A Non-Parametric Method For Graphical Analysis of Survey Data: an Application to Consumer Satisfaction Research

Hooman Estelami, Fordham University
ABSTRACT - Data collection through surveys is a common practice in consumer research. In order to obtain precise consumer response measures, researchers often utilize continuous and multi-item scales in surveys. A critical piece of information captured by these scales is the shape of the probability density function of the consumer responses. A graphic inspection of this distribution may help one determine the number and size of consumer segments. However, the existing methods for graphically analyzing the shape of the probability density function of survey measures obtained through multi-item and continuous scales are restrictive. This paper will introduce a non-parametric density estimation technique as a convenient method for approximating the probability density function of measures obtained in survey data. The benefits of the approach are demonstrated using consumer satisfaction survey data.
[ to cite ]:
Hooman Estelami (1998) ,"A Non-Parametric Method For Graphical Analysis of Survey Data: an Application to Consumer Satisfaction Research", in NA - Advances in Consumer Research Volume 25, eds. Joseph W. Alba & J. Wesley Hutchinson, Provo, UT : Association for Consumer Research, Pages: 116-123.

Advances in Consumer Research Volume 25, 1998      Pages 116-123


Hooman Estelami, Fordham University


Data collection through surveys is a common practice in consumer research. In order to obtain precise consumer response measures, researchers often utilize continuous and multi-item scales in surveys. A critical piece of information captured by these scales is the shape of the probability density function of the consumer responses. A graphic inspection of this distribution may help one determine the number and size of consumer segments. However, the existing methods for graphically analyzing the shape of the probability density function of survey measures obtained through multi-item and continuous scales are restrictive. This paper will introduce a non-parametric density estimation technique as a convenient method for approximating the probability density function of measures obtained in survey data. The benefits of the approach are demonstrated using consumer satisfaction survey data.


Use of consumer surveys is a common practice in both applied and academic consumer research. Surveys are often used to gauge constructs such as consumer satisfaction, quality perceptions, involvement, and brand image (Bearden, Netemeyer, and Mobley 1993). Once the survey data has been collected, one has to rely on the emerging basic statistics such as the mean and the variance to make the necesary judgements on the levels of key constructs being measured. However, in addition to the mean and the variance, one can often rely on an equally vital measure: the distribution of the ratings. The shape of this distribution provides for a better understanding of the market place. For example, the existence of a skewed or multi-modal distribution of consumer responses may signal the potential existence of multiple consumer segments. Such information may prompt additional managerial attention, and may thereby initiate a segment-based marketing program.

As will be demonstrated in this paper, using the existing techniques for obtaining the probability density function of consumer ratings often results in ambiguous interpretations. This is especially true for measures obtained through continuous and multi-item scales. While a large number of multi-item scales have been developed in order to measure a variety of marketing constructs (e.g., Bearden, Netemeyer, and Mobley 1993), and continuous scales are also often utilized to increase measurement precision, the existing marketing literature offers no directions on how the probability density function of the obtained measures should be estimated. In this paper, a non-parametric approach to estimating consumer response densities-the kernel density estimation method-is introduced.

The kernel method has several key advantages over the common approaches used today. For example, the traditional approach of building histograms to estimate the shape of consumer response distributions is widely known to yield unstable results (Silverman 1986). Moreover, unless strong a priori information exists, the application of parametric density estimation methods-which by their nature force the estimated distribution to take on particular shapes-is inappropriate. Parametric density estimation is limited in its ability to provide density estimates that reflect the subtle fluctuations in consumer response data. The following section discusses the applications of multi-item and continuous scales in consumer survey research, following which the histogram, parametric, and non-parametric estimation methods are discussed in detail. The section following that presents empirical results which help demonstrate the benefits of the proposed approach using consumer satisfaction survey data.


The primary objective of most consumer surveys is to provide observed measures which closely resemble the true magnitude of the construct under study. Such measures are typically arrived at through the consumers’ own subjective estimates on related survey questions. To obtain such measures, single-item scales have often been utilized (e.g., Cezepiel and Rosenberg 1976, Day 1977, LaBarbera and Mazursky 1983). However, the limitations of single-item scales has given way to the development and use of standardized multi-item scales in consumer research to increase measurement reliability (e.g.,Bearden, Netemeyer, and Mobley 1993; Parasuraman, Zeithaml and Berry 1988), and to the use of continuous scales to increase measurement precision (e.g., Anderson 1982).

Several key psychometric advantages are gained by using multi-item and continuous scales in consumer surveys. Multi-item scales allow for the measurement errors of the individual items in the scale to cancel each other, reducing the random error attached to the scale, and therefore increasing its overall reliability (Churchill 1979). In addition, by using a single-item scale, the observed values of the measure are clearly limited to the range of that single item. For example with a single-item Likert scale at 7 levels, 7 possible values can be observed, and as a result, consumer responses can only be categorized into 7 groups. However, with a multi-item scale the range of the observed measures multiplies. For example, with a 5-item Likert scale, with each item at 7 levls, 31 possible responses may be obtained, thereby increasing the precision of the obtained measure. This increased precision, is also an advantage gained by using continuous scales. Moreover, many continuous scales have the added advantage of not restricting the consumers’ possible responses and thereby increasing the level of flexibility granted to the respondents in providing their responses.

Application of multi-item satisfaction scales among academics and practitioners has grown in recent years. For example, in consumer satisfaction research where consumer surveys are frequently used, Oliver (1980) utilized a multi-item scale, made up of 6 Likert-format items and found the scale to have high measurement reliability. In the area of service quality measurement, Parasuraman, Zeithaml and Berry (1988) have developed the SERVQUAL scale which is made up of 22 items measuring consumers’ expectations and performance perceptions on a variety of service attributes. Bearden, Netemeyer, and Mobley (1993) have also compiled a large array of multi-item scales used to gauge a variety of key marketing constructs.

With reliable consumer responses at hand, the researcher may very likely be interested in some of the key characteristics of the measure-characteristics such as its mean and standard deviation. Equally important, one may be interested in graphically inspecting the shape of the distribution of the obtained measure. This may facilitate the identification of underlying consumer segments. For example, the presence of a dense area in the distribution indicates a high concentration of consumers, and may signal the existence of a consumer segment As shown by Kumar and Rust (1989), such graphic inspections can not only aid one in identifying the relevant market segments, but are the most preferred mode of segment identification by practicing managers. Therefore, the ability to appropriately identify the relevant consumer segments based on the shape of the probability density function of consumer responses seems to be a valuable concern. However, as will be shortly demonstrated, the common approaches currently used to estimate the probability density function of consumer responses typically result in unreliable and unstable distribution estimates. In the next two sections, the existing approaches for assessing the distribution of consumer response measures in survey data will be discussed, following which the proposed method will be introduced, and its benefits demonstrated using consumer satisfaction survey data.


The oldest and most commonly used density estimator is the histogram (Silverman 1986). A histogram is defined by an origin X0, and a bin width h. Given X0 and h, a bin is then defined as the interval [X0+kh,X0+(k+1)h), where k takes on integer values. The histogram then provides an estimate of the density for each bin by presenting the proportion of observations which fall in the bin:


where yi is the total number of observations within the bin containing xi, and n is the total number of observations in the sample.

In constructing a histogram, the researcher needs to make two choices: (a) choice of the origin X0, and (b) choice of the bin width h. The shape of the histogram primarily depends on the choice of thse two parameters. While the choice of h determines the degree of smoothness in the histogram, X0 determines the reference point, based on which consecutive bins are defined. As a result, depending on one’s chosen value for the origin and the bin width, drastically different density estimates may often be obtained, leading to potentially conflicting interpretations of the same data.

To demonstrate the instability of distributions obtained through histograms, an example using consumer satisfaction data collected through a survey of over 300 consumers will be used. A total of five single-item satisfaction scales ranging from a low of 1 to a high of 10 were used to develop the multi-item satisfaction scale. The measure used is obtained by averaging the five items. In Figure 1a, X0 is set at 5, and h is set at 1. Figure 1a suggests that the consumer responses are uni-modal in distribution, and slightly loaded on the positive end of the scale. The peak density is in the 6-7. However, a simple change of the bin-width from 1 to 1/2 produces Figure 1b, which suggests a different interpretation of the same data. Figure 1b indicates that the consumer satisfaction distribution is actually multi-modal, suggesting the potential existence of multiple consumer segments. Choosing a larger bin width of 3 would yield Figure 1c, which suggest a highly homogenous distribution, and no identifiable segments. Choosing a fine bin-width of 0.2 results in Figure 1d which suggests yet another drastically different interpretation of the exact same data. Similar ambiguous results would emerge simply from different choices of the origin.

From the above illustration it may be clear that the shape of consumer response histograms is directly influenced by the researcher’s choice of the bin width and the origin. The choice of one bin width or origin over the other, may drastically change one’s assessment of the shape of the probability density function of the consumer ratings.


An alternative approach to histogram building is the parametric approach. In parametric density estimation, one has to assume that the consumer responses have a particular distribution function. Using the data obtained from the survey, one can then estimate the parameters of that distribution function. The main drawback of this approach is that it constrains the shape of the estimated distribution to the one assumed by the researcher. This is especially troubling in the context of consumer survey research where one often can not make any a priori assumptions about the shape of the distribution of responses.

Two commonly used distribution functions are the normal and the beta. Both of these distributions have two parameters. The normal probability density at a point x, is defined by:


where m and s are the population mean and standard deviation. The normal distribution is symmetric and unimodal, and allows x to range from -$ to +$. The beta distribution function on the other hand can take on more flexible forms. The beta distribution at the point x is defined by:


where a and b are the distribution parameters. Depending on the values of a and b, the beta distribution function can take on forms such as a U, an inverted U, or a J shape, among others, and is therefore a convenient distribution function for many marketing applications (Morrison 1981, Heckman and Willis 1977, Sabavala and Morrison 1981). Similar to the normal distribution, the beta distribution suffers from the fact that it is constrained in its shape. With the exception of a U or J shape, the beta distribution is unable to reflect other cases of multi-modality in data.

Figures 2a and 2b show the normal and beta distributions estimated on the consumer satisfaction data mentioned earlier. As can be seen from Figure 2a, by using the normal distribution, much of the details of the data seem to have vanished. Due to the uni-modal nature of this distribution, the suggested high-density area in the 6-7 range of Figure 1a is no longer observable. Also, due to its symmetric nature, the positive skewness in the data can no longer be noticed. Figure 2b shows the beta distribution estimated on the same data. To obtain this estimate, the satisfaction data was re-scaled to the 0-1 range and the method of moments was then applied to obtain the parameter estimates for a and b. As in the case of the normal density estimate, the beta density estimate seems to also be constrained. Due to the shape of the probability density function enforced by the beta distribution, much of the details have disappeared. Since the currently used methods for estimating consumer response distributions are limited in their ability to display subtle fluctuations in consumer ratings, a more flexible method is needed. In the following section, the kernel density estimation is introduced as an approach for conveniently estimating the underlying probability density of the consumer responses obtained from surveys.


Kernel density estimation is a non-parametric approach to density estimation. It has been widely used in a variety of econometric applications and is currently the most commonly used method for non-parametrically estimating probability densities (Silverman 1986). Due to its non-parametric nature, Kernel estimation requires no priors on the shape of the distribution function, and is therefore an attractive option in consumer research where no strong assumptions about the distribution of consumer responses can be made. In the marketing literature, the method has been used to a limited extent. Prior applications include the pioneering work by Rust (1988), using kernel density estimation as a means for relaxing the restrictive assumptions of classical regression. In a later work, Donthu and Rust (1989) used the method to estimate the geographic distribution of a city’s population, and to identify the optimal location for a new retail outlet. More recently, Abe (1995) has applied the method for modeling consumers’ brand choice behavior. Formally, the kernel density estimate at a point x is defined by:


where n is the sample size, h is the smoothing parameter, K is the kernel function, and the xi are the data points. Often the kernel function is chosen such that it is non-negative, symmetric and:


The smoothing parameter h is also chosen such that as n->oo, h->0.



Despite the moderate complexity of the above definition, the concept of kernel density estimation is somewhat simpler. The kernel density estimate at a particular point x is simply the sum of n normalized kernel function values. Each kernel function value at a point x depends on the distance between x and a corresponding observation xi. If x is close to many observations, the value of the individual kernel functions K(x-xi/h), are large, and therefore their sum is large, resulting in a high estimate of the probability density at the point x. In contrast, if x is far from most sample observations, the individual kernel function values are small, resulting in a low probability density estimate at x. Note that since the kernel function was chosen to be non-negative at all points and integrable to 1, it too is a probability density function. Therefore the kernel density estimate, which is an average of the individual kernel values is by definition also a probability density function. In addition, the continuity and differentiability properties of the kernel function are passed on to the kernel density estimate (Silverman 1986).

Given an optimal choice of the smoothing parameter-which will be discussed shortly-kernel density estimates have been proven to be pointwise consistent (Rosenblatt 1956). In addition, provided that the underlying distribution is continuous, the kernel density estimate is uniformly consistent, and asymptotically unbiased (Parzen 1962). Therefore, with a sufficiently large sample size, we are guaranteed to obtain good estimates of the true probability density function.



From the above discussion, it may be evident that obtaining kernel density estimates requires one to make a choice of a kernel function K(.), and a smoothing parameter h. Many choices of the kernel function, such as the normal, the Epanechnikov, and the uniform exist. Interestingly, it has been shown that the choice of the kernel function-even with small sample sizes-does not greatly affect the resulting kernel density estimate (Silverman 1978). The resulting kernel density estimate is uniformly consistent, regardless of the choice of the kernel function. On the other hand, the choice of the smoothing parameter h can drastically influence the resulting density estimate. While a large value of h results in a smooth density estimate, a small h results in a "bumpy" estimate. The ideal value of h is found by minimizing the mean integrated square error (Silverman 1986, Parzen 1962):


and is given by:


Note that as n->oo, h->0.

It is important to point out that although the kernel method benefits from attractive statistical characteristics, it does have some minor limitations. When applied to data which are drawn from a long-tailed distribution, due to the constant window width, the estimates in the tails may become very unstable. In such cases, the nearest-neighbor method (Silverman 1986) can be used. Some of the other limitations of the method relate to the size of the sample used to conduct the estimation. With samples smaller than one hundred observations, the density estimates tend to be unreliable (Donthu and Rust 1994). In such cases, the histogram approach is likely to be preferable. On the other hand, when applied to large data sets (i.e., one thousand or more observations), due to its data intensive procedure, the kernel estimation process becomes very slow. Fast Fourier transforms can be applied in such cases to speed up the estimation process (Hardle 1993). Fortunately, the above concerns often do not apply to most consumer survey data, as response scales are often limited in their range of possible values, and most consumer surveys utilize moderate sample sizes.


This section will demonstrate the merits of the kernel approach using consumer satisfaction survey data. The data was collected through a standard annual satisfaction survey in an East coast educational institution, and is discussed in more detail in Estelami and De Maeyer (1997). A total of 315 graduate business students completed the survey which had yielded a 63% response rate. The sample was 31 % female, 69 % male, with an average age of 27.5 years. Table 1 outlines the five questions used in the analysis, and their corresponding basic statistics. A multi-item scale was constructed by taking an average of the five items, resulting in a coefficient alpha of 0.92. The value of the coefficient alpha indicates a high degree of measurement reliability. The mean and standard deviation for the multi-item scale were found to be 5.57 and 1.79, respectively. To facilitate the identification of programs that can be implemented for improving consumer satisfaction, additional measures regarding the performance of the institution on individual aspects of its services were also obtained. These include multi-item scales for performance perceptions of the student service offices (coefficient alpha=0.80), the physical facilities (coefficient alpha=0.74), the administration (coefficient alpha=0.89), and teaching quality (coefficient alpha=0.87). These measures were obtained in order to assist the management in identifying areas where specific quality improvement initiatives can be directed.

Figure 1a-discussed earlier-provides a histogram of the consumer satisfaction measure. As was mentioned earlier, with its origin at 5, and a bin width of 1 unit, Figure 1a suggested that the consumer satisfaction measure’s distribution is unimodal and slightly skewed to the right. However, a simple change of the bin width from 1 to 1/2 produces Figure 1b, which suggested that the consumer satisfaction distribution is actually multi-modal. Fiures 1c and 1d show similar ambiguous results from simple changes in the bin width. At this point, one is left with contradicting interpretations of the exact same data-one suggesting the potential of multiple consumer segments, and the others refuting it.

Figures 2a and 2b, which were also discussed earlier, show the normal and beta probability density estimates of the data. Both lack much of the detail observed using the histograms. Moreover, they are forced-by their functional form-to be uni-modal, and in the case of the normal density, symmetric. In order to assess the appropriateness of using normal density estimation on the consumer satisfaction data, the Shapiro-Wilk test was conducted. Similar to the Kolmogorov-Smirnov test, the Shapiro-Wilk statistic tests the null hypothesis that the sample data is a random variable drawn from a normal distribution, and is a common test of normality for small samples. The results of the test rejected the null hypothesis at the p<0.0001 level, thereby confirming that the data is not drawn from a normal distribution. Therefore, the normal approximation seems to be an inappropriate density estimate for the consumer satisfaction data.





To resolve the above issues, namely the potential bi-modality, and non-normality of the satisfaction responses, kernel density estimation was then conducted on the data. A commonly used kernel function-the normal kernel-was utilized in this application:


Since f(.) in equation 7 is unknown, the optimal smoothing parameter is chosen such that the mean integrated square error (MISE) is minimized. The optimal smoothing parameter for the normal kernel-one that minimizes equation 6-is approximated by (4/3)1/5n-1/5s, where s is the standard deviation of the satisfaction measure (Silverman 1986, p. 87).

At this point, it is important to note that one possibility when applying kernel density estimation to multi-item scales is that the domain of the scale values for which the density is being estimated may be bounded. For example in our case, the satisfaction scale measures range from 1 to 10 and no values beyond this range exist. As a result-unless corrected for-the density estimate near the boundaries (i.e., 1 and 10) would be biased downwards. The bias results from the fact that the window width will force the kernel estimate to take into account scale ranges-such as x<1 and x>10-for which there are no scale values. To deal with this issue, the raw data was transformed to create "reflections" around the boundary points (Silverman 1986, pp. 30-31). In other words, the data is augmented by adding reflections of the original data around the upper and lower boundaries. So for example, for a data point with a rating of x=3, a lower reflecting value of x=-1 (i.e., 2-x) is created. This represents a "reflection" of the x=3 data point around the scale’s lower bound of 1. Similar reflections are created around the upper bound of the scale. The resulting density estimate for the bounded region [1,10], is then multiplied by 3 to restore the unity criterion of density estimation. The Gauss programming language was used to conduct the estimation, the results of which are shown in Figure 3. As can be seen, compared to the histograms of Figure 1, inspection of Figure 3 reveals a multi-modal distribution. One of the modes is located at 4.3, and the other at 6.5. Moreover, the kernel density estimate seems to look very diffeent from the parametric distributions shown in Figure 2.

To assess the relative performance of the various methods, their fit to the data was compared. To do so, 150 responses were randomly chosen and used as the estimation sample, and the remaining 165 responses were used as a hold-out to test the obtained estimates. For each of the methods discussed, the cumulative density obtained from the estimation sample was compared to the cumulative density of the hold-out sample using the following fit index:


where Festimate(x) and Fsample(x) are the cumulative density functions at point x of the density estimate and the sample, respectively, and L and U represent the lower and upper bounds of the scale. The higher the index, the better the fit. The intuition behind this fit measure is that a good fit would imply a cumulative distribution estimate which closely follows the actual distribution of the hold-out sample. As a result, the better the fit, the smaller the denominator, and the higher the GFI. This approach for comparing the fits was used since conventional fit measures such as the AIC or BIC would enforce penalties for fit deviations which would artificially favor non-parametric estimates such as the kernel. Therefore the appropriate way to establish fit superiority is to apply the results of an estimation sample to a hold-out sample and to compare the fits of the competing methods. As expected, the kernel density estimate produced the best fit to the hold-out sample data. The GFI for the kernel estimate was 18.52, compared to 11.41 and 3.03 for the normal and beta estimates, respectively, and 2.10 for the histogram (with an origin of 5 and bin width of 1).

Since the kernel density estimate is an unbiased and consistent estimator of the underlying distribution, one can reasonably conclude that the fluctuations observed in the probability density graph are likely to be due to the underlying distribution of the consumers’ satisfaction responses. Graphic inspection of Figure 3 identifies two consumer segments, one low-satisfaction segment with satisfaction ratings centered at 4.3, and the other high-satisfaction segment with ratings centered at 6.5, with the midpoint separating the two segments. To identify the differences between these segments, the varying satisfaction dynamics of the two segments were analyzed. A standard regression analysis was conducted by regressing the respondents’ overall satisfaction rating on the component level performance measures (i.e., physical facilities, student services, administration, and teaching quality), and is outlined in Table 2. Two separate regressions were run for each of the two segments. The standardized coefficients were then used to gain insights on the differences between the two segments. Table 2 presents the results of the regression analysis

As can be seen from Table 2, the two segments vary in their satisfaction dynamics. For the low-satisfaction segment, attributes such as teaching quality and the physical facilities are the strongest drivers of satisfaction, as reflected by the corresponding high t-values. On the other hand, student services and perceptions of the administration do not seem to have any significant influence on the satisfaction ratings of this segment. For the high satisfaction segment however, the effect of teaching quality on satisfaction is considerably lower than for the low-satisfacton segment. For this segment, the effect of the physical facilities is negligible, and in contrast to the low satisfaction segment, the perceptions of the administration are an important factor in their satisfaction responses.

The above findings have direct managerial implications. They suggest that in order to improve consumer satisfaction levels, there is a need for addressing the two segments differently. Each segment seems to be more sensitive to particular aspects of the service, and therefore the management can prioritize and focus quality improvement efforts, based on which segment is considered more critical to the school’s objectives. While addressing the low-satisfaction segment’s needs would require improvements to the physical facilities and improved teaching quality, the high-satisfaction segment would also benefit from improvements in the perceptions of the administration. Physical facility improvements would however not significantly improve the high-satisfaction segment’s ratings.


In this paper, we have offered kernel density estimation as a tool for facilitating a graphical analysis of ratings obtained from consumer survey data. Using an application in consumer satisfaction research, we have demonstrated that kernel density estimation enables one to better visualize and interpret the distribution of consumer responses. Several advantages are gained by using kernel estimation. For example while the traditional approach of building histograms is highly sensitive to one’s choice of the origin or the bin width, parametric density estimation methods tend to constrain the shape of the estimated distribution. Kernel density estimation on the other hand, not only provides stable estimates, but also allows the consumer response data to determine the shape of the estimated distribution. Moreover, kernel density estimation results in asymptotically unbiased and consistent estimates and is easy to program and implement. Many common statistical software now provide the capability of readily estimating kernel distributions on data, eliminating the need for any elaborate programming.

As mentioned earlier, certain limitations apply to the kernel method. The most important one relates to sample size, and the requirement that the sample consist of at least a couple of hundred observations. Smaller samples are likely to yield unreliable estimates, and in such cases one might benefit from the use of histograms instead (Donthu and Rust 1994). The graphical approach discussed also has a drawback in that the segments are split based on the mid-point separating the two segment centers on the response scale. While this is a convenient way of separating to independent distributions, future research could compare the proposed approach to latent class models which allow for a mixing of independent distributions. However, replication of the kernel approach on other consumer surveys has indicated the robustness of the method in identifying distinct consumer segments (Estelami and DeMaeyer 1997).



The application presented in this paper demonstrated the ability of the kernel method to identify two consumer segments, which were then contrasted in terms of their satisfaction dynamics. Use of such an approach is not only appealing from a managerial perspective where segment-based marketing programs can be initiated, but also from an academic perspective as it facilitates the study of consumer heterogeneity. While the presented application utilized consumer satisfaction data, the proposed methodology is applicable to most consumer survey data in which multi-item or continuous scales are utilized. For example, the technique can be applied to marketing scales designed to measure consumer involvement, emotions, or degree of information processing (Bearden, Netemeyer, and Mobley 1993). In such cases, the kernel density estimate provides the consumer researcher with an effective method for estimating the distribution of the construt of interest, and helps in gaining analytical insights by graphically inspecting the distribution of consumer responses.


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