Local Smoothing: a Method of Controlling Error and Estimating Relationships in Consumer Research

The object of smoothing is to estimate the expected value of the criterion given the vector of predictors, or E(y|x). This is very similar to regression except that no specific functional form, such as linearity, is assumed. What is assumed is that the function is smooth and differentiable about each point in the relevant range. Smoothing estimates the value of the function for observation i as the average over y_j 's which have x_j close to x_i. Taking the Taylor series expansion of Equation 1 about observation j,

f(x_i) = f(x_j) + f'(x_j)(x_i-x_j) + f"(x_i-x_j)²/₂+...   (2)

+ higher ordered terms of (x_i - x_j).

Thus, given that a function exists, for small values of x.-x., the error from using the Taylor is relatively insignificant. The gain series expansion from estimating the function as the average of proximate points derives from the canceling effect of averaging the uncorrelated error. Thus, the problem is co find the relative weights to be given to observations in estimating nearby observations. Too much weight on points that are very close means that there will be few observations over which to smooth disturbances. On the other hand, a very broad net provides plenty of points to minimize the effect of random error, but substantial deviation from the Taylor series approximation. The optimal weighting obviously depends on the characteristics of a data set. In particular, it depends on the number of observations; the more points available for smoothing, the less distant points are needed. The optimal weighting also depends on the absolute value of the higher order derivatives of the function; the greater these are, the greater the need to put the heaviest weight close to the point being estimated.

This paper presents L-SMOOTH as an algorithm to determine the optimal weighting scheme for locally smoothing any given data set. Emphasis in this paper, however, is not so much on the particular algorithm as on the concept of local smoothing as it applies co the kinds of data used in consumer research. In the next sections, the algorithm is described and examples are provided of its use. Then a general section follows discussing the values and pitfalls of any such smoothing scheme.

L-SMOOTH: A GENERAL SMOOTHING PROGRAM

L-SMOOTH estimates each value of each observation, y., as a weighted average of the yj values at other points. The weight, w..-, on any point j to predict i, is inversely related to the standardized distance between -i and j on predictor dimension k. Thus:

EQUATION   (3)

where:

EQUATION   (4)

EQUATION   (5)

EQUATION   (6)

is a minimum. This formulation does not have an analytic solution so a search routine is used. In the case of L-SMOOTH, a routine using Hooke and Jeeves (1961) pattern search is used in an algorithm SDRMIN by Buffa and Taubert (1972). This search routine is quite flexible and efficient.

FIGURE 1

R²_smooth = 1 - var(e)/var(y),   (7)

where var(e) is defined in Equation 6 and var(y) is the variance of the criterion. Thus, R²_smooth measures the percent of the original criterion variance that is accounted for by the smoothing. Since the smoothing function is not limited in its functional form, this measure provides an asymptotic upper bound to the fit of any parametric form. This is clearly seen if the number of observations is very large. Then, the smoothing estimate of E(y|x) is simply the average of the y values at a given x value. Obviously, no parametric form will be able to do better than smoothing in the asymptotic case.

In the case where the number of observations is not arbitrarily large, a parametric form may have a higher R² than smoothing for two reasons. First, the large number of parameters in, say, a quadratic regression may produce a high R² due to the large numbers of degrees of freedom. Second, smoothing assumes that the same weighting scheme is appropriate throughout the range of the data, so that a weighting scheme might produce large errors in certain regions which a parametric model might avoid. Fortunately, the work done in exponential smoothing of time series (Winter, 1960) indicates that average error is relatively insensitive to the exact smoothing constant used. Therefore, given that the data is not unduly perverse, it is fairly safe to assert to that one can expect little improvement in accuracy moving from a smoothing solution to a parametric one.

Notice that the claim is not being made that a smoothing model is better than a parametric one such as linear regression. Indeed, as a model, it is somewhat worse. Smoothing cannot predict a criterion from a knowledge of the predictor variables alone; it requires the values of other criterion observations with weights defined by the predictors. By contrast, a regression model provides a compact summary of the functional form which allows direct prediction from any set of predictor values. Therefore, a routine such as L-SMOOTH should be used to suggest functional forms and to lessen the effects of high levels of error, but not as a model in itself.

Examples of Smoothing

The first example of local smoothing involves the area where the technique is perhaps of most value to those in consumer research -- dealing with probabilities. The following data set represents a hypothetical set which was used in the initial testing of L-SMOOTH. It illustrates both the advantages and disadvantages of smoothing.

Imagine the following set of data. Twenty consumers are asked to evaluate a product they have just purchased for the first time. The only subsequent data is whether the product was repurchased in the next shopping trip. Smoothing attempts to estimate the relationship between evaluating the product and the probability of subsequent repurchase. The results of the L-SMOOTH analysis are given in Figure 2. Conceptually, what the routine has done is find that weighting of adjacent points that best predicts each point. The least squares solution to this problem is given by the dotted line in Figure 2. The optimal weighting is

w_ij = 1/D²_ij + .49

where D²_ij is the distance from the point to be estimated. This weighing scheme produces an improvement in variance accounted for of about 50%. The standard error about each estimate is about .35.

There are several things to notice about the result. Except for the part of the curve in the lower left hand corner, the curve takes a normal sigmoid shape. On examination, the "hump" in the lower left hand corner can be seen to be due to the one subject who rated it poorly but still repurchased. While it might be tempting to remove this outlier, the data as presented provide no credible way to discriminate between a logistic and, say, a linear representation of the data. Like any model that uses a squared error criterion, smoothing is strongly affected by outliers, particularly at the edges of the data set and in regions where the density of observations is low. The issue of outliers is raised again later in this paper. Fortunately their effect becomes minimal as the number of observations increases.

FIGURE 2

SMOOTHING HYPOTHETICAL 0-1 DATA: PROBABILITY OF REPURCHASE V. EVALUATION AFTER TRIAL

An example of smoothing actual data is provided in Figure 3. In this case, data represent fifty tipping occasions at a local restaurant. The dollar tip is smoothed with respect to the total cost of the meal including drinks. The smoothed function represents a good fit with R²_smooth = 0.79. The shape of the smoothed function indicates that a linear approximation should work well with this data and further that the intercept is greater than zero. Both of these hypotheses were verified by linear regression. This illustrates how local smoothing one variable on another can provide a good source of hypotheses for future analysis.

Using the same data set, age and the number of people at the table were used to smooth tip paid as a percent of the total bill. The results provided the following information:

TABLE

FIGURE 3

DOLLAR TIP SMOOTHED BY TOTAL BILL

The importance of age contradicted a linear analysis of the same data. Therefore, age alone was used to smooth percent tip. The results are provided in Figure 4. Notice the strong nonlinearities indicating peaks of tipping at ages 30 and 50, but otherwise lower. In this case, local smoothing provided a good way to understand what is obviously a rather complex phenomena, and in particular, why a linear function understated the importance of age.

DISCUSSION

In consumer behavior, a smoothing routine such as L-SMOOTH can be used in three ways. It can be used as a method of analysis in its own right. It can be used as a means of conducting preliminary research by suggesting hypotheses and functional forms for more rigorous analysis. And, finally, it can be used as a means of preparing data for later analysis by reducing the effect of errors and especially outliers. This section discusses the value of smoothing in these contexts and some of the problems or pitfalls that might arise.

FIGURE 4

PERCENT TIP SMOOTHED BY AGE OF PATRON

Currently the routine lacks measures of statistical significance which severely limit the use of smoothing as a terminal method of analysis.

There are, however, several aspects to smoothing that make it ideal for preliminary estimation of relationships between sets of variables. Since smoothing is not limited to any parametric form, the resulting visual display can suggest various transformations of the data. For example, in dealing with binary or multinomial criterion data, smoothing against the dependent variables may indicate whether a logistic transform is going to be needed or whether a linear approximation is adequate within the range of the data. In a case where smoothing might have been useful, Bettman (1974) found probit analysis produced identical results as regression with a 0-1 criterion. This result might have been predicted by smoothing the data and noting that it is approximately linear in the appropriate range.

When smoothing is used to suggest the direction of further analysis, the raw rather than the smoothed data should serve as input to the later analysis. Smoothed data produces an artificially high index of fit since much of the variance about each point has been removed. Furthermore, the parametric routines generally have well established measures of significance that would be overstated with smoothed data.

In using smoothing to suggest relationships, caution must be taken to avoid distortion from outliers. One large outlier will produce a continuous hump or valley in the smoothed solution. Since it is continuous, there is a temptation to infer that it is "really there." Furthermore, when the raw data is placed in a parametric routine that models the hump, it will indeed predict better then those that do not, in that it can account for the outlier. Therefore, the smoothing routine should be used to identify outliers and possibly remove them from the analysis. This is easily done by examining the residuals from the smoothing solution. An outlier should form a pattern of one large residual surrounded by smaller errors of the opposite sign. Outliers can be tentatively identified by looking for residuals whose absolute values are more than three times the standard error of estimate. The analysis can then be rerun with these values deleted.

Because of its effect on significance tests, one would not generally input smoothed data into an analysis such as regression. There may, however, be some benefits from such a move. First, where one's concern is estimation of parameters rather than their statistical significance, presmoothing the dependent variable would be expected to stabilize the estimates of the coefficients and be less affected by outliers. Thus, presmoothing should be used where the primary focus is on robust parameter estimation. Second, when working with binary or multinomial data, smoothing helps remove the sampling error to enable the researcher to concentrate on the functional form of the relationship. Transformations such as the arcsine or the logistic can be applied directly to the smoothed data since mathematically the smoothed probability estimates are constrained to be greater than zero and less than one.

Presmoothing data prior to subsequent analysis can cause problems of interpretation if caution is not used. The effect of any smoothing or grouping by the predictor variable is to heighten the relationship between criterion and predictor while at the same time reducing the relationship between criterion and other variables. Blalock (1964, p. 105) provides an excellent discussion of this problem with respect to different ways to group variables prior to linear analysis. Suppose, for example, one is concerned with the effect of the age of a customer and his distance from a store, on purchases from that store. By presmoothing with respect to location, different age groups are averaged and their effect diminished. A subsequent analysis of age against location-smoothed purchases would result in lower correlation than against the raw data. Thus, smoothing with respect to one set of predictor variables generally increases the strength of the relationship with that set and decreases it with other variables.

The lesson here is that if smoothed or clustered data is to be used in a subsequent analysis, all of the attributes that will go into the final model should be included in the smoothing model. Nakanishi (1976) clustered purchase probabilities by region and then found that ease of travel (distance) is the most important variable in the selection of a store. He also found that the linear coefficients of some of the other attributes, such as perceived quality, were non-significant or had counter-intuitive signs. What happened is that by clustering regionally, the quality ratings were averaged by location so that their effect became lessened or distorted. Thus, smoothing can be used to highlight certain relationships, but at a cost of lessening other relationships.

In summary, local smoothing represents a generalization of the concepts of exponential smoothing of time series and spatial smoothing of geographic data to the needs of those working in consumer research. As a tool, smoothing provides substantive data in its own right; it can suggest functional relationship and identify outliers in a data set to prepare the way for further analysis; and finally, it can, with certain interpretive caveats, be used to prepare data for subsequent analysis. No particular claim is made in this paper for the virtues of L-SMOOTH. Other routines will undoubtedly be developed which will be more efficient and have more useful output. The claim here is that smoothing has not been much used in consumer research, and we are the worse for it.

REFERENCES

James R. Bettman, "A Threshold Model of Attribute Satisfaction Decisions," Journal of Consumer Research, l (September 1974), 30-35.

Hubert M. Blalock, Causal Inference in Nonexperimental Research (New York City: Norton, 1964).

Robert G. Brown, Smoothing, Forecasting and Prediction of Discrete Time Series (New Jersey: Prentice Hall, 1963).

E. S. Buffa and W. H. Taubert, Production-Inventory Systems: Planning and Control (Homewood, Illinois: Irwin, 1972).

R. Hooke and T. A. Jeeves, "Direct Search Solution of Numerical and Statistical Problems," Journal of the Association of Computing Machinery, (April 1961).

Masao Nakanishi, Attitudinal Influence on Retail Patronage Behavior," Advances in Consumer Research, V.III, Beverlee B. Anderson, Editor, Association for Consumer Research, (1976).

Donald S. Shepard, "SYMAP Interpolation Characteristics Computer Mapping as an Aid in Air Pollution Studies, Report L, Lab for Computer Graphics, Harvard University, Cambridge, Massachusetts, 2(1970).

Peter R. Winters, "Forecasting Sales by Exponentially Weighted Moving Averages," Management Science, 6(1960), 324-342.

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