Durable Accumulation: an Examination of Priority Patterns

Robert F. Lusch, University of Oklahoma
Edward F. Stafford, Jr., University of Oklahoma
Jack J. Kasulis, University of Oklahoma
ABSTRACT - This paper examines the order of accumulation of discretionary durable goods by consumers. Guttman scalogram analysis was used to identify the underlying priority patterns. A multi-measure procedure was developed to evaluate the reliability and item homogeneity of the derived scale.
[ to cite ]:
Robert F. Lusch, Edward F. Stafford, Jr., and Jack J. Kasulis (1978) ,"Durable Accumulation: an Examination of Priority Patterns", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 119-125.

Advances in Consumer Research Volume 5, 1978      Pages 119-125

DURABLE ACCUMULATION: AN EXAMINATION OF PRIORITY PATTERNS

Robert F. Lusch, University of Oklahoma

Edward F. Stafford, Jr., University of Oklahoma

Jack J. Kasulis, University of Oklahoma

ABSTRACT -

This paper examines the order of accumulation of discretionary durable goods by consumers. Guttman scalogram analysis was used to identify the underlying priority patterns. A multi-measure procedure was developed to evaluate the reliability and item homogeneity of the derived scale.

INTRODUCTION

Over a person's life cycle he or she consumes many goods. Some of these are nondurables, and therefore the consumer derives satisfaction from them over a relatively short span of time. In addition, since these goods are non-durable, the consumer must purchase them with relative frequency. On the other hand, satisfaction from durable goods is derived over an extended period of time and as a consequence these goods are not purchased as frequently.

Most consumers cannot purchase all the nondurable and durable goods they would desire at any given time. This is especially true of durables since they generally involve a larger financial commitment. Therefore consumers will need to decide the order in which they will acquire durable goods. The order of acquisition of durable goods has received little attention by the various parties studying consumer behavior. Instead, the focus has been on how a single good is acquired and on how a consumer selects a brand from among a field of competing brands. Although these two types of investigations have considerable value in the study of consumer behavior, they will not be addressed here. Rather, the purpose of this paper is to explore the order of accumulation of discretionary durable goods by consumers. The order of acquisition of discretionary durable goods is an important area of inquiry since knowledge of acquisition patterns can: (1) aid managers in forecasting sales for consumer durables, (2) provide managers with a framework for directing marketing efforts at consumers at various points in the acquisition process, and (3) provide a framework for more theoretical areas of inquiry such as the adoption of innovations.

THEORETICAL BACKGROUND

There are at least three types of models that have been used to explain the acquisition of durable goods. These three models are: (1) growth models; (2) behavioral models; and (3) income consumption models.

Growth Models

Growth models attempt to predict the acquisition of new products by a population. The epidemic growth model which has been adopted from the biological sciences is one such example. This model, which was developed by biometricians to depict the spread of disease, basically postulates a logistics growth curve. When this model is adapted to the consumption of durable goods, the result is a model which can be used to predict the adoption of a new durable good by the population. Briefly, the model suggests that at first only a few people have contact or exposure to a new type of durable good. These few people interact with others and as a result the sales of the product spreads as does an infectious disease spreading through a closed population. This type of model has been validated for use in predicting the demand for durable goods by Derksen and Rombouts (1937), Roos and Von Szeliski (1939), and DeWoolf (1938). In addition, Bass (1969) has developed a growth model which is a variant on the classic epidemiological model. The model produced fairly good forecasts for the eleven appliance innovations that Bass tested the model with. Several other growth models are discussed by Kotler (1971, pp. 519-560). All of these growth models attempt to depict how the population as a whole adopts new durable goods. Furthermore these growth models depict how a population acquires a single durable good over time. No attention is given to how populations prioritize their acquisition of a set of durables over time.

Behavioral Models

Many constructs and models have been borrowed from sociology and psychology to help understand and predict the purchase patterns of consumers. A large number of these constructs have been combined into comprehensive models of consumer behavior such as the Howard-Sheth model (1969) and the Engel, Kollat and Blackwell model (1968). The major constructs that these models include are: attitudes, motives, personality, perceptual processes, learning processes, and social influences. A complete review of the consumer behavior literature on each of these is obviously unnecessary here. However it is important to point out the lack of evidence on the performance of these models in their entirety. Even more disheartening is the fact that the majority of the available evidence on parts of the models deal with nondurable goods. Consumer behavioralists in general have examined how consumers purchase beer, cigarettes, catsup and cereal, and have de-emphasized the study of how consumers purchase durable goods such as houses, air conditioners, and televisions. More important to the discussion at hand, no behavioral research has been directed at how consumers acquire a series of assets (i.e., what are the behavioral determinants of how consumers decide upon which asset to acquire first, second, third, and so forth).

Income Consumption Models

Perhaps the most formal and rigorous theory of consumer behavior available is the classical microeconomic theory of consumption behavior. This theory postulates that given a set of prices for a group of products, consumers will allocate their income to the purchase of these products so as to maximize utility. Importantly, microeconomic theory deals with perishable goods and does not address the problem of how individuals purchase durable goods, such as houses, in which utility is derived not only from consuming the product but from holding stocks of the product. Theil (1954) has suggested that classical economic theory could be reconstructed so that the individual consumer's utility function would depend both on the consumption during a period and stocks held at the end of it.

Derived from the classical economic theory of utility are income consumption curves which allow one to derive the income elasticity of demand. These curves, also frequently referred to as "Engel Curves," have led to the development of macroeconomic models that quantify the relationship between increases in income and the consumption of goods (durables and nondurables). Predictably, the decision to purchase a durable depends on other factors such as consumer stocks and prices; nonetheless, over the long run there is generally a fairly strong relationship between income and the demand for durable goods. Some examples of the specific relationships for several commodity groups are given in Shubin (1961, pp. 423-441) and Spencer (1968, pp. 146-157). The available evidence suggests that income-consumption models perform better over longer periods of time than for short periods of time and generally do not do well during cyclical periods. In addition, they do a fair job at predicting industry demand for durables but are inadequate for predicting market share within the industry. They do not address the issue of how consumers prioritize their wants for durable goods over time as their income increases.

ORDER OF ACQUISITION

Three types of models that could be used to explain the acquisition of durable goods have been briefly mentioned. Importantly none of these models has attempted to theorize how consumers go about developing a priority pattern for the acquisition of durable goods. Additionally none have attempted to investigate whether members of a population have similar priority patterns. Obviously, given limited income, consumers will not be able to acquire all the durables they desire at a given moment. Therefore it follows that an order of acquisition will need to exist which will by definition be a temporal concept.

Although classical economic theory does not deal with the consumption of durables, one could reasonably hypothesize that consumers acquire goods over time so as to maximize the present value of their utility function. Thus, given knowledge of future prices, consumer incomes and the utility function, one could theoretically determine the order of acquisition of consumer durables for an individual consumer. Assuming further that all consumers have similar utility structures, one could predict that all consumers would acquire goods in a stated order. For example if there were k durable goods then there would be k! possible patterns of acquisition. Nonetheless, given the above scenario one could predict the order of acquisition; possibly it could be: D1, D2, D3, ... Dk. Thus, first D1 would be acquired, then D2 and so on until Dk is acquired. Initially this assumes that not more than one of a given product could be acquired; but this need not be the case since additional units of a durable good could be treated as separate products. If we wanted to allow for the consumer to purchase up to four of any given durable then given k durables a consumer would have to prioritize 4k items.

This notion of the necessity for consumers to prioritize their acquisition of durable goods leads to the hypothesis that populations can be characterized as having a common order of acquisition for many types of consumer durable goods. This implies that consumers will have similar enough utility structures that in general they will acquire consumer durables in the same order.

Several researchers, using a variety of techniques, have examined whether consumers tend to have similar priority of acquisition patterns for sets of consumer durables (Paroush, 1965; McFall, 1969; Hebden and Pickering, 1974). Analysis techniques have included the Guttman coefficient of reproducibility (Paroush, 1965; McFall, 1969), the point correlation matrix (Paroush, 1965), and a matrix of conditional probabilities (Hebden and Pickering, 1974). Generally, it has been found that all three of these methods can be used to construct priority patterns for consumer durables. In addition priority patterns have been generated for different segments of a market (McFall, 1969; Hebden and Pickering, 1974). The results from this type of analysis suggests that priority or acquisition patterns vary by market segment, however, some overlap tends to exist among the patterns. Finally, priority patterns have been generated based on both intentions to purchase and cross sectional ownership data (McFall, 1969). These two types of analysis tended to generate different patterns of acquisition or priority.

The study at hand will use Guttman scaling, however, importantly multiple measures of scale quality will be computed. Thus although the research reported here is not totally novel, it is unique in that it subjects the Guttman scale to more stringent statistical criteria.

METHODOLOGY

The empirical analysis of this paper focuses on the household's acquisition of major discretionary kitchen durables. Included for study are washing machines, dryers, dishwashers, freezers, and microwave ovens. Refrigerators and ranges are the only two major kitchen durables excluded from the analysis. These two kitchen appliances are excluded because they are viewed as necessities for modern household operation, with ownership being inordinately dependent on home ownership. It should further be noted that apartment dwellers may not own washing machines and dryers due to community washers and dryers being provided in the apartment complex. However apartment dwellers were included in the analysis since they were relatively few in number. Furthermore the inclusion of apartment dwellers could not possibly strengthen the scale quality since the presence of community washers and dryers in apartments would induce error into the scale. Thus our procedure is conservative (i.e., by excluding apartment dwellers our measures of scale quality would improve).

Subjects

The data for analysis was obtained from the DRP/OPUBCO Continuing Consumer Audit. The Distribution Research Program (DRP) at the University of Oklahoma and the Oklahoma Publishing Company (OPUBCO) collaborate in the collection of data on the purchasing behavior of individuals in the Oklahoma City SMSA. [The authors wish to thank Phil Stout and Jim Williams of the Oklahoma Publishing Company for their continued cooperation in the collection and use of the Continuing Consumer Audit data base.] Reported in this paper are some findings of the 1975 Audit which includes 1854 respondents from a stratified random sample. The sample is representative of the populations in geographic regions in the OKC SMSA which includes subjects from urban, suburban, and rural areas. It includes all types of dwelling units--houses, apartments, condominiums, trailers, etc. New samples are drawn each year with the distribution of the sample reflecting population changes in the strata.

Procedure

The Consumer Audit questionnaire is administered in a personal interview with both the male and female heads of the household responding. Each visit is a "cold" call with no pre-visit contact made with the potential respondent to request cooperation; i.e., the first contact is when the doorbell is rung. Not-at-home families are revisited at different hours of the day four times before a substitute respondent is designated for the interview. The data is collected continuously throughout the year by professional interviewers under close supervision. Each week completed questionnaires are returned for processing. Telephone callbacks are made within three days of the return to verify the data collection. In addition, subjects may be telephoned again to obtain clarification of responses if need be.

Analysis

The stock of durables that a household possesses can be characterized by a multivariate distribution of 0's and l's. A value of "one" would depict possession of the durable. A value of "zero" would represent non-ownership. If it is true, as hypothesized, that consumers have relatively similar utility structures for discretionary kitchen durables, the data on durable consumption behavior can be described in terms of a unidimensional "scaling" model.

The approach used in this paper is similar to that of Paroush (1965). In dealing with five household durables, a logically consistent conceptual model would be theoretically characterized by the pattern exhibited in Table 1. This indicates the order of acquisition to be D1, D2, D3, D4, D5. If it is observed that each consumer fits into any one of these patterns (rows), then we could transform the multivariate data into a unidimensional scale. Thus, by only knowing the last durable acquired, one can perfectly predict a consumer's total stock of durables. In other words, if the last durable added to one's stock of durables is D4, then we would know that this consumer also possesses D1, D2, D3 and not D5.

TABLE 1

THEORETICALLY PERFECT PATTERNS OF DURABLE OWNERSHIP

Now that the theoretically pure or ideal situation has been developed, it should be clear that not all consumers will acquire a set of durables in the same pattern. For whatever the reasons, the perfect durable acquisition scale will not be able to characterize all people since deviants will inevitably exist. The task is to determine whether divergence from the ideal model is a function of relatively unimportant aberrations or whether the divergence is sufficiently large for the perfect model to be considered unrealistic for the real world. Thus, some measure of scalability is needed to empirically test the appropriateness of the model.

Guttman (1971) developed a scaling model called scalogram analysis which may be applied to this task even though it was originally devised for a different purpose (attitude measurement). In the discussion of attitudinal dispositions, and the role of scalogram analysis, Guttman has stated that "...the universe is said to be scalable for the population if it is possible to rank the people from high to low in such a fashion that from a person's rank alone we can reproduce his response to each of the items in a simple fashion." (Guttman, 1971, p. 188).

From the Guttman perspective, an attitude scale should possess two properties: (1) unidimensionality and (2) cumulativeness. A unidimensional scale is one in which the component items measure movement toward or away from a single underlying object. A cumulative scale is one in which the components can be ordered by degree of difficulty; i.e., if a respondent replies positively to a more difficult component of the scale, then he would always respond positively to each of the less difficult scale components. For example, consider the following two simple attitudinal statements: (1) I would use Brand X coffee if it were given to me as a sample. (2) I would buy Brand X coffee in the supermarket if I could. A positive response to both statements would indicate a stronger disposition toward Brand X than a positive response to only one of the statements (unidimensionality). A positive response to Statement 2 should always result in a positive response to Statement 1, but not necessarily vice versa (cumulativeness).

Guttman scaling has been traditionally used with cross sectional data in order to rank an individual's attitude toward an object. In the case at hand, it is desired to use Guttman scaling to model the temporal phenomenon of consumers acquiring discretionary durable goods. Although the data used in this study is cross sectional, it is possible to scale the underlying temporal phenomenon Since the sample represents a true cross section of the entire population of the Oklahoma City SMSA, individuals at all stages in the order of acquisition process will be present in the sample. Thus, because this cross section of individuals is at various stages in the acquisition process, conclusions can be drawn about the order of acquisition over time.

From the above discussion, the appropriateness of scalogram analysis in examining our hypothesis is evident. The thesis of this paper is that the five discretionary kitchen durables mentioned earlier tend to be acquired in a designated priority pattern with the "more difficult" durables being acquired only after the "less difficult" appliances. In this context, a lesser degree of difficulty is synonymous with higher levels of expected utility derived from the ownership of the appliances. Among the various scaling techniques available, the Guttman approach is almost unique in its possession of this cumulative property (Nie, et al., p. 529).

The various techniques used to analyze the data collected for this study are detailed in the Appendix to this paper. These techniques include the original measures suggested by Guttman (1971); the Kuder-Richardson Equation 20 (Kuder and Richardson, 1937); the approach of Loevinger (1948); the extensions of the Kuder-Richardson and Loevinger works, suggested by Horst (1953); and the unique approach suggested by Green (1956). (A Technical Appendix, detailing the calculations performed using these techniques in order to derive the statistics reported in Table 3, is available on request from the authors.) The results of applying these techniques are presented in the next section of this paper.

RESULTS AND ANALYSIS

As mentioned earlier, for the five discretionary kitchen durables, there are 32 ownership situations, yet only one pattern of acquisition represents the perfect scale. (There are k + l = 6 ownership situations represented by this perfect scale.) To examine the extent of deviation from the perfect scale pattern, a variety of techniques (described in the Appendix) were used. A statistical summary of the data used to generate the Guttman scalogram, and to generate the scale reliability measures associated with these analytical techniques, is presented in Table 2.

A brief interpretation of Table 2 is in order. The scale rankings (row labels) are determined, for each respondent, according to the Guttman "perfect scale" technique. That is, a respondent receives a scale score equal to the number of positive responses (1's) he gives to the items on the scale. For this study, a ranking of 3 indicates that the respondent possessed three of the five kitchen durables. Each column of Table 2 actually consists of two columns of numbers; the left-hand sub-column represents the number of respondents who did not possess the item represented by the overall column, for a given scale ranking, while the right-hand sub-column represents the number of respondents who did possess that item, for the given ranking. For example, of the 344 respondents who received a scale ranking of 4, two did not possess a dishwasher, while 342 did. Note that the sum of the sub-column entries, for any column, equals the total number of respondents in this study. Further, the sum of the sub-column elements for any row equals the number of respondents receiving that row's scale ranking.

TABLE 2

STATISTICAL SUMMARY OF THE GUTTMAN SCALE ANALYSIS FOR CONSUMER ACQUISITIONS OF DISCRETIONARY KITCHEN DURABLES

The "easiest" items (most possessed) start at the right of Table 2, and progress to the "most difficult" items (least possessed) at the left of this table. Thus the "ideal" order of acquisition of kitchen durables would be: washer, dryer, dishwasher, freezer, and finally microwave oven. The values in the box of each sub-column of Table 2 represent the number of respondents that deviated from the perfect scale, for the item represented by the overall column. For example, of the 181 respondents scaled as a '1', 60 did not own the first item in the scale, a washer. Of these 60, five owned a dryer as their single durable; 18, a dishwasher, and 37, a freezer. These item errors occurred either by respondents possessing an item of greater difficulty than their scale value suggests, or by not possessing an item that would normally be owned, given their assigned scale value. The various row and column totals from Table 2 are used to generate the reliability measures, presented in Table 3, and discussed below.

Examination of the performance measure values presented in Table 3 provides strong support for the hypothesis of this paper that consumers tend to acquire discretionary kitchen durables in a pattern that can be detected through scalogram analysis. The Guttman measure of scale reproducibility exceeds 0.85, the minimum level suggested by Guttman for claiming a valid scale. Further, Green's index of consistency substantially exceeds his suggested minimum value of 0.5; and the coefficient of scalability is well above 0.6. These results suggest that the proposed scale, based on the hypothesis stated earlier, is both reproducible and consistent.

TABLE 3

RELIABILITY MEASURES OF THE DISCRETIONARY KITCHEN DURABLES SCALE

Drawing conclusions from the Loevinger Index, the Green Index, or the corrected Kuder-Richardson Equation 20 measure is considerably more difficult. To date no suggestions of reasonable minimum levels for any of these performance measures has been found anywhere in the literature. Despite this apparent lack of historical/theoretical guidance, the procedure described below for combining the Guttman, Green, Loevinger, and Horst measures for assessing the validity and reliability of scales developed from a set of dichotomous variables has been chosen. Future research efforts need to identify reasonable bounds of the Loevinger, Green, and Horst measures.

First since both the Guttman and the Green Reproducibility Coefficients measure the same phenomenon, both should approach, if not exceed 0.85. Second, the Coefficient of Scalability should exceed 0.6, and/or Green's Index of Consistency should exceed 0.5. Further because the Loevinger measure and the Green Index of Consistency both attempt to evaluate the homogeneity of a scale, the Loevinger Index should also exceed 0.5 as a minimum. The value of the Horst measure, then, will also exceed 0.5 because it is always a multiple ($ 1.0) of Loevinger's Index. If all of these conditions are met, it is concluded that a valid, reliable scale has been identified.

For this current study, all sample values of the various reliability and homogeneity measures exceeded the stated minimums. Hence it was concluded that a reliable and valid scale has been identified for the acquisition of discretionary kitchen durables.

To further test the internal validity of the scaling procedures reported above, the sample of 1854 respondents was split into two halves. The respondents were split according to the odd or even last digit of their sequential identification number to provide two subsamples. Since the completed questionnaires, gathered over many months of 1975, were numbered as they were collected from the interviewers, neither split-half sample was biased by seasonal factors. The reliability statistics calculated for each split-half are reported in Table 3. Inspection of these statistics strongly suggests that the sampling procedures used in this study were indeed random, unbiased and internally valid.

For the present, Paroush's notion of examining an item correlation matrix for simplex structure has been excluded from this scale evaluation procedure. This was done because it is not clear just which of the many available procedures for calculating correlations should be used. The method used by Paroush provides a significantly different matrix than the Yule's procedure employed in SPSS; nonetheless, each method produced a matrix that had a nearly perfect simplex structure. The Paroush version of the point-correlation matrix for the items examined in this study is presented in Table 4. The only flaws in this matrix are both associated with the item "freezer." This same phenomenon also appears in the data shown in Table 1.

TABLE 4

POINT-CORRELATION MATRIX FOR THE DISCRETIONARY KITCHEN DURABLES SCALE ITEMS

DISCUSSION AND CONCLUSIONS

This paper expanded earlier work on an intuitively appealing application of Guttman scaling. Is the application of any value; and if this application has value, is its value theoretical, managerial or both? We believe that the ability to characterize the population's multivariate acquisition of goods by a single scale offers both immediate managerial and theoretical possibilities.

Managerially, it can be said that the ability to scale the order of acquisition of a set of durables allows managers a convenient way to estimate market potential, or forecast sales. For example, the potential purchasers of microwave ovens are those consumers that have already acquired a washer, dryer, dishwasher, and freezer. Those consumers that have only purchased a washer are not presently potential customers for microwaves until they first purchase a dryer, dishwasher and freezer. Obviously, therefore, the market potential changes as the population progresses through the order of acquisition process. In regard to forecasting sales, Brown, Buck and Pyatt (1965) have developed a technique utilizing priority patterns that can be used to improve the sales forecasts for consumer durables.

Second, it would be possible for managers to use the order of acquisition phenomenon to direct marketing efforts. In short, if microwaves were being sold, then marketing dollars should not be wasted on consumers who have only purchased a washing machine. In this case, direct mail could be focused on recent freezer purchasers, inasmuch as their next most likely new discretionary durable would be a microwave oven.

Turning to the more theoretical applications, a researcher could use the order of acquisition scale to investigate its potential for studying the adoption of innovations. Perhaps a person's scale score could be a determinant of him adopting a new innovation. For example if one examined those consumers that have adopted trash compactors, one would possibly find that the adopters were the individuals that had a high scale score (i.e., had already acquired all other discretionary kitchen durables).

Also on a theoretical plane, one could investigate the concurrent and/or predictive validity of the scale. The scale could be investigated to see how it correlates with, and its ability to predict, other behavioral constructs.

Finally, theoretical research could be directed at techniques for minimizing the error in the scale. This may be possible by segmenting the population to see if order of acquisition varies by segments. For example, individuals entering adulthood may acquire durables in several different orders depending on such things as marital status and social class. By constructing separate Guttman scales for each segment more perfect scales should result.

In summary, it can be concluded that the study of the order of acquisition of durable goods offers a promising area to all the respective parties interested in the study of consumer behavior. Coupled with this, it appears as though the Guttman scale can be a fruitful technique for the study of the order of acquisition of durables.

APPENDIX

This Appendix presents a discussion of the various techniques available for analyzing the data gathered in conjunction with the identification of a unidimensional scale. Except for the Guttman approach, which is presented first, these techniques are presented in the order in which they appeared in the literature.

Guttman

The most frequent analytical technique used in scalogram analysis and the most common one operationalized in the more popular computer statistical packages is the Coefficient of Reproducibility (CR). Related measures are the Minimum Marginal Reproducibility (CMMR), Percent Improvement (%I), and Coefficient of Scalability (CS). Guttman's CR indicates how well the data approximates a perfect scale by examining the extent to which positive replies to "more difficult" scale items are associated with the positive replies to those which are "less difficult''. Mathematically, this statistic is as follows:

EQUATION  (1)

Errors are said to occur when a positive response is given to a "more difficult" item and not to a "less difficult'' one. As a rule of thumb, Guttman assumes a valid scale if CR exceeds 0.85 (Guttman, 1971, p. 176).

Subsequent research noted that CR is biased upward to the extent that the dichotomized items have extreme distributions. The CMMR measure was therefore devised to indicate the minimum coefficient of reproducibility that could occur for a scale given the proportion of respondents replying positively to each of the items. It is calculated as follows:

EQUATION  (2)

It should be noted that CMMR is in itself, a meaningless statistic. Its value comes from an examination of its relationship to CR. This relationship is examined through the %I statistic. It indicates the extent to which CR is due to response patterns rather than to the underlying associations of the scale items. In this regard, the percent improvement is akin to the error or measurement variance of variance analyses techniques. Mathematically it is as follows, with the larger the value of %I, the stronger the indication of a valid scale.

EQUATION  (3)

The final Guttman statistical ratio is CS. It is calculated as follows:

EQUATION  (4)

The denominator of this formula indicates the largest amount of percent improvement which can be obtained and the numerator represents the actual amount obtained. Therefore CS is a measure of how close the scale has come towards achieving the maximum improvement possible. Values above 0.6 are considered necessary for scale validity (Nie, et al., 1975, p. 533).

Kuder-Richardson

Several years prior to Guttman's original reports on his scalogram procedures, Kuder and Richardson (1937) described a method of estimating the reliability of tests and scales. They described several special cases, one of which has proven useful for testing for this current study. As with their other cases, the Kuder-Richardson Equation 20 (KR20) is based on the variances of the individual items of a test or scale, and on the total variance of the entire scale. This equation was given as:

EQUATION  (5)

where:

k = the number of items in the scale

VI = the sum of the variances associated with each individual item of the scale

VT = the total variance associated with the entire scale

Loevinger

Subsequent research has noted two defects in the KR20 approach. Loevinger (1948) argued that scale reliability formulas, like KR20, estimate item homogeneity as well as scale or test reliability. Furthermore, Loevinger (1947) pointed out that KR20 has an upper limit of 1.0 only when all items of the scale were of equal difficulty. As a consequence, Loevinger developed an alternate measure of test reliability called the Coefficient of Homogeneity (LI). LI produces identical results as KR20 when all items are of equal difficulty, but corrects for situations where there is a dispersion of item difficulty. A computationally useable formula for LI is as follows (Horst, 1953):

EQUATION  (6)

where:

VM = maximum possible variance given the distribution of item difficulty

VT,VI = same as for KR20

Horst

Later discussion by Horst (1953) demonstrated that LI really estimated the average item correlation for a test or scale, corrected for dispersion of item difficulty. Additional sophistication by Horst resulted in a better measure of reliability in that it incorporated total scale reliability. His Corrected Kuder-Richardson Equation 20 (KR*20) is calculated as follows:

EQUATION  (7)

Green

A different approach to estimating the reliability of a scale was introduced by Green (1956). It depends not on the calculations of item and test variances, but rather on the summary statistics of the sampling results obtained when a scale is designated. A researcher first estimates a scale for the items of interest, ranking them in order of ascending difficulty with item 1 being the least difficult. Next, the number of positive and negative responses are identified for each item. Although there are k-1 orders of possible error responses for a scale of k items, Green argued that, at most, just the first two orders contributed significantly to the reliability coefficients of a scale. A first-order error would be one where a respondent answered positively on an item of greater difficulty, say item g+1, while he responded unfavorably on item g. The index g is calculated over the ranked set of items. A count of all such first-order errors for the gth item would be denoted as ng+1, g,, meaning this many respondents possessed the g+1st item, but not the gth item. Extension of this discussion and notation to second-order error counts is obvious.

Green presented two alternative estimates of the reliability of a scale which he called Rep" and RepB. He showed the equivalence of these measures with the assumption that item error counts are independent statistics. Because it is computationally much less difficult, the RepB estimate has become more popular among social scientists. The Green-B Index of Reproducibility (RGB) is as follows:

EQUATION  (8)

where:

N = the number of respondents to the test (scale)

k = the number of items in the test (scale)

n = the number of errors for the particular item-order interaction

g = the item index across the set of ranked items for the scale

Although not explicitly stating so, Green implied that this index should be tested as though it were the original Guttman CR. Thus, a value of RGB in excess of 0.85 suggests that a valid scale has been identified.

Green also introduced the Index of Consistency (I) to serve as a measure of the homogeneity of the developed scale. This index is calculated as follows:

EQUATION  (9)

where:

RGB  = Green B Index of Reproducibility (defined above)

RI = Expected coefficient of Reproducibility (defined below)

RI is the scale reproducibility that is expected by chance if a set of items possessed their observed popularities, and they are also mutually independent. Thus, this index indicates the extent to which an estimated RGB is representative of a valid scale for the observed items. A negative value for I indicates a degree of negative correlation among some or all of the items in the scale. The equation for RI is as follows:

EQUATION  (10)

where:

N, k, n, g = same as for RGB

Paroush

In addition to the approaches just discussed, another method for judging the validity of a scale has been suggested by Paroush (1965). In this approach, the scale items are also ordered by their ranked popularity. A point-correlation matrix for each pair of ranked items is then calculated. If this matrix has a simplex structure, the hypothesized scale is accepted. A simplex structure is one where the values of the matrix entries decrease as one moves away from the matrix diagonal, either horizontally or vertically.

SUMMARY

Discussed have been several measures of scale reliability or reproducibility found in the literature of the social sciences. These measures vary not only in how they are constructed, but also in their ability to deal with scales where items are believed to differ in degree of difficulty. It is not the intent of this Appendix to discuss in detail the relative merits of these measures; rather, each of them has been calculated for this current study as a measure of scalability of discretionary kitchen durables.

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