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Nonmetric Multidimensional Scaling • input data are ranks: most similar pair AB<CB<AC most different pair, or input data are rating scales: (very similar 1. . .9 very different), but we don't believe in assuming interval level property of data. • This is actually astounding!! We begin with rank/ordinal-level data and result of model is a set of distances or coordinates (both ratio-level). Nonmetric Multidimensional Scaling Some basics • We will be attempting to estimate coordinates Xik's that result in model-fitted distances to match our dissimilarity data as best as possible. • for metric mds dij = linear function (δij) • for nonmetric dij = monotonic function (δij) a. monotonically increasing function (always increases or stays same, never goes down): actually -- 2 definitions of monotonicity i. strong monotonicity: whenever δij < δkl then d^ij < d^kl ii. weak monoticity: whenever δij < δkl then d^ij d^kl We'll go with "b" (less restrictive, less demanding assumptions on data that could be errorful). Nonmetric Multidimensional Scaling b. input dissimilarities data {δij's} are immediately translated to ranks (thus input data can certainly be ranks). example: --ranked like: --.20 --1 --.53 .41 --3 2 --- c. How to handle "ties" in the data (δij = δkl ranks?) example: --ranked like: --.20 --1 --.41 .41 --2?3? 2?3? --i. primary approach to ties: if δij = δkl then d^ij may or may not equal d^kl ii. secondary approach to ties: ^kl if δij = δkl then d^ij d We'll go with i because again it's more flexible and less restrictive for data we know are likely errorful. Nonmetric Multidimensional Scaling 3. Monotonic (or "isotonic") regression 3 values to watch for each pair of points (i & j): * * * data point = dissimilarity = δij , immediately translated to ranks. distances = dij computed at each iteration from model (estimate coordinates, compute dij's, etc.) disparities = d^ij (in k & w notation) values needed in the monotonic regression These are close to dij values, but d^ij's are not distances (i.e., they will not neccessarily satisfy axioms like triangle inequality). Nonmetric Multidimensional Scaling 4. An example of what's done in monotonic regression 1 a. Begin with data ij s : b. Translate to ranks: 2 3 4 5 1 - 2 .34 - .47 .39 - 3 .92 .45 .50 4 5 .25 .56 .72 .23 - - 1 2 3 4 5 1- 2 3 - 6 4 - 3 10 5 7 4 5 2 8 9 1 - - c. We'll have dij's at each iteration (those are computed using formula for Euclidean distance based on estimated coordinates Xik's at that iteration). (The Xik's are estimated via algorithm of "steepest descent"). Nonmetric Multidimensional Scaling d. Estimate disparities d^ij's stimulus pair ordered from max to min δij (i,j) δij rank of δij dij estimated at "current" iteration 5,4 .23 1 3 5,1 .25 2 6 2,1 .34 3 3* 3,2 .39 4 5 4,2 .45 5 8 3,1 .47 6 10 4,3 .50 7 13 5,2 .56 8 11* 5,3 .72 9 9* 4,1 .92 10 15 These are distances (so they satisfy our distance axioms), but we want a model to obtain values that are monotonically increasing for our dissimilarities. Note there are deviations (i.e., decreases) at points denoted by "*". dij's = 0 Rank of δij Nonmetric Multidimensional Scaling dij's do not form monotonically increasing function, but disparities dij's do: d ij 3 6 3 5 8 10 13 11 9 15 dˆ ij 3 4.5 4.5 5 8 10 11 11 11 15 ^ dij Rank δij Shepard Diagram Model Assessment: Kruskal’s “Stress” badness-of-fit: ( d ij - dˆ ij )2 all pairs 2 d ij all pairs