# MCA -- MULTIPLE CLASSIFICATION ANALYSIS

MCA examines the relationships between several categorical independent variables and a single dependent variable, and determines the effects of each predictor before and after adjus­tment for its inter-correlations with other predictors in the analysis. It also provides informa­tion about the bivariate and multivariate relationships between the predictors and the dependent variable.

The dependent variables must be measured on an interval scale or must be a dichotomy. Predictor variables must be categorical, preferably with six or fewer cate­gories.

See Andrews, F. M., J. N. Morgan, J. A. Sonquist and L. Klem. Multiple Classification Analysis. Second edition. Ann Arbor: Institute for Social Research, The University of Michigan, 1973 for a complete description of the methodology used.

MCA produces:

Dependent Variable Statistics:  For the dependent variable (Y):

Grand mean

Standard deviation (square root of unbiased estimator of the population variance.)

Sum of Y

Sum of Y-squared

Total sum of squares

Explained sum of squares

Residual sum of squares

Number of cases used in the analysis

The sum of weights

Independent Variable Category Statistics:  For each category of an independent variable:

The number of cases (raw, weighted, and percentages)

Mean and standard deviation

Eta and eta squared

Partial beta and beta-squared coefficients

Bivariate frequency tables for every pair of predictors (optional)

One-Way Analysis of Variance Summary Statistics:  If only one independent variable is specified, the following are printed:

Eta squared

Total sum of squares

Between-mean sum of squares

Within-groups sum of squares

F value (degrees of freedom are printed)

## Interpretation of Results(from Multiple Classification Analysis, Andrews, Morgan, et al, 1973)

The major interpretation in a MCA is of the adjusted and unadjusted coefficients printed out for each subclass. In a population where there was no correlation among the predictors, the observations in one class of characteristic A would be distributed over all classes of the other characteristics in a fashion identical to the way in which those in other classes of A were distributed. Hence, the unadjusted mean Y for each subclass of A would be an unbiased estimate of the effect of belonging to that class of characteristic A. In the real world, however, characteristics are correlated. Young people are more likely to be in lower income groups, and in higher education groups than are older people. The multivariate process is essentially one of adjusting for these "non-orthogonalities." The adjusted means are estimates of what the mean would have been if the group had been exactly like the total population in its distribution over all the other predictor classifications. It is useful not only to have the "pure" effects of each class adjusted for all the other characteristics, but also to see how these adjusted effects differ from the unadjusted effects.

The adjusted coefficients for any predictor may be considered an estimate of the effect of that predictor alone "holding constant" all other predictors in the analysis. Differences between the adjusted and unadjusted coefficients can be analyzed, and explanations for these differences may often be found in the two-way tables of predictors. It is often valuable to compare the coefficients within a predictor to see whether there is a pattern or, possibly, a lack of pattern which is of theoretical interest.

Presentation of Results

It is most informative to present first the etas and betas, measures of the relative importance of each predictor singly and in competition with the others, and then to present the unadjusted and adjusted sub-group averages, together with a detailed description of what the subclasses represent and with the number of cases in each. The number of cases it is an indicator of the potential variability of the estimates. Multiple R2 unadjusted and multiple R2 adjusted are also usually reported.

Examples of presentation of MCA results can be found in Barfield and Morgan (1969), Blumenthal, Kahn, Andrews and Head (1972), Johnston and Bachman (1972), Johnston (1973), Katona, Strumpel and Zahn (1971), Morgan. David, Cohen and Brazes' (1962), Mueller (1969), and Pelz and Andrews (1966).

Example:   Predicting income (V268) from  occupation, marital status, and education.

DEPENDENT VARIABLE (Y) =   V268   TOTAL FAMILY INC

MEAN                         10528.32

STANDARD DEVIATION           7553.407

SUM OF Y                     3147968.

SUM OF Y SQUARE              .5014490E+11

TOTAL SUM OF SQUARES         .1700208E+11

EXPLAINED SUM OF SQUARES     .8352816E+10

RESIDUAL SUM OF SQUARES      .8649263E+10

NUMBER OF CASES                       299

PREDICTOR  V251   OCCUPATION B

NO OF  SUM OF             CLASS        DEVIATION FROM                                           STANDARD

CLASS  CASES  WEIGHTS   %        MEAN           GRAND MEAN       COEFFICIENT       ADJUSTED MEAN      DEVIATION

0     68       68  22.7     4592.206         -5936.115         -4256.094           6272.228          4161.586

1     30       30  10.0     16396.07          5867.746          1165.547           11693.87          9158.358

2     22       22   7.4     19716.09          9187.770          7577.927           18106.25          6896.417

3     14       14   4.7     15615.71          5087.393          3987.124           14515.45          11944.88

4     22       22   7.4     9988.636         -539.6847          547.4017           11075.72          5269.902

5     42       42  14.0     12596.05          2067.727          1663.999           12192.32          5372.033

6     36       36  12.0     10407.06         -121.2655          461.7471           10990.07          4254.318

7     36       36  12.0     7910.333         -2617.988         -1574.841           8953.480          5063.992

8     21       21   7.0     11960.00          1431.679          1774.740           12303.06          6163.097

9      8        8   2.7     4009.000         -6519.321         -5901.890           4626.431          2196.427

ETA-SQUARE =     .380238        BETA-SQUARE       .195452

ETA =     .616634        BETA              .442099

PREDICTOR   V30   MARITAL STATUS

NO OF  SUM OF             CLASS        DEVIATION FROM                                        STANDARD

CLASS  CASES  WEIGHTS   %        MEAN           GRAND MEAN       COEFFICIENT    ADJUSTED MEAN      DEVIATION

1    221      221  73.9     12449.90          1921.575          1123.470         11651.79         7563.060

2     17       17   5.7     7115.882         -3412.439         -2828.932         7699.389         4465.809

3     41       41  13.7     3732.463         -6795.858         -2956.380         7571.941         2752.520

4     16       16   5.4     5748.750         -4779.571         -4603.841         5924.480         4340.339

5      4        4   1.3     7640.000         -2888.321         -1330.495         9197.826         8306.206

ETA-SQUARE =     .194470        BETA-SQUARE       .658475E-01

ETA =     .440988        BETA              .256608

ADJUSTED DEVIATION SS =     .111955E+10PREDICTOR SUMMARY STATISTICS

NO OF  SUM OF             CLASS        DEVIATION FROM                                            STANDARD

CLASS  CASES  WEIGHTS   %        MEAN           GRAND MEAN       COEFFICIENT        ADJUSTED MEAN      DEVIATION

1     16       16   5.4     5973.375         -4554.946         -564.7311            9963.590          6006.004

2     71       71  23.7     6579.493         -3948.828         -2085.182            8443.139          4868.404

3     44       44  14.7     11013.86          485.5426          397.8526            10926.17          8730.284

4     70       70  23.4     10257.70         -270.6211         -789.0604            9739.261          6009.121

5     37       37  12.4     11210.03          681.7060         -1273.955            9254.365          5760.727

6     30       30  10.0     14161.87          3633.546          2836.744            13365.06          7470.542

7     17       17   5.7     16022.71          5494.385          3034.737            13563.06          6769.267

8     14       14   4.7     19327.71          8799.393          7518.277            18046.60          12470.24

ETA-SQUARE =     .203802        BETA-SQUARE       .949135E-01

ETA =     .451445        BETA              .308080

ANALYSIS SUMMARY STATISTICS

DEPENDENT VARIABLE (Y) =   V268   TOTAL FAMILY INC

R-SQUARED(UNADJUSTED) = PROP. OF VARIATION EXPLAINED BY FITTED MODEL:  .49128

ADJUSTMENT FOR DEGREES OF FREEDOM =   1.07194

LISTING OF BETAS IN DESCENDING ORDER

RANK     VAR. NO.      NAME                                 BETA

1     V251          OCCUPATION B                       .442099

2      V32          EDUC OF HEAD                       .308080

3      V30          MARITAL STATUS                     .256608