David A. Kenny
December 9, 1997

Constraints on Parameters:  Phantom Variables

Linear Constraints (parameter a linear function of other parameters)
    zero:  a = 0
    equality: a = b
    negative: a = -b
    proportionality: a/b = c/d or a = kb and c = kd
    additive: a + b = 1

Non-linear Constraints (less than means less than or equal)
    a > 0
    a > k
    a < k
    a > b
    a > b + k
    a = bc
    a = b2

When Constraints Are Needed
    Statistical
        Force variances to be non-negative (i.e., prevent Heywood cases)
        Force correlations to be in range
        Test of effects of latent products requires numerous constraints
        Repeated measures with latin squares can require constraints 
    Theoretical  
        Proportionality:  For example, if one looks at the effects of mother and 
           father on daughters and sons, on might want to force the following 
           constraint:

                              M --> D   F --> S
                              ------- = -------
                              M --> S   F --> D

           That is, a parent has proportionally more influence on a same- than an
           opposite-gender child.
        Greater than zero:  Over-time stabilities in most models cannot be negative. 
        Complex constraint:  Certain circumplex models can require that a2 + b2 = k. 
        Proportionality:  Kenny and Cohen (1980) describe a situation in over-time 
           analyses in which the effect of measures on the pretest versus the posttest
           increase by a proportional constant.  This is a version of Campbell's
           fan-spread hypothesis.

Phantom Variables
    Variables that have no substantive meaning, but that are created to force 
constraints on the model. Normally phantom variables have no disturbances. (See D. 
Rindskopf (1984).  Using phantom and imaginary latent variables to parameterize
constraints in in linear structural models.  Psychometrika, 49, 37-47.  Some 
computer programs such as LISREL 8 allow the user to force these constraints without 
using phantom variables.  

How to Use of Phantom Variables to Force Constraints
    (The material below is difficult to follow and requires close study.) 
    The following notation is used:  The effect of X on Y equals a is denoted 
    by X --> Y = a.  Phantom variables are denoted as P.

    a = -b:  Given X --> Z = a and Y --> Z = b, the following model is estimated:
             X --> Z = a, Y --> P = -1, P --> Z = a.  

    a/b = c/d or   
    a = kb and c = kd:  Given X --> W = a, X --> Z = b, Y --> W = c, and 
                        Y --> Z = d, the following model is estimated:  X --> Z = b, 
                        X --> P1 = k, P1 --> W = b, Y --> Z = d, Y --> P2 = k, P2 --> W = d.

    a + b = k:  Given X --> Z = a and Y --> Z = b, the following model is estimated:  
                X --> Z = a, Y --> P = -1, P --> a, and Y --> Z = k.  

    a > 0:  Given X --> Y = a, the following model is estimated:  X --> P = b and 
            P --> Y = b where a = b2.  

    a > k:  Given X --> Y = a, the following model is estimated:  X --> Y = k and
            X --> P = b and P --> Y = b where a = b2 + k.  

    a > b:  Given X --> Z = a and Y --> Z = b, the following model is estimated:  
            X --> Y = a, Y --> Z = b, X --> P = c, and P --> Z = c where a = b + c2.  

    a > b + k:  Given X --> Z = a and Y --> Z = b, the following model is estimated:  
            X --> Y = a, Y --> Z = b, X --> P = c, P --> Z = c, and X --> Z = k where 
            a = b + c2 + k.  

    a = bc:  Given X --> W = a, Y --> W = b, and Z --> W = c, the following model is 
             estimated: Y --> W = b, Z --> W = c, X --> P = a, P --> W = b.

    a = b2:  Given X --> Z = a and Y --> Z = b, the following model is estimated:  
             X --> Y = a, Y --> Z = b, X --> P = b, P --> Z = b where a = b2.  

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