Multiple Regression Y = b 0 + b 1 X 1 + b 2 X 2 + …+ b n X n Same as Simple Regression in principle New Issues: – Each X i must represent something unique – Variable selection
Multiple Regression Example 1: – Spending = a + b income + c age Example 2: – weight = a + b height + c sex + d age
Real Estate Example How is price related to the characteristics of the house?
SAS Code proc reg; model price = section lotsize bed bath age other; run;
Interpreting the Regression Output Parameter Estimates or Slope Coefficients capture the marginal impact of explanatory variable on price Example: the coefficient of the variable beds represents the impact of increasing the number of bedrooms by one on price
Significance of the Coefficients Are they significantly different from zero? – Look at the T values and p values T value higher than 1.8 or p<0.05 good Sometimes p<0.10 is considered reasonably significant Overall Goodness of Fit – Look at R 2 (also refer to note in Session 1)
Where are we Now? Behavior Segment 1 Segment 2 Secondary Data Distinguishing Characteristics Targeting Factor Analysis Cluster Analysis Discriminant /Logit Analysis
Web Browsing Identified two groups of consumers – One that visits your website frequently – One that doesn’t Can the differences in behavior be related to socio-demographic variables? Can we use these discriminators to classify prospects into one of these two groups?
Catalog Business Identified two consumer segments – One which buys a lot – Other which does not buy as much Can we find variables that help discriminate the behavior of these two groups? Can we use these discriminators to classify other consumers into one of these two groups?
Promotional Campaigns Identify groups based on their response to promotional campaigns – One group purchases a lot on promotion – Other does not Identify characteristics that distinguish these two groups Can we use these discriminators to identify price sensitive prospects from the not so price sensitive ones?
Segmentation Analysis General Problem – Identified segments in the population based on behavior – Want to find targetable characteristics that discriminate these groups – Classify prospects into different groups
Identifying the Best Discriminators Two groups appear to be well separated on each ratio: ROI and GE/A Also well separated in two dimensional space But this need not always be the case!
Discriminating Variables X1X1 X2X2
Discriminant Analysis Identify a set of variables that best discriminate between the two groups Does so by choosing a new line that maximizes the similarity between members of the same group and minimizing the similarity between members belonging to different groups
Discriminant Function Z = w 1 GEA + w 2 ROI Between-Group Sum of Squares – SS b Within-Group Sum of Squares – SS w =(SS b /SS w )
More on the Criterion For Z to provide maximum separation between the groups, the following must be satisfied: – The means of Z for the two groups should be as far apart as possible (or high SS b ) – Values of Z for each group should be as homogenous as possible (or low SS w )
Classification Discriminant Function: The line that separates the members of the two groups Methods of Classification – Cut-Off Value Method – Decision Theory Approach – Classification Function Approach – Mahalanobis Distance Method
Cut-Off Value Method Uses the Discriminant Function line to score new observations (prospects) and classify them into one of two groups based on a cut-off value
Classification Z Cut-off Value R2R2 R1R1
Classification Function Approach Classifications based on this approach are identical to those done by Decision Theory approach Classification functions are computed for each group: C 1 = *GEA *ROI C 2 = *GEA – 1.404*ROI
Basic Idea Score each new observation using these two scoring functions The observation gets assigned to the group with the higher score
What To Look For In The Results? Significance of the Discriminating Variables – Idea is to test whether the means of the discriminating variables are statistically different across the two groups – Statistic: Wilks’ Lamda must be small (Look for the p value/significance level)
Estimate of The Discriminant Function Canonical Discriminant Function Z = *GEA *ROI It is possible that the group means are statistically different even though for all practical purposes, the differences between the groups may not be large Look at the squared Canonical Correlation: ratio of between group SS/Total SS (High is good)
Importance of the Discriminant Variables and the Discriminant Function How important is a variable to the Discriminant Function? Look at the structure loadings: Pooled Within Canonical Structure – Variable with the higher loading is relatively more important – Caution: If the variables are highly correlated relative importance of the variables can change with sample
Classification Summary Look at Cross-Validation results
Web Browsing Can use the Discriminant function to classify prospects into one of these two groups Target Appropriately
Catalog Business Classify other consumers into one of these two groups Do stuff!
Promotional Campaigns Classify Prospects into price sensitive and not so price sensitive segments Target appropriately
Summary Discriminant Analysis Extremely Useful Segmentation Analysis tool Intermediate step in the overall picture – helps classify prospects and devise the appropriate targeting strategies