Row and column contributions for Simple Correspondence Analysis

Find definitions and interpretation for every statistic that is provided for row and column contributions for simple correspondence analysis.

In This Topic

Qual
Mass
Inert
Coord
Corr
Contr

Qual

Quality (Qual) is the squared distance of the point from the origin in the chosen number of dimensions divided by the squared distance from the origin in the space defined by the maximum number of dimensions. Minitab calculates a quality value for each row and each column in the contingency table.

Interpretation

Use the quality values to determine the proportion of the row inertia or column inertia represented by the components. Quality is always a number between 0 and 1. Larger quality values indicate that the row or column is well represented by the components. Lower values indicate poorer representation. The quality values for the rows and columns can help you interpret the components.

Row Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	Geology	0.916	0.107	0.137	-0.076	0.055	0.016	-0.303	0.861	0.322
2	Biochemistry	0.881	0.036	0.119	-0.180	0.119	0.030	0.455	0.762	0.248
3	Chemistry	0.644	0.163	0.021	-0.038	0.134	0.006	-0.073	0.510	0.029
4	Zoology	0.929	0.151	0.230	0.327	0.846	0.413	-0.102	0.083	0.052
5	Physics	0.886	0.143	0.196	-0.316	0.880	0.365	-0.027	0.006	0.003
6	Engineering	0.870	0.111	0.152	0.117	0.121	0.039	0.292	0.749	0.310
7	Microbiology	0.680	0.046	0.010	-0.013	0.009	0.000	0.110	0.671	0.018
8	Botany	0.654	0.108	0.067	0.179	0.625	0.088	0.039	0.029	0.005
9	Statistics	0.561	0.036	0.012	-0.125	0.554	0.014	-0.014	0.007	0.000
10	Mathematics	0.319	0.098	0.056	-0.107	0.240	0.029	0.061	0.079	0.012

Column Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	A	0.587	0.039	0.187	-0.478	0.574	0.228	-0.072	0.013	0.007
2	B	0.816	0.161	0.110	-0.127	0.286	0.067	-0.173	0.531	0.159
3	C	0.465	0.389	0.094	-0.083	0.341	0.068	-0.050	0.124	0.032
4	D	0.968	0.162	0.347	0.390	0.859	0.632	-0.139	0.109	0.103
5	E	0.990	0.249	0.262	0.032	0.012	0.006	0.292	0.978	0.699

In the Row Contributions table, the highest quality values occur for Zoology (0.929) and Geology (0.916). Therefore, these two rows are best represented by the two components. Mathematics has the poorest representation, with a quality value of 0.319. In the Column Contributions table, the highest quality values are for funding categories B (0.816), D (0.968), and E (0.990). Therefore, the two components explain most of the deviation from the expected values in funding categories B, D, and E.

Mass

Mass is the total of the matrix of relative frequencies for a row or column in the contingency table. The mass of a row is the sum of all the frequencies in the row divided by the sum of all the frequencies in the contingency table. The mass of a column is the sum of all the frequencies in the column divided by the sum of all the frequencies in the contingency table.

Interpretation

Use the mass to determine the proportion of each row or column category. Larger mass values indicate that the row or column has a higher relative frequency. The total mass for all the row categories or all the column categories equals 1 (100%).

In the Column Contributions table, funding category C has the highest mass (0.389), and accounts for 38.9% of the data. Funding category A has the lowest mass (0.039), and accounts for 3.9% of the data.

Row Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	Geology	0.916	0.107	0.137	-0.076	0.055	0.016	-0.303	0.861	0.322
2	Biochemistry	0.881	0.036	0.119	-0.180	0.119	0.030	0.455	0.762	0.248
3	Chemistry	0.644	0.163	0.021	-0.038	0.134	0.006	-0.073	0.510	0.029
4	Zoology	0.929	0.151	0.230	0.327	0.846	0.413	-0.102	0.083	0.052
5	Physics	0.886	0.143	0.196	-0.316	0.880	0.365	-0.027	0.006	0.003
6	Engineering	0.870	0.111	0.152	0.117	0.121	0.039	0.292	0.749	0.310
7	Microbiology	0.680	0.046	0.010	-0.013	0.009	0.000	0.110	0.671	0.018
8	Botany	0.654	0.108	0.067	0.179	0.625	0.088	0.039	0.029	0.005
9	Statistics	0.561	0.036	0.012	-0.125	0.554	0.014	-0.014	0.007	0.000
10	Mathematics	0.319	0.098	0.056	-0.107	0.240	0.029	0.061	0.079	0.012

Column Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	A	0.587	0.039	0.187	-0.478	0.574	0.228	-0.072	0.013	0.007
2	B	0.816	0.161	0.110	-0.127	0.286	0.067	-0.173	0.531	0.159
3	C	0.465	0.389	0.094	-0.083	0.341	0.068	-0.050	0.124	0.032
4	D	0.968	0.162	0.347	0.390	0.859	0.632	-0.139	0.109	0.103
5	E	0.990	0.249	0.262	0.032	0.012	0.006	0.292	0.978	0.699

In the Row Contributions table, the highest mass values occur for Chemistry (0.163) and Zoology (0.151). Therefore, these two rows (fields) account for 0.163 + 0.151 = 0.314, or 31.4% of the data in all the rows. Biochemistry (0.036) and Statistics (0.036) have the lowest mass. Each of these rows accounts for only approximately 3.6% of the data.

Inert

Cell inertia is the chi-squared value in the cell divided by the total frequency for the contingency table. The row inertia (Inert) is the sum of the cell inertias for the row. The column inertia (Inert) is the sum of the cell inertias for the column. The sum of all the cell inertias is the total inertia, or simply the inertia.

Interpretation

Use row inertia and column inertia to assess associations between categories and contributions to variation in the data. Higher values generally indicate a stronger association and greater deviation from the expected values.

Row Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	Geology	0.916	0.107	0.137	-0.076	0.055	0.016	-0.303	0.861	0.322
2	Biochemistry	0.881	0.036	0.119	-0.180	0.119	0.030	0.455	0.762	0.248
3	Chemistry	0.644	0.163	0.021	-0.038	0.134	0.006	-0.073	0.510	0.029
4	Zoology	0.929	0.151	0.230	0.327	0.846	0.413	-0.102	0.083	0.052
5	Physics	0.886	0.143	0.196	-0.316	0.880	0.365	-0.027	0.006	0.003
6	Engineering	0.870	0.111	0.152	0.117	0.121	0.039	0.292	0.749	0.310
7	Microbiology	0.680	0.046	0.010	-0.013	0.009	0.000	0.110	0.671	0.018
8	Botany	0.654	0.108	0.067	0.179	0.625	0.088	0.039	0.029	0.005
9	Statistics	0.561	0.036	0.012	-0.125	0.554	0.014	-0.014	0.007	0.000
10	Mathematics	0.319	0.098	0.056	-0.107	0.240	0.029	0.061	0.079	0.012

In the Row Contributions table, the column labeled Inert is the proportion of the total inertia contributed by each row. Thus, Geology contributes 13.7% to the total chi-squared statistic. Zoology has the highest inertia value (0.230). Therefore, Zoology contributes 23% to the total chi-squared value and has a stronger association with the column categories (funding) than the other row categories.

Coord

Minitab calculates row and column principal coordinates (Coord) for each component. The row principal coordinates are the principal coordinates of each row profile in terms of the principal component. The column principal coordinates are the coordinates of each column profile in terms of the principal component.

To visually display the points defined by the row and column principal coordinates, use a symmetric plot or asymmetric plot.

Corr

Minitab calculates row and column correlation values. The row correlation value represents the contribution of the component to the inertia of the row. The column correlation value represents the contribution of the component to the inertia of the column. Correlation values range from 0 to 1.

Interpretation

Use the correlation value to interpret each component in terms of its contribution to inertia. Values close to 1 indicate that the component accounts for a high amount of inertia. Values close to 0 indicate that the component contributes little to inertia.

Row Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	Geology	0.916	0.107	0.137	-0.076	0.055	0.016	-0.303	0.861	0.322
2	Biochemistry	0.881	0.036	0.119	-0.180	0.119	0.030	0.455	0.762	0.248
3	Chemistry	0.644	0.163	0.021	-0.038	0.134	0.006	-0.073	0.510	0.029
4	Zoology	0.929	0.151	0.230	0.327	0.846	0.413	-0.102	0.083	0.052
5	Physics	0.886	0.143	0.196	-0.316	0.880	0.365	-0.027	0.006	0.003
6	Engineering	0.870	0.111	0.152	0.117	0.121	0.039	0.292	0.749	0.310
7	Microbiology	0.680	0.046	0.010	-0.013	0.009	0.000	0.110	0.671	0.018
8	Botany	0.654	0.108	0.067	0.179	0.625	0.088	0.039	0.029	0.005
9	Statistics	0.561	0.036	0.012	-0.125	0.554	0.014	-0.014	0.007	0.000
10	Mathematics	0.319	0.098	0.056	-0.107	0.240	0.029	0.061	0.079	0.012

In these results, Component 1 accounts for most of the inertia of Zoology (Corr = 0.846) and Physics (Corr = 0.880), but explains little of the inertia of Microbiology (Corr = 0.009).

Contr

The contribution (Contr) of each row or column to the inertia of each component.

Interpretation

Use the contribution values for the rows and/or columns to interpret the components.

Row Contributions

					Component 1			Component 2
ID	Name	Qual	Mass	Inert	Coord	Corr	Contr	Coord	Corr	Contr
1	Geology	0.916	0.107	0.137	-0.076	0.055	0.016	-0.303	0.861	0.322
2	Biochemistry	0.881	0.036	0.119	-0.180	0.119	0.030	0.455	0.762	0.248
3	Chemistry	0.644	0.163	0.021	-0.038	0.134	0.006	-0.073	0.510	0.029
4	Zoology	0.929	0.151	0.230	0.327	0.846	0.413	-0.102	0.083	0.052
5	Physics	0.886	0.143	0.196	-0.316	0.880	0.365	-0.027	0.006	0.003
6	Engineering	0.870	0.111	0.152	0.117	0.121	0.039	0.292	0.749	0.310
7	Microbiology	0.680	0.046	0.010	-0.013	0.009	0.000	0.110	0.671	0.018
8	Botany	0.654	0.108	0.067	0.179	0.625	0.088	0.039	0.029	0.005
9	Statistics	0.561	0.036	0.012	-0.125	0.554	0.014	-0.014	0.007	0.000
10	Mathematics	0.319	0.098	0.056	-0.107	0.240	0.029	0.061	0.079	0.012

In this Row Contributions table, Zoology (0.413) and Physics (0.365) contribute the most to the inertia of Component 1. Geology (0.322), Biochemistry (0.248), and Engineering (0.310) contribute the most to the inertia of Component 2.