Output options

Data

Original Multivariate Data

The complete input data matrix is displayed in the output.

Proximities Histogram

Generates a histogram that shows the distribution of the proximities.

Model

The Model tab also includes options that are related to model visualization and interpretation, specifically graphical output of the common space configuration and biplots.

The focus is on generating interpretable spatial maps and understanding variable influence through biplot interpretation. The following output components were selected for Tables:

Coordinates

Shows the coordinates of the final configuration (common space).

Variable Directions

Gives the directions of the variables in the common space.

Other graphing options such as Reduced Rank Biplot, Individual Space, and Space Weights are not available for the Identity model. No output was requested for Residuals or Shepard diagrams in this case. Because the configuration is mapping the transformed proximities, the Transformation plot was also selected, together with the Fit plot.

The graph for Common space depicts the 50 states in a common space. States that are close together in this space are similar in their profiles on the variables used. States that are far apart differ strongly in these profiles

Biplot Arrows: Interpreting Variable Directions

To interpret the position of the states in the configuration, see the table that lists the direction coordinates of the original variables. These coordinates, with the state coordinates in the common space, define the biplot.

Each arrow in the biplot represents the direction of increase for a variable. Arrows extend through the origin in both directions, indicating both increase and decrease. The length of the arrow indicates how strongly the variable is represented in the configuration, and is directly proportional to the Variance Accounted For (VAF) in the selected dimensions. The plotted vectors in the biplot are proportional to the direction coordinates provided in the table.

Variables such as income, fail, illit, homic, and school are well represented, with VAF values exceeding 75%. The variable popul is less well represented, with a VAF below 50%. This can be observed from the projections of states on the arrow of popul: Although high-population states such as CA, NY, TX, IL, and DW project highly, states like AK and HI also show high projections, despite their relatively small populations. This illustrates the lower representational quality of the variable popul in the biplot.

Variable	Direction in Biplot	High-Projecting States	Low-Projecting States
fail	right	SC, MS, LA, AL, GA, NC	ND, NV, MN, IA, AK, NE
illit	right	SC, MS, LA, AL, GA, NC	ND, MN, IA, SD, NE, UT
homic	lower-right	LA, SC, MS, CA, AL, GA	ND, SD, MN, UT, IA, NE
income	lower left	AK, CA, HI, NV, WA, NY	MS, SC, LA, AL, AR, NM
school	lower-left	AK, NV, HI, WA, FL, MN	MS, SC, LA, AL, GA, NC
life	left	ND, NV, MN, AK, IA, NE	SC, MS, LA, AL, GA, NC
freeze	upper-left	ND, SD, MN, UT, IA, NE	LA, SC, MS, CA, AL, GA

To summarize the overall structure of the biplot, the configuration of the variable directions is examined in a clockwise order, starting from the right. The directions of fail and life are nearly opposite, indicating a strong negative correlation between these two variables. The directions of fail and illit lie close together, suggesting a strong positive association. In contrast, homic and freeze point in opposite directions, reflecting a negative relationship.

The arrows for income and school are aligned, indicating that these variables are positively correlated and share similar explanatory structure in the configuration. The direction of income is approximately orthogonal to that of illit, while school is roughly orthogonal to homic, indicating little correlation between these pairs of variables.

The biplot with original state names gives a clear socio-geographic mapping of the US, consistent with broader health, education, and economic divides. A clear regional grouping is seen in the common space.

Position	Interpretation
Right side	These states are high on failing and illiteracy rate, and low on life expectancy. Top right side: Also high on homicide rate, and low on income and school. Socioeconomically challenged states.
Left side	These states are low on failing and illiteracy rate, and high on life expectancy. Upper left: in addition, high on freeze, and low on homic.More affluent, well-educated, and healthier states. And also colder.

The main division (left-right) separates states by socioeconomic advantage versus disadvantage.

Because fail contributes most strongly to dimension 1, and income to dimension 2, these two variables are examined in greater detail. The states are plotted in the common space and labeled by the variable BRegion, which defines broad regional categories: North-East (NE), South (SO), Mid-West (MW), and West (WE).

Special attention is given to the variables fail and income by adding them as supplementary variables in the analysis, allowing for additional output options. The variable region is included as supplementary, which categorize the states in more precise regions:

Abbreviation	Census Division Name	States Included
NEW	New England	ME, NH, VT, MA, RI, CT
MAT	Middle Atlantic	NY, NJ, PA
ENC	East North Central	OH, IN, IL, MI, WI
WNC	West North Central	MN, IA, MO, ND, SD, NE, KS
SAT	South Atlantic	DE, MD, VA, WV, NC, SC, GA, FL
ESC	East South Central	KY, TN, MS, AL
WSC	West South Central	AR, LA, OK, TX
MTN	Mountain	MT, ID, WY, CO, UT, NV, AZ, NM
PAC	Pacific	WA, OR, CA, AK, HI

Supplementary Output: Property Directions and Biplot

In this analysis, the original variables income and fail are also included as supplementary variables, each with a linear transformation. In addition, the supplementary categorical variable region is specified with a nominal transformation.

Output for the property directions table, the corresponding biplot, and the Projection Plot are requested. The projection plot gives the one-dimensional projections of objects on the selected (supplementary, non-nominal) variables.

Because the transformation functions for income and fail were specified as linear, the property directions and Variance Accounted For (VAF) values in this output are identical to those shown for the original variables. In addition, the output for property directions also includes the multiple coordinates for the categories of the nominal variable region.

In the biplot, the states are labeled by the variable BRegion, which classifies each case into one of the broader regional categories: Midwest (MW), West (WE), Northeast (NE), and South (SO). The supplementary region category points reflect the nine, more detailed Census divisions.

The biplot displays the REGION categories as points located at the centroids of the corresponding states. Each category is represented by a square marker, annotated with a category number. The table of multiple nominal coordinates includes both the region number and its corresponding label, facilitating identification and interpretation of the plotted categories.

The states are scattered around their region centroids, with 1=New England, 4=West NC, and 8=Mountain in the upper left corner, 2=MAT, 3=East NC and 9= Pacific in the bottom center and 5, 6, and 7 at the upper right.

When projecting the region categories onto the direction of fail, the following order is obtained (from high to low projection):

6 (ESC–SO) → 7 (WSC–SO) → 5 (SAT–SO) → 2 (MAT–NE) → 3 (ENC–MW) → 8 (MTN–WE) → 9 (PAC–WE) → 1 (NEW–NE) → 4 (WNC–MW)

For projections on the direction of income, the resulting order is:

9 (PAC–WE) → 2 (MAT–NE) → 3 (ENC–MW) → 4 (WNC–MW) → 8 (MTN–WE) → 1 (NEW–NE) → 5 (SAT–SO) → 7 (WSC–SO) → 6 (ESC–SO)

Criteria

The Criteria subtab in Output contains options for generating diagnostic summaries and graphical tools related to the convergence and fit of the PROXMAP solution.

The following options were selected in the Criteria subtab:

• Summary: A concise overview of the analysis
• Diagnostics: Detailed information including stress, variance accounted for, and other fit statistics.
• Iteration History (table and graph): Displays the progression of the optimization, showing changes in the loss function over iterations.
• Stress Decomposition: Breaks down total stress by object.
• Stress Plot: A visual depiction of the stress function across iterations.

The Stress Decomposition and Stress Plot allows closer inspection of objects that have poor fit.

The normalized stress values have been diminished approximately by factor 2.

To compare the solution with the Classical MDS solution, which is the default starting point for the computation of the optimal solution, output from the Initial subtab in Output is selected.

The scree plot indicates that the data is well represented in two dimensions. Since this is true for the classical scaling configuration, it is even more true for the PROXMAP common space.

Note: The coordinates that are shown for the common space are based on two SPSS output tables that have been combined for this case study: output_model coordinates and output_criteria stress decomposition.

The stress plot for the initial configuration, which displays stress per object, highlights three states that warrant special attention: Alaska (AK), Hawaii (HI), and Nevada (NV). These states exhibit relatively high values of normalized stress, indicated by a noticeably large area surrounding their object points in the plot.

When these elevated stress values are compared with those obtained in the common space solution, it is observed that the values remain relatively high but have substantially decreased, on average by a factor of four. This reflects an improved fit for these outlying states in the final configuration: AK, HI, and NV moved to the far lower left in Common Space, reflecting their high values on income and school.

For the last table and figure in this case study, all eight original variables are included as supplementary properties with linear transformation to obtain correlations and projections for all of them.