**David A. Kenny
August 22, 2011**

**Page recently revised.**

**Please send comments and
corrections.**

*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 = b ^{2}**

**Proportionality:
For example, if one looks at the effects of mother (M) and father (F) on
daughters (D) and sons (S), 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 different-gender child.**

**Greater than zero: Over-time stabilities in most models cannot
be negative. **

**Complex constraint: Certain circumplex models can require that a ^{2}
+ b^{2} = k.**

**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 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: Y → Z = b, X → P = -1, P →
Z = b. **

**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 → P _{1} = k,
P_{1} → W = b, Y → Z = d, Y → P_{2} = k, P_{2}
→ 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, making a
= b ^{2}. **

**a ≥ k:
Given X → Y = a, the following model is estimated: X → Y = k and X → P = b and P →
Y = b, making a = b ^{2} + k.**

**a ≥ b:
Given X → Z = a and Y --> Z = b, the following model is
estimated: X → Y = b, Y → Z
= b, X → P = c, and P → Z = c, making a = b + c ^{2} and
because c^{2} cannot be negative then a ≥ b.**

**a ≥ b + k:
Given X → Z = a and Y → Z = b, the following model is
estimated: X → Y = b, Y → Z
= b, X → P = c, P → Z = c, and X → Z = k, making a = b + c ^{2}
+ 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, making a =
bc.**

**a = b ^{2}:
Given X → Z = a and Y → Z = b, the following model is
estimated: Y → Z = b, X → P
= b, P → Z = b, making a = b^{2}.
**