Nonmetric Multidimensional Scaling input data are ranks: most similar pair AB<CB<AC most different pair, or input data are rating scales: (very similar 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).

Some basics We will be attempting to estimate coordinates X ik 's that result in model-fitted distances to match our dissimilarity data as best as possible. for metric mdsd ij = linear function (δ ij ) for nonmetric d ij = monotonic function (δ ij ) a.monotonically increasing function (always increases or stays same, never goes down): actually -- 2 definitions of monotonicity i.strong monotonicity: whenever δ ij < δ kl then d ij < d kl ii.weak monoticity: whenever δ ij < δ kl 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 {δ ij 's} are immediately translated to ranks (thus input data can certainly be ranks). example: ---ranked like: c.How to handle "ties" in the data (δ ij = δ kl  ranks?) example: ---ranked like: ?3? 2?3? --- i.primary approach to ties: if δ ij = δ kl then d ij may or may not equal d kl ii.secondary approach to ties: if δ ij = δ kl then d ij  d kl 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 = δ ij, immediately translated to ranks. *distances = d ij computed at each iteration from model (estimate coordinates, compute d ij 's, etc.) *disparities = d ij (in k & w notation) values needed in the monotonic regression These are close to d ij 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 a.Begin with data b. Translate to ranks: c.We'll have d ij 's at each iteration (those are computed using formula for Euclidean distance based on estimated coordinates X ik 's at that iteration). (The X ik 's are estimated via algorithm of "steepest descent"). Nonmetric Multidimensional Scaling

d.Estimate disparities d ij 's 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 "*". d ij 's = 0 Nonmetric Multidimensional Scaling stimulus pair ordered from max to min δ ij (i,j) δ ij rank of δ ij d ij estimated at "current" iteration 5, , ,1.3433*3* 3, , , , , * 5,3.7299*9* 4, ^ Rank of δ ij

d ij 's do not form monotonically increasing function, but disparities d ij 's do:     11 9  Nonmetric Multidimensional Scaling Shepard Diagram Model Assessment: Kruskal’s “Stress” badness-of-fit: