As discussed in , the linear regression model is fit by the ordinary least squares (OLS) procedure. In OLS regression, the parameters are estimated so as to minimize the sum of squared errors of prediction. The error terms in regression can take on any values and are assumed to follow a normal distribution when conducting statistical tests. In contrast, in the binary logit model, each error can assume only two values. If Y ‘” 0, the error is p. and if Y = I, the error is I – p, Therefore, we would like to estimate the parameters in such a way that the estimated values of p would be close to 0 when Y = 0 and close to I when Y = L The procedure that is used to achieve this and estimate the parameters of the binary logit model is called the maximum likelihood method. This method is so called because it estimates the parameters so as to maximize the likelihood or probability of observing the actual data.
In multiple regression the model fit is measured by the square of the multiple correlation coefficient, R2, which is also called the coefficient of multiple determination In logistic regression (binary 10git~, commonly use~ r:neasures of model fit are based on the likelihood function and are CtJ ..•~ Snell R sguare and Nagelkerke R square. Both these measures are similar to R2 in multiple regression. The.~ox & Snell R square is constrained in such a way that it cannot equal 1.0, even if the model perfectly fits the data. This limitation is overcome by the Nagelkerke R square
Interpretation of the Coefficients and Validation
The interpretation of the coefficients or estimated parameters is similar to that in multiple regression, of course taking into account that the nature of the dependent variable is different. In logistic regression, the log odds, that is, 10g'(1 ~ p), is a linear function of the’estimated parameters. Thus, if Xi is increased by one unit, the log odds will increase by aj units, when the effect of other independent variables is held constant. Thus aj is the size of the increase in the
log odds of the dependent variable event when the corresponding independent variable I is· increased by one ‘unit and the effect of the other independent variables is held constant. The sign of aj will determine whether the probability increases (if the sign is positive) or decreases (if the sign is negative).
The validation process is very similar to that discussed for discriminant analysis. The analysis sample is used for estimating the model coefficients; the validation sample is used for developing the classification matrix. As before, the hit ratio is the percentage of cases correctly classified.
An Illustrative Application of Logistic Regression
We illustrate the logit model by analyzing the data of Table 18.6. This table gives the data for 30 respondents, 15 of whom are brand loyal (indicated by 1) and 15 of whom are not (indicated by 0). We also measure attitude toward the-brand (Brand), attitude toward the product category (Product), and attitude toward shopping (Shopping), all on a 1 (unfavorable) to 7 (favorable) scale. The objective is to estimate the probability of a consumer being brand loyal as a function of attitude toward the brand, the product category, and shopping
F~st we run an OLS regressiop on the data of Table 18.6 to illustrate the limitations of this procedure for analyzing binary data. The estimated equation is given by
Only the constant term and Brand are significant at the 0.05 level. It can be seen from the estiinated regression equation that the estimated values of p are negative for lo~ values of the independent variables (e.g., when Brand = I, Product = I, and Shopping = I, and for many other values of Brand = 1,2, or 3). Likewise, the estimated values of p are greater than I’ for high values of the independerit’variables (e.g., when Brand = 7,Product = 7,and Shopping = 7). This is intuitively and conceptually unappealing because p is a probability and must’lie between °and 1.
This limitation of OLS regression is overcome by logistic regression. The output for togistic regression when analyzing the data for Table 18.6 is shown in Table 18.7. The Cox & Snell R square and Nagellcerke R square measures indicate a reasonable fit of the model to the data. This is further verified by the classification table that reveals that 24 of the 30, that is, 80 percent of the cases: are correctly classified. The significance of the estimated coefficients is based on Wald’s statistic. We note that only attitude toward the brand is significant in explaining brand loyalty. Unlike discriminant analysis, logistic regression reSults in standard error estimates for the estimated coefficients and hence their signifacance can be assessed. The positive sign for the coefficient indicates that positive attitude toward the brand results in higher loyalty toward the brand. Attitude toward the product category and attitude toward shopping do not inflllence brand loyalty. Thus, a manager seeking to increase brand loyalty should focus on fostering more positive attitude toward the brand and not worry about attitude toward the product category or attitude toward shopping
The logit model can also be used when the dependent variable has more than two categories. In this case, the model is termed the multinomial logit. This procedure is discussed elsewhere by the author.
Boston Market: Sizing the Market
Richard Arras, president and CEO of Boston Market, is well aware of the fact that according 10 syndicated data, home meal replacement (HMR) will be the family dining business of this century. HMR is portable, high-quality food that’s meant for takeout, and it is the fastest-growing and most significant opportunity in the food industry today. According 10 Nielsen’s consumer panel data S5 percent of respondents purchased a meal for at-home consumption several limes a month. Convenience and type of food were the two most influential factors when purchasing HMR. Also, 77 percent of the respondents preferred their meals ready 10 eat.
Satisfactory Results of Satisfaction Programs in Europe
These days, more and more computer companies are emphasizing customer service programs rather than their erstwhile emphasis on computer features and capabilities. Hewlett-Packard learned this lesson while doing business in Europe. Research conducted on the European market revealed that there was a difference in emphasis on service requirements across age segments. Focus groups revealed that customtn above 40 years of age had a hard time with the technical aspects of the computer and greatly required the customer service programs. On the other hand, younger customers
appreciated the technical aspects of the product, which added to their satisfaction. Further research in the form of a large single cross-sectional survey was done to uncover the factors leading to differences in the two segments. A two-group discriminant analysis was conducted with satisfied and dissatisfied customers as the two groups and several independent variables such as technical information, ease of operation, variety and scope of customer service programs, and so on. Results confirmed the fact that the variety and scope of customer satisfaction programs was indeed a strong differentiating factor. This was a crucial finding because HP could better handle dissatisfied customers by focusing more on customer services than technical details. Consequently, HP successfully started three programs on customer satisfaction-customer feedback, customer satisfaction surveys, and total quality control. This effort resulted in increased customer satisfaction. After seeing the successful results of these programs in Europe, HP developed a goal to earn and keep customers’ satisfaction, trust, and loyalty and to enable them to apply technology to meet their business and personal needs. To achieve this goal, HP established and implemented a total customer experience and quality (TCE&Q) leadership framework in 200S. The details of this framework were documented in HP’s 200S Global Citizenship report. This framework was still in operation in 2000
Two-Group Discriminant Analysis
In the department store project two-group discriminant analysis was used to examine whether those respondents who were familiar with the stores,versus those who were unfamiliar,attached different relative importance to the eight factors of the choice criteria.The dependent variable was the two familiarity groups, and the independent variable swere the importance attached to the eight factors of the choice criteria.The overall discriminant function was significant,indicating significant differences between the two groups.The results indicated that, as compared to the unfamiliar respondents, the familiar respondents attached greater relative importance to quality of merchandise,return and adjustment policy service of store personnel,and credit and billing polici
Download the SPSS or SAS data file Sears Data 17 from the Website for this book. for a description of this file,
1. Recode preferencefor Sears into two groups: I to 4 = 1; 5 to 6 = 2. Can these two groups be explained in terms of the evaluationsof Sears on the eight factors of the choice criteria? Compare these results to the regressionresults in
2. Recode preferencefor Sears into three groups: I to 3 = 1;4 = 2; 5 to 6 =3. Can these three groups be explained in terms of the evaluations of Sears on the eight factors of the choice criteria? Compare these results to the regressionresults in Chapter