MULTIVARIATE ANALYSIS CHAPTER 18 MULTIVARIATE ANALYSIS

What the Experts Say …I’ve never considered myself a ‘quant jock,’ for reasons probably due to genetic ancestry. I find it difficult to get excited about the inner workings of optimization algorithms or exploring the sensitivity of [multivariate] ANOVA to violations in the independence assumption, for example. Rather, my interest in the various multivariate tools arises from their usefulness as a means for examining phenomena that do interest me. --Rob Kleine, A Seminar in Multivariate Statistics, http://www.gentleye.com/research/ multivar/index.html, September 23, 2000.

Learning Objectives Discuss the basics of multivariate statistical analyses Explain which technique is appropriate given the type of variables involved Describe the usefulness of multivariate statistics

Get This! Fore! Golfers Benefit from Conjoint Analysis Every golfer has two things in common. They’re all looking to drive the ball farther and to hit it with more accuracy. Sawtooth Technologies, a well-known company providing software for research data collection and analysis, uses conjoint analysis to examine the extent to which average driving distance, average ball life, and price are concerns of golfers when selecting their golf balls.

Get This! Fore! Golfers Benefit from Conjoint Analysis – cont’d Conjoint analysis encompasses three critical steps: Collecting trade-offs Estimating buyer value systems Making choice predictions The trade-offs might deal with paying a little extra for a ball that travels farther. A golfer might value a long drive more than a highly durable ball. And a choice prediction might be that a golfer prefers the long-life ball over the distance ball since it has the larger total value. All of these findings would be based on computations from conjoint analysis.

Now Ask Yourself Does conjoint analysis make intuitive sense to you? If so, why is it needed? What other multivariate techniques are available to researchers? Do I need to be a statistical expert to understand multivariate statistical analysis?

Multivariate Statistical Analysis Multivariate Statistics: Investigates more than two variables at a time. Many times, multivariate techniques are a means of performing in one analysis what used to take multiple analyses using univariate techniques. The techniques can be used to summarize data and reduce the number of variables necessary to describe the data. Several of the more common multivariate techniques: Multiple Regression Analysis Multiple Discriminate Analysis Factor Analysis Cluster Analysis Multidimensional Scaling Conjoint Analysis

Multiple Regression Analysis The premise behind multiple regression analysis is consistent with that of simple regression analysis: to determine the association or relationship between dependent and independent variables. In multiple regression analysis, more than two variables are included in examinations. The dependent and independent variables must be interval-scaled to use this technique. The general form of the multiple regression model is as follows: where = Y intercept of the regression model = slope of the regression model

Multiple Regression Analysis – cont’d or the computed multiple regression model is where computed value of the dependent variable a = y intercept when x equals zero and partial regression coefficients independent variables Partial Regression Coefficient: Denotes the change in the computed value, , per one unit change in when all other independent variables are held constant.

Multiple Regression Analysis – cont’d The association between the dependent and independent variables is referred to as the coefficient of multiple determination, denoted by . It is interpreted in a similar manner as we did when we referred to bivariate data. The coefficient of multiple determination is computed as follows: where TSS = total sum of squares = RSS = regression sum of squares = ESS = error sum of squares =

Multiple Discriminant Analysis (MDA) Multiple Discriminant Analysis (MDA): The appropriate tool for predicting the membership of observations in two or more groups. Similar to multiple regression analysis except different types of variables are involved. Appropriate if the dependent variable is nominal, categorical, or multichotomous and the independent variables are interval data. When two classifications are being examined, it is referred to as a two-group discriminant analysis. When three or more classifications are identified, then multiple discriminant analysis is used.

Multiple Discriminant Analysis (MDA) – cont’d MDA is useful in situations where the total sample can be divided into groups, based on a dependent variable characterizing several known classes. The intent of this technique is twofold: To understand group differences. To predict the likelihood that a variable will belong to a particular group, based on several independent variables. The linear combination is known as the discriminant function, and is derived from the following equation: where Z = discriminant score = discriminant weight for variable i = independent variable i

Multiple Discriminant Analysis (MDA)  cont’d By averaging the discriminant scores for all the individuals within a certain group, we create a group mean, also referred to as a centroid. An important function of discriminant analysis is to create a classification matrix, which shows the number of correctly and incorrectly classified cases. The total number of properly classified cases divided by the total number of cases is used to determine the hit ratio—the percentage of properly classified cases.

Factor Analysis MDA identifies groups of attributes on which individual objects differ. Factor analysis groups attributes that are alike. This technique can be used to examine interrelationships among many variables and to explain these variables in terms of their common underlying and unobservable dimensions (called “factors”). Marketing researchers use factor analysis to reduce the information contained in several original variables into a smaller, more manageable set of variables while losing as little information as possible.

Factor Analysis – cont’d While there is no distinction between dependent and independent variables when using this analysis technique, data must be gathered from interval scales. The factor model that is used for calculations is:  where = estimate of the ith factor = weight or factor score coefficient = number of variables

Cluster Analysis Cluster Analysis: Involves grouping data into “clusters” such that elements in the same group are similar to each other and elements in different groups are as different as possible. It is a statistical method that classifies or segments a sample into homogeneous classes. Marketers often use cluster analysis to identify market segments—groups of consumers with relatively similar needs. They also use the technique to design products and establish brands, target direct mail, make decisions about customer conversion and retention, and decide on marketing cost levels.

Cluster Analysis – cont’d Unlike factor analysis, which seeks to identify constructs that underlie several variables, cluster analysis seeks to identify constructs that underlie objects. Like factor analysis, though, in order to use cluster analysis, interval scales must be used during data gathering. While cluster analysis is similar to factor analysis in that it is often used to reduce complexity in a data set, factor analysis is concerned with reducing the number of variables; cluster analysis tries to reduce the number of objects (e.g., individuals, products, advertisements). Cluster analysis differs from discriminant analysis in that cluster analysis actually creates groups of like items, whereas discriminant analysis assigns elements to groups that were defined beforehand.

Multidimensional Scaling Multidimensional Scaling: (a.k.a., perceptual mapping) Is a technique used to identify important dimensions underlying respondents’ evaluations of test objects. The objective is to convert judgments of similarity or preference into distances represented in multidimensional space. Allows the researcher to illustrate relationships within data using pictures (a spatial representation of data) rather than only numbers.

Multidimensional Scaling – cont’d There is no distinction between dependent and independent variables. Marketing researchers tend to use multidimensional scaling techniques to identify important dimensions underlying customer evaluations of products, services, or companies.

Conjoint Analysis Conjoint Analysis: Provides information about the relative importance respondents place on individual attributes when choosing from multiple products or brands. Appropriate tool for nominal independent variables and an ordinal dependent variable. Conjoint analysis estimates the value of each attribute based on the choices respondents make along product concepts that are systematically differed. So respondents’ preferences toward the attributes are inferred from their choices rather than from self-reporting.

Conjoint Analysis – cont’d This technique is built on the assumption that consumers make complex decisions based not on one factor at a time but on several factors “jointly” (hence the term conjoint). Consumers make trade-offs in their decisions that will create the most satisfaction. Conjoint analysis predicts what products and services consumers will select and evaluates the weight people give to various factors that underlie their decisions. Utility: Is the number that represents the value consumers place on an attribute. Conjoint analysis creates a part-worth function that describes the utility respondents give to the levels of each attribute.

Choosing the Appropriate Test Multivariate Tests According To Scaled Data Multivariate Test Independent Variable Dependent Variable Multiple Regression Interval Multiple Discriminant Analysis Nominal Factor Analysis Cluster Analysis -- Multidimensional Scaling Ordinal or Interval Conjoint Analysis Ordinal

Decision Time! You are a marketing manager of a mid-sized company, and your marketing researcher has recently returned from a two-day seminar on multivariate statistics. He starts using some of the techniques he learned, but you feel that the research results he presents you with contradict your knowledge of the market. What are you going to do? Confront him and admit that you do not know anything about multivariate statistics, but you are uncomfortable with the research results? Or do you educate yourself before confronting him? Is it your responsibility to learn statistical techniques?

Net Impact The Internet: Will not help researchers with statistical analyses. Can lend qualitative support for the research findings obtained from the quantitative analyses. Can inform researchers about advancements made in statistical analyses through published manuscripts, clipboards, and chat groups.

Chapter 18 End of Presentation