The interpretation of the results is aided by an examination of the standardized discriminant function coefficients, the structure correlations, and certain plots. The standardized coefficients indicate a large coefficient for income on function I, whereas function 2 has relatively larger coefficients for travel, vacation, and age. A similar conclusion is reached by an examination of the structure matrix To help interpret the functions, variables with large coefficients for a particular function are grouped together. These groupings are shown with asterisks. Thus, income and household size have asterisks for function I because these variables have coefficients that are larger for function I than for function 2. These variables are associated primarily with function 1. On the other hand, travel, vacation, and age are predominantly associated with function 2, as indicated by the asterisks.
Figure 18.3 is a scatter gram plot of all the groups on function I and function 2. It can be seen that group 3 has the highest value on function I, and group I the lowest. Because function”! is
primarily associated with income and household size, one would expect the three groups to be ordered on these two variables. Those with higher incomes and higher household size are likely to spend large amounts of money on vacations. Conversely, those with low incomes and smaller household size are likely to spend small amounts on vacations. This interpretation is further strengthened by an examination of group means on income and household size
A similar interpretation is obtained by examining a territorial map, as shown in Figure 18.4. In a territorial map, each group centroid is indicated by an asterisk. The group boundaries are shown by numbers corresponding to the groups. Thus, group I centroid is bounded by Is, group 2 centroid by 2s, and group 3 centroid by 3s
The Home Is Where the Patient’s Heart Is
As of 2009. the largest industry sector in the u.s. economy was the health services industry. Through 2015. it is expected that spending on health care services will grow significantly faster than the economy. Contributing to the positive outlook for this industry are the current demographics. especially with demand for long-term care increasing as the population ages. It is expected that the number of Americans who are 8S and older will increase greatly by 2020. and with such a large increase, it is crucial that the health care system be portrayed positively to this segment of the population. Consumers were surveyed to determine their attitudes toward four systems of health care delivery (home health care, hospitals. nuning homes, and outpatient clinics) along 10 attributes. A total of 102 responses were obtained. and the results were analyzed using multiple discriminant analysis (Table I). Three discriminant functions were identified. Chi-square tests performed on the results indicated that all three discriminant functions were significant at the 0,0 level. The flnt function accounted for 63 percent of the total discriminating power. and the remaining two functions contributed 29.4 percent and 7.6 percent. respectively
The four group centro ids are shown in Table 2. This table shows that home health care was evaluated most favorably along the dimension of personalized care, and hospitals least favorably. Along the dimension
of quality of medical cap:, there was 8 substantial separation between nursing homes and tile other three systems. Also. home health care received higher evaluations on the quality of medical care than did outpatient clinics. Outpatient clinics, on the other hand, were judged to offer the best value
Classification analysis of the 102 responses, reported in Table 3, showed correct classifications from 86 percent for bospitals to 68 percent for outpatient clinics. The mis classifications for hospitals 6 percent each to nursing homes and outpatient clinics, and 2 percent to home health care. Nursing showed mis classifications of 9 percent to hospitals, IO percent to outpatient clinics, and 3 percent to honie health care. For outpatient clinics, 9 percent misclassifications were made to hospitals, 13 ~t to nursing homes, and IO percent to home health care. For home health care, the misclassifications were 5 percent to hospitals, 4 percent to nuning homes, and t3 percent to outpatient clinics. The results demonstrated that the discriminant functions were fairly accurate i. pledietins sroup membenhip
Stepwise Discriminant Analysis
Stepwise discriminant analysis is analogous to stepwise multiple regression in that the predictors are entered sequentially based on their ability to discriminate between the groups. An F ratio is calculated for each predictor by conducting a univariate analysis of variance in which the groups are treated as the categorical variable and the predictor as the criterion variable. The predictor with the highest F ratio is the first to be selected for inclusion in the discriminant function, if it meets certain significance and tolerance criteria. A second predictor is added based on the highest adjusted or partial F ratio, taking into account the predictor already selected
The selection of the stepwise procedure is based on the optimizing criterion adopted. The Mahalanobis procedure is based on maximizinl a generalized measure of the distance between the two closest groups. This procedure allows marketing researchers to make maximal use of the available information
The Mahalanobis method was used to conduct a two-group stepwise discriminant analysis on the data pertaining to the visit variable in Tables 18.2 and 18.3. The first predictor variable to be selected was income, followed by household size and then vacation. The order in which the variables were selected also indicates their importance in discriminating between the groups.
This was further corroborated by an examination of the standardized discriminant function coefficients and the structure correlation coefficients. Note that the findings of the stepwise analysis agree with the conclusions reported earlier by the direct method
The Logit Model
When the dependent variable.is binary and there are several independent variables that are metric, in addition to two-group discriminant analysis one can also use ordinary least squares (OLS) regression, the logit, and the probit models for estimation. The data preparation for running OLS regression, logit, and probit models is similar in that the dependent variable is coded as 0 or I. OLS regression was discussed in The probit model is less commonly used and will not be discussed, but we give an explanation of the binary logit model.
Conducting Binary Logit Analysis
The steps involved in conducting binary logit analysis are given in Figure
Formulate the Problem
As discussed earlier under the basic concept of discriminant analysis, there are several instances in marketing where we want to explain a binary dependent variable in terms of metric independent
variables. (Note that logit analysis can also handle categorical independent variables when these are recoded using dummy variables, as discussed in Discriminant analysis deals with the issue of which group an observation is likely to belong to, On the other hand, the binary logit model commonly deals with the issue of how likely an observation is to belong to each group. It estimates the probability of an observation belonging to a particular group. Thus, the logit model falls somewhere between regression and dis¢rninant analysis in application. We can estimate the probability of a binary event taking place using the binary logit model, also called logistic regression. Consider an event that has two outcomes: success and failure. The probability of success may be modeled using the logit model as
p = probability of success
Xj = independent variable i
Qj = parameter to be estimated
It can be seen from the third equation that although Xi may vary from -00 to +00, p is constrained to lie between 0 and I. When Xi approaches -00, p approaches 0, and when Xi approaches +00, p approaches I. This is desirable because p is a probability and must lie between 0 and 1. On the other hand, whenOl.S regression is used the estimation model is
Thus, when OLS regression is used, p is not constrained to lie between 0 and I; it is possible to obtain estimated values of p that are less than 0 or greater than \1. These values are, of course, conceptually and intuitively unappealing: We demonstrate this phenomenon in our illustrative application. As in the case of discriminant analysis, the researcher should specify the objectives and clearly identify the binary criterion variables and the independent variables that will be considered in the analysis. Moreover, the sample may have to be divided into the analysis and validation subsamples.