# Measuring Cohort Role on Husband-Wife Differences in Temporal Behavior

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Jacob Hornik (1993) ,"Measuring Cohort Role on Husband-Wife Differences in Temporal Behavior", in NA - Advances in Consumer Research Volume 20, eds. Leigh McAlister and Michael L. Rothschild, Provo, UT : Association for Consumer Research, Pages: 527-533.

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http://acrwebsite.org/volumes/7502/volumes/v20/NA-20

Recent studies demonstrated inconsistencies in husbands and wives responses to various social issues. This paper presents a measurement procedure for comparing the responses of husbands and wives while also considering one of the groups cohort. The procedure was employed in a study concerning consumers' temporal orientation. The method is based on D^{2} distances. In addition to constructing the differences between the groups, the paper compares the statistical procedures for assessing significant differences among the measured distances. Formal statistical tests are employed, as well as graphical methods for presenting results. The suggested approach seems useful in a variety of consumer research areas over and beyond the area of husband-wife differences.

INTRODUCTION

The literature in the social sciences and particularly in consumer behavior contains a growing number of studies concerned with similarities and differences between husbands and their wives with respect to their attitudes, preferences and decision-making processes. [For a recent comprehensive review see Menasco and Currey (1989).] Notable examples are the studies on sex-role orientation and its influence on the household decision process (e.g. Qualls 1987); differences in product evaluation (Sung-Tai and Toner 1989); inconsistencies in contraceptive use responses and sexual behavior (Koening, Simmons and Misra 1984); differences in general life-style determinants (Roberts and Wortzel 1979); media usage (Hornik and Schlinger 1981); and, more recently, differences between husbands and wives in their perception of time (Umesh, Weeks and Golden 1988) their time use, time orientation and preferences (e.g. Hornik 1985).

These findings led many researchers to challenge the use of the household as a unit of analysis and to call for more methodological studies as well as measurement procedures to compare husbands and their wives on various behavioral domains (Blau and Ferber 1986).

The present paper was part of a more comprehensive project investigating individuals' time allocation, perception and orientation. The primary objective of this paper is to offer a general statistical framework based on Mahalanobis distances (Kshirsager and Arseren 1975; Morrison 1988) to compare husbands with their wives with regard to their time orientation, while considering cohort influences, too. This paper is, therefore, more concerned with the methodological than the comparison issue.

Time Orientation

The many facets of time as a concept central to household behavior have been much discussed by economists, psychologists, and consumer behavior researchers. These investigations are primarily concerned with three aspects of household time: time allocation/behavior, time perception, and time orientation. Time orientation (or perspective) refers to the relative dominance of past, present, or future in a person's thoughts and its perceived pace (Reichler and Brickman 1989). If people make daily decisions on the basis of their expectations concerning the present, and the future, their time perspective is obviously important. Individual differences in time orientation was used, for example, to explain decision differences in adopting new products (Lee and Ferber 1977) and in conducting negotiations (Qualls 1987). Recently Bergadaa (1990) found that those with present time orientations selected vacations with an emphasis on relaxation, while those with a future time orientation were more likely to select a vacation by considering the enrichment dimension. Complete individual orientation to and preference for the past, present or future do not necessarily reflect a stable inner state (Rakowski 1986). They are, rather personal judgments which, like other judgments, are subject to a variety of influences, including those of the family, age cohort and other social groups (Holahan and Gilbert 1979). The evolving status of women, fuelled by their increasing education and occupational attainment coupled with the feminist movement has been regarded in the literature as a major force underlying cohort influence on married women's temporal attitudes and preferences. Some investigators (e.g. Holahan and Gilbert 1979) have argued that recent social trends made wives closer to their cohort than to their husbands on some dimensions of their life style and time orientation. Therefore, reported temporal inconsistencies between individual members are suggested to be coused, primarily, by cohort influences, especially on wives. Indeed, recent studies beg the question of whether wives were closer to their husbands or their cohort with respect to their time orientation (e.g. Reichler and Brickman 1989). However, previous research on these issues has been hampered by a variety of methodological problems, including data limitations (Settle, Alreck and Glasheen 1977), lack of statistical rigor, and possible bias in parameter estimation (Feldman and Hornik 1981). Kiker and Ling Chu Ng (1990) as well as Reichler and Brickman (1989) called, independently, for the development of analytical methods to compare spouses on their temporal behavior while controlling for possible social influences.

METHODS

As noted before, the data for this study came from two recent experiments that were part of a larger study on consumers' intertemporal choices and preferences. A full description of data and procedure is provided in a companion paper (Hornik and Meir 1992). For brevity, we will not repeat all details of the data collection methodology employed. To review very briefly, participants were married females attending a child development session in a University extension program. Subjects were asked to respond to a two-page form containing a time-orientation task.

For this investigation a cohort group was obtained using Glenns (1976) "cohort table". Specifically, from the women who participated in the two experiments, in this study were included 397 married women who were born between 1953 and 1960. Also, a group of husbands (n=73), selected randomly from the interviewed women, was also included. This group was smaller than the women's group for practical reasons. Husbands followed the same procedure and completed the same questionnaire (without the presence of their wives). The comparison of each wife with her husband, regarding time perspective, was obviously performed on the data relating to the couples only. However, it should be emphasized that the larger data set, which was available for the entire group of wives, was also utilized in the study. This extra information was incorporated in the construction of the statistical distances.

Instrument: Of the time-orientation instruments suggested by researchers, the Time Reference Inventory (TRI) was used in the present study. The TRI was developed by Roos and Albers (1965) and further modified by others (e.g., Fitzpatrick, Donovan and Johnson 1980). The modified TRI is a 32-item paper and pencil instrument. Items refer to different life events and their perceived speed of movement. It is based on four subscales: future orientation (FO), present orientation (PO), past orientation (ST), and pace perception (PP). The TRI is intuitively appealing and scored. [An early analysis of the TRI instrument showed the following internal consistency reliabilities (Chronbach's alpha) for the four time-orientation subscales: past .85; present .91; future .87; and pace .84.]

ANALYTICAL FRAMEWORK

Consider any population, and let x_{i} denote the k-dimensional vector of the values of the random variable X for the i^{th} subject in the population. Let S be the covariance matrix of X. The Mahalanobis squared distance (D^{2}) between subjects i and j is defined as:

where x_{i}, x_{j }are the respective X values for subjects i and j. Unlike the Euclidean distance, D^{2} takes into account the correlations and variances of the components of X. In this study, the components of X are the time perspective items, and they take any integer value between 1 and 5. We need to define three distances: between the husband and his wife, between the wife and her cohort (other wives), and between the husband and his wife's cohort.

For constructing the distance between any two wives, there is no question of which S to use in D^{2} because both belong to the same population. However, for the distance between a husband and his wife we could use for S either the covariance matrix of the husbands, or that of the wives, or a pooled estimate. In the present case, it seemed more appropriate to use the wives' covariance matrix because we were interested in distances with respect to the wife population. Thus, all the Mahalanobis distances that were constructed were expressed in terms of the covariance matrix of the wife population. For estimating this matrix we took advantage of the large sample of wives (N=397).

Algebraic Formulation

Let E denote the estimated covariance matrix based on the N observations. E is usually a consistent estimate of E3. [For large N (as N=397), the estimated E can be considered as being practically equal to E.] Denote each paired observation of (wife, husband) by (x_{i}, x_{j}), i = l,...,n. The squared distance between the wife and her husband, denoted by D^{2}(x_{i}, y_{i}), is defined according to equation 1 by the following:

where E is the sample covariance matrix of X, based on the X values of the N wives.

We now define the distance between wife i(i = l,...,n) and her cohort, denoting it by D^{2}(x_{i}, X). Again, we take advantage of the N available observations. According to equation 1 the squared distance between any two wives is D^{2}(x_{i}, x_{j}), j = l,...,N; j _ i. Thus we can define as D^{2}(x_{i}, X) the average:

Similarly, we define the squared distance between the i^{th} husband and his wife's cohort as:

The question of whether wives are closer to their husbands or to their cohort is answered by comparing the paired data values of (D^{2}[x_{i}, y_{i}], D^{2}[x_{i}, X]). Our analysis on the wife-husband relationship is based on the three distances defined by equations 2-4.

Note that another distance should be considered - between an individual (husband or wife) and the wife population. As the squared distance -between wife i and her cohort we could consider D^{2}(x_{i}, ), namely, the squared distance of wife i from the centroid of the wives population. x is defined as the average:

Given that D^{2}(x_{i}, x_{j}) (and D^{2}[x_{i}, y_{i}]) are squared distances between individuals, that is, between two random observations, whereas D^{2}(x_{i}, ) is the squared distance between an individual and a mean, these distances should not be compared without a correcting factor. The expected value of D^{2}(x_{i}, x_{j}) is:

E (D^{2}[xi, xj]) = 2k

Therefore, the average D^{2}(x_{i}, X), defined by equation 3, also has an expected value of 2k. Similarly, under the null hypothesis that there is no correlation between x_{i} and y_{i} and that there is no difference between husbands and wives, y_{i} is a random observation from the X population and

E (D^{2}[y_{i, xi}]) = 2k [6]

On the other hand,

E (D^{2}[y_{i, }i]) = k [7]

Indeed, one may expect a much larger distance between two observations drawn at random from a population than between a random observation and the sample mean. Therefore, to answer whether wives are closer in their time perspective to their husbands or their cohort, we could have considered comparing the three statistics: 2D^{2}(x_{i}, x), 2D^{2}(y_{i}, x), and D^{2}(x_{i}, y_{i}). Our choice of comparing instead the three measures, D^{2}(x_{i}, X), D^{2}(y_{i}, X), and D^{2}(x_{i}, y_{i}), was arbitrary.

One should also note the relationship between the two possible measures under discussion: {2D^{2}(x_{i}, x)} and D^{2}(x_{i}, X). Their averages over the N wives are essentially equal because

This formula is analogous to the well-known relationship for univariate data: X_{i},...,X_{N} (Anderson 1958):

The Empirical Model

For each of the n paired observations (x_{i}, y_{i}), three distances were constructed. Our main purpose was to compare D(x_{i}, y_{i}) with D(x_{i}, X), though we can also learn about the husband-wife relationships by investigating D(y_{i}, X).

The first part of comparing D(x_{i}, y_{i}) with D(x_{i}, X) is a location comparison for paired observations. The Wilcoxon signed rank test is suitable for the comparison.

A graphical summary of the three distances can be performed by drawing a triangle whose edges are proportional to the three "typical" distances. For each of the subscales one can draw a triangle whose edges are proportional to the average of the square root of the Mahalanobis distances formally presented:

The triangle was drawn in this case such that its base was proportional to the first of the three averages in formula 10, namely,

Because

E(D^{2}[xi,X])=2k

the value of D should be approximately equal to [the square root of 2k]. Given that

2k = ED^{2} _ (ED)^{2}

one should expect

D<[the square root of 2k]

Indeed, as shown by the data obtained, the bases of the triangles were less than [the square root of 2x8 = the square root of 16]. Figure 1 displays the triangles of two subscales.

The size of the triangle gives a picture of the magnitude of the distances; its shape shows the relationships between these distances. The null hypothesis is that for each wife, the husband is neither closer nor further than any other individual from the population. Under this hypothesis the triangle should be equilateral. If husbands were closer to their wives, we would find that the edge corresponding to D(x_{i}, y_{i}) will be smaller than the others. A visual comparison can be made between the subscales on the basis of their triangles.

Graphical Display of Ranking

Another graphical representation of the distances is based on principles similar to the biplot (Gabriel 1971) and as presented by Cohen and Mallows (1980). The method enables one to display data matrices graphically by using their singular decomposition. For our particular problem, the data matrix will be of rank r = 2. [Cohen and Mallows (1980) discuss this particular case in detail but they also generalize the graphic method for data matrices with r>2.]

For each of the n pairs we rank the three respective distances: D(x_{i}, X), D(x_{i},y_{i}) and D(y_{i}, X). Thus if (1,2,3) is the ranking obtained for the eighth pair, then D(x_{i}, X) < D(x_{i}, y_{i}) < D(y_{i}, X). Under the null hypothesis, all 3! = 6 possible permutations are equally probable. The data of the rankings for the n pairs are summarized by an n x 3 matrix whose rows are permutations of the integers 1,2, and 3. Denote this matrix by P. The sum of each row in P is a constant, so that its rank is not larger than 2. Thus, it can be fully described in a two-dimensional space.

For displaying P in the plane, we consider a matrix Q of dimension 6 x 3 whose rows are the six possible permutations of (1,2,3). We express Q by the spectral decomposition:

where l_{i}(i=1,2) are eigenvalues of Q (and of Q'Q), U_{i}(i=1,2) are the respective normalized eigenvectors of QQ', and V_{i}(i=1,2,) are the respective normalized eigenvectors of Q'Q. The normalization is such that the length of each vector is 1. Let U_{ij} denote the i^{th }element of U_{j}(i=1,...,6; j=1,2) and let V_{ij} denote the eighth element of V_{j}(i=1,2,3; j=1,2). We plot the six vectors EQUATION as points in the plane, as well as the three vectors EQUATION. The six points corresponding to g_{i} form the vertices of a symmetric hexagon, while the three points corresponding to f_{i} form the vertices of an equilateral triangle. The graph provides an exact representation of Q: the (i,j)'th element of Q is the scalar product g_{i}'f_{i}.

A geometric model of rank correlation has been introduced by Schulman (1979). When k objects are ranked, his model yields an equilateral polyhedron with k! vertices. We consider here the particular case of k=3. It has been shown by Schulman (1979) that the distance between any two vertices in the hexagon obtained for k=3 is a function of the rank correlation between the permutations corresponding to the two vertices.

The technique of displaying data matrices by using the singular decomposition is the basis of various methods: principal components, biplot, correspondence analyses. In general, the matrix is of rank r > 2 and the singular decomposition consists of r terms. For the graphical display, the terms of the decomposition are ignored so that the display is only an approximation of the data matrix. For our problem, the n x 3 ranking matrix P is of rank r = 2, so that it can be displayed exactly. Each of the rows in P is one of the six possible permutations of (1,2,3). Because n = 73 (>> 6), several subgroups of rows in P must be equal. Actually, P consists of the same rows as in Q, except that the frequency of each permutation (row) in P is not necessarily 1. The important information for our study is the frequency (n_{i}, i=1,...,6) of each of the six permutations in P. This information can be displayed as follows. First, we draw the hexagon and triangle that describe the matrix Q. Recall that each permutation corresponds to a vertex of hexagon. Then, around each vertex (i=1,...,6) we draw a circle the area of which is proportional to n_{i}. Having the area proportional to the variable of interest seems to provide the proper visual impact (Cleveland et al. 1980). If all rankings were equally probable, we would obtain six circles with approximately equal areas. The graphical display shows which rankings are more frequent or, in other words, which distances tend to be smaller than others.

GRAPHICAL PRESENTATION OF GROUPS DIFFERENCES FOR TWO SUBSCALES

ANALYTICAL COMPARISONS

The purpose of this section is to compare our method, which is based on the distances, with classical procedures that could be applied for the comparison of wives and their husbands. First, we consider the method based on the Hotelling T^{2} test. As before we denote by x_{i}, y_{i}(i=l,...,n) the time perspective of the eighth wife and her husband, respectively. Let

We can test the null hypothesis of no differences in means between wives and their husbands by using the Hotelling T^{2} statistic (e.g., Morrison 1988):

T^{2}=n(d'S^{-1}d)

Under the null hypothesis,

The question that is answered by using T^{2} is whether wives differ from their husbands in their mean time perspective. However, this does not entirely answer the question raised by the present research, of whether wives are closer to their husbands or their cohort. The question as posed in the present research involves not only means. We shall illustrate this point in the following example. For simplicity, our example will be concerned with k=l. Suppose that for half of the sample the husband's time perspective is y=4 while his wife's perspective is x=2. For the other half of the sample x=4, y=2. In other words, there is a strong negative correlation between x and y. In this case, there will be no difference in means between husbands and wives, though wives are further removed from their husbands than from their cohort. Our method of comparing distances would indeed show the cases where wives are closer to their age-group than to their husbands with respect of some of the subscales.

The T^{2} test is based on the assumption that the data are multivariate normal whereas our method is nonparametric. Normally, this would imply that if the data were indeed normal and if wives differed from their husbands in their mean time perspective, then the T^{2} test should be more powerful than our test. It is not clear that for our particular study we could draw such a conclusion. Recall that our data have an unusual structure: there are values for n pairs of wives and their husbands, but there are also temporal reports for a large number of wives whose husbands were not interviewed. Unlike the T^{2} tests, our method takes advantage of this added information and increases the power of our test.

Other classical multivariate methods could be considered by rewriting the data in a MANOVA framework (Anterson 1958), but our above arguments would still apply.

ILLUSTRATIVE ANALYSIS

The following analysis consists of the correlation scores between wives and husbands on each scale item as well as the mean differences between them. Given that the intent of this paper was to present the measurement procedures, in this section we shall only present the results in a brief form and for only two subscales: Pace perception (PP), and present orientation (PO). These two subscales were selected because the results are different and illustrative.

GRAPHICAL DISPLAY OF THE RANKINGS FOR PACE-PERCEPTION

Pace perception and present orientation were each measured with k=8. In both subscales none of the husband-wife correlations were statistically significant. The overall F statistics for PP and PO were 5.12 and 3.23, respectively. Both were highly significant (P < .001). Thus the classical method shows that there are significant differences in the mean results, for the two subscales, between wives and their husbands. However, there are notable differences between the two subscales.

Comparing the two triangles in Figure 1 we can see the average distances of D(X_{i}, y_{i}), D(x_{i}, X), D(y_{i}, X). The PP triangle is not equilateral and its edges are larger than the base. This indicates that wives tend to be closer to their cohort than to their husbands. The husbands are almost close to their wives as they are to the entire age group of their wives. The Wilcoxon signed-rank test indicates (with a<.001) that the distance between the wife and her cohort is smaller than the distance between the wife and her husband.

The triangle for PO is also not equilateral, but it differs in shape from the one corresponding to PP. Here, the small edge corresponds to the distance between the husband and his wife. Also, the area of the triangle for PO is smaller than the area of the triangle corresponding to PP. This shows that the distances for PO tend to be smaller than for PP. The Wilcoxon signed rank test indicates that the distances between wives and their husbands tend to be smaller (a<.001) than between the wives and their age-group.

Figures 2 and 3 present the matrix ranking (P) for the subscales. The asymmetric structure indicates that the six permutations are not equally likely. For definiteness, in the center of each circle (corresponding to a permutation), we also provided their frequency (n_{i}) in P. The two largest circles in Figure 2 correspond to rankings for which the distances between the wife and her cohort were smaller than the other distances. The smallest circles correspond to permutations where the smallest distance among the three was between the husband and his wife's cohort. The ranking plot for PO (Figure 3) differs from the one shown for PP. In the case of the present-orientation subscale the large circles are placed where distances are smaller between the husband and wife.

SUMMARY

It is generally recognized that the "family" rather than the "individual" should be the basic unit of analysis in the study of individual time behavior and preferences. Studying family decision making and response is difficult, however, in as much as it entails the synthesis of the goals, preferences, and perceptions of all family members, who also belong to broader social groups. Important differences characterize family members, and important changes occur (especially in wives) as a result of their interaction with their work and age cohort.

The major intent of this paper was to introduce a statistical procedure for comparing husbands and wives, considering also the wives' cohort. The methodology was utilized in a study concerning individuals' temporal orientation. The data provided the opportunity to demonstrate the procedure's applications and significance. The findings are of both theoretical and practical value. Theoretically, they provide further empirical support to the notion that the information provided by different family members might be incongruent. Separate responses by husbands and wives should be obtained therefore, if one is attempting to investigate "family" temporal preferences. Furthermore, studying the time orientation of both spouses can be used to better understand their attitudes towards work and household production (Robinson 1987); shopping time preferences (Hornik 1985; 1992); the timing of childbirth (Happel, Hill and Low 1984); and to predict the future activities that will be performed jointly or separately by husbands and their wives.

GRAPHICAL DISPLAY OF THE RANKING FOR PRESENT ORIENTATION

The statistical procedure introduced in the present study can be easily applied to other husband-wife comparisons such as predicting spousal preferences (Davis, Hoch and Ragsdale 1986). It may also be used in other group comparisons, such as, investigating whether adolescents are more influenced by their parents or peers (Shah and Zalnik 1981). [Another important aspect to consider is the effect of measurement errors on the method. To obtain an Appendix on this issue please write to the author.]

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