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1 General Structural Equation (LISREL) Models Week 3 #2 A.Multiple Group Models with > 2 groups B.Relationship to ANOVA, ANCOVA models C.Introduction to mean & intercept models

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2 LISREL PROGRAMMING: MULTIPLE GROUPS Special considerations for 3 or more groups Group 1:[specification for matrix] Group 2: LY=IN Group 3: LY=PS You would think this means, LY[1]=LY[2] but LY[3] ≠ LY[1] = LY[2] But this is not the case: LY[1] = LY[2] = LY[3] (PS in group 3 “copies” the group 2 specification, which is IN!)

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3 LISREL PROGRAMMING: MULTIPLE GROUPS Special considerations for 3 or more groups Group 1:[specification for matrix] Group 2: LY=IN Group 3: LY=PS You would think this means, LY[1]=LY[2] but LY[3] ≠ LY[1] = LY[2] But this is not the case: LY[1] = LY[2] = LY[3] (PS in group 3 “copies” the group 2 specification, which is IN!) Possibilities: Re-organize input so Group 3 is now group 1 Group 1: [specification for matrix] Group 2: LY=PS Group 3: LY=IN

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4 LISREL PROGRAMMING: MULTIPLE GROUPS Possibilities: Re-organize input so Group 3 is now group 1 Group 1: [specification for matrix] Group 2: LY=PS Group 3: LY=IN OR: If Group1=Group2≠Group 3 Group 2 LY=IN Group 3 LY=IN or LY=PS (will do same thing) THEN use a FR statement for all parameters in matrix

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5 LISREL PROGRAMMING: MULTIPLE GROUPS If Group1=Group2≠Group 3 Group 2 LY=PS Group 3 LY=IN or LY=PS (will do same thing) THEN use a FR statement for all parameters in matrix Eg: LY=IN FR LY 2 1 LY 3 1 LY 4 1

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6 A three group example: Religion & Sexual Morality Data USA Canada Britain See file: /Week3Examples/ThreeGroupLISREL

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7 Quick notes on more complex multiple-group models Any number of groups can be modeled, subject to software limitations: –EQS: version 5 max. of 10 (version 6???) –AMOS: no apparent max. –LISREL: had a “Fortran file maximum” restriction of 17 but could be worked around if covariance matrix pasted into program itself: CM * (INSERT MATRIX)

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8 Quick notes on more complex multiple-group models Four group models could be 4 categories of one variable OR 2 x 2 design Could consider the equivalent of a 3-way interaction Eg: Sex (male/female) Country (Canada/US) Example: effect of education on attitudes, each of 4 groups Interested in Gender*Educ*Country interaction

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9 Quick notes on more complex multiple-group models Eg: Sex (male/female) Country (Canada/US) Example: effect of education on attitudes, each of 4 groups Interested in Gender*Educ*Country interaction Coefficients: Gamma1[1] US male Gamma1[2] US female Gamma1[3] Cdn male Gamma1[4] Cdn female

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10 Notes on more complex multiple- group models Coefficients: Gamma1[1] US male Gamma1[2] US female Gamma1[3] Cdn male Gamma1[4] Cdn female TEST for male/female differences in effect of education: Model 1 all gammas free Model 2 gamma1[1]=gamma1[2] gamma1[3]=gamma1[4] Other tests possible (e.g., all gammas fixed, then allow ga1[1]≠ga1[2] and ga1[3]≠ga1[4]

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11 Notes on more complex multiple- group models Coefficients: Gamma1[1] US maleGamma1[3] Cdn male Gamma1[2] US female Gamma1[4] Cdn female TEST for male/female differences in effect of education: Model 1 all gammas free Model 2 gamma1[1]=gamma1[2] gamma1[3]=gamma1[4] Other tests possible (e.g., all gammas fixed, then allow ga1[1]≠ga1[2] and ga1[3]≠ga1[4] SIMILAR TEST FOR effect of Country Three way interaction !!! ga1[1] – ga1[2] = ga1[3]-ga1[4] allows males, females to be different but extent of difference must be the same in each country Vs. a model where these constraints are freed. LISREL CO statement could be used to program this (more difficult in AMOS) CO GA 1 1 1 = ( GA 3 1 1 – GA 4 1 1 ) + GA 2 1 1 [re-expression of: ga 1 1 1 – ga 2 1 1 = ga 3 11 – ga 4 1 1

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12 Means and intercepts in SEM models If we work with X d and y d in a regression model instead of X and y, then the intercept drops out.

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13 Means and intercepts in SEM Models

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14 Means and intercepts in SEM Models

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15 Means and intercepts in SEM Models

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16 Means and intercepts in SEM Models This is the variance-covariance matrix of the X’s

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17 Means and intercepts in SEM Models By contrast, the X’X matrix is: divide by N, “Moment Matrix”

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18 Means and intercepts in SEM Models But the X matrix in a regular regression model has a vector of 1s:

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19 Means and intercepts in SEM Models Augmented Moment Matrix This matrix has k more pieces of information

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20 Means and intercepts in SEM Models Working from this matrix instead of working from S, we can add intercepts back into equations (reproduce M instead of S).

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21 Means and intercepts in SEM Models Conventional Model: X1 = 1.0 LV1 + e1 X2 = b2 LV1 + e2 X3 = b3 LV1 + e3 Extended to include intercepts: X1 = a1 + 1.0 LV1 + e1 X2 = a2 + b2 LV1 + e2 X3 = a3 + b3 LV1 + e3 [LV1 = a4] EQS calls this “V999”. Other programs do not explicitly model “1” as if it were a variable

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22 Means and intercepts in SEM Models Three new pieces of information: Means of X1, X2, X3 Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 Other parameters:Var(e1) Var(e2) Var(e3) Var(L1) Mean(L1) One of the following parameters needs to be fixed: a1,a2,a3, mean(L1)

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23 Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 Conventions: a1 = 0 Then Mean(L1) = Mean(X1) and a2 is difference between means X1,X2 (not usually of interest) a3 is difference between means X1, X3 (not usually of interest)

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24 Means and intercepts in SEM Models Conventions: Mean(L1) = 0 Then a1=mean of X1 a2 = mean of X2 a3 = mean of X3 Not particularly useful: means of LV’s by definition =0 Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3

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25 Means and intercepts in SEM Models Construct equation now: L2 = a1 + b1 L1 + D1 (also: new parameter: mean of L1)

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26 Means and intercepts in SEM Models In longitudinal case, more interesting possibilities: Constrain measurement models: b1=b3 b2=b4 Constrain intercepts: a1 = a4 a2 = a5 a3 = a6 Fix Mean(L1) to 0 Can now estimate parameter for Mean (L2) Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 X4 = a4 + 1.0 L2 + e4 X5 = a5 + b3 L2 + e5 X6 = a6 + b4 L2 + e6

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27 Means and intercepts in SEM Models Constrain measurement models: b1=b3 b2=b4 Constrain intercepts: a1 = a4 a2 = a5 a3 = a6 Fix Mean(L1) to 0 Can now estimate parameter for Mean (L2) Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 X4 = a4 + 1.0 L2 + e4 X5 = a5 + b3 L2 + e5 X6 = a6 + b4 L2 + e6 Example: X1 X2 X3 X4 X5 X6 Means: 2 3 2.5 3 4 3.5 X4 = a4 + 1.0 L2 + e4 (E(L2)=a7 Estimate: a7=1.0 X4 = 2 + 1.0*1 + 0 (expected value of L2=1.0) X5 = 3 + b3*1 + 0 (expected value of L2 = 1.0) New parameter:a7

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28 Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 X4 = a4 + 1.0 L2 + e4 X5 = a5 + b3 L2 + e5 X6 = a6 + b4 L2 + e6 There can be a construct equation intercept parameter in causal models L2 = a7 + b5 L1 + D2 If mean(L1) fixed to 0 E(L2) = a7 + b5*0 = a7 As before, a7 represents the expected difference between the mean of L1 and the mean of L2

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29 Means and intercepts in SEM Models L2 = a7 + b1 L1 + D2 If mean(L1) fixed to 0 E(L2) = a7 + b1*0 = a7 In practice, if L1 and L2 represent time 1 and time 2 measures of the same thing, we would expect correlated errors:

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30 Means and intercepts in SEM Models Same principle can be applied to multiple group models: Group 1 Group 2 X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 a1[1] = a1[2] a2[1]=a2[2] a3[1]=a3[2] Mean(L1)=0 Mean(L1) = a4 We usually constrain measurement coefficients: b2[1]=b2[2] & b3[1]=b3[2]

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