DSC 410/510 - Multivariate Statistical Methods

Chapter 10

Suggested Solutions

1. HOW DOES MDS DIFFER FROM CLUSTER ANALYSIS?
   
   Multidimensional scaling (MDS) is a family of techniques which helps the analyst to identify key dimensions underlying respondents evaluations of objects. MDS techniques enable the researcher to represent respondents perceptions spatially; that is, to create visual displays that represent the dimensions perceived by the respondents when evaluating stimuli (e.g., brands, objects).
   MDS differs from cluster analysis in that it provides a visual representation of individual and group respondents' perceptions of the object(s), while cluster analysis provides a classification of objects or variables so that each object is very similar to others in its cluster.
   
   
5. HOW CAN THE ANALYST DETERMINE WHEN THE "BEST" MDS SOLUTION HAS BEEN OBTAINED?
   
   The objective of the analyst should be to obtain the best fit with the smallest number of dimensions, which requires a trade-off between the fit of the solution and the number of dimensions. Interpretation of solutions derived in more than three dimensions is extremely difficult and is usually not worth the improvement in fit.
   The analyst may also use an index of fit to determine the number of dimensions. The index of fit (or R-square) is a squared correlation index that can be interpreted as indicating the proportion of variance of the disparities that can be accounted for by the MDS procedure. Measures of .60 or better are considered acceptable; the higher the R-square, the better the fit.
   A third approach is to use a measure of stress. Stress measures the proportion of the variance of the disparities that is not accounted for by the MDS model.
   
   
7. COMPARE AND CONTRAST CORRESPONDENCE ANALYSIS TO THE MDS TECHNIQUES.
   
   Correspondence analysis is a compositional perceptual mapping technique which relies on the association among nominally scaled variables. Measures of similarity are based on the chi-square metric derived from a cross-tabulation table. It has the unique feature of spatially representing both objects and attributes on the same spatial map.
   
   
8. DESCRIBE HOW "CORRESPONDENCE" OR ASSOCIATION IS DERIVED FROM A CONTINGENCY TABLE.
   
   Correspondence analysis allows the representation of the rows and columns of a contingency table in joint space. Using the totals for each category an expected value is calculated for each cell. Then the difference between the expected and actual is calculated. Using this value a chi-square statistic is formed for each cell as the squared difference divided by the expected value. The chi-square values can be converted to similarity measures by applying the opposite sign of their difference. The similarity measure provides a standardized measure of association that can be plotted in an appropriate number of dimensions (number of rows or columns minus one).