Homework No. 4: __H__ and __C__

Outline of Psychological Measurement

Response = J[C(H(A), H(B))]

A_{j} and B_{i} are the stimuli for Column j and Row i.

s_{Aj} and s_{Bi} are the subjective, scale values for the stimuli.

Y_{ij} is the subjective impression of the combination of A_{j} and B_{i}.

R_{ij} is the overt response to this combination.

**H** are the functions that assign subjective values to the stimuli.

**C** is the combination function combining the subjective values.

**J** is the judgment function that assigns responses to impressions.

In this assignment, we let **J** be the identity function;
i.e., R_{ij} = Y_{ij}

*Homework:*

Make predictions for a 4 x 4, A x B, symmetric, factorial design using integers from 1 to 4
for levels of A and B. (s_{Aj} = A_{j} = j; s_{Bi} = B_{i} = i).
Plot predictions as a function of A with a separate curve for each level of B.

Part A: __H__ and __J__ are identity functions.

1. Additive: T_{ij} = s_{Aj} + s_{Bi}

2. Multiplicative: P_{ij} = s_{Aj} s_{Bi}

3. Subtractive: D_{ij} = s_{Aj} - s_{Bi}

4. Ratio: R_{ij} = s_{Aj}/s_{Bi}

Part B: Repeat Part A, but now let __H__ be a power function (square):

s_{Aj} = H(A_{j}) = A^{2}
Similarly, let s_{Bi} = H(B_{i}) = B^{2}

Part C: Repeat Part A, but substitute A^{.5} and B^{.5} for H(A) and H(B)

(H(B) = square root of B).

Part D: Repeat Part A, substituting log(A+1) and log(B+1) for H(A) and H(B).

Part E: What remains the same, irrespective of __H__, for the additive model?
What changes? How can you use these graphical properties to separate __H__ and __C__?
Given a new set of data, how would you decide what model is appropriate? How would you determine __H__?
Before you answer this question, organize your sixteen graphs for Parts A through D and additive, multiplicative,
subtractive, and ratio models.

by Michael H. Birnbaum, © 1974-2001, all rights reserved.