1. Multidimensional Scaling

2. Agenda Multidimensional Scaling Goodness of fit measures Nosofsky, 1986

3. Proximities AmherstBelchertownHadleyLeverettPelhamShutesburySunderland Amherst09.944.327.296.819.947.81 Belchertown014.0614.948.2513.9617.66 Hadley011.0210.9314.499.5 Leverett012.577.455.18 Pelham05.7116.16 Shutesbury011.04 Sunderland0 p Amherst, Hadley

4. Configuration (in 2-D) xixi

5. Configuration (in 1-D)

6. Formal MDS Definition f: p ij  d ij (X) MDS is a mapping from proximities to corresponding distances in MDS space. After a transformation f, the proximities are equal to distances in X. AmherstBelcherto wn HadleyLeverettPelhamShutesbu ry Sunderla nd Amherst09.944.327.296.819.947.81 Belcherto wn 014.0614.948.2513.9617.66 Hadley011.0210.9314.499.5 Leverett012.577.455.18 Pelham05.7116.16 Shutesbu ry 011.04 Sunderla nd 0

7. Distances, d ij d Amherst, Hadley (X)

8. Distances, d ij

9. d Amherst, Hadley (X)=4.32

10. Proximities and Distances AmherstBelchertownHadleyLeverettPelhamShutesburySunderland Amherst09.944.327.296.819.947.81 Belchertown014.0614.948.2513.9617.66 Hadley011.0210.9314.499.5 Leverett012.577.455.18 Pelham05.7116.16 Shutesbury011.04 Sunderland0 Proximities AmherstBelchertownHadleyLeverettPelhamShutesburySunderland Amherst0 10.05776.33257.47387.93137.83197.8328 Belchertown0 12.045516.83326.795912.721517.6600 Hadley0 12.035013.149214.16328.1892 Leverett0 12.20977.35916.6429 Pelham0 6.336015.4250 Shutesbury0 12.7366 Sunderland0 Distances

11. The Role of f f relates the proximities to the distances. f(p ij )=d ij (X)

12. The Role of f f can be linear, exponential, etc. In psychological data, f is usually assumed any monotonic function. –That is, if p ij <p kl then d ij (X)  d kl (X). –Most psychological data is on an ordinal scale, e.g., rating scales.

13. Looking at Ordinal Relations AmherstBelchertownHadleyLeverettPelhamShutesburySunderland Amherst09.944.327.296.819.947.81 Belchertown014.0614.948.2513.9617.66 Hadley011.0210.9314.499.5 Leverett012.577.455.18 Pelham05.7116.16 Shutesbury011.04 Sunderland0 Proximities AmherstBelchertownHadleyLeverettPelhamShutesburySunderland Amherst0 10.05776.33257.47387.93137.83197.8328 Belchertown0 12.045516.83326.795912.721517.6600 Hadley0 12.035013.149214.16328.1892 Leverett0 12.20977.35916.6429 Pelham0 6.336015.4250 Shutesbury0 12.7366 Sunderland0 Distances

14. Stress It is not always possible to perfectly satisfy this mapping. Stress is a measure of how closely the model came. Stress is essentially the scaled sum of squared error between f(p ij ) and d ij (X)

15. Stress Dimensions Stress “Correct” Dimensionality

16. Distance Invariant Transformations Scaling (All X doubled in size (or flipped)) Rotatation (X rotated 20 degrees left) Translation (X moved 2 to the right)

17. Configuration (in 2-D)

18. Rotated Configuration (in 2-D)

19. Uses of MDS Visually look for structure in data. Discover the dimensions that underlie data. Psychological model that explains similarity judgments in terms of distance in MDS space.

20. Simple Goodness of Fit Measures Sum-of-squared error (SSE) Chi-Square Proportion of variance accounted for (PVAF) R 2 Maximum likelihood (ML)

21. Sum of Squared Error DataPrediction(Data-Prediction) 2 75.033.88 86.971.06 12.121.25 88.910.83 66.970.94 SS E 7.97

22. Chi-Square DataPrediction (Data- Prediction) 2 (Data - Prediction) 2 /Predicti on 7540.80 8710.14 1210.50 8910.11 6710.14 Chi-Square1.70

23. Proportion of Variance Accounted for DataMean PredictionModel Prediction MeanErrorError 2 PredictionErrorError 2 76115.031.973.88 86246.971.031.06 16-5252.12-1.121.25 86248.91-0.910.83 66006.97-0.970.94 SS T 34SS E 7.96 (SST-SSE)/SST = (34-7.96)/34 =.77

24. R2R2 R 2 is PVAF, but… DataMean PredictionModel Prediction MeanErrorError 2 PredictionErrorError 2 76115.91.11.21 862410.1-2.14.41 16-5254-39 86245.92.14.41 66001525 SS T 34SS E 44.03 (SST-SSE)/SST = (34-44.03)/34 = -0.295

25. Maximum Likelihood Assume we are sampling from a population with probability f(Y;  ). The Y is an observation and the  are the model parameters. Y  =[0] N(-1.7; [  =0])=0.094

26. Maximum Likelihood With independent observations, Y 1 …Y n, the joint probability of the sample observations is: Y1Y1  =[0] 0.094 x 0.2661 x.3605 =.0090 Y2Y2 Y3Y3

27. Maximum Likelihood Expressed as a function of the parameters, we have the likelihood function: The goal is to maximize L with respect to the parameters, .

28. Maximum Likelihood Y1Y1  =[0] 0.094 x 0.2661 x.3605 =.0090 Y2Y2 Y3Y3 Y1Y1  =[-1.0167] 0.3159 x 0.3962 x.3398 =.0425 Y2Y2 Y3Y3 (Assuming  =1)

29. Maximum Likelihood Preferred to other methods –Has very nice mathematical properties. –Easier to interpret. –We’ll see specifics in a few weeks. Often harder (or impossible?) to calculate than other methods. Often presented as log likelihood, ln(ML). –Easier to compute (sums, not products). –Better numerical resolution. Sometimes equivalent to other methods. –E.g., same as SSE when calculating mean of a distribution.