The SIMPLIS Command Language - Springer978-1-4612-3974-1/1.pdf · 15. When using the SIMPLIS command language, one still can obtain the traditional LISREL output by including the

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.


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


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


Table A.3(a). Partial SIMPLIS Output from the Analysis of the Model in Figure 1.1


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


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


Table A.4(a). SIMPLIS PATH DIAGRAM Output from an Analysis of the Model in Figure 1.6

FaEd

HSRonk


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


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)


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


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


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


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


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


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


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


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


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;


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}.


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


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


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.


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.

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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


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


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

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