mds presentation natalia angelina

8/11/2019 MDS Presentation Natalia Angelina

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MultidimensionalScaling (MDS)

Angelina Anastasova

Natalia Jaworska

PSY5121 March 18/2008


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Multidimensional Scaling (MDS):

What Is It?

Generally regarded as exploratory data analysis(Ding, 2006).

Reduces large amounts of datainto easy-to-visualizestructures.

Attempts to find structure(visual representation) in a set ofdistance measures, e.g. dis/similarities, between objects/cases.

Shows how variables/objects are related perceptually.

How? By assigning cases to specific locations in space.

Distances between points in space match dis/similarities asclosely as possible:

Similar objects: Close pointsDissimilar objects: Far apart points


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MDS Example: City Distances

Distances

Matrix:

Symmetric

Spatial Map

Dimensions

1: North/South

2: East/West

Cluster


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Data Collection for MDS (1)

Direct/raw data: Proximities values directly obtained

from empirical, subjective scaling. E.g. Rating or ranking dis/similarities (Likert scales).

Indirect/derived data: Computed from other measurements:correlations or confusion data (based on mistakes) (Davidson, 1983).

E.g. Letters of alphabet presented briefly and must be identified. Rarelyconfused letters given high dissimilarity values, those that are confusedget low values.

Data collection: Pairwise comparison, grouping/sorting tasks,direct ranking, objective method (e.g. city distances).

Pairwise comparisons: All object pairs randomly presented:# of pairs = n(n-1)/2, n= # of objects/cases

Can be tedious and inefficient process.


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Types of MDS (2)

More typical in Social Sciences is the classification of

MDS based on nature of responses:

1) DecompositionalMDS: Subjects rate objects on an

overall basis, an impression, without reference toobjective attributes.

Production of a spatial configuration for an individual and

a composite map for group.

2) CompositionalMDS: Subjects rate objectson a variety of specific, pre-specified attributes(e.g. size).

No maps for individuals, only composite maps.


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Classical MDS uses Euclidean principles to model

data proximities in geometrical space, where distance(dij) between points iandjis defined as:

xiand xjspecify coordinates of points i

and j on dimension a, respectively.

The modeled Euclidean distances are related to the observed

proximities, ij, by some transformation/function (f).

Most MDS models assume that the data have the form:

ij =f(dij) All MDS algorithms are a variation of the above (Davidson,

1983).

The MDS Model


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Output of MDS

MDS Map/Perceptual Map/Spatial Representation:

1) Clusters: Groupings in a MDS spatialrepresentation.

These may represent a domain/subdomain.2) Dimensions: Hidden structures in data. Orderedgroupings that explain similarity between items.

Axes are meaningless and orientation is arbitrary.

In theory, there is no limit to the number ofdimensions.

In reality, the number of dimensions that can be

perceived and interpreted is limited.


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Diagnostics of MDS (1)

MDS attempts to find a spatial configuration Xsuchthat the following is true:f(ij) dij(X)

Stress(Kruskals) function: Measures degree ofcorrespondence between distances among points on the

MDS map and the matrix input.Proportion of variance of disparities

notaccounted for by the model:

Range 0-1: Smaller stress = better representation.

None-zero stress: Some/all distances in the map aredistortions of the input data.

Rule of thumb: 0.1 is excellent; 0.15 not tolerable.


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R2(RSQ): Proportion of variance of the disparitiesaccounted for by the MDS procedure.

R20.6 is an acceptable fit.

Weirdness Index: Correspondence of subjects map and the

aggregate map outlier identification. Range 0-1: 0 indicates that subjects weights are proportional to the

average subjects weights; as the subjects score becomes moreextreme, index approaches 1.

Shepard Diagram: Scatterplot of input proximities (X-axis)against output distances (Y-axis) for every pair of items.

Step-line produced. If map distances fall on the step-line thisindicates that input proximities are perfectly reproduced by the MDSmodel (dimensional solution).

Diagnostics of MDS (2)


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Interpretation of Dimensions

Squeezing data into 2-D enables readability but maynot be appropriate: Poor, distorted representation of thedata (high stress).

Scree plot: Stress vs.

number of dimensions.E.g. cities distance

Primary objective in dimension interpretation: Obtain

best fit with the smallest number of possibledimensions.

How does one assign meaning to dimensions?


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Meaning of Dimensions

Subjective Procedures:Labelling the dimensions by visual

inspection, subjective

interpretation, and informationfrom respondents.

Experts evaluate and identify thedimensions.


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Validating MDS Results

Split-sample comparison:

Original sample is divided and a correlationbetween the variables is conducted.

Multi-sample comparison: New sample is collected and a correlation is

conducted between the old and new data.

Comparisons are done visually or with a simple

correlation of coordinates or variables.Assessing whether MDS solution(dimensionality extraction) changes in asubstantial way.


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MDS Caveats

Respondents probably perceive stimulidifferently. In non-aggregate data, differentdimensions may emerge.

Respondents may attach different levels ofimportance to a dimension.

Importance of a dimension may change over time.

Interpretation of dimensions is subjective.

Generally, more than four times as many objectsas dimensions should be compared for the MDSmodel to be stable.


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Advantages of MDS An alternative to the GLM.

Does not require assumptions of linearity,metricity, or multivariate normality.

Can be used to model nonlinear relationships.

Dimensionality solution can be obtained fromindividuals; gives insight into how individualsdiffer from aggregate data.

Reveals dimensions without the need for definedattributes.

Dimensions that emerge from MDS can beincorporated into regression analysis to assess

their relationship with other variables.


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Disadvantages of MDS

Provides a global measure of dis/similarity but

does not provide much insight into subtleties (Streetet al., 2001).

Increased dimensionality: Difficult to representand decreases intuitive understanding of the data.

As such, the model of the data becomes as

complicated as the data itself.

Determination of meanings of dimensions is

subjective.


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A Tiny Break . . .


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SPSSing MDS

In the SPSS Data Editor window, click: Analyze>

Scale> Multidimensional Scaling


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Select four or more Variablesthat you want to test.

You may select a single variable for the Individual

Matrices forwindow (depending on the distances optionselected).


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If Data are distances(e.g. cities distances) option is

selected, click on the Shapebutton to define

characteristic of the dissimilarities/proximity matrices.

If Create distance from data

isselected, click on the

Measurebutton to control the

computation of dissimilarities,

to transform values, and to

compute distances.


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In the Multidimensional Scaling dialog box, click on the

Modelbutton to control the level of measurement,

conditionality, dimensions, and the scaling model.

Click on the Optionsbutton to control the

display options, iteration criteria, and

treatment of missing values.


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MDS: A Psychological Example

Multidimensional scaling modelling approach to latentprofile analysis in psychological research (Ding, 2006)

Basic premise: Utilize MDS to investigate types orprofiles of people.

Profile: From applied psych where test batteries areused to extract and construct distinctivefeatures/characteristics in people.

MDS method was used to:

Derive profiles (dimensions) that could provide informationregarding psychosocial adjustment patterns in adolescents.

Assess if individuals could follow different profile patternsthan those extracted from group data, i.e. deviations fromthe derived normative profiles.


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Data for MDS

Scored data for MDS profile analysis Sample data for 14 individuals:

BI=body image, PR=peer relations, FR=family relations, MC=mastery & coping,VE=vocational & educational goal, SA=superior adjustment, PMI-1=profile matchindex for Profile 1, PMI-2=profile match index for Profile 2, LS=life satisfaction,

Dep=depression, PL=psychological loneliness


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MDS map

Euclidean distance model

Profile 1

3210-1-2

2.0

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

save mc

fr

pr

bi

The Analysis: Step by Step

Step 1: Estimate the number of profiles(dimensions) from the latent variables.

Kruskal's stress = 0.00478Excellent stress value.

RSQ = 0.9998

Configuration derived in 2

dimensions.


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MDS map

Euclidean distance model

Profile 1

3210-1-2

2.0

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

save mc

fr

pr

bi

Scale values of two MDS profiles (dimensions) in

psychosocial adjustment.

Normative profiles of

psychosocial adjustments

in young adults.

Each profile represents

prototypical individual.


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Step 2: Using the estimated scale values as

independent variables and observed variables

as dependent variables estimate: Individual profile match index (PMI):

The extent of individual variability along a profile.

Intra-individual variability across profiles.

PMI-1=profile match index for Profile 1, PMI-2=profile

match index for Profile 2, LS=life satisfaction,

Dep=depression, PL=psychological loneliness

Fit index:

The proportion of variance

in the individuals observed

data that can be accounted

for by the profiles.


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Individual Profiles vs. Aggregate

PMI-1 PMI-2 FIT

Subject 1 -0.73 0.29 0.94

Subject 2 -0.38 0.23 0.99

Subject 4 -0.16 0.24 0.32

Profile 1 Profile 2 Subject 1 Subject 2 Subject 4

Body Image (BI) 2.28 -0.5 2.82 3.82 5.09

Peer Relations (PR) 0.23 1.49 5.1 5 5.3

Family Relations (FR) 0.7 -1.2 5 4.71 4.69

Mastery & Coping (MC) -0.25 0.14 4.6 4.9 6

Voc-Ed Goals (VE) -1.49 0 5.7 5.4 6

Superior Adjust. (SA) -0.08 0.08 4.3 4.9 5.5

-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6

Profile 1

Profile 2

Subject 1

Subject 2

Subject 4


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Step 3: Assess the association between profiles

and other factors by regression.

Profile 1: -High scores on Body Image - higher degree of life satisfaction.

-High scores on the Vocational-Educational Goal - higher degree of depression.Profile 2: -Higher scores on the family relationships profile - higher degree of psychological

loneliness.

Level: -Average scores of individuals psychosocial adjustment.

-Overall positive psychosocial adjustment scores suggest less depression or

psychological loneliness and higher degree of life satisfaction.


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Commentary on MDS Profile

Analysis

Strength of MDS profile analysis:

Provides representation of what typical

configurations or profiles of variables exist in the

population and how individuals differ with respect

to these profiles.

Enables identification/analysis of:

Individuals who develop in an idiographic (specific

and subjective) manner; not consistent with

aggregate profiles.


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Thank You!

Questions?


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References

Davidson, M. L. (1983).Multidimensional scaling. New York: J. Wileyand Sons.

Ding, C. S. (2006). Multidimensional scaling modelling approach to latentprofile analysis in psychological research.International Journal ofPsychology41 (3), 226-238.

Kruskal, J.B. & Wish M.1978.Multidimensional Scaling. Sage.

Street, H., Sheeran, P., & Orbell, S. (2001). Exploring the relationshipbetween different psychosocial determinants of depression: amultidimensional scaling analysis.Journal of Affective Disorders 64,5367.

Takane, Y., Young, F.W., & de Leeuw, J. (1977). Nonmetric individualdifferences multidimensional scaling: An alternating least squares methodwith optimal scaling features,Psychometrika42 (1), 767.

Young, F.W., Takane, Y., & Lewyckyj, R. (1978). Three notes onALSCAL,Psychometrika43 (3), 433435.

http://www.analytictech.com/borgatti/profit.htm

http://www2.chass.ncsu.edu/garson/pa765/mds.htm

http://www.terry.uga.edu/~pholmes/MARK9650/Classnotes4.pdf
http://www.analytictech.com/borgatti/profit.htmhttp://www2.chass.ncsu.edu/garson/pa765/mds.htmhttp://www.terry.uga.edu/~pholmes/MARK9650/Classnotes4.pdfhttp://www.terry.uga.edu/~pholmes/MARK9650/Classnotes4.pdfhttp://www2.chass.ncsu.edu/garson/pa765/mds.htmhttp://www.analytictech.com/borgatti/profit.htm

mds presentation natalia angelina

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