Algorithm for Least-Squares
Fit to Multiplicative and Ratio Models

Michael H. Birnbaum
California State University, Fullerton
Irvine Research Unit in Mathematical Behavioral Science

file: multfit algorithm 9- 6-9-1992
Prof. Michael H. Birnbaum
Dept. of Psychology
C.S.U.F.
Fullerton, CA 92634

e-mail: mbirnbaum@fullerton.edu
phone: 714-773-2102

Abstract
This paper describes an algorithm for finding least-squares estimates of scales and the additive constant in multiplicative and ratio models, using an iterative system of equations.  Examples are presented of the method, comparing it with previous techniques that employ scales estimated from marginal means or sums.

	Multiplicative models and ratio models have arisen in a number of areas of psychological inquiry (e.g., Anderson, 1974; Birnbaum, 1980; 1982; Cliff, 1954; Gollob, ???: Gulliksen, 1959; Krantz, Luce, Suppes, & Tversky, 1971; Saaty, 1979; Shanteau, 1974).  Techniques for fitting these models have been discussed by Anderson (1983), Birnbaum (1980), Gulliksen (1959), Shanteau (1977), and Weiss (1991), among others.  Although these authors part company on some issues, they all agree that measurement of subjective values can proceed independent of physical measurements of the stimuli and that behavior can provide the basic data for scaling in the multiplicative or ratio model.
	The multiplicative model can be written for a factorial experiment as follows:
			Pij = a(sitj)c + b					(1)
where Pij is the predicted value of the data array corresponding to treatment combination xi and yj; si is the scale value for stimulus (treatment) xi; tj is the scale value of the jth level of the column factor, yj; a and c are constants; and b is the additive constant.  The constants, a and c cannot be identified unless the scale values, si and tj, are known or constrained by some other empirical relationships.  Therefore, we can simplify Equation 1 as follows:
			Pij = SiTj + b					(1a)
where Si = a(si)c and Tj = (tj)c.
	There are several models that are also equivalent to Equation 1.  One of the special cases is the ratio model,
								(2)
where si and tj are scale values; a is a multiplier; c is an exponent, and b is an additive constant.  Equation 2 is equivalent to Equation 1, substituting Si = a(si)c; Tj = (1/tj)c.  From a factorial matrix of data alone, the exponent and multiplier, c and a, cannot be identified, nor can a ratio model be distinguished from a multiplicative model.  Such distinctions would require additional information, such as an independent way of scaling the stimuli (Birnbaum, 1978; 1980; 1982).  
In Case of Perfect Fit: Model Implications
	Suppose that Equation 1a is true.  The additive constant and the scale values can be computed from the data.  The matrix of data will consist of family of linear functions that intersect at the point b.  The point of crossover is the functional zero point.  When either scale value is zero, the other has no effect on the response.  Although Tj and/or Si may or may not be linear functions of prior, physical values (yj and xi), any two rows of the array are linearly related to each other.  Let us consider two rows, row i and row r.  For row r we have,
			Prj = SrTj + b.
It follows that Prj - b = SrTj; thus, 
			Tj = [Prj - b]/Sr.  
Substituting for Tj in Equation 1, we have, 
						(3)
Equation 3 shows that any two rows are linearly related with a slope equal to the ratio of the row scale values; Equation 3 also shows that whenever Prj = b, then so also does Pij.  (By a similar derivation, it can be shown that any two columns should also be linearly related to each other, with intersection point equal to b).
	 Figure 1 illustrates how four entries of the form Pij, Prj, Pic, Prc, can be used to find b.   In Figure 1, all four entries are plotted as a function of the values for Row i.  The values of Row i then plot as the identity function (by construction), and the values of Row r can  be represented as a linear function with slope = A, given as follows:
								(4)
Two points always define a straight line, hence, we can use the point-slope formula to express Row r as a linear function of Row i, and the two linear functions can be solved simultaneously to find the point of intersection, as follows:
								(5)
where A is the slope as defined in Equation 4, and b is the intersection point, projected onto the ordinate.  [If the curves are parallel, A = 1; in such a case, A is set to 1.000001, as if the curves intersect very far off.  Parallel curves can thus be fit (to any specified degree of accuracy) by a multiplicative model by placing the intersection point far enough off].

Figure 1.  The additive constant can be estimated from four points that create two intersecting lines.

	We can now subtract b from the data, which translates the data so that the curves cross at zero in the transformed matrix:
								(6)
This new matrix can then be used to calculate scale values that will reproduce the matrix, using the following expressions:
									(7)
and
						(8)
where the number of rows and columns is n and m, respectively.  Predictions are calculated from the equation,
								(9)
Predictions calculated using Equations 4 through 9 will be equal to data entries, even though the scale values are not necessarily equal to the theoretical values (assuming that Equations 1 or 2 is true).  The scale values cannot be uniquely identified.  For example, the scale values for the rows could be multiplied by any positive constant if the columns were divided by the same constant, and the products would still be the same.
	If some scale values are not positive, the grand total of the matrix might be zero, in which case Equation 8 cannot be used.  In some cases, negative rows and/or columns in the translated matrix need to be reflected (multiplied by minus one), and re reflected in calculating predictions.  The procedure for reflection is directly analogous to the corresponding technique for extracting the second factor by the centroid method (Guilford, 1954, p. 485-500).  After suitable reflections, the grand total should be positive.
Least-Squares Solution
	Now suppose that the data do not fit the model exactly.  We define the loss function as the sum of squared deviations (SSD) between the theory and the data, as follows:
			(10)
Where L is the loss function.  We can minimize L by taking the partial derivatives with respect to each of the parameters, setting them equal to zero, and solving for the parameter values.  The equations result in the following system,

							(11)
There are n equations for the row scale values, as follows:
							(12)
and there are m equations for the column scale values:
							(13)
The equations for each unknown assume knowledge of the other unknowns; for example, Equation 11 defines b in terms of the data and the values of Si and Tj whereas Equations 12 and 13 compute Si and Tj in terms of b and the other unknowns.  
	The general approach to solving Equations 11-13 is similar to that of finding a location of minimal distance from a set of cities (Noble, 1967, p. 22-33): the trick is to find good initial values, then use the equations iteratively.  The computer program, MULTFIT, uses Equations 5-8 to define initial estimates and then applies Equations 11-13 iteratively.  New estimates are calculated in the right sides of Equations 11-13 to replace the old estimates on the left.  Iterations cease when the values of the unknowns do not change from iteration to iteration (i.e., when Equations 11-13 are satisfied).  In practice, convergence is determined by establishing a tolerance criterion, e, such that the sum of the absolute changes in the parameters from iteration to iteration is less than that criterion.  It is also a good idea in practice to continue iterations to one tenth the value of the tolerance criterion, checking to make sure that the value of the sum of squared deviations cannot be further reduced.
	It is worth noting that the calculations never make use of any prior or physical measures of the stimuli.  The relationship between the psychological values estimated from the multiplicative model are therefore purely empirical relationships.  The scale values can be plotted against the physical measures to obtain a graph of the psychophysical functions.
Examples
Salary Problem
	Figure 2 shows judgments of the "salary deserved" by faculty members in 1978 (Birnbaum, 1983, Experiment 3).  Faculty members at University of Illinois, Urbana-Champaign judged salary deserved as a function of years of experience and a rating of merit by a committee that evaluated scholarship, teaching, service, and recognition.  Merit =1 was defined as "satisfactory," and 0 was defined as "unsatisfactory."  Levels of Merit varied from .5 to 2.5 ("very meritorious").  Median judgments are shown in Figure 2 as a function of Years of Experience, with a separate type of symbol for each level of Merit.  Curves show the best-fit solution of MULTFIT.  The data appear to show the divergent pattern characteristic of multiplication, and though the curves are not a linear function of years of experience, they fit the multiplicative model quite well, 
			Salary = S(Merit)T(Years) + b.		(1c)

Figure 2.  Median judgments of salary deserved by professors in 1978, plotted as a function of years of experience with a separate symbol for each level of merit, ranging from .5 to 2.5.  Lines are predictions of the multiplicative model; symbols are data.

	The initial solution to these 30 data points using Equations 4-9 left a sum of squared deviations between the data and model of 4.89.  The iterative solution using a criterion of .01 gave a sum of squared deviations of 2.13, and this value was not further reduced by further iterations to a criterion of .001.  The least-squares solution in this case has less than half of the average squared error of the marginal sums.  The value of b = 15.09 ($thousand).
	The estimated scale values are shown in Figures 3 and 4.  Figure 3 shows the estimated scale values for Merit as a function of the Merit Rating.  The data appear to fall on a straight line through the origin, as if the subjects treated the Merit ratings as a ratio scale.  According to the model, subjects would have judged that the salary should have been flat for zero Merit.  Apparently, a person with zero Merit and 32 years of experience would earn about $15,090, since zero Merit has a scale value of zero.  Scale values for Years of Experience are a negatively accelerated function of Years.  Apparently, subjects believe that cross-sectional salaries should be closer together for people of 16 to 32 years of experience than between people of 1 to 16 years of experience.  
Figure 3.  Estimated scale values of Merit.  Data are nearly a similarity function of prior Merit values.
Figure 4.  Estimated scale values of experience as a function of Years.

	Since the scale values appear to be a linear function of the square root of years of experience, another graph of the data was drawn in Figure 5 to check this assumption.  The curves in Figure 5 are least-squares linear functions of the square root of years, fit separately to each curve of Merit.  These functions appear to intersect near the previous estimate of b on the ordinate.  The projection of the intersection point on the abscissa shows that a person who has taught for about one semester has a subjective value of zero in experience (i.e., until the person has completed a semester, merit should not make a difference).  
Figure 5.  Salary deserved plotted against square root of years of experience.  Lines are separately best-fit to each curve to check whether curves intersect at a common point.  Projection of intersection shows b; projection of intersection onto abscissa shows the zero point on the scale of square root of years.
Pizza Problems
	The prices of pizzas of different sizes with different numbers of toppings (goodies) usually fit the multiplicative model, 
		Price = S(Size)T(goodies) + b. 			(1d)
	A typical structure of pizza prices is shown in Figure 6, which plots the 1976 prices of Flying Tomato Brothers' pizzas as a function of the estimated value of T(goodies), with a separate curve for each size of pizza.  The curves intersect at a common point ($.26).  The sum of squared deviations using the marginal means was .00456, and the least squares solution had SSD = .00431.  
Figure 6.  Flying Tomato Brothers' pizza prices in 1976.  Symbols show prices; lines show predictions of multiplicative model.
	The psychophysical function for the sizes of the pizzas is a nearly linear function of the pizza's area with a positive constant, as shown in Figure 7.  Such a function causes the price per square inch to decrease as the pizza is increased, which makes sense with respect to both the utility of the consumer and the costs to produce the pizza (mostly labor and overhead).  The scale for the number of goodies, shown in Figure 8, is negatively accelerated.  The smooth curve is a fitted quadratic polynomial, consistent with the property that successive differences in T decrease as a nearly linear function of number of goodies.  The negatively accelerated curve is also consistent with a diminishing marginal utility of the number of added ingredients.   
Figure 7. Scale of Pizza Area
Figure 8. Scale of Pizza Goodies

	The data in Figure 6 can also be fit by the non optimal theory that the intercept is zero; in this interpretation, the largest deviations are 5 cents, and appear to be due to underpricing of pizzas that would otherwise fall on even dollar amounts.  For example, when the predicted value was $5.00, the actual price was $3.95, as if the proprietor thinks that will make the price seem psychologically much lower.
	An exception to the multiplicative model is shown in Figure 9, which plots the price structure of a real restaurant under a fictitious name.  Although the data look orderly in a table, the graph shows that the curves do not intersect at a common point.  The price per square inch of the smallest pizza is actually lower than that of medium sized pizzas.  These prices may have been determined by Bistromathics (Adams, 1982) rather than good judgment.  Although this establishment has an award-winning pizza recipe, it has been under three different owners in the last six years, all of whom used the same price structure.  
Figure 9. Pizza Prices at Gina Maria's.
Adverbs as Multipliers
	Cliff (1954) found that adverbs act as multipliers of adjectives to determine psycholinguistic meaning.  He asked people to judge the evaluative content of combinations such as "very good" or "slightly average."  The multiplicative model can be written as follows:
		Evaluation = S(adverb)T(adjective) + b		(1e)
A 3 by 5 subset of Cliff's data are shown in Figure 10 for illustration.  The adverbs "slightly" and "very" were combined with adjectives of "wicked", "bad," "average," "nice," and "admirable."   When the raw data were submitted to the program, the convergence was very slow.  The initial fit using Equations 4-9 was not very accurate (SSD=35.33) and a Mac SE (without math co-processor) required 8 minutes to reach a solution with a tolerance of .01, which yielded a sum of squares of 6.95.  A much faster solution was reached by subtracting 20 from each number, and reflecting the first two columns for negative adjectives.
Figure 10.  Reanalysis of adjective-adverb combinations.  Evaluative judgments are plotted as a function of the scale values for the adjective with a separate curve for each adverb.  Unmodified means there was no adverb.  See Cliff (1959).
Ratios of Distances
	When subjects are asked to judge "ratios" and "differences" of stimulus magnitudes, the ratio and subtractive models fit the data; however, the two judgments are monotonically related, consistent with the theory that both judgments are monotonic functions of the same comparison operation (Birnbaum, 1978; 1980; 1982).  The evidence is consistent with the theory that subjects in most cases compare stimuli by subtraction, but the "ratio" task can involve an exponential judgment function.  This theory can be written as follows:
				Rij = aRexp[Tj - Si] + bR			(14)
				Dij = aD[Tj - Si] + bD				(15)
where  Rij and Dij are "ratio" and "difference" judgments; aR, bR, aD, and bD are constants; and exp(x) represents the exponential function.  Equation 14 implies that "ratio" judgments should fit the ratio model (Equation 2) even though they are mediated by subtraction.
	However, when asked to judge "ratios of distances" and "differences of distances", the data do not conform to Equations 14 and 15, but instead fit the theory that subjects use the two operations as instructed.  This theory can be written as follows:
			RDijkl = aRD[(Tj - Si)/(Tk - Sl)] + bRD		(16)
			DDijkl = aDD[(Tj - Si) - (Tk - Sl) + bDD		(17)
where RD and DD represent "ratios of distances" and "differences of distances" between stimulus i and j relative to the distance between stimulus k and l.  To fit Equations 14-17 simultaneously requires techniques beyond the scope of this paper (see Birnbaum, 1980).  For the present purposes, "ratios of distances" from Birnbaum, Anderson, and Hynan (1989) will be used to illustrate the use of MULTFIT to estimate the ratio model.
	Figure 11 shows the "ratios of distances" plotted against the estimated values of T(j) for the distances between San Francisco, CA (SF), Salt Lake City, UT (SLC), Champaign-Urbana, IL (CU), and Philadelphia, PA (Phil).  Separate symbols represent different denominator distances.  In this case, the values of S(i) estimated from the program represent reciprocals of the subjective distances.  The sum of squared deviations in this matrix was .122.  The program actually used by Birnbaum, et al (1989) minimized a more complex loss function and also was constrained to fit several matrices at once. 
Figure 11.  "Ratios of Distances" as a function of the estimated scale value of the first distance, with a separate symbol for each level of the second distance.  Symbols show data, and lines show predictions.
 	The estimated values in the present analysis were 2.59, 4.59, 6.14, 8.60, 11.21, and 13.74.  Since these six values are distances between four cities, it is possible to check if they fit in one dimension.  For example, the distance from SF to SLC plus the distance from SLC to CU plus the distance from CU to Phil should add up to the distance from SF to Phil, if the cities are on a line (in this case 2.59 + 6.14 + 4.59 = 13.32, which is close to but smaller than 13.74).
Utility Theory
	Consider gambles with a probability of pi to win an amount xj, otherwise nothing.  If the utility of nothing is zero, then most utility theories state that the utility of a gamble should be a product of the utility of the money and the weight of the probability.  If the subject's task is to state the value of the gamble in money, then the response can be theorized to be the inverse of the utility function applied to a product,
				Price = U-1[s(pi)U(xj)]				(18)
where Price is the value of the gamble.  If it is further supposed that the utility function is a power function, U(x) = axc, then Price can be represented as the product of two functions, one of which will be a linear function of money (Birnbaum & Sutton, 1992).
				Price = [s(p)xc](1/c)				(19)
or Price = S(p)x, where S(p) = [s(p)](1/c).  According to Birnbaum, Coffey, Mellers, and Weiss (1992), judgments from different points of view can be represented using a different subjective weighting function for each point of view and a linear function of money for the utility scale.  Thus, their theory can be written as follows for these gambles from different viewpoints:
				Price(i,j,k) = S(i,k)T(j) = S(i,k)xj		(20)
where S(i,k) is the weight of probability pi to win xj from point of view k in a gamble that otherwise pays nothing.
	The gambles consisted of a six by six array of probability and amount values in geometric steps.  Probability values were .05, .09, .16, .29, .52, and .94; monetary amounts were $3, $5.40, $9.70, $17.50, $31.50, $56.70.  This six by six was judged by 38 undergraduates at University of California, Berkeley.  Each subject was instructed to judge the value of the gambles from each of three points of view, buyer's, neutral's, and seller's. (In the buyer's point of view, what is the most a buyer should pay to buy the chance to play the gamble?  In the seller's point of view, what is the least that the seller should accept to give up the gamble rather than play it?).
	According to the model, the data can be considered an 18 by 6 design, in which there are 18 levels of S(i,k) in Equation 20 and six levels of amount.  
	Figure 12 shows the mean judgments as a function of the amount to win, with 18 separate curves for each probability of winning and point of view.  For example, the top three curves are the gambles with a .94 probability to win the amount shown on the abscissa from the Seller's (highest), Neutral's, and Buyer's (lowest of those three) points of view.  The data are nearly linear functions of the amount to win, and show the characteristic pattern of the multiplicative model.  When the data are fit using marginal means, the SSD was 20.22.  After iteration to a tolerance of .01, the least squares solution yielded SSD = 15.25.  The best-fit value of b was .54, as if there is value even in a worthless gamble.  Similar results have been found by Mellers, Ordóñez, & Birnbaum (1992), by Mellers, Chang, Birnbaum, & Ordóñez (1992), and using different techniques by Komorita (1955) and Tversky (1967).  See
Figure 12. Judgments of the value of gambles from different points of view. Judgments plotted as a function of estimated utility of monetary amounts with separate curves for different levels of probability from different points of view.  
	The estimated scale of money is shown in Figure 13, plotted as a function of numerical value.  The curve is nearly linear, consistent with Equation 20.  The estimated values of the weights of probability, S(i,k), are shown in Figure 14.  These weighting functions resemble those derived by Birnbaum, et al (1992).

Figure 13.  Estimated scale of amount to win, plotted against the amount to win.  If utility is a power function of money, the curves should be linear.
Figure 14.  Estimated probability weighting functions, plotted as a function of the objective probability with a separate curve for each point of view.  Each curve represents the weight of the amount to win as a function of its probability.
Violations of Monotonicity
	The multiplicative model can also be used to estimate the point of crossover of a set of curves that are nonlinearly related to a physical measure but theorized to be linearly related to each other.  According to most utility theories, the utility of a gamble to receive y with probability p otherwise receive x should be a linear function of S(p), a function of p.  When y is fixed and p and x are varied, the curves can be expressed as a set of curves that should intersect when S(p) =1.  However, a number of recent articles have explored a situation in which one can obtain systematic violations of monotonicity (Birnbaum, 1992; Birnbaum, Coffey, Mellers, & Weiss, 1992; Birnbaum & Sutton, 1992; Mellers, Weiss, & Birnbaum, 1992).
	Figure 15 shows judgments of the selling prices of gambles with probability of p to receive $72 otherwise x, with a separate curve for x = $0 and another one for x = $24.  According to utility theories that satisfy monotonicity, the point of intersection should exceed the highest value.  In other words, the curve for x = $24 should exceed the curve for x = $0 for all values of probability.  Instead, at high levels of probability, judgments are actually lower for dominant gambles.  
Figure 15.  Judged price to sell gambles to win $72 or x, plotted as a function of the scale of probability, with a separate curve for x = $0 and for x = $24.  The multiplicative model finds the least-squares solution to the intersection point.  The fact that the functional scales of probability change sign is a violation of monotonicity.
Discussion
	One early method for fitting a multiplicative model works in the special case where b = 0.  In that method, the model is reduced by logarithmic transformation to an additive model, and a least-squares solution can be found to the logs of the response.  Birnbaum (1980) describes how the logarithmic index of fit can also be used even in the case where b is not zero, using a function minimizing subroutine, such as STEPIT (Chandler, 1969).
	A closely related approach has been developed by Weiss (1991), based on earlier programs by Shanteau (1974; 1977; see also Weiss & Shanteau, 1982; Anderson & Shanteau, 1970).  In this approach, marginal means are used to fit the additive main effects to minimize the interaction (under the additive model, marginal means are least-squares estimates).  Marginal means are then used in an analysis of variance to calculate the bilinear interaction component and the residual from bilinearity.  The residual from bilinearity is used as a test of the assumptions of the multiplicative model.  That approach can be described as a compromise that uses least-squares estimation for the additive model, but nonoptimal estimation for the multiplicative model.  
	To compare that technique with the least-squares method, the program of Weiss (1991) was applied to the data studied here, and MULTFIT was applied to example data presented by Weiss (1991).  In each case, the final solution of MULTFIT achieved a better fit than the program of Weiss (1991).  For example, for the salary problem, the Weiss program yielded SSD = 2.42, compared with 2.12 for MULTFIT.  For Subject 1 of Anderson & Shanteau (1970), the Weiss (1991) program yielded a sum of squared residuals from bilinearity of 20,474.6, which is about 21% higher than the MULTFIT solution of 16,930.7.   From a listing of MULTFIT, Weiss has now incorporated the least-squares algorithm into his program (Weiss, personal communication, 1992); with the new algorithm, his program achieves better fits.
	The MULTFIT algorithm appears to work successfully to find better solutions to multiplicative models than are provided by scale values estimated from marginal means.  Although the algorithm appears to be slow in certain applications, it may be worthwhile for purposes in which the best fit is required, such as for the development of statistical tests.

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Footnotes
	Author’s address: Department of Psychology, California State University, Fullerton, CA 92634.  E-mail address:  mbirnbaum@fullerton.edu.
	Support was received from National Science Foundation Grant, SES 8921880.   Thanks are due James Shanteau and David Weiss for helpful suggestions.
Figure Captions
Figure 1.  The additive constant can be estimated from four points that create two intersecting lines.
Figure 2.  Median judgments of salary deserved by professors in 1978, plotted as a function of years of experience with a separate symbol for each level of merit, ranging from .5 to 2.5.  Lines are predictions of the multiplicative model; symbols are data.
Figure 3.  Estimated scale values of Merit.  Data are nearly a similarity function of prior Merit values.
Figure 4.  Estimated scale values of experience as a function of Years.
Figure 5.  Salary deserved plotted against square root of years of experience.  Lines are separately best-fit to each curve to check whether curves intersect at a common point.  Projection of intersection shows b; projection of intersection onto abscissa shows the zero point on the scale of square root of years.
Figure 6.  Flying Tomato Brothers' pizza prices in 1976.  Symbols show prices; lines show predictions of multiplicative model.
Figure 9. Gina Maria's pizza prices.  Curves do not intersect at a common point, unlike most pizza price matrices.  Straight lines are extrapolated through prices.
Figure 10.  Reanalysis of adjective-adverb combinations.  Evaluative judgments are plotted as a function of the scale values for the adjective with a separate curve for each adverb.  Unmodified means there was no adverb.  See Cliff (1959).
Figure 11. Judgments of "ratios of distances" between U. S. Cities, plotted as a function of the estimated value of the numerator distance with separate symbols for each denominator distance.  After Birnbaum, et al (1989).
Figure 12. Judgments of the value of gambles from different points of view. Judgments plotted as a function of estimated utility of monetary amounts with separate curves for different levels of probability from different points of view.  
Figure 13.  Estimated scale of amount to win, plotted against the amount to win.  If utility is a power function of money, the curves should be linear.
Figure 14.  Estimated probability weighting functions, plotted as a function of the objective probability with a separate curve for each point of view.  Each curve represents the weight of the amount to win as a function of its probability.
Figure 15.  Judged price to sell gambles to win $72 or x, plotted as a function of the scale of probability, with a separate curve for x = $0 and for x = $24.  The multiplicative model finds the least-squares solution to the intersection point.  The fact that the functional scales of probability change sign is a violation of monotonicity.