111, GFI

 2002 Robert A. Ping, Jr. 9/20/02 ii
= .91, AGFI = .88, CFI = .96, RMSEA = .05)
Table N3- Results of T 2nd Order and UxT with all 60 Kenny and Judd Indicators
(χ2/df = 28519/3591, GFI = .14, AGFI = .13, CFI = .29, RMSEA = .17)
Table N4- Results of T 2nd Order and UxT with 4 Arbitrary Kenny and Judd
Indicators (χ2/df = 893/372, GFI = .78, AGFI = .75, CFI = .89, RMSEA =
.07)
Table N5- Results of T 2nd Order and UxT with 4 Different but Arbitrary Kenny
and Judd Indicators (χ2/df = 793/372, GFI = .79, AGFI = .76, CFI = .91,
RMSEA = .07)
Table N6- Results of T 2nd Order and UxT with 4 Consistent Kenny and Judd
Indicators (χ2/df = 637/372, GFI = .84, AGFI = .81, CFI = .94, RMSEA =
.05)
Table N7- Results of T 1st Order and UxT with all 15 Kenny and Judd Indicators
(χ2/df = 3402/454, GFI = .44, AGFI = .39, CFI = .56, RMSEA = .17)
Table N8- Results of T 1st Order and UxT with 4 Arbitrary Kenny and Judd
Indicators (χ2/df = 371/168, GFI = .86, AGFI = .82, CFI = .92, RMSEA =
.07)
Table N9- Results of T 1st Order and UxT with 4 Different but Arbitrary Kenny
and Judd Indicators (χ2/df = 439/168, GFI = .83, AGFI = .78, CFI = .89,
RMSEA = .08)
Table N10- Results of T 1st Order and UxT with 4 Consistent Kenny and Judd
Indicators (χ2/df = 353/168, GFI = .87, AGFI = .83, CFI = .92, RMSEA =
.07)

 2002 Robert A. Ping, Jr. 9/20/02 iii
Table 1- The Number of Cases per Unique Covariance Matrix Element, Based on the Number of Variables in the
Covariance Matrix and the Number of Cases per Variable in a Data Seta

Number of
Number of Unique Covariance                                Number of Cases per Variable
Variables Matrix Elements         4      5      6      7       8     9 10       11 12          13     14      15
2              3          2.67   3.33   4.00   4.67    5.33 6.00 6.67 7.33 8.00         8.67   9.33   10.00
3              6          2.00   2.50   3.00   3.50    4.00 4.50 5.00 5.50 6.00         6.50   7.00    7.50
4             10          1.60   2.00   2.40   2.80    3.20 3.60 4.00 4.40 4.80         5.20   5.60    6.00
5             15          1.33   1.67   2.00   2.33    2.67 3.00 3.33 3.67 4.00         4.33   4.67    5.00
6             21          1.14   1.43   1.71   2.00    2.29 2.57 2.86 3.14 3.43         3.71   4.00    4.29
7             28          1.00   1.25   1.50   1.75    2.00 2.25 2.50 2.75 3.00         3.25   3.50    3.75
8             36          0.89   1.11   1.33   1.56    1.78 2.00 2.22 2.44 2.67         2.89   3.11    3.33
9             45          0.80   1.00   1.20   1.40    1.60 1.80 2.00 2.20 2.40         2.60   2.80    3.00
10             55          0.73   0.91   1.09   1.27    1.45 1.64 1.82 2.00 2.18         2.36   2.55    2.73
11             66          0.67   0.83   1.00   1.17    1.33 1.50 1.67 1.83 2.00         2.17   2.33    2.50
12             78          0.62   0.77   0.92   1.08    1.23 1.38 1.54 1.69 1.85         2.00   2.15    2.31
13             91          0.57   0.71   0.86   1.00    1.14 1.29 1.43 1.57 1.71         1.86   2.00    2.14
14            105          0.53   0.67   0.80   0.93    1.07 1.20 1.33 1.47 1.60         1.73   1.87    2.00
15            120          0.50   0.63   0.75   0.88    1.00 1.13 1.25 1.38 1.50         1.63   1.75    1.88
16            136          0.47   0.59   0.71   0.82    0.94 1.06 1.18 1.29 1.41         1.53   1.65    1.76
17            153          0.44   0.56   0.67   0.78    0.89 1.00 1.11 1.22 1.33         1.44   1.56    1.67
18            171          0.42   0.53   0.63   0.74    0.84 0.95 1.05 1.16 1.26         1.37   1.47    1.58
19            190          0.40   0.50   0.60   0.70    0.80 0.90 1.00 1.10 1.20         1.30   1.40    1.50
20            210          0.38   0.48   0.57   0.67    0.76 0.86 0.95 1.05 1.14         1.24   1.33    1.43
21            231          0.36   0.45   0.55   0.64    0.73 0.82 0.91 1.00 1.09         1.18   1.27    1.36
22            253          0.35   0.43   0.52   0.61    0.70 0.78 0.87 0.96 1.04         1.13   1.22    1.30
23            276          0.33   0.42   0.50   0.58    0.67 0.75 0.83 0.92 1.00         1.08   1.17    1.25
24            300          0.32   0.40   0.48   0.56    0.64 0.72 0.80 0.88 0.96         1.04   1.12    1.20
25            325          0.31   0.38   0.46   0.54    0.62 0.69 0.77 0.85 0.92         1.00   1.08    1.15
26            351          0.30   0.37   0.44   0.52    0.59 0.67 0.74 0.81 0.89         0.96   1.04    1.11
27            378          0.29   0.36   0.43   0.50    0.57 0.64 0.71 0.79 0.86         0.93   1.00    1.07
28            406          0.28   0.34   0.41   0.48    0.55 0.62 0.69 0.76 0.83         0.90   0.97    1.03
29            435          0.27   0.33   0.40   0.47    0.53 0.60 0.67 0.73 0.80         0.87   0.93    1.00
30            465          0.26   0.32   0.39   0.45    0.52 0.58 0.65 0.71 0.77         0.84   0.90    0.97
───────────────────────
a
The table shows the cases per unique input covariance matrix element for combinations of variables in a model (the
rows) and the number of cases per model variable (columns 3 through 14). For example for 5 variables (row entry) 6
cases per variable (column entry) produces 2 cases per unique input covariance matrix element. Column 2 shows the
number of unique input covariance matrix elements, in this case 15 (=n(n+1)/2, where n is the number of variables).

 2002 Robert A. Ping, Jr. 9/20/02 74
Table 2a- The Number of Cases Required for 4 Indicators per Latent Variable (LV), Based on the Number of Latent
Variables (LV's), and the Number of Cases per Indicator Variable in a Data Set a
# of                                                Cases per
of MM                                               Unique Cov
Number      Param.       Cases   per Parameter   Estimated      Matrix Element
of LV's     Estimated       1        2     3       4       5         1      2
2            17        17      34     51      68     85        12      24
3            27        27      54     81     108    135        24      48
4            38        38      76    114     152    190        40      80
5            50        50     100    150     200    250        60     120
6            63        63     126    189     252    315        84     168
7            77        77     154    231     308    385       112     224
8            92        92     184    276     368    460       144     288
9          108        108     216    324     432    540       180     360
10          125        125     250    375     500    625       220     440

Table 2b- The Number of Cases Required for 5 Indicators per Latent Variable (LV), Based on the Number of Latent
Variables (LV's), and the Number of Cases per Indicator Variable in a Data Set a

# of                                                 Cases per
of MM                                                Unique Cov
Number      Param.        Cases   per Parameter   Estimated      Matrix Element
of LV's     Estimated       1        2     3        4      5          1      2
2            21        21       42    63       84   105         55     110
3            33        33       66    99      132   165        120     240
4            46        46       92   138      184   230        210     420
5            60        60      120   180      240   300        325     650
6            75        75      150   225      300   375        465     930
7            91        91      182   273      364   455        630 1260
8          108        108      216   324      432   540        820 1640
9          126        126      252   378      504   630       1035 2070
10          145        145      290   435      580   725       1275 2550

Table 2c- The Number of Cases Required for 6 Indicators per Latent Variable (LV), Based on the Number of Latent
Variables (LV's), and the Number of Cases per Indicator Variable in a Data Seta
# of                                                Cases per
of MM                                               Unique Cov
Number      Param.        Cases per Parameter Estimated         Matrix Element
of LV's     Estimated       1      2     3      4      5             1      2
2            25        25     50    75    100   125            78     156
3            39        39     78   117    156   195           171     342
4            54        54    108   162    216   270           300     600
5            70        70    140   210    280   350           465     930
6            87        87    174   261    348   435           666 1332
7          105        105    210   315    420   525           903 1806
8          124        124    248   372    496   620          1176 2352
9          144        144    288   432    576   720          1485 2970
10          165        165    330   495    660   825          1830 3660
───────────────────────
a
The table shows the number of cases required for 4, 5 and 6 indicators per latent variable using combinations of the
number of latent variables in a model (the rows), the number of cases per parameter estimated (columns 3 through 7),
and the number of cases per unique covariance matrix element (columns 8 and 9). For example 5 latent variables
(row entry) produces 70 measurement model parameters to be estimated (column 2), and requires 140 cases if 2
cases per parameter are desired (column 4), but 930 cases if 2 cases per unique covariance matrix element are
desired (column 9).

 2002 Robert A. Ping, Jr. 9/20/02 75
Appendix A- Interaction Specification Using a Single Product-of-Sums Indicator
and Structural Equation Analysis

The following summarizes the seminal article on structural equation analysis interactions, Kenny and Judd
(1984), then it summarizes an alternative technique proposed by Ping (1995).
Kenny and Judd (1984) proposed for latent variables X and Z with multiple indicators xi and zj, the
interaction XZ could be specified in structural equation analysis using multiple indicators that are all possible
product indicators xizj. However, this technique has proven difficult for researchers to implement (Aiken and West,
1991). While LISREL 8 reduces the effort involved by providing constraint equations, specifying all possible
product indicators still requires considerable effort (Jöreskog and Yang, 1996). In addition, the set of all possible
product indicators is usually inconsistent and thus a measurement or structural model with an interaction specified
with all possible product indicators will usually not fit the data.
In addition, measurement and structural models with an interaction are per se nonnormal because products
of indicators (i.e., xizj) are nonnormal (Kenny and Judd, 1984). While maximum likelihood parameter estimates are
robust to departures from normality (see the citations in Ping, 1995), their model fit and significance statistics may
not be (Bollen, 1989). However model fit and significance statistics may be robust to the addition of a few
(nonnormal) product indicators (i.e., four or fewer) (Jaccard and Wan, 1995; Ping, 1995).
The Ping (1995) technique requires a single (nonnormal) product-of-sums indicator for a structural equation
analysis interaction. Under the Kenny and Judd (1984) normality assumptions, (i.e., the latent variables X and Z with
indicators xi and zj are independent of the error terms for their indicators ε xi and εzj, the error terms are independent
of each other, and xi, zj, εxi and εzj are normally distributed) an interaction can be specified with one indicator that is
the product of sums of the indicators of the linear latent variables. For example the indicator for XZ, comprised of X
and Z with indicators x1, x2, z1, and z2 respectively, would be x:z = (x1+x2)(z1+z2), or, if equivalently sized elements
in the resulting covariance matrix are desired, x:z = [(x1+x2)/2][(z1+z2)/2]. The loading and error variance of x:z are
given by
λx:z = ΓXΓZ                                                                                           (A1
and                  θεx:z = ΓX2Var(X)θZ + ΓZ2Var(Z)θX + θXθZ ,                                                            (A2
where λx:z is the loading of x:z on XZ, θεx:z is the variance of the error term (εx:z) for x:z, Var(a) is the variance of a,
and for equivalently sized elements, ΓX = (λx1 + λx2)/2, θX = (Var(εx1) + Var(εx2))/22, ΓZ = (λz1 + λz2)/2, and θZ =
(Var(εz1) + Var(εz2))/22 (see Ping, 1995). The loading and error variance of x:z could then be specified subject to the
constraint equations A1 and A2 using LISREL 8.
For example the loading and error of u:t (= [(u1+u2+u3+u4+u5)/5][(t1 + t2 + t3)/3], see Figure A and the
example below), the single indicator of UxT in the example below, are given by
λu:t = ΓUΓT                                                                                           (A3
and       θεu:t = ΓU2Var(U)θT + ΓT2Var(T)θU + θUθT ,                                                                       (A4
where λu:t is the loading of u:t on UxT, θεu:t is the variance of the error term (εu:t) for u:t, Var(a) is the variance of a,
ΓU = (λu1 +...+ λu5)/5, θU = (Var(εu1) +...+ Var(εu5))/52, ΓT = (t1 + t2 + t3)/3, and θT = (Var(εt1) + Var(εt2) +
Var(εt3))/32.
An Example
A Marketing survey involving the latent variables T, U, V, W, the interaction UxT, and their indicators t i,
vk, up, wq, and u:t, produced more than 200 usable responses. The measures for the latent variables were judged to be
unidimensional, valid and reliable. The code to estimate the Figure A structural model using the Ping (1995)
technique and LISREL 8 and the covariance matrix are available from the authors. The results are shown in Table
C1, and the significant UxT effect is discussed there as well. The indirect and total effects are discussed in Appendix
D.
Each indicator of the independent and dependent variables was mean centered by subtracting the
indicators average from its value in each case (centering independent variables is important to reduce collinearity,
and centering dependent variables is important to compensate for not estimating intercepts-- see Jöreskog and Yang,
1996). The value for the interactions single-indicator u:t was added to each case. Next the structural model was
specified using PAR variables (Jöreskog and Sörbom, 1996b:347) and constraint equations (Jöreskog and Sörbom,
1996b:346) for ΓU, ΓT θU, and θT. Constraint equations (CO statements) were written for Equations A3 and A4 using
PAR variables, then the structural model was estimated using maximum likelihood. For example the constraint code

 2002 Robert A. Ping, Jr. 9/20/02 76
for this model was
co par(1)=.2*ly(1,1)+.2*ly(2,1)+.2*ly(3,1)+.2*ly(4,1)+.2*ly(5,1)
co par(2)=.33333*ly(6,2)+.33333*ly(7,2)+.33333*ly(8,2)
co ly(9,3)=par(1)*par(2)
co par(3)=.04*te(1,1)+.04*te(2,2)+.04*te(3,3)+.04*te(4,4)+.04*te(5,5)
co par(4)=.11111*te(6,6)+.11111*te(7,7)+.11111*te(8,8)
co te(9,9)=par(4)*ps(1,1)*par(1)^2+par(3)*ps(2,2)*par(2)^2+par(3)*par(4),
where ly(1,1) through ly(5,1) and ly(6,2) through ly(8,2) were the loadings of U and T respectively, te(1,1) through
te(5,5) and te(6,6) through te(8,8) were the measurement errors of U and T respectively, and ps(1,1) and ps(2,2)
were the variances of U and T respectively.
This use of PAR variables is sensitive to the sequence and location of the PAR and CO statements in the
LISREL program. In general PAR variables should not be used recursively (Jöreskog and Sörbom, 1996b:346). In
this application they are used recursively and they should appear at the end of the program. In addition, these PAR
variables and the variables constrained in the CO statements should be defined in their natural numerical order (e.g.,
PAR(1), PAR(2), etc.) and a PAR variable should be used in a CO statement as soon after it is defined as possible.
Starting values for the loading, error, and variance terms of the interaction must be provided and were estimated
using a measurement model involving all the variables except the interaction. The resulting measurement parameters
estimates were substituted into Equations A3 and A4 to produce a starting value for λu:t and θεu:t (see Ping, 1995). The
starting values for the structural coefficients and the structural disturbances (ζV and ζW) must also be provided and
were approximated using OLS regression coefficients and R2's.

Figure A- An Abbreviated Structural
Modela

───────────────────────
a
V, and W had indicators vi and wi (see Table C1). U, T and UxT were correlated, indicator error terms were uncorrelated, and the ζs
were uncorrelated.

 2002 Robert A. Ping, Jr. 9/20/02 77
Appendix B- OLS Regression and Structural Equation Analysis

Even with very reliable survey data (e.g., α in the .85-.95 range), models with several correlated
independent variables can produce regression coefficients that are significant or nonsignificant using OLS
regression, when the structural equation analysis results (which should be unbiased) suggest the reverse is true, as the
following example shows.
An Example
A marketing survey involving the latent variables Q, T, Z, R, V, X, Y, U, and W produced more than 200
usable responses in the final test. The multiple item measures for each latent variable were judged to be
unidimensional, valid and reliable (reliabilites were .92, .94, .94, .91, .80, .91, .85, .94, and .92, respectively). The
results of estimating a model with Y as the dependent variable and the other variables as independent variables using
OLS regression (and summed indicators for each latent variable) then LISREL 8 with the indicators specified in the
usual way (see Figure A for example) and maximum likelihood are shown in Table B.
While the significance of several coefficients is similar between OLS and structural equation analysis, the U
coefficient was significant using structural equation analysis but nonsignificant using OLS regression. Similarly, the
V coefficient was nonsignificant using structural equation analysis but significant using OLS regression. Assuming
structural equation analysis, with its ability to specify measurement error and its use of maximum likelihood
estimation, is more likely to produce unbiased structural coefficient estimates, OLS regression, which is well known
to be biased for variables measured with error, may produce false negative (Type I) and false positive (Type II)
errors even for variables that would be judged highly reliable.

Table B - Structural Equation Analysis and OLS Regression Coefficient Estimates

Abbreviated Structural Equation
Analysis Results:                                                                  OLS Regression Results:

Endogenous Variable = Y                             Dependent Variable = Y
(χ2 = 1091, df = 666, CFI = .944, RMSEA = .053)      (F = 9.84, p = .000) (R2 = .270)
Unstandardized                                      Unstandardized
Variable    Coefficient       SE     t-value           Variable      Coefficient      SE      t-value (p-value)
Q           .092         .064     1.43              Q           .067          .061       1.09 (.276)
T           .076         .054     1.40              T           .074          .051       1.45 (.147)
Z           .178         .086     2.07              Z           .182          .078       2.32 (.021)
R          -.067         .066    -1.00              R          -.061          .062      -0.98 (.327)
V           .115         .059     1.96              V           .137          .058       2.34 (.020)
X           .251         .092     2.73              X           .310          .092       3.35 (.000)
U           .184         .086     2.12              U           .156          .081       1.91 (.056)
W          -.137         .113    -1.21              W          -.132          .107      -1.23 (.218)
Const.       -1.4D-16        .038      -0.00

 2002 Robert A. Ping, Jr. 9/20/02 78
Appendix C- Interaction Interpretation

The Appendix A example produced parameter estimates shown in Table C1. The UxT interaction effect on V
was significant (see βV,UxT in Table C1), and Table C2 shows the resulting contingent (i.e., interacting) effects of U
and T on V, along with their significances. Contingent (i.e., interaction) coefficients were explained in Step I, and
because the interaction UxT involves two constituent variables (i.e., U and T), when it is significant the effects of
both its constituent variables on V are contingent. Thus, Table C2 shows the contingent effects of U and T on V.
Using a range of values for the interacting variable reveals any changes in significance or sign in the contingent
effect as the interacting variable ranges from its low to its high in the study. The contingent coefficients and their
standard errors are calculated by hand using the equations shown in the notes to Table C2.
Interpreting Table C2, a significant UxT interaction means the effects of both U and T on V are dependent on
the levels of their interacting variable (T and U respectively) (also see Aiken and West, 1991 for an accessible
discussion of interaction interpretation in survey research). For example, when U was low, the T coefficient was
positive and significant. However as U increased to the study average, Ts effect on V weakened, and it became non
significant near the study average and at higher levels of U. Thus T affected V only for below average U.
Similarly T moderated the effect of U on V. At low levels of T the U coefficient was positive and significant,
and as it approached the study average it weakened and became nonsignificant. However, when T was high the U
coefficient was negative and significant. Thus U affected V positively for low T and negatively for high T.
Several comments are of interest here. The contingent effects of U and T on V were significant over part(s) of
the ranges of their interacting variable. Had the interaction been excluded from the Figure A model, the effects of U
and T on V would have been nonsignificant. This can be seen in the nonsignificance of the U and T coefficients at
the study averages of the interacting variables (see Table C2). In fact, had the interaction been excluded, the
coefficients and significance of U and T would have been close to the values shown for the Table C2 study averages
of the interacting variables (see Aiken and West, 1991).

Table C1- Appendix A Structural Model Estimation Resultsa

V = βV,UUb+ βV,TT + βV,UxTUxT + ζV
-0.130 0.158 -0.339         0.603
(-1.13)c (1.37) (-2.73)      (7.52)

W = βWUU + βWTT + βWTT + ζV
0.145 0.100 -0.176 0.150
(2.96) (1.85) (-4.63) (8.22)

Fit Statistics:d
Chi-Square Statistic Value                                                  189
Chi-Square Degrees of Freedom                                               112
p-Value of Chi-Squared Value                                               .000
Bentler (1990) Comparative Fit Index                                       .966
Steiger (1990) RMS Error of Approximation (RMSEA) e                        .056
GFIf                                                                       .913
AGFIf                                       V  W                           .882

Squared Multiple Correlation:                           .065 .218

───────────────────────
a
Maximum likelihood.
b
βV,U is the effect of U on V.
c
t-value.
d
Overall, adequate fit is suggested. Chi-square, GFI, and AGFI suggest inadequate fit, but may not be appropriate. CFI and RMSEA
e
.05 suggests close fit, values through .08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993).
f
Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing,
1984).
* t-value > 2.

 2002 Robert A. Ping, Jr. 9/20/02 79
Table C2- Table C1 UxT Interaction Statistical Significance

T-V Effect                              U-V Effect
T      SE of                              U      SE of
U     Coef- T Coef-                t-    t-    Coef- U Coef-                   t-
Valuea ficient b ficientc          value Valued ficiente ficientf             value
1.20     1.16    0.41             2.79 1.83      0.43     0.18                2.35
2      0.89    0.32             2.73     2     0.37     0.16                2.23
3      0.55    0.22             2.49     3     0.03     0.11                0.34
4      0.21    0.14             1.45    3.5 g -0.13     0.12                -1.02
4.16g 0.15      0.13              1.13    4    -0.30     0.16                -1.82
5     -0.12    0.15             -0.83    5    -0.64     0.26                -2.40

───────────────────────
a
The values ranged from 1.20 (=low) to 5 in the study.
b
The coefficient of T is given by (.158-.339U)T with U mean centered (e.g., the coefficient of T at U = 1.20 is given by (.158-
.339(1.20-4.16))T).
c
The Standard Error (SE) of the T coefficient is given by
____________          ______________________________
Var(bT+bUxTU) = (Var(bT)+U2Var(bUxT)+2UCov(bT,bUxT) (see Friedrich, 1982).
d
The values ranged from 1.83 (=low) to 5 in the study.
e
The coefficient of U is given by (-.130-.339T)U with T mean centered (e.g., the coefficient of U at T = 1.83 is given by (-.130-
.339(1.83-3.5))U).
f
The Standard Error (SE) of the U coefficient is given by
____________         ______________________________
Var(bU+bUxTT) = Var(bU)+T2Var(bUxT)+2TCov(bU,bUxT) (see Friedrich, 1982).
g
Mean value.

 2002 Robert A. Ping, Jr. 9/20/02 80
Appendix D- Indirect and Total Effects

The Appendix A example also had significant indirect and total effects, shown in Tables D1 and D2. In Figure
A, UxT, for example, significantly affected V directly via the path from UxT to V (i.e., β V,UxT was significant, see
Table C1), and indirectly via the significant indirect path from UxT to V, then from V to W (see Table D1). As a
result, the sum of all the paths from UxT to W (i.e., the total effect of UxT on W) was significant (see Table D2).
The Stability Index (Note b in Table D1) should be less than 1 for indirect and total effects to be meaningful (see
Bollen, 1989).

Table D1- Figure A Model Standardized Indirect Effectsa b c

V        W          U        T       UxT

V

W                                 .038 -.047 .084
(1.09) (-1.30) (2.37)

───────────────────────
a
The table is read from column to row (e.g., the indirect effect of U on W is
.038).
b
Stability Index = .158
c
t-values are shown in parentheses.

Table D2- Figure A Model Standardized Total Effectsa b

V        W             U        T       UxT

V                                    -.011   .147 -.261
(-1.13) (1.37) (-2.73)
W              -.323                  .281 .123 .084
(-4.63)                (3.21) (1.28) (2.37)

───────────────────────
a
The table is read from column to row (e.g., the total effect of V on W is -.323).
b
t-values are shown in parentheses.

 2002 Robert A. Ping, Jr. 9/20/02 81
Appendix E- Consistency Improvement using Summed First Derivatives

A measure of the latent variable X had eight items in a Marketing survey that produced more than 200 usable
responses. The first derivatives with respect to the error terms (Var(e)s in Equation 3), and their sum without regard
to sign for each item, from a single construct measurement model of X are shown in Table E1. The item with the
largest Table E1 sum (x4) was deleted, and the measurement model was re-estimated to produce Table E2. This
process was repeated until RMSEA was .08 or less (see Table E4). An investigation of all other measurement models
with of five items (not shown) produced combinations of items that were less consistent (i.e, they had worse model
fit statistics), suggesting the Table E4 items were maximally consistent.
However, maximizing consistency does not necessarily maximize reliability or Average Variance Extracted
(AVE). The items with maximum reliability and AVE were x4, x5, x6, x7, and x8 (Reliability = .884 and AVE = .606,
but χ2 = 25, df = 5, p-value = .0001, RMSEA = .135).
There is no guidance for trading off reliability and consistency in cases where they diverge. In the present case
the reliabilities of both itemizations would likely be judged acceptable. However AVE for the Table E4 itemization
is only slightly above the suggested cutoff (i.e., .5), and x4 through x8 are marginally consistent. In cases where
reliability and consistency diverge, I suggest using the higher reliability itemization(s) first.

Table E1- First Derivatives for the Eight Item Measure
x1       x2       x3       x4      x5       x6 x7     x8
x1        0.000 -0.439 -0.025 -0.086 0.047 0.006 0.010 0.371
x2       -0.439 0.000 -0.272 0.287 0.217 0.042 -0.200 0.143
x3       -0.025 -0.272 0.000 -0.527 0.184 0.364 0.422 -0.207
x4       -0.086 0.287 -0.527 0.000 -0.943 0.505 0.534 0.144
x5        0.047 0.217 0.184 -0.943 0.000 0.222 0.359 0.019
x6        0.006 0.042 0.364 0.505 0.222 0.000 -0.929 -0.187
x7        0.010 -0.200 0.422 0.534 0.359 -0.929 0.000 -0.113
x8        0.371 0.143 -0.207 0.144 0.019 -0.187 -0.113 0.000
Suma      0.983 1.600 2.000 3.027 1.991 2.254 2.565 1.184
χ2 = 86 df = 20 p-value = 0 RMSEA = .123 Reliability = .860 AVE = .442

Table E2- First Derivatives with x4 Deleted
x1        x2      x3       x5     x6       x7 x8
x1        0.000    -0.442 -0.064 -0.057 0.037           0.044 0.354
x2       -0.442     0.000 -0.287 0.129 0.214 -0.067 0.195
x3       -0.064    -0.287 0.000 -0.172 0.319            0.382 -0.313
x5       -0.057     0.129 -0.172 0.000 0.090            0.231 -0.252
x6        0.037     0.214 0.319 0.090 0.000 -0.544 0.012
x7        0.044    -0.067 0.382 0.231 -0.544 0.000 0.112
x8        0.354     0.195 -0.313 -0.252 0.012 0.112 0.000
Suma      0.998     1.334 1.537 0.933 1.217 1.381 1.239
χ2 = 56 df = 14   p-value = .44D-6 RMSEA = .117 Reliability = .828 AVE = .416

Table E3- First Derivatives with x3 and x4 Deleted
x1     x2        x5       x6     x7       x8
x1        0.000 -0.445 -0.086 0.045 0.054 0.304
x2       -0.445 0.000 0.036 0.190 -0.103 0.107
x5       -0.086 0.036 0.000 0.114 0.270 -0.383
x6        0.045 0.190 0.114 0.000 -0.252 -0.013
x7        0.054 -0.103 0.270 -0.252 0.000 0.096
x8        0.304 0.107 -0.383 -0.013 0.096 0.000
Suma      0.937 0.883 0.891 0.616 0.776 0.904
χ2 = 36 df = 9 p-value = .38D-4 RMSEA = .116 Reliability = .814 AVE = .433

(Continued)

 2002 Robert A. Ping, Jr. 9/20/02 82
Table E4- First Derivatives with x1, x3 and x4 Deleted
x2      x5       x6        x7     x8
x2        0.000 -0.026 0.110 -0.180 0.079
x5       -0.026 0.000 0.104 0.252 -0.352
x6        0.110 0.104 0.000 -0.233 0.064
x7       -0.180 0.252 -0.233 0.000 0.173
x8        0.079 -0.352 0.064 0.173 0.000
χ2 = 5.89 df = 5 p-value = .136 RMSEA = .028 Reliability = .835 AVE = .509
───────────────────────
a
Without regard to sign

 2002 Robert A. Ping, Jr. 9/20/02 83
Appendix F- Ordered Similarity Coefficients and Consistency

Ordered similarity coefficients for the eight items analyzed in Appendix E are shown in Table F, in descending
similarity. However, ordered similarity coefficients do not necessarily suggest maximally consistent item clusters. The
most similar items are x4, x5, x6, x7, and x8, (each had .9 or higher coefficients with the others). Item x3 is less similar with
.8 coefficients, and x2 and x1 have one or more coefficients below the suggested .8 cutoff. The five items with the highest
consistency were x2, x5, x6, x7, and x8 (see Appendix E). Items x4 through x8, had maximal reliability but marginal
consistency (again see Appendix E).

Table F- Ordered Similarity Coefficients for the Appendix E Itemsa
x4 x5 x6 x7 x8 x3 x2 x1
x4 100 97 94 93 93 91 80 71
x5 97 100 94 93 93 87 79 69
x6 94 94 100 97 93 84 80 69
x7 93 93 97 100 92 83 82 69
x8 93 93 93 92 100 88 76 60
x3 91 87 84 83 88 100 81 66
x2 80 79 80 82 76 81 100 81
x1 71 69 69 69 60 66 81 100
───────────────────────
a
Table entries are coefficients times 100

 2002 Robert A. Ping, Jr. 9/20/02 84
Appendix G- Error Adjusted Regression Estimates for the Figure A Model

A Marketing survey involving the latent variables A, B, C, D, E, and their indicators a i, bj, ck, dm, and ep
produced more than 200 usable responses. The measure for A was an established measure, and item omission in
several other measures to attain acceptable model-to-data fit for LISREL degraded their content validity. Thus the
measures were used with no item omissions. The unidimensionality of A, B, C, D, and E was gauged using
exploratory factor analysis. Each measure produced one factor with an eigenvalue greater than one, which suggested
their unidimensionality.
Next the indicators for A, B, C, D, and E were zero centered. This was accomplished by subtracting the mean of
each indicator from each case value of that indicator. Zero centering produces indicators with means of zero, which
was assumed in deriving the adjustment equations (e.g., Equation 5 and 6). After zero centering was accomplished,
the indicators of each latent variable were averaged to form the variables A, B, C, D, and E, then these variables
were added to the data set. Next an unadjusted covariance matrix of A, B, C, D, and E was produced using SPSS
(see Table G1).
Then the coefficient alpha reliabilities for A, B, C, D, and E were obtained using SPSS (see Table G1), and the
(see Table G2) was used as input to the ordinary least squares regression procedure in SPSS to produce the adjusted
regression results also shown in Table G2.
The coefficient standard errors (SEs) for the adjusted regression coefficients were computed as follows. The
regression standard error of the estimate (SSE) (SSE = Σ[yi - i]2, where yi and i. are observed and estimated ys
respectively). In SPSS the SSE is presented with the regression R2, adjusted R2, etc. values. The adjusted regression
SEs were then calculated by multiplying each unadjusted regression coefficient SE by the ratio of the unadjusted
regression SSE to the adjusted regression SSE, in a manner similar to two-stage least squares.
Table G2 also shows LISREL 8 results. Although the model fit was unacceptable, the structural equation
estimates were interpretationally equivalent to the adjusted regression results.

Table G1-- Unadjusted Covariances for A, B, C, D, and E:

A          B               C            D              E
A         0.3426156 -0.2333447       0.1806126    0.1877566    -0.2961835
B        -0.2333447 0.5458926       -0.1310060   -0.2834373     0.3407495
C         0.1806126 -0.1310060       0.6173532    0.4911174    -0.1347000
D         0.1877566 -0.2834373       0.4911174    1.0055032    -0.1772019
E        -0.2961835 0.3407495       -0.1347000   -0.1772019     0.6664125

A       B       C       D       E
Reliabilities (α)                   0.9325  0.9262 0.9271 0.9494 0.9622
Measurement Errors (θ)b             1.13320 1.45032 1.62018 1.27196 2.51903

───────────────────────
a
= α1/2 .
b
= Var(X)(1-α).

 2002 Robert A. Ping, Jr. 9/20/02 85
Table G2-- Adjusted Covariances for A, B, C, D, and E, with Coefficient Estimates

A                  B              C         D          E
A        0.3426156        -0.2510852      0.1942496 0.1995474 -0.3126826
B       -0.2510852         0.5458926     -0.1413759 -0.3022595 0.3609525
C        0.1942496        -0.1413759      0.6173532 0.5234767 -0.1426171
D        0.1995474        -0.3022595      0.5234767 1.0055032 -0.1854005
E       -0.3126826         0.3609525     -0.1426171 -0.1854005 0.6664125

Dependent Variable= E
A       B         C         D
bi       -0.673 0.386       0.025    0.052
SEa       0.0912 0.0715 0.0724 0.0570
t-value -7.37     5.39      0.34     0.91
SSEb                                       0.57917
Kc                                         1.04030

LISREL 8 Coefficient Estimates:
Dependent Variable= E
A          B        C                    D   χ 2/df  GFId AGFId CFIe RMSEAf
bi      -0.655 0.383 -0.028                      0.087 1412/517 .723 .681 .886 .088
SEa      0.1010 0.0764 0.0787                    0.0639
t-value -6.48      5.01    -0.36                 1.36
_________
a
b
Standard error of the estimate.
c
K is the adjustment factor used to obtain the pseudo latent variable coefficient SEs, and is equal to the ratio of the SEEs.
d
Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing, 1984).
e
.90 or better indicates acceptable fit (see McClelland and Judd, 1993).
f
.05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993).

 2002 Robert A. Ping, Jr. 9/20/02 86
Appendix H- Scenario Example

The following scenario was composed of: the instructions (see Exhibit H), the scenario (titled "Research
Material" in Exhibit a), a questionnaire containing measures for the study constructs which was attached to the
instructions (not shown), and student subjects. The Research Materials manipulated the independent variables, and
the questionnaire measured the manipulations and the dependent variables.
Each student received the Instructions/Research Materials sheet with the questionnaire attached. The
Research Material in Exhibit a has been truncated at the ellipses to conserve space, and each student received a
Research Material section showing only one of the two possible choices in each parenthesis. This experiment had 8
treatments (see the last paragraph of the Research Material), each with two levels (represented by the alternatives in
parentheses), so there were 256 (= 28) different Research Materials, one for each treatment group. While ideally
treatment groups should be homogenous, and there should be more than one subject per treatment, this particular
scenario involved one subject per treatment and nonhomogeneous student subjects.

Exhibit H- A Scenario

responses are anonymous and very important to the development of a study of personal selling.
RESEARCH MATERIAL
Please attempt to place yourself in the position of X, the major character in the following short story. Try to
imagine that person's feelings and attitudes as vividly as you can, considering what it would be like to be in their
situation. You may need to read the story several times before you are completely familiar with the details of the
situation. Then respond to the statements that follow the story, indicating how you would react if you were in that
situation. There are no "right" or "wrong" answers. It is your own, honest opinion of how X would feel and act that
I want.
Imagine that you are X. You are working for a financial services company. The company sells mutual
funds and other investments. It helps clients manage their personal and family assets using offices located around the
country. Clients seek the company's advice and investment products to maintain and build their net worth for
retirement, college for their children, etc.
You are an account representative for, among other things, the company's mutual fund products that include
stock and bond funds, and funds made up of securities from foreign companies. You are very good at advising
clients regarding their financial planning. You and the company have (a common, different) goal-- (satisfied
customers, you want satisfied customers and they want brokerage fees). You are also paid a very (attractive,
unattractive) combination of salary and commissions that (generously, does not) compensate(s) you for all the
preparation and work that you do for the company. The company's policies and procedures regarding performance
evaluation and feedback, promotion, vacation, health care, etc. are (very, not) fair compared to other companies.
These policies and procedures are administered very (fairly, unfairly): you see (no) favoritism in promotions, for
example, (and, or) inconsistent administration of these policies and procedures (any-, every-)where. You are treated
with (great, no) respect (and, or) concern for your feelings by company management.
You have worked for the company for (seven years, three months) now, and have devoted many of these
years to developing your client base. (You have spent many nights and weekends, Some of this time has been
devoted to) learning the company's products and services, and how to serve clients with these products and services,
(that could have been spent having fun; Some of this time has been spent developing your client base).
.
.
.

Things at work had been fine, but in the past week a problem developed. Your manager called you to say
that you will be asked to give several of your best clients to the newly hired account representatives. They currently
go too long without commissions. In addition, you will be asked to help train these new account representatives.
This would reduce the available time you have to find replacement clients, and reduce your ability to serve your
existing clients.
Remember, you have worked for this company a (long, short) time. You and the company have (the same,
very different) goals. Your compensation is very (fair, unfair). The company's policies and procedures are very
(fair, unfair). These policies and procedures are administered very (fairly, unfairly). You are treated with (great, no)
respect (and, or) concern for your feelings by your company's management. Other potential employers are very
(attractive, unattractive). Changing jobs would require (a lot of, little) effort (and, or) risk.

 2002 Robert A. Ping, Jr. 9/20/02 87
Appendix I- Structural Equation Analysis with Summed Indicators

The Appendix G structural equation analysis model was re-run with a single summed indicator for each of
the variables A, B, C, D, and E. To use this summed indicator approach, the indicators for each latent variable X
should be unidimensional using exploratory factor analysis and criteria such only one factor with an eigenvalue
greater than one. Next the indicators for X, x1, x2, ... , xn, should be summed and x1+x2+ ... +xn should be added to
each case. Then the variance of X should be determined using SAS, SPSS, etc. Next coefficient alpha (α) for X
should be determined using the items of X, x1, x2, ... , xn. Finally, the summed indicator should be specified in the
structural equation analysis model (i.e., measurement or structural) with a fixed loading equal to α 1/2, and a fixed
measurement error equal to Var(X)(1-α). A through E were also mean centered, although this is not necessary unless
an interaction is being specified.
These steps were taken for the LVs A through E, using loadings that were the square root of the Table L
SPSS reliabilities, and errors that were the product of the Table G1 variances and 1 minus the Table L SPSS
reliabilities. The results of the structural model estimation is shown in Table I, and they are practically identical to
the Table G error-adjusted regression results. Thus error-adjusted regression or structural equation analysis with
summed indicators could be used to estimate the Appendix G model, and preference would depend on researchers
familiarity with structural equation analysis software or its availability.

Table I- Coefficient Estimates

LISREL 8 Coefficient Estimates:
Dependent Variable= E
A        B        C                       D         χ 2/df    GFIb AGFIb CFIc RMSEAd
bi      -0.673 0.386      0.025                   0.052          0/0     -------(not applicable)-------
SEa      0.1014 0.0797 0.0828                     0.0642
t-value -6.64    4.84     0.30                    0.81
_________
a
Coefficient standard error.
b
Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing, 1984).
c
.90 or better indicates acceptable fit (see McClelland and Judd, 1993).
d
.05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993).

 2002 Robert A. Ping, Jr. 9/20/02 88
Appendix J- Second-Order Construct Example

The constructs B, C and D were respecified as a second-order construct F that was plausibly face or content
valid with the "items" B, C and D. It also fit the data adequately for these purposes ( χ 2/df = 366/116, GFI = .84,
AGFI = .79, CFI = .92, RMSEA = .09) in a second-order measurement model (i.e., B, C and D were specified with
their observed indicators, and F was specified with B, C and D as indicators-- see the left half of Figure J). An
exploratory factor analysis of B, C and D with each construct specified as a sum of its items produced a single factor
(Factor 1 eigenvalue = 1.85 (61% of variance), loadings = .66 (B), .82 (C), and .85 (D)), which suggested B, C and
D can be combined into a single second-order measure for regression composed of the sum of the items for B, C and
D.
To address whether the specification of F as a sum of the indicators for regression is equivalent to the
structural equation analysis specification, if this specification of F for regression is correct, the variance of F from its
second-order measurement model should approximately equal the adjusted SPSS, SAS, etc. variance of F (see
Equation 5). The SPSS variance of F was 3.918, which when adjusted to its error-free variance value was .601
(loadings = .504, .610, 1, errors = .642, .319, .265), and the measurement model variance of F was .686. This
suggests F specified as a sum of the indicators of B, C and D is appropriate specification of a second-order construct
for regression.

Figure J- Second-Order Constructs

 2002 Robert A. Ping, Jr. 9/20/02 89
Appendix K- Scenario Analysis Results Comparison

The Appendix H scenario was administered to more than 200 students, and its questionnaire was sent as a
survey to more than 200 respondents. A psychometric comparison is shown in Table K1, and regression results for
several of the variables are shown in Table K2. The results of the scenario analysis, when compared with survey
data using the same questionnaire, are similar enough to suggest that scenario analysis may be useful for measure
debugging, and preliminary model evaluation.

Table K1- Comparison of Scenario and Survey Data from a Common Questionnaire Using Factor Analysis

Scenario Data:                                                Field Survey Data:

FACTOR 1         2     3      4      5                            1       2         3      4        5

EX6 .858                                                EX7     .841
EX4 .851                                                EX3     .829
EX2 .839                                                EX2     .821
EX7 .839                                                EX4     .815
EX5 .826                                                EX5     .814
EX1 .770                                                EX1     .807
EX8 .769                                                EX6     .778
EX3 .730                                                EX8     .771
IN8          .933                                       SA8             .850
IN3          .897                                       SA7             .848
IN5          .897                                       SA4             .809
IN6          .887                                       SA6             .794
IN1          .869                                       SA3             .747
IN4          .861                                       SA2             .746
IN7          .683                                       SA1             .703
IN2          .601                                       SA5             .675
AL5                  .820                               IN5                     .906
AL6                  .768                               IN8                     .901
AL3                  .743                               IN3                     .879
AL2                  .739                               IN4                     .876
AL4                  .732                               IN6                     .873
AL7                  .729                               IN1                     .823
AL1                  .701                               IN7                     .680
SA7                  .814                               IN2                     .646
SA3                         .780                        AL5                              .778
SA2                         .771                        AL1                              .771
SA8                         .750                        AL3                              .768
SA6                         .721                        AL2                              .761
SA4                         .718                        AL7                              .759
SA1                         .657                        AL4   -.415                      .752
SA5                         .518                        AL6                              .646
SC2                                .823                 SC5                                     .797
SC4                                .778                 SC4                                     .784
SC5                                .721                 SC6                                     .768
SC6 -.406                          .711                 SC3                                     .743
SC1                                .692                 SC2                                     .637
SC3 -.443                          .642                 SC1                                     .635

Eigen-
value 13.24 5.93 3.10 2.35         2.06                        16.23   5.87     2.79     1.85   1.79
Pct.
Var    35.8 16.0 8.4 6.4            5.6                         43.9   15.9        7.6    5.0    4.9

 2002 Robert A. Ping, Jr. 9/20/02 90
Table K2- Comparison of Scenario and Field Survey Data from a Common Questionnaire Using Regression

Scenario:                                                Survey:
Dependent Variable= J                                      Dependent Variable= J
F       G        H         I                               F      G       H        I
bi      -0.169 0.365      0.147    -0.303                  bi      -0.573 0.551    0.037   -0.255
SEa      0.102   0.100    0.089     0.092                  SEa      0.100 0.108    0.104    0.088
t-value -1.66    3.64     1.65     -3.27                   t-value -5.73   5.06    0.358   -2.88

 2002 Robert A. Ping, Jr. 9/20/02 91
Appendix L- Average Variance Extracted

Table L presents average extracted variance estimates for the Appendix A variables T, U, V, W, and UxT.,
and the Appendix G variables A, B, C, D, and E. Since the fit of the structural equation analysis model for A through
E was slightly below acceptability, both the structural equation analysis AVEs and the reliability estimates are
approximations of the measures AVE.

Table L- AVE and Reliability Estimates for the Appendix A Variables T, U, V, W, and UxT., and the Appendix G
Variables A, B, C, D, and E

Measure                  U            T UxT V               W       A      B      C      D    E
LV Reliabilitya        .946         .635 .686 .817        .928    .933   .926   .927   .949 .962
SPSS Reliabilityb .942 .609         .749e .818 .925       .941    .929   .917   .947   .968
SEA AVEc               .781         .384 .136 .534        .765    .672   .692   .671   .792 .721
Reliability AVEd       .737         .341 .166 .475        .731    .643   .637   .588   .760 .710

_____________________________
a
Using Equation 2.
b
Reliabilities using raw data and SPSS, except for the reliability of UxT.
c
Using Equation 4.
d
AVE calculated using the Footnote 10 formula

where X is the sum of the n items, Var(X) is the SPSS, SAS, etc. variance of X, ci are the item-to-total correlations for the items of X, and α is the
coefficient alpha reliability of X (except for UxT-- see Footnote e below).
e
SAS, SPSS, etc. reliability is incorrect for interactions. The reliability of an interaction is determined by solving for α in the Footnote d AVE
equation. Thus Interaction Reliability (α) is

where

 2002 Robert A. Ping, Jr. 9/20/02 92
Appendix M- Nonrecursive Analysis Example

The variables A through E were respecified as the nonrecursive or bi-directional model shown in Figure M
that was identified using Berrys (1984) algorithm and it fit the data adequately for these purposes (χ2/df = 1558/522,
GFI = .70, AGFI = .66, CFI = .86, RMSEA = .09). The nonrecursive relationship between A and E was significant
from A to E but not significant from E to A. This suggests that changes in A are likely to negatively affect E, but
changes in E are likely to have no effect on A. If directionality from A to E had been hypothesized, these results
suggest it was not disconfirmed. Because this is a necessary condition for directionality, confirmation is not
demonstrated, and it can be established only by repeated observation, longitudinal studies, or experiments.

Figure M- A Nonrecursive Model

_________________
a
Nonstandardized coefficients.
b
T-value.

 2002 Robert A. Ping, Jr. 9/20/02 93
Appendix N- Second-Order Interactions

The following presents the results of an investigation of several respecifications of the UxT interaction in
Appendix A. The range of possible first-order by second-order interaction specifications is considerable, but most of
them are impractical (e.g., interaction indicators composed of a first-order observed variable and a second-order
construct cannot be reflected in case values). Shown below are several single product-of-sums indicator
specifications of UxT, and variations on the Kenny and Judd (1984) approach. Tables N1 through N10 show the
results involving the specification of U with multiple indicators. The results of U specified with a single summed
indicator, as in single indicator structural equation analysis, are discussed after Table N10.
The first specification involved U as a first-order construct with multiple indicators (rather than a single
summed indicator), T as a first-order construct with multiple summed indicators, and UxT with a single product-of-
sums indicator, and it used the Ping (1995) estimation approach. The results were shown in Table C1, and are
summarized in Table N1.

Table N1- Results for T First-Order and UxT as Single Indicator with Ping (1995) Specification
(χ2/df = 189/112, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)

V       W           U        T     UxT

V                              -.130     .158 -.339
(-1.13)   (1.37) (-2.73)
W           -.176               .145     .100
(-4.63)             (2.96)   (1.85)

The next specification was identical to the first except the loading and error of the interaction were fixed at
their single product structural equation analysis values (i.e. λUxT = α1/2 and θε = Var(UxT)(1-α), where α is the
reliability of UxT calculated as shown in Footnote e of Table L and Var(UxT) is the SPSS, SAS, etc. variance of
UxT with U and T averaged and mean centered). The results shown in Table N2 were interpretationally equivalent to
the Table N1 results above.

Table N2- Results for T First-Order and UxT as Single Indicator with
Single Indicator structural equation analysis Specification
(χ2/df = 189/111, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)

V       W           U        T     UxT

V                              -.120     .155 -.384
(-1.05)   (1.30) (-2.75)
W           -.176               .145     .103
(-4.63)             (2.92)   (1.83)

The third specification involved U as a first-order construct with multiple indicators (rather than a single
summed indicator), T as a second-order construct with first-order constructs as "indicators" (each in turn with their
respective observed indicators), and the Kenny and Judd (1984) approach of specifying UxT with all possible unique
products of the indicators of U and T. Since T was a second-order construct, this involved products of the indicators
of U and the indicators of the first-order constructs comprising T. Since there were 5 indicators for U and T had
three first-order construct, each with four indicators, this produced 60 product indicators for UxT. The resulting
model did not fit the data (because the 60 product indicators were inconsistent, and the addition of 60 nonnormal
indicators substantially increased the nonnormality of the model), but surprisingly the results were similar to those in
Tables N1 and N2, and are shown in Table N3. Parenthetically, no difficulty was encountered in estimating this

 2002 Robert A. Ping, Jr. 9/20/02 94
model, aside from the long execution time.

Table N3- Results of T 2nd Order and UxT with all 60 Kenny and Judd Indicators
(χ2/df = 28519/3591, GFI = .14, AGFI = .13, CFI = .29, RMSEA = .17 )

V       W          U        T      UxT

V                           -.102     .144 -.294
(-1.22)   (1.36) (-2.69)
W         -.177              .138     .102
(-4.61)            (3.85)   (2.00)

Because Jaccard and Wan (1995) suggested using an arbitrary subset of four product indicators instead of all
the indicators, Tables N4 through N6 show the results of two arbitrarily chosen subsets of four product indicators,
and one set of four chosen because they were consistent. These results suggest there is no guarantee the resulting
structural model will fit the data when an arbitrary set of four indicators is chosen, and the results vary by the set of
four indicators chosen, suggesting product indicators are unique, and construct validity is changed by dropping
product indicators.

Table N4- Results of T 2nd Order and UxT with 4 Arbitrary Kenny and Judd Indicators
(χ2/df = 893/372, GFI = .78, AGFI = .75, CFI = .89, RMSEA = .07 )

V       W          U        T      UxT

V                           -.162     .209 -.360
(-1.59)   (1.31) (-2.70)
W         -.174              .125      .143
(-4.55)            (3.20)   (1.89)

Table N5- Results of T 2nd Order and UxT with 4 Different but Arbitrary Kenny and Judd Indicators
(χ2/df = 793/372, GFI = .79, AGFI = .76, CFI = .91, RMSEA = .07)

V       W          U        T      UxT

V                           -.041    .211 -.268
(-0.42) (1.29) (-1.69)
W         -.174              .135 .160
(-4.55)            (2.89) (1.77)

Table N6- Results of T 2nd Order and UxT with 4 Consistent Kenny and Judd Indicators
(χ2/df = 637/372, GFI = .84, AGFI = .81, CFI = .94, RMSEA = .05 )

V       W          U        T      UxT

V                           -.061      .256 -.182
(-0.58)   (1.21) (-1.69)
W         -.174              .134       .164
(-4.54)            (2.77)    (1.65)

 2002 Robert A. Ping, Jr. 9/20/02 95
Similarly, the next specification involved U as a first-order construct with multiple indicators (rather than a
single summed indicator), T as a first-order construct with multiple summed indicators, and the Kenny and Judd
(1984) approach of specifying UxT with all possible unique products of the indicators of U and T. Since T was a
first-order construct, this involved products of the indicators of U (5) and the three summed indicators comprising T,
for a total of 15 product indicators. The resulting model did not fit the data (because the 15 product indicators were
inconsistent, and their addition to the model substantially increased its nonnormality), and this time the results were
not similar to those in Tables N1 and N2 (see Table N7). Parenthetically, no difficulty was encountered in estimating
this model.

Table N7- Results of T 1st Order and UxT with all 15 Kenny and Judd Indicators
(χ2/df = 3402/454, GFI = .44, AGFI = .39, CFI = .56, RMSEA = .17 )

V      W          U         T     UxT

V                           -.067      .161 -.212
(-0.71)   (1.70) (-2.34)
W         -.175              .158       .076
(-4.56)            (3.60)    (1.70)

As before the Jaccard and Wan (1995) approach of using an arbitrary subset of four product indicators
suggest there is no guarantee the resulting structural model will fit the data, unless the indicators are chosen to be
consistent (see Tables N8 through N10). Again, the results vary by the set of indicators, suggesting product
indicators are unique, and construct validity is changed by dropping product indicators.

Table N8- Results of T 1st Order and UxT with 4 Arbitrary Kenny and Judd Indicators
(χ2/df = 371/168, GFI = .86, AGFI = .82, CFI = .92, RMSEA = .07 )

V      W          U         T     UxT

V                           -.043     -.110 -.633
(-0.36)   (-0.29) (-2.29)
W         -.169              .237     .193
(-4.41)            (4.20)    (1.04)

Table N9- Results of T 1st Order and UxT with 4 Different but Arbitrary Kenny and Judd Indicators
(χ2/df = 439/168, GFI = .83, AGFI = .78, CFI = .89, RMSEA = .08 )

V      W          U         T     UxT

V                            .012 .053 -.395
(0.15) (0.43) (-2.90)
W         -.171              .132 .154
(-4.54)            (3.21) (2.51)

Table N10- Results of T 1st Order and UxT with 4 Consistent Kenny and Judd Indicators
(χ2/df = 353/168, GFI = .87, AGFI = .83, CFI = .92, RMSEA = .07 )

V      W          U         T     UxT

V                           -.044    .140     -.233
(-0.47) (1.76)    (-2.05)
W         -.176              .173 .072
(-4.59)            (3.87) (1.90)

The results for U specified as a single summed indicator, as in single indicator structural equation analysis,
and the above combinations of specifications of T and UxT produced similar results. The single product-of-sums
indicators produced estimates interpretationally equivalent to those shown in Tables N1 and N2, while the Kenny

 2002 Robert A. Ping, Jr. 9/20/02 96
and Judd (1984) results with T specified as a second-order construct did not fit the data because of the resulting 9
product indicators. As before, four indicator subset results varied by the set of indicators, and suggested construct
validity is changed by dropping product indicators. However, the Kenny and Judd results with T specified as a first-
order construct did fit the data and produced results that were interpretationally equivalent to Tables N1 and N2
results.
Thus the single indicator approaches produced models that fit the data, while the Kenny and Judd (1984)
approaches, with one exception, did not. The Jaccard and Wan (1995) approach of selecting a subset of the Kenny
and Judd product indicators produced results that suggested the construct validity of UxT is changed by dropping
product indicators, and this is not an efficacious approach. Because the population values were unknown, it is
impossible to conclude more from this investigation any more than that it seems advisable to specify a second-order
interaction with all the Kenny and Judd (1984) product indicators, but in a form that produces as few indicators as
possible (i.e., the single product-of-sums indicator is a factored form of the sum of the Kenny and Judd product
indicators).

(end of Exhibits)

 2002 Robert A. Ping, Jr. 9/20/02 97

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