The SIMPLIS Command Language - Springer978-1-4612-3974-1/1.pdf · 15. When using the SIMPLIS...
Transcript of The SIMPLIS Command Language - Springer978-1-4612-3974-1/1.pdf · 15. When using the SIMPLIS...
APPENDIX A
The SIMPLIS Command Language
Overview and Key Points
The SIMPLIS (SIMPle LISrel) command language within the LISREL package gives the user the option of conducting path, confirmatory factor, or full structural equation model analyses without having to specify explicitly the 0 and non-zero elements in each of the basic matrices B, r, <1>, '1', Ax, e,h A y , and e G • An English-like syntax is used to easily specify a wide variety of models, and, with the MS Windows version of LISREL, output options include drawings of path diagrams with attached parameter estimates, t
values (the nonsignificant ones are distinguished from the significant ones by being displayed in a different color), modification indices, and expected parameter change statistics. One of the most advanced SIMPLIS options after requesting a path diagram and estimating a model is the possibility of model modification by freeing (or fixing) parameters on-screen through "pointing," "clicking," and "dragging" in the diagram. A pull-down menu then gives the option of reestimating and displaying the modified model. Although very convenient and user-friendly, the researcher should be aware that these options can be abused easily: With an ill-conceived and ill-fitting initial model, it becomes all too tempting to "go fishing" in search of a model-any model -that, by chance, will fit a particular data set. As I have stressed throughout the book, the user of SEM techniques again is urged to conceptualize theoretically sound models prior to data analysis and adjust initial models only if the modification is substantively justified. If this is not possible, tools such as exploratory factor analysis could be used to uncover possible structures underlying the variables in the current data set, and, with a different data set, these structures subsequently could be evaluated with the confirmatory methods discussed here.
The tables in this appendix contain the SIMPLIS input files and selected output corresponding to each of the LISREL examples discussed in the
179
180 Appendix A. The SIMPLIS Command Language
book. The reader should consult Joreskog and Sorbom (1993b) for a detailed description of the SIMPLIS command language. However, before the input files are presented, some key points regarding the SIMPLIS syntax are listed.
1. A typical SIMPLIS program is divided into sections by certain header lines such as OBSERVED VARIABLES, COVARIANCE MATRIX, SAMPLE SIZE, and RELATIONSHIPS. Optionally, each such header can end with a colon (:) to increase readability.
2. The first line in a SIMPLIS program usually is a title line that can contain any information except start with the strings of characters Observed Variables, Labels, or DA. To avoid possible problems, one should start the title line with an exclamation point (!), the character used in LISREL to indicate a comment line (i.e., everything in a line typed after "!" anywhere in the input is ignored by the program).
3. After the title, unique names (up to eight characters in length) must be given to the observed variables in a model. These labels can be listed in free format after the SIMPLIS headers OBSERVED VARIABLES or LABELS.
4. Information regarding the input data must be given next. SIMPLIS accepts raw data or a covariance or correlation matrix together with means and/or standard deviations. Correspondingly, appropriate header lines are RAW DATA, COVARIANCE MATRIX, CORRELATION MATRIX, MEANS, and/or STANDARD DEVIATIONS.
5. After specification of the input data, the sample size (n) is given following the header SAMPLE SIZE.
6. Observed variables may be reordered to increase the readability of the output by listing the variables in their new order after the key words Reorder Variables.
7. If the model contains latent variables, they are identified by descriptive labels (up to eight characters in length; different from those for the observed variables) after the header LA TENT VARIABLES or UNOBSERVED VARIABLES.
8. The section entitled RELATIONSHIPS (or RELATIONS or EQUATIONS) contains all model-implied equations linking observed and latent variables. The general format of a statement in this section is
dependent (latent or observed) variable(s)
= independent (latent or observed) variable(s)
Structural coefficients linking a dependent to an independent variable can be fixed to a constant by writing the constant-followed by an asterisk (*)-in front of the appropriate independent variable. For example, if MoEd is one of three indicators of the latent variable PaSES, the unit of measurement of the independent variable PaSES can be set equal to that of MoEd by the statement
MoEd = 1 *PaSES
Overview and Key Points 181
9. If no reference variables are specified for the purpose of assigning a unit of measurement to the latent variables, SIMPLIS assumes that the latent variables are standardized to unit variance.
10. All measurement error terms of observed variables are free parameters by default. The user can override this default and specify an error variance for a variable, Var, to equal some value, a, with the statement
Let the Error Variance of Var be a
or
Set the Error Variance of Var equal to a.
11. Covariances between any error terms in a model are 0 by default. However, co variances between (a) measurement errors (j of observed exogenous variables X, (b) measurement errors 8 of observed endogenous variables Y, and (c) disturbance terms ( of latent endogenous variables 1]
can be set free by statements of the form
Let the Errors between VarA and VarB Correlate
or
Set the Error Covariance between VarA and VarB Free.
12. The latent exogenous variables ~ are assumed to be correlated. To override this default, specify, for example,
Set the Co variances of Ksil - Ksi2 to 0
or
Set the Correlation of Ksi 1 - Ksi2 to O.
13. Various options such as the estimation method, number of decimals printed in the output, or the maximum number of iterations can be specified with the key words Method, Number of Decimals, and Iterations, respectively.
14. A graphic representation of an estimated model [and access to the advanced features mentioned above (e.g., on-screen model modification)] can be obtained by specifying PATH DIAGRAM in a SIMPLIS input file.
15. When using the SIMPLIS command language, one still can obtain the traditional LISREL output by including the header LISREL OUTPUT in the SIMPLIS program. Now all LISREL output options such as SC (Standardized Completely) or EF (total and indirect EFfects) are available.
16. The optional header END OF PROBLEM indicates the end of the input file.
182 Appendix A. The SIMPLIS Command Language
Table A.I. SIMPLIS Input File for the Simple Linear Regression in Example 1.1
!Example 1.1. SIMPLIS: Simple Linear Regression 2 OBSERVED VARIABLES: Degree FaEd 3 CORRELATION MATRIX: 4 5 .129 1 6 MEANS: 7 4.535 3.747 8 STANDARD DEVIATIONS: 9 .962 1.511 10 SAMPLE SIZE: 3094 11 RELATIONSHIPS: 12 Degree = FaEd 13 Number of Decimals = 3 14 END OF PROBLEM
Table A.l(a). Partial SIMPLIS Output from the Simple Linear Regression in Example 1.1
LISREL ESTIMATES (MAXIMUM LIKELIHOOD) Degree = 4.227 + 0.0821 *FaEd, Errorvar. = 0.910, R2 = 0.0166
(0.0459) (0.0114) (0.0231) 92.153 7.234 39.319
Table A.2. SIMPLIS Input File for the Multiple Linear Regression in Example 1.2
1 !Example 1.2. SIMPLIS: Multiple Linear Regression 2 OBSERVED VARIABLES: Degree Fa Ed DegreAsp Selctvty 3 COV ARIANCE MATRIX: 4 .925 5 .1882.283 6 .247.187 1.028 7 .486 .902 .432 3.960 8 MEANS: 9 4.5353.7474.0035.016 10 SAMPLE SIZE: 3094 11 RELATIONSHIPS: 12 Degree = Fa Ed DegreAsp Selctvty 13 Number of Decimals = 3 14 END OF PROBLEM
Overview and Key Points 183
Table A.2(a). Partial SIMPLIS Output from the Multiple Linear Regression in Example 1.2
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
Degree = 3.170 (0.0768) 41.288
+ 0.0289*FaEd + O.l95*DegreAsp + 0.0949*Selctvty, (0.0114) (0.0165) (0.00876) 2.543 11.804 10.823
Errorvar. = 0.825, (0.0210) 39.306
R2 = 0.108
Table A.3. SIMPLIS Input File for the Path Analysis Model in Figure 1.1, Example 1.3
1 !Example 1.3. SIMPLIS: Path Analysis With One Exogenous Variable 2 OBSERVED VARIABLES: Degree FaEd DegreAsp Selctvty 3 COV ARIANCE MATRIX: 4 .925 5 .1882.283 6 .247.187 1.028 7 .486 .902 .432 3.960 8 SAMPLE SIZE: 3094 9 Reorder Variables: DegreAsp Selctvty Degree FaEd 10 RELATIONSHIPS: 11 DegreAsp = FaEd 12 Selctvty = FaEd DegreAsp 13 Degree = Fa Ed DegreAsp Selctvty 14 LISREL OUTPUT: SC ND = 3 15 END OF PROBLEM
184 Appendix A. The SIMPLIS Command Language
Table A.3(a). Partial SIMPLIS Output from the Analysis of the Model in Figure 1.1
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
BETA DegreAsp Selctvty Degree
DegreAsp
Selctvty 0.354 (0.033) 10.612
Degree 0.l95 0.095 (0.017) (0.009) 11.808 10.827
GAMMA Fa Ed
DegreAsp 0.082 (0.012) 6.839
Selctvty 0.366 (0.022) 16.374
Degree 0.029 (0.011) 2.543
PHI FaEd
2.283
PSI DegreAsp Selctvty Degree
1.013 3.477 0.825 (0.026) (0.088) (0.021) 39.319 39.319 39.319
SQUARED MULTIPLE CORRELATIONS FOR STRUCTURAL EQUATIONS
DegreAsp Selctvty Degree
0.015 0.122 0.108
Overview and Key Points 185
Table A.4. SIMPLIS Input File for the Path Analysis Model in Figure 1.6, Example 1.4
1 !Example 1.4. SIMPLIS: Path Analysis With Two Exogenous Variables 2 OBSERVED VARIABLES: DegreAsp Selctvty Degree Fa Ed HSRank 3 CORRELATION MATRIX: 4 1 5 .214 1 6 .253.2541 7 .122.300.1291 8 .194.372 .189 .1281 9 STANDARD DEVIATIONS: 10 1.014 1.990.962 1.511 .777 11 SAMPLE SIZE: 3094 12 RELATIONSHIPS: 13 DegreAsp = Fa Ed HSRank 14 Selctvty = FaEd HSRank DegreAsp 15 Degree = Fa Ed HSRank DegreAsp Selctvty 16 PATH DIAGRAM 17 LISREL OUTPUT: SC EF ND = 3 18 END OF PROBLEM
186 Appendix A. The SIMPLIS Command Language
Table A.4(a). SIMPLIS PATH DIAGRAM Output from an Analysis of the Model in Figure 1.6
FaEd
HSRonk
Overview and Key Points
Table A.5. SIMPLIS Input File for the Overidentified Model in Figure 1.10, Example 1.5
1 !Example 1.5. SIMPLIS: An Over-Identified Model 2 OBSERVED VARIABLES: DegreAsp Degree FaEd 3 COV ARIANCE MATRIX: 4 1.028 5 .247 .925 6 .187.1882.283 7 SAMPLE SIZE: 3094 8 RELATIONSHIPS: 9 DegreAsp = FaEd 10 Degree = DegreAsp 11 Number of Decimals = 3 12 END OF PROBLEM
Table A.5(a). Partial SIMPLIS Output from an Analysis of the Model in Figure 1.10
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
187
DegreAsp = 0.0819*FaEd, Errorvar. = 1.013, R2 = 0.0149 (0.0120) (0.0258) 6.839 39.319
Degree = 0.240*DegreAsp, Errorvar. = 0.866, R2 = 0.0642 (0.0165) (0.0220) 14.560 39.319
GOODNESS OF FIT STATISTICS
CHI-SQUARE WITH 1 DEGREE OF FREEDOM = 32.691 (P = 0.0)
188 Appendix A. The SIMPLIS Command Language
Table A.6. SIMPLIS Input File for the CFA Model in Figure 2.1, Example 2.1
1 !Example 2.1. SIMPLIS: CFA of Parents' SES and Academic Rank 2 OBSERVED VARIABLES: MoEd FaEd PalntInc HSRank 3 CORRELATION MATRIX: 4 1 5 .610 1 6 .446.5311 7 .115.128.055 1 8 STANDARD DEVIATIONS: 9 1.229 1.511 2.649.777 10 SAMPLE SIZE: 3094 11 LATENT VARIABLES: PaSES AcRank 12 RELATIONSHIPS: 13 MoEd = 1 *PaSES 14 FaEd PalntInc = PaSES 15 HSRank = I*AcRank 16 Set the Error Variance of HSRank to 0 17 Number of Decimals = 3 18 END OF PROBLEM
Table A.6(a). Partial SIMPLIS Output from an Analysis of the Model in Figure 2.1
LISREL ESTIMATES (MAXIMUM LIKELIHOOD)
MoEd= 1.000*PaSES, Errorvar. = 0.737, R2 = 0.512 (0.0285) 25.827
FaEd= 1.467*PaSES, Errorvar. = 0.618, R2 = 0.729 (0.0483) (0.0488) 30.355 12.681
PalntInc = 1.870*PaSES, Errorvar. = 4.312, R2 = 0.386 (0.0628) (0.133) 29.796 32.361
HSRank = 1.000* AcRank, R2 = 1.000
COVARIANCE MATRIX OF INDEPENDENT VARIABLES
PaSES AcRank
PaSES 0.774 (0.040) 19.419
AcRank 0.098 0.604 (0.014) (0.015) 7.055 39.326
Overview and Key Points
Table A.7. SIMPLIS Input File for the HB] Model in Figure 2.6, Example 2.2
!Example 2.2. SIMPLIS: Validity and Reliability of the HBI 2 OBSERVED VARIABLES: TfTc Fa Fe At Ac 3 COVARIANCE MATRIX: 4 .436 5 .045 .196 6 -.349 -.048.468 7 -.145.126.112.243 8 -.037.013 -.117.037.284 9 .029.165 -.112.127.100 .280 10 SAMPLE SIZE: 167 11 LATENT VARIABLES: Thinking Feeling Acting 12 RELATIONSHIPS: 13 Tf = Thinking Feeling 14 Tc = Thinking 15 Fa = Feeling Acting 16 Fc = Feeling 17 At = Acting Thinking 18 Ac = Acting 19 PATH DIAGRAM 20 Number of Decimals = 3 21 END OF PROBLEM
189
190 Appendix A. The SIMPLIS Command Language
Table A.7(a). SIMPLIS PATH DIAGRAM Output from an Analysis of the H BI Model in Figure 2.6
.034 If b762
.02" Te
_024 Fa
_074 Fe
.114 At
.084 Ae ~U8
Overview and Key Points 191
Table A.S. SIMPLIS Input File for the General Structural Equation Model in Figure 3.1, Example 3.1
1 !Example 3.1. A Structural Equation Model of Parents' on Respondent's SES 2 Observed Variables: 3 MoEd FaEd PaJntInc HSRank FinSucc ConColIg AcAbiIty DriveAch SelfConf 4 DegreAsp ColContr SeIctvty Degree OcPrestg Income 5 Correlation Matrix: 6 1 7 .610 1 8 .446.531 1 9 .115.128.055 1 10 -.077 -.097 -.016 -.0521 11 -.203 -.216 -.393.002 -.018 1 12 .192.216.154.493 -.086 -.0791 13 -.042 -.017 -.023 .205 .063 .010 .251 1 14 .090 .112 .068 .269 .021 - .043 .487 .327 1 15 .116.122.101.194 -.008.021.236.195.2061 16 .139.205.170.049 -.125.011 .119.018.056.1061 17 .255.300.293 .372 -.Ill - .114.382.152.216.214.294 1 18 .117.129.141.189 .025 -.067.242.184.179.253.144.2541 19 .057.084.059.153 -.002.017.163.098.090.125.110.155.4811 20 .012 -.008.093.037.157 -.060.064 .096 .040 .025 -.020.074.106.1361 21 Standard Deviations: 22 1.229 1.511 2.649.777 .847 .612 .744 .801 .782 1.014.475 1.990.962 23 1.591 1.627 24 Sample Size: 3094 25 Reorder Variables: 26 AcAbilty SelfConf DegreAsp SeIctvty Degree OcPrestg MoEd FaEd PaJntInc HSRank 27 Latent Variables: AcMotiv ColgPres SES PaSES AcRank 28 Relationships: 29 AcAbiIty = 1 * AcMotiv 30 SelfConf DegreAsp = AcMotiv 31 SeIctvty = 1 *ColgPres 32 Degree = 1 *SES 33 OcPrestg = SES 34 MoEd = 1 *PaSES 35 FaEd PaJntInc = PaSES 36 HSRank = 1 * AcRank 37 AcMotiv = PaSES AcRank 38 ColgPres = PaSES AcRank AcMotiv 39 SES = PaSES AcRank AcMotiv ColgPres 40 Set the Error Variance of HSRank to 0 41 Set the Error Variance of SeIctvty to 0 42 Let the Errors between AcAbilty and SelfConf Correlate 43 Let the Errors between DegreAsp and Degree Correlate 44 Path Diagram 45 Number of Decimals = 3 46 LISREL Output: EF 47 End of Program
192 Appendix A. The SIMPLIS Command Language
Table A.8(a). Partial SIMPLIS PATH DIAGRAM Output from an Analysis of the Model in Figure 3.1: The Structural Portion
tOO~ MoEd
·'4 F.Ed .. ~
~.o.
PaJntlnc r Oo~
HSRank
110,
AcAbilty
SeffConf
DegreAlj)
SeicMy
Degree
OcPre5tg
194 Appendix A. The SIMPLIS Command Language
Table A.9. SIMPLIS Input File for the General Structural Equation Model in Figure 3.5, Example 3.2
!Example 3.2. A Structural Equation Model of Sex, SES, and Situation on T, F, and A 2 Observed Variables: 3 Tf Tc Fa Fc At Ac Sex MoEd FaEd FaOcc Sit 4 Correlation Matrix: 5 1 6 .153 1 7 -.773 -.1571 8 - .447.579 .332 1 9 -.106.054 -.320.142 1 10 .083.704 -.310 .487 .3541 11 - .213 - .003 .086 .188 .136 .056 1 12 .042.009 -.012 -.059.036.031.0521 13 -.041 .Oll -.026 -.022.061.025.081.5081 14 .054.077 .052 .034 .056 .057 -.011 .363.5261 15 -.323 -.176.495.096 -.291 -.276.004 -.046 -.020 -.083 1 16 Standard Deviations: 17 .660.443 .684 .493 .533 .529 .500 1.991 2.059 1.578 .501 18 Sample Size: 167 19 Latent Variables: Thinking Feeling Acting BioSex SES Situatin 20 Relationships: 21 Tc = 1 *Thinking 22 Tf = Thinking Feeling 23 Fc = 1 * Feeling 24 Fa = Feeling Acting 25 Ac = 1 * Acting 26 At = Acting Thinking 27 Sex = 1 *BioSex 28 MoEd = 1 *SES 29 Fa Ed = SES 30 FaOcc = SES 31 Sit = 1 *Situatin 32 Thinking = Situatin 33 Feeling = Situatin 34 Acting = Situatin 35 Set the Error Variance of Sex to 0 36 Set the Error Variance of Sit to 0 37 Let the Errors of Thinking and Feeling Correlate 38 Let the Errors of Thinking and Acting Correlate 39 Let the Errors of Feeling and Acting Correlate 40 Path Diagram 41 Method of Estimation = Generalized Least Squares 42 Number of Decimals = 3 43 Admissibility Check = OfT 44 End of Program
Overview and Key Points
Table A.9(a). Partial SIMPLIS PATH DIAGRAM Output from an Analysis of the Model in Figure 3.5: The Structural Portion
Sex ~OO~ ~
MoEd ~tOOO FaEd F-t8
FaOec ra70
Sit r-too tOOO~
195
Tf
Tc
Fa
Fe
At
Ac
Table A.9(b). Partial SIMPLIS PATH DIAGRAM Output from an Analysis of the Model in Figure 3.5: The Measurement Portions
Sex ~toOO .213
2bb~ hOM MoEd
%4 FaEd ~" 1.1~6
1.254 FaOee r870
.245
Sit r1.000
1.6se~ Tf f-019
.17 Te r019
Fa r010
.15
Fe f- 068
.19 At f-119
lOO~ Ae
rOe1
APPENDIX B
Location, Dispersion, and Association
Overview and Key Points
A meaningful study of structural equation modeling partially depends on a thorough understanding of some very fundamental statistical concepts. Clearly, not all pertinent issues can be reviewed within a short appendix such as this. However, as an introduction to some of the notation used throughout the book and a reminder of some basic statistical concepts, this appendix contains a brief review of the definitions and central properties of statistical expectation, variability, covariation, and standardization~all concepts of central importance to any area of applied statistics. Readers not familiar or comfortable with applying or interpreting the reviewed topics should consult appropriate sections within any of the recommended books listed at the end of this appendix. Specifically, the six key points briefly addressed in this appendix are as follows:
1. The expected value of a continuous variable can be viewed as the estimation of the value of a randomly selected score from the variable's distribution.
2. The mean of a distribution of scores from a continuous variable is used as a measure of the distribution's location. The mean is defined as the expected value of the variable.
3. The variance of a distribution of scores from a continuous variable is used as a measure of the distribution's dispersion. Variance is defined as the expected value of the squared deviations of the scores from their mean. The standard deviation of a distribution is the positive square root of the vanance.
4. The covariance between two continuous variables is used as a measure of association between two variables. Covariance is the expected value of the products of deviations of the variables' scores from their respective means.
197
198 Appendix B. Location, Dispersion, and Association
5. A standardized variable is a variable that has a distribution with a mean of 0 and a variance of 1. A continuous variable can be standardized by dividing each score's deviation from the distribution's mean by the distribution's standard deviation.
6. The Pearsonian correlation between two continuous variables can be viewed as the covariance between the corresponding two standardized variables.
Statistical Expectation
A Measure of a Distribution's Location
Given a distribution of N scores, X k , k = 1, ... , N, of a variable X, the "best guess" at the value of X k is defined as the expected value of X; formally,
N
E(X) = I XkP(Xk), (B.1) k=l
where p(Xk) is the probability of X k being chosen, Le., p(Xk) = !tIN with !t being the frequency of occurrence of the value X k • If the values of the variable X are listed individually, E(X) is one way to express the location of the distribution of the variable X. That is, using equation (B.l), the mean J1.x of X can be defined as
N N ,,\,N X "\' "\' L...k=l k J1.x = E(X) = L... Xk(!t/N) = L... (XdN) = . k=l k=l N
(B.2)
For example, suppose that variable X takes on the values {4, 3, 5, 8, 1O}. Then, the mean of this set of scores is given by
J1.x = E(X) = [4(1/5) + 3(1/5) + 5(1/5) + 8(1/5) + 10(1/5)]
= (4 + 3 + 5 + 8 + 10)/5 = 6.
A Measure of a Distribution's Dispersion
How far spread out are the values of the variable X in the distribution? Usually, the variance ui of the variable X is used to measure the dispersion of scores and is defined as the mean squared deviation of scores from their mean, that is,
IN (X _ )2 ui = var(X) = E([X - E(X)]2) = E([X - J1.X]2) = k=l ~ J1.x , (B.3)
where the numerator usually is referred to as the sum-oj-squares (SSx) associated with variable X.
Statistical Expectation 199
Since the variance measures dispersion in squared units of the variable X, a related measure of dispersion is defined to enhance interpretability: The standard deviation of X, ax, is defined as the positive square root of the variance of X,
ax = sd(X) = ~ (B.4)
and, thus, expresses score dispersion in the same units of measurement as the variable X.
For the above set of values of X, {4,3,5,S, iO}, the variance and standard deviation can be computed as
ai = [(4 - 6)2 + (3 - 6)2 + (5 - 6)2 + (S - 6f + (10 - 6)2J/5 = 6.S
and
ax = J6~8 = 2.61.
A Measure of Association Between Two Variables
To numerically assess the direction and strength of the relationship or association between two continuous variables, say, X and Y, define the covariance aXY between X and Y as the expected value of the products of the deviations of the variables from their respected means, as in
aXY = cov(XY) = E([X - E(X)] [Y _ E(Y)J) = If=l (Xk - ~x)(Y" - f.1y),
(B.5)
where the numerator usually is referred to as the cross-product (CPXY ) associated with variables X and Y. For the variable X with values {4, 3, 5, s, lO} and mean f.1x = 6, and the variable Y with values {O, 2, 6, 7, iO} and mean f.1y = 5, the covariance between X and Y is
aXY = [(4 - 6)(0 - 5) + (3 - 6)(2 - 5) + (5 - 6)(6 - 5)
+ (S - 6)(7 - 5) + (10 - 6)(10 - 5)]/5
= [10 + 9 + (-1) + 4 + 20J/5 = S.4.
Five identities are very helpful when dealing with co variances and are used throughout the book (as an exercise, the reader is encouraged to use the above data to numerically verify these identities and then try to prove them mathematically). Consider variables X, Y, and Z, and let c be any constant. Then,
1. cov(XY) = cov(YX); that is, a change in variable order does not change the value of the covariance between two variables;
2. cov(cX) = 0; a variable does not covary with a constant; 3. cov(X X) = var(X); the covariance of a variable with itself is its variance;
200 Appendix B. Location, Dispersion, and Association
4. cov[(cX)Y) = (c)cov(XY); the multiplication ofa variable by a constant c changes the variable's covariance with another variable by a factor of c; and, finally
5. cov[X(Y + Z)] = cov(XY) + cov(XZ); that is, the covariance operator is distributive with respect to addition.
Now consider a variable Y that is a linear combination of another variable X; that is, Y = Co + ClXl ' where Co and Cl are constants. Some algebraic manipulations using the definitions in equations (B.2), (B.3), and (B.5) and the identities just mentioned show that
(B.6)
and
(B.7)
Thus, if a variable Y is a linear function of a variable X then its mean can be expressed as a linear function of the mean of X. In addition, its variance is a nonlinear function (with respect to the coefficient cl ) of the variance of X.
Similarly, if Y = Co + ClXl + C1Xl ' i.e., a linear combination of two variables, Xl and Xl' then its mean and variance are given by
(B.8)
and
(B.9)
For example, consider a variable Xl with values {4, 3, 5, 8, lO}, mean f.1XI = 6, and O'i l = 6.8, and Xl with values {0,2,6, 7, lO}, f.1X2 = 5, and O'i 2 = 12.8. As was shown above, O'X l X 2 = 8.4. If, for example, Co = 1, Cl = 2, and Cl = 3, then the mean and variance of Y = Co + ClXl + C1Xl = 1 + (2)Xl + (3)Xl are given by
E(Y) = 1 + 2(6) + 3(5) = 28
and
0'; = 21(6.8) + 32 (12.8) + 2(2)(3)(8.4) = 243.2.
In general, if the variable Y is expressed as a constant plus a linear combination of other variables, Xko that is,
NX Y = Co + C1X1 + C2 X 2 + ... + CNXXNX = Co + I CkXk'
k=l (B.I0)
where each Cb k = 0, 1, 2, ... , N X, is a constant and N X is the total number of X variables, then the mean and variance of Y can be written as
NX E(Y) = Co + I CkE(Xk) (B.ll)
k=l
Statistical Standardization
and (Ji = L CkCI(JXkX,
(allk,l)
= L cf(Jik + L L CkCI(JXkX" (k=l) (k;<l)
where k, I = 1,2" .. , NX.
Statistical Standardization
Standardized Variables
201
(B.12)
Let X represent a variable with a given mean Ilx and variance (Ji. How can X be transformed into a variable Zx with mean equal to 0 and variance equal to unity? Let Zx = Co + cX, where Co and C are constants. Then, using equations (B.6) and (B. 7),
E(Zx) = Co + cE(X) = 0
and 2 2 2 1 (Jzx = C (Jx = .
Solving equations (B.13) and (B.14) for C and Co yields
C = l/(Jx
and
Co = -E(X)/(Jx·
Thus, the standardized variable Zx is given by
(B.13)
(B.14)
X - E(X) X - Ilx Zx = Co + cX = (-E(X)/(Jx) + (l/(Jx)X = = . (B.15)
(Jx (Jx
This transformed variable has a mean of 0 and a variance (and standard deviation) of 1. Similarly, given variable X, if a transformed variable D is to have a mean of 0 but an unchanged variance (Ji, then D = [X - E(X)] = (X - Ilx) is the appropriate transformation.
Again consider the variable X with values {4, 3, 5, 8, 10}, Ilx = 6, and (Jx = 2.61. The set of standardized scores Zx, computed using equation (B.15),
{(4 - 6)/2.61, (3 - 6)/2.61, (5 - 6)/2.61, (8 - 6)/2.61, (10 - 6)/2.61}
= {-0.77, -1.15, -0.38,0.77, 1.53}
has a mean of 0 and a standard deviation of 1, as can be verified easily. Similarly, for the variable Y with values {O, 2, 6, 7, 10}, Ily = 5, and (Jy = 3.58, the set of associated standardized scores Zy is
{ -1.40, -0.84,0.28,0.56, 1.40}.
202 Appendix B. Location, Dispersion, and Association
A Standardized Measure of Association Between Two Variables
The computation of the covariance between two standardized continuous variables leads to the concept of Pearson ian correlation. Let X and Y be two unstandardized continuous variables with their corresponding standardized counterparts Zx and Zy. Then,
(JZXZy = cov(ZxZy) = cov([X ~xjtxJ[Y ~yjtYJ) and, after some algebraic manipulations using the above covariance identities,
(B.l6)
For example, using the definition formula of covariance in equation (B.5) to calculate the left side of equation (B.16) for variables Zx and Zy in the above example leads to cov(ZxZy) = 0.90. This value equals the result of calculating the right side with (JXY = 8.4, (Jx = 2.61, and (Jy = 3.58.
The term on the right side of equation (B. t 6) is one way to define Pearson's product-moment correlation coefficient between two continuous variables X and y, denoted here as PXy; that is,
(B.17)
Recommended Readings
For a thorough introduction to concepts mentioned in this appendix, any elementary statistics text can be consulted. For the social scientist, books like the following might be particularly helpful:
Hays, W.L. (1988). Statistics (4th ed.). New York: Holt, Rinehart and Winston. Hinkle, D.E., Wiersma, W., and Jurs, S.G. (1994). Applied Statistics for the Behavioral
Sciences (3rd ed.). Boston: Houghton Miffiin. Howell, D.C. (1992). Statistical Methods for Psychology (3rd ed.). Boston: PWS-Kent. Keppel, G. (1991). Design and Analysis: A Researcher's Handbook (3rd ed.). Englewood
Cliffs, NJ: Prentice-Hall. Kirk, R.E. (1982). Experimental Design (2nd ed.). Belmont, CA: Brooks/Cole. Marascuilo, L.A., and Serlin, KC. (1988). Statistical Methods for the Social and Behav
ioral Sciences. New York: Freeman.
APPENDIX C
Matrix Algebra
Overview and Key Points
The mathematical foundations of many statistical techniques, including structural equation modeling, can be presented and discussed rather easily when using matrix formulations. In every textbook on elementary linear algebra and most books on intermediate applied statistics, matrix algebra is thoroughly discussed. Thus, it suffices here to review pertinent elementary definitions and properties that are used throughout the book. Specifically, in this appendix four key points are reviewed:
1. Matrix addition is an elementwise operation that is commutative, associative, and has an identity and inverse element.
2. Matrix multiplication is not an elementwise operation. It is not commutative in general, but it is associative and distributive with respect to matrix addition. An identity and, under certain conditions, an inverse element exists.
3. Determinants are unique numbers assigned to square matrices that are used throughout the more technical parts of this book.
4. The analysis of variance/covariance matrices of observed variables is at the center of structural equation modeling. Thus, an understanding of this type of matrix is of great importance.
Some Basic Definitions
A matrix is defined as a collection of numbers (called the elements of the matrix) organized by rows and columns. The order of a matrix gives the number of rows and columns. For example, the matrix A, given by
203
204 Appendix C. Matrix Algebra
[5 17]
A = 6 3 ,
o 11
is a matrix of order (3 x 2) since there are three rows and two columns. In general, if A is a (p x q) matrix, then A has p rows and q columns; the element that is in the ith row and the jth column of A is denoted by aij • If p = q, the A is said to be a square matrix. The transpose of a (p x q) matrix A, denoted by A', is a (q x p) matrix obtained by interchanging the rows and columns. Thus, for the above example,
A ' [560J = 17 3 11 .
Note that (A')' = A. If A = A', then A is called a symmetric matrix, and, if in addition, all off-diagonal elements are 0, then A is said to be a diagonal matrix. A (p x p) diagonal matrix with only ones on the diagonal is called the (p x p) identity matrix, denoted by I.
If a matrix A is symmetric and contains only zeros above or below the main diagonal, then A is called a triangular matrix. The trace ofthe matrix A, denoted by tr(A), is defined to be the sum of the diagonal elements in A. Finally, a (p x 1) matrix is called a column vector, while a (1 x q) matrix is called a row vector.
Algebra with Matrices
Matrix addition and subtraction are elementwise operations in the sense that adding or subtracting two matrices, A and B, results in a third matrix, C = (A ± B), whose elements are obtained by adding or subtracting corresponding elements in A and B. For example, if
[5 17]
A = 6 3
o 11
and
then
[5 + 0 17 + 3] [5 20]
C = A + B = 6 + 10 3 + 9 = 16 12 o + 5 11 + 12 5 23
or
[ 5 -0 17 - 3 ] [5 14 ] C=A-B= 6-10 3-9 = -4 -6 .
0-511-12 -5-1
Note that only matrices of the same order can be added or subtracted.
Algebra with Matrices 205
If A, B, and C are all (p x q) matrices, then the following three properties are preserved under matrix addition:
1. the commutative law, i.e., A + B = B + A; and 2. the associative law, i.e., (A + B) + C = A + (B + C); furthermore, 3. (A + B)' = Af + Bf.
A (p x q) matrix, 0, consisting only of zeros, serves as the identity element for matrix addition, i.e., A ± 0 = A. Finally, the additive inverse of A, denoted by - A, is ( - l)A [obtained by multiplying each element in A by the constant (-1)] with A + (-A) = O.
Matrix multiplication, as opposed to matrix addition and subtraction, is not an elementwise operation. Instead, the (p x r) product, AB, of a (p x q) matrix A with a (q x r) matrix B is defined as follows: let aik and bkj denote elements in A and B, respectively. Then, the elements (ab)ij of AB are defined by
for i = 1,2, ... , p and j = 1,2, ... , r.
Note that the number of columns in A must be equal to the number of rows in B for the product AB to be defined. Consider the following example: Let
and B=[109J. 5 2 '
then
AB = (6)(10) + (3)(5) (6)(9) + (3)(2) = 75 60 . [(5)(10) + (7)(5) (5)(9) + (7)(2)] [85 59]
(0)(10) + (1)(5) (0)(9) + (1)(2) 5 2
In general, if A, B, and C are matrices of the appropriate orders, then the following three properties are preserved under matrix multiplication:
1. the associative law, i.e., (AB)C = A(BC); and 2. the distributive law with respect to matrix addition, i.e., A(B + C) =
AB + AC and (A + B)C = AC + BC; furthermore, 3. (AB)' = Bf Af.
Note, however, that matrix multiplication is not commutative, that is, in general, AB is not equal to BA. The identity matrix I serves as the identity element for matrix multiplication:
AI = IA = A.
The multiplicative inverse of a matrix A, denoted by A -1, does not always exist; if it does, then A is called nonsingular or invertible; otherwise, A is called
206 Appendix C. Matrix Algebra
singular. If A is invertible, then AA -I = A -I A = I (there are a variety of algorithms available to calculate a matrix inverse-if it exists; consult any elementary textbook on linear algebra). If A and B are both invertible matrices of orders (p x q) and (q x r), respectively, then
1. (A -I )-1 = A; and 2. (AB)-I = B-1 A -I; furthermore, 3. (A -1)' = (AT!.
Finally, the concept of the determinant of a square matrix A, denoted by det(A) or IAI, is important in statistics. Loosely defined, IAI is a unique number assigned to A that must satisfy certain properties. Depending on the order of A, IAI is calculated by a certain algorithm. For example, the determinant of a (2 x 2) matrix is calculated as follows: If
then IAI is defined as IAI = ad - be. Thus, if
A = [2 5J 1 3 '
then IAI = (2)(3) - (5)(1) = 1. In general, if A and B are two matrices of the appropriate orders, then the
following five properties of determinants hold:
1. IA ± BI = IAI ± IBI; 2. IABI = IAIIBI; 3. lA-II = l/IAI, provided A is nonsingular; 4. if IAI = 0, then A is singular; otherwise, A is invertible; furthermore, 5. IA'I = IAI.
The Variance/Covariance Matrix
A central concept underlying structural equation modeling is the analysis of a variance/covariance matrix based on data from N individuals on N X observed variables, XI' ... , X NX ' Define the (N x NX) data matrix X as the matrix of deviation scores from variable means of the N individuals on N X observed variables. First, note that X'X is the (N X x N X) matrix that has the sum-of-squares (SSi = I.f;1 (Xik - E(Xi))2) of the NX variables Xi as its diagonal elements and the cross-products (CPij(i#j) = I.f;1 (Xik - E(XJ) X (Xjk - E(X))) of the variables as its off-diagonal elements (also see Appendix B for the definitions of sum-of-squares and cross-products). Second, the expected value E(A) of a matrix A containing variables as its elements is the matrix containing the expected values of each of the elements of A; that is, the expected value operator is an element wise operator with respect to matrices (see Appendix B for a definition of the expected value of a variable). Now, it
Recommended Readings 207
follows that the matrix
[[
SSl
E(X'X) = E C~2l
CPNXl
CP12 ... CP1NX]] SS2 ... CP2NX
· . · . · . CPNX2 SSNX
~;:::;Z] = [:;' ;i ••• ::::], CPNxz/N SSNx/N aNXl aNX2 a~x
r SSl/N
CP2l/N
= CPN~dN called the variance/covariance matrix L of the N X observed variables, contains the variances of the N X variables on its diagonal and the covariances between the variables as its off-diagonal elements. When all variables are standardized, the variance/covariance matrix L becomes a correlation matrix with ones on its diagonal and the Pearsonean correlations between variables as its off-diagonal elements.
Consider an example: Suppose three individuals obtain scores of {2, 4, 6} on some variable Xl and scores {8, 2, 5} on another variable, say, X 2 • Clearly, I1x, = E(Xd = 4 and I1X2 = E(X2) = 5 (see Appendix B). Then the (3 x 2) data matrix X consisting of deviation scores from the means is given by
x ~ [~2 ~3l Now,
X'X~[ ~2 ~3 ~]D2 ~+[ ~6 ~86]~U::, ~j and
E(X'X)
[ 8 -6J [8/3 -6/3J [2.67 -2.00J [ai ax X2J = (1/3) -6 18 = -6/3 18/3 = -2.00 6.00 = aX2~' ai,
is the variance/covariance matrix with ax,x2 = COV(X1 X2 ) = -2.00, ai, = 2.67, and aiz = 6.00, as can be verified easily by using the formulae presented in Appendix B.
Recommended Readings
For a thorough introduction to the topics reviewed in this appendix, any basic text on linear algebra can be consulted. See, for example,
Anton, H. (1991). Elementary Linear Algebra (6th ed.). New York: John Wiley & Sons.
208 Appendix C. Matrix Algebra
Kolman, B. (1993). Introductory Linear Algebra with Applications (5th ed.). New York: Macmillan.
Very useful references for statisticians are the two listed below. Both deal exclusively with statistics-related matrix algebra; the latter is introductory while the former is a more advanced text in differential matrix calculus.
Magnus, lR., and Neudecker (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: John Wiley & Sons.
Searle, S.R. (1982). Matrix Algebra Useful for Statistics. New York: John Wiley & Sons.
Finally, some applied multivariate statistics texts have good summaries of fundamentals of matrix algebra. In particular, see
Stevens, J. (1992). Applied Multivariate Statistics for the Social Sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Tatsuoka, M.M. (1988). Multivariate Analysis (2nd ed.). New York: Macmillan.
AP
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IX D
Des
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Sta
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Ana
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Tab
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Var
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Var
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Var
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Not
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.0
04
.037
.0
07
-.0
20
.0
65
.084
. 0
75
.123
-.
01
5
.032
.3
35
('i .
on
15.
Inco
me
3.81
6 1.
371
-.0
01
.0
19
.040
.0
61
.090
-.
01
5
.080
.1
15
.070
.0
90
.022
.1
07
.226
.1
68
0'
....
Not
e: D
ata
in T
able
s 0
.2 a
nd 0
.3 a
re t
aken
fro
m M
uell
er (1
988)
with
per
mis
sion
fro
m t
he p
ubli
sher
. ;.
("1
)
CIl tTl
CIl ~
::s P> -<
~.
on
IV -...,
AP
PE
ND
IX E
Des
crip
tive
Sta
tist
ics
for
the
HB
I A
naly
sis
Tab
le E
.l.
Cod
ing
Sch
ema
for
Var
iabl
es i
n th
e H
BI
Ana
lysi
s
Var
iabl
e
HB
I Sc
ales
1.
T
f*
2.
Te
3.
Fa*
4.
F
e 5.
A
t*
6.
Ae
Cod
e
see
the
H B
I m
anua
l,
Hut
chin
s &
Mue
ller
(1
992)
Var
iabl
e
8.
Mot
her'
s E
duca
tion
(M
oEd)
; 9.
F
athe
r's
Edu
cati
on
(FaE
d)
Cod
e
1 =
le
ss t
han
high
sc
hool
2
=
high
sch
ool
grad
uate
3
=
less
tha
n 2
year
s of
vo
cati
onal
, tra
de, o
r bu
sine
ss s
choo
l 4
=
two
year
s o
r m
ore
of v
ocat
iona
l, t
rade
, o
r bu
sine
ss s
choo
l or
less
tha
n 2
year
s of
col
lege
5
= tw
o ye
ars
or
mor
e of
col
lege
6
=
fini
shed
col
lege
7
=
Mas
ter'
s de
gree
or
equi
vale
nt
8 =
P
hD, M
D, o
r ot
her
adva
nced
deg
ree
Var
iabl
e
11.
Sit
uati
on s
peci
fici
ty
(Sit
uati
n)
Cod
e
o =
"H
ow d
o I
view
m
ysel
f as
a st
uden
t?"
1 =
"H
ow d
o I
view
m
ysel
f whe
n co
nfro
nted
wit
h a
clos
e fr
iend
in
emot
iona
l di
stre
ss?"
Tab
le E
.l (c
ont.)
Var
iabl
e C
ode
7.
Sex
0=
mal
e 1
= f
emal
e
Var
iabl
e
10.
Fat
her'
s oc
cupa
tion
(F
aO
cc)
*Res
cale
d b
y a
fac
tor
of 0
.1 (
see
Ben
tler
, 19
93, p
. 20
)
Cod
e
SES
(Dun
can,
196
1)*
Var
iabl
e C
ode
N +> >
'0
"g
::l
0- ;;;.
~ o t1
> en
() ..., ~.
~.
rJ) g, ~ n'
en 0'
..., :;.
t1> ::t:
tx:l ...., >
::l
po ~
[:!.:.
en
Tab
le E
.2.
Mea
ns.
Sta
ndar
d D
evia
tion
s. a
nd C
orre
lati
ons
for
the
H B
I A
naly
sis
(n =
16
7 ba
sed
on p
airw
ise
dele
tion
)
Var
iabl
es
(1 Ii
2
3 4
5 6
7 8
9
I.
Tj'
1.
09
.660
2.
T
c 2.
01
.443
.1
53
3.
Fa
1.51
.6
84
-.7
73
-.
15
7
4.
Fe
2.13
.4
93
-.4
47
.5
79
.332
5.
A
t 1.
07
.533
-.
10
6
.054
-.
32
0
.142
6.
A
e 1.
87
.529
.0
83
.704
-.
31
0
.487
.3
54
7.
Sex
.4
6 .5
00
-.2
13
-.
00
3
.086
.1
88
.136
.0
56
8.
Mo
Ed
4.
37
1.99
1 .0
42
.009
-.
01
2
-.0
59
.0
36
.031
.0
52
9.
Fa
Ed
5.
50
2.05
9 -.
04
1
.011
-.
02
6
-.0
22
.0
61
.025
.0
81
.508
10
. F
aO
ee
6.06
1.
578
.054
.0
77
.052
.0
34
.056
.0
57
-.0
11
.3
63
.526
11
. S
itu
ati
n
.49
.501
-.
32
3
-.1
76
.4
95
.096
-.
29
1
-.2
76
.0
04
-.0
46
-.
02
0
Not
e: D
ata
in T
able
s E
.l a
nd E
.2 a
re t
aken
fro
m M
uell
er (1
987)
with
per
mis
sion
fro
m t
he a
utho
r.
10
-.0
83
» '0
'0 " ::l c..
>< m
0 " vo
(") :l.
'S
< " [/J g §.. n'
'" 0'
.., ;:. " ::r: tI:l - » ::
l 1'0
~
(/0
in'
N
VI
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Index
Assumptions, underlying confirmatory factor analysis, 68 general structural equation modeling,
137~139
path analysis, 25~26 regression, 14
Bentler-Weeks notation, 1O~ 11
Carry-over effects. See Reliability Cause, xii~xiii, 65 Chi-square, 51~52
and the distribution of Lagrange multipliers. See Lagrange multiplier
and the distribution of Wald tests. See Wald test.
expected increase or decrease of. See Modification, indices; Wald test
fit index. See Fit indices and the fitting function, 82, 153 test of difference in, 87
Coefficient of determination, 5, 15~ 16, 85
bias in, 16 computation from EQS output, 13,
21,32 as a data-model fit index, 15~ 16, 43,
57
interpretation and testing of, 5~6, 15~16,43
relation to the GFI and AGFI, 85~ 86
as reliability estimates, 77 ~ 79, 105, 113,121
path. See Coefficient, structural regression, 4, 13
estimation of. See Estimation method, ordinary least squares
interpretation of, 5, 15 metric versus standardized, 5, 15
reliability. See Reliability structural, 24, 67, 135
metric versus standardized, 77~ 79, 112
validity. See Validity Confirmatory factor analysis (CF A), 64~
125 assumptions underlying. See
Assumptions data-model fit assessment. See Fit
indices equations in. See Equation error in. See Error and exploratory factor analysis (EF A),
62,68,100, Ill, 124~125 matrix representation of. See
Specification model
identification. See Identification modification. See Modification
223
224
Confirmatory factor analysis (CF A) (cont.)
specification. See Specification parameter estimation. See Estimation
methods reliability assessment with. See
Reliability as a special case of general structural
equation modeling, 137 validity assessment with. See Validity
Construct. See Variable, latent Correlation, 202
attenuated versus disattenuated, 79, 107
between errors. See Error matrix. See Matrix
Covariance, 199-200 decomposition of. See Effect
components Cross-product, 199 Cross-validation index. See
Modification
Data-model fit. See Fit indices Deviation scores, 25, 201 Direct effect. See Effect components Disturbance. See Error
Effect components, direct (DE), indirect (IE), total (TE), 32--47,141-144
definition of, 36 interpretation of, 42-43
EQS examples
confirmatory factor analysis model modification ( # 2.1), 96-
110 validity and reliability ( # 2.2),
121-124 general structural equation
modeling generalized least squares (# 3.1),
155-159 model modification ( # 3.2), 168-
174 path analysis
Index
direct, indirect, and total effects (# 1.4), 44-47
overidentified model (# 1.5), 52-54
simple model (# 1.3), 30-32 under identified model ( # 1.6), 56
regression multiple linear (# 1.2),19-21 simple linear (# 1.1), 9-13
matrix, input versus to be analyzed, 12,44
notation. See Bentler-Weeks notation order of variables, 11,30 syntax
Analysis =, 44 Apriori =, 121 asterisk (*), 11, 98 Cases =,11 jCOVARIANCES, 20, 98,158 D,155 Digit =,100 E,l1 Effects =, 44 jEND,12 jEQUATIONS, 11,20, 158 F,98 (LABELS, 11 jLMTEST, 100, 168 (MATRIX, 12 Matrix =,11,19 Method =, 158 jPRINT,44, 100 semicolon (;), 10-
12 Set =,100, 168 (SPECIFICATIONS, 11 (STANDARD DEVIATIONS, 12 jTITLE,ll V, 11 Variables =, 11 (VARIANCES, 11,32,98, 158 jWTEST, 100, 121
Equality constraint. See Identification, in path analysis
Equation regression, 4-5, 13-15 structural
in confirmatory factor analysis, 66, 77
226
Lagrange multiplier, 100, 168 distribution of, 100 interpretation of, 105-106, 171 See also Modification, indices
LISREL examples
confirmatory factor analysis data-model fit (# 2.1), 75-79 validity and reliability ( # 2.2),
117-121 general structural equation
modeling direct, indirect, and total effects
(#3.1),144-150 generalized least squares ( # 3.2),
164-168 path analysis
direct, indirect, and total effects (# 1.4), 36-43
overidentified model ( # 1.5), 51-52
simple model (# 1.3), 26-30 underidentified model ( # 1.6), 55
regression multiple linear ( # 1.2), 17-19 simple linear ( # 1.1), 6-9
matrix, input versus to be analyzed, 7-8,38
notation. See loreskog-Keesling-Wiley notation
order of variables, 7, 28 PRELIS,155 SIMPLIS,6, 179-196 syntax
AD, 165 AL,8 BE, 28 CM,7,17 DA,7 DI,28 EF,38 FI, 51-52, 75 FR, 8, 75 FU, 7, 28 GA,28 KM,7 LA, 7 LE,145 LK,76
LX, 75 LY,145 MA,38 ME (mean), 7 ME (method), 164 MO,8 ND,8 NE,146 NI,7 NK,75 NO, 7 NX, 8, 28 NY, 8, 28 OU,8 PH, 28, 75 PS,28 SC,8 SD (standard deviation), 8, 18 SD (subdiagonal), 28 SE,28 SS, 77 SY,7 TD,75 TE,146 VA, 76
Matrix, 203
Index
addition and subtraction, 204-205 correlation, 207 data, 206 determinant of, 206 diagonal, 204 element, 203
additive inverse, 205 identity, 205 multiplicative inverse, 205-206
expected value of, 206 identity, 204 input versus to be analyzed. See EQS,
matrix; LISREL, matrix invertible. See Matrix, singular multiplication, 205-206 order of, 203-204 representation of
confirmatory factor analysis model. See Confirmatory factor analysis
general structural equation model.
Index
See Structural equation modeling
path analysis model. See Path analysis
regression model. See Regression singular versus nonsingular, 205-206 square, 204 symmetric, 204 trace of, 204 transpose of, 204 triangular, 204 variance/covariance, 206-207
unrestricted versus model-implied, 69-72,82,95,151-152
Maximum likelihood (ML). See Estimation method
Mean, 198 structures, 25, 68, 139
Measurement error. See Error model. See Model
Model alternative. See Model, comparison baseline, 87-89 comparison
in confirmatory factor analysis, 87-90,93,95-96,101-107
in general structural equation modeling, 146-150; see also Model, comparison, in confirmatory factor analysis; in path analysis
in path analysis, 42-43 in regression, 16 See also Fit indices; Modification
confirmatory factor analysis. See Confirmatory factor analysis
evaluation. See Fit indices; Model comparison; Modification
fit. See Fit indices full versus sub, 16 general structural equation. See
Structural equation modeling identification of. See Identification independence, 87-89 indicators, cause versus effect, 65 measurement, 133-137, 161-163; see
also Confirmatory factor analysis
modification. See Modification
nested, 87-88; see also Model, comparison
overfitted, 94, 120
227
parsimonious versus complex, 87-89, 90-92
path analysis. See Path analysis recursive versus nonrecursive, 23 regression. See Regression saturated, 87 specification. See Specification specification error in, identification
and elimination, 94 structural influences and order of
variables in, 22-24 structural portion of, 134-135, 161-
162 Modification, 93-96, 163
consequences of, 95 cross-validation index (CVI), 95
expected value of (ECVI), 96 example of, 101-107 expected parameter change statistics
(EPC), 94 indices (MI), 94, 100; see also
Lagrange multiplier strategies, 94 see also Model, comparison
Multicolinearity, 21-22
Non-recursive. See Model
Ordinary least squares (OLS). See Estimation method
Overidentifying restriction, 51-52; see also Identification, in path analysis
Parameter estimation. See Estimation method free versus fixed, 8, 51-52, 75-77, 100 identification. See Identification number to be estimated, 48, 73, 139 See also Coefficient
Parsimony. See Fit indices, in confirmatory factor analysis; Model, parsimonious versus complex
228
Path analysis, 22-56 assumptions underlying. See
Assumptions data-model fit assessment. See Fit
indices diagram. See Path diagram equations in. See Equations error in. See Error intercept term. See Intercept matrix representation of. See
Specification model
identification. See Identification modification. See Modification specification. See Specification
parameter estimation. See Estimation methods
as a special case of general structural equation modeling, 137
Path diagram, 22-23, 64-65, 131; see also Model
Prediction error. See Error Product-moment correlation coefficient.
See Correlation
Recursive. See Model Regression, 3-22
assumptions underlying. See Assumptions
coefficient. See Coefficient data-model fit assessment. See Fit
indices equations. See Equation error in. See Error intercept term. See Intercept matrix representation of. See
Specification model
identification. See Multicolinearity modification. See Multicolinearity specification. See Specification
parameter estimation. See Estimation methods
Reliability, 77-78,105,112-113,146 assessment with confirmatory factor
analysis, 112-113
Index
Sample size requirement. See Structural equation modeling
SIMPLIS. See LISREL S pecifi ca ti 0 n
of confirmatory factor analysis models, 64-69
summary figure, 70 error. See Error of general structural equation models,
131-137,175-176 summary figure. 138
of path analysis models, 25 summary figure, 27
of regression models, 4, 13-14 Standard deviation, 199 Structural equation modeling (SEM),
general assumptions underlying. See
Assumptions data-model fit assessment. See Fit
indices equations in. See Equation error in. See Error matrix representation of. See
Specification model
identification. See Identification modification. See Modification specification. See Specification
parameter estimation. See Estimation as a research process, 159 sample size requirements in, 26, 57
Structure. See Model; Causality Sum-of-squares, 198
Thinking-feeling-acting (TF A) behavior orientation. See Hutchins Behavior Inventory
Total effect. See Effect components t-value,8
Underidentified. See Identification
Validity, 110-112, 171 assessment with confirmatory factor
analysis, 111-112, 119-121
Index
Variable dependent versus independent, 4 endogenous versus exogenous, 23 indicator, 4, 13, 65, 112
of more than one latent construct, 114-115
latent versus observed, 64-67, 111-112
units of measurement, 73-74, 76, 116-117,121,139-140,161
reference, 73; see also Variable, latent
versus observed, units of measurement
standardized, 201 Variance, 198
229
Variance/covariance matrix. See Matrix Vector, row versus column, 204
Wald test, 100, 120-121, 168 distribution of, 100 interpretation of, 105, 124, 171
Springer Texts in Statistics (continued from page ii)
Noether: Introduction to Statistics: The Nonparametric Way Peters: Counting for Something: Statistical Principles and Personalities Pfeiffer: Probability for Applications Pitman: Probability Rawlings, Pantula and Dickey: Applied Regression Analysis Robert: The Bayesian Choice: A Decision-Theoretic Motivation Santner and Duffy: The Statistical Analysis of Discrete Data Saville and Wood: Statistical Methods: The Geometric Approach Sen and Srivastava: Regression Analysis: Theory, Methods, and
Applications Shao: Mathematical Statistics Whittle: Probability via Expectation, Third Edition Zacks: Introduction to Reliability Analysis: Probability Models and Statistical
Methods