GENERAL DESCRIPTION
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. See Andrews, et al., Multiple Classification Analysis, for a complete description of the methodology used.
COMMAND FEATURES
Missing Data: Cases with missing data on the independent variables may be eliminated (see DELETE option). Cases with missing data on the dependent variable are automatically excluded from the analysis. MCA produces RECODE control statements for computing residuals if requested.
PRINTED OUTPUT
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
Deviation of the category mean (unadjusted and adjusted)
Adjusted class mean MCA coefficient
Eta and eta squared
Partial beta and beta-squared coefficients
Unadjusted and adjusted sum of squares
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
Adjustment factor
Adjusted eta and eta squared
Total sum of squares
Between-mean sum of squares
Within-groups sum of squares
F value (degrees of freedom are printed)
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.
Both the adjusted and unadjusted coefficients are expressed by the program as deviations from the overall mean, and are constrained so that their sum, weighted by the proportion in each subclass, is zero.
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.
The coefficients for the predictors do not provide definitive information about logical priorities, chains of causation, or about interaction effects. It is possible for the program to assign considerable explanatory power to a variable late in a causal chain, such as an attitude, when much of the credit "really" belongs to a logically prior, but not as powerful variable, such as race.
Interaction effects of two or more predictors on the dependent variable will not be revealed by the program, since the assumption is that the effects of all the predictors are additive, i.e. the effect for predictor A is assumed to be the same for one class of predictor B as it is for every other class.
A difficulty in using the adjusted coefficients as a presentational device is that the additivity assumptions may lead to absurd adjusted means for some groups (less than zero, for instance) if the assumption is inappropriate for the data being analyzed. This is particularly likely when the dependent variable is a dichotomy, such as home ownership. Clearly, it is not sensible to predict that less than 0 percent of a subgroup own a home.
Presentation of Results
It is most informative to the reader 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 should be included because it is an indicator of the potential variability of the estimates.) Multiple R2 unadjusted and multiple R2 adjusted are also usually reported.
We recommend that the results be given in the form of unadjusted and adjusted subgroup averages rather than in the form of deviations because the user finds it easier to scan unadjusted and adjusted subgroup averages than positive and negative deviations. However, the adjusted deviations can be included for convenience in seeing the net effects of each predictor. As noted above, a complication of subgroup averages is that occasionally the expected value is impossible (e.g. negative although the dependent variable is a variable with no negative values); if impossible expected values are presented, a short explanatory note should be included.
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).
RESIDUAL RECODE CONTROL STATEMENT OUTPUT
RECODE control statements to compute predicted and residual values based on the MCA regression may be written to the file assigned to RESIDUAL (option RESIDUALS). These statements may be used with LISTDATA to list the residuals or with TRANS to create a permanent residuals dataset.
INPUT DATA
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. When using more than one predictor all codes must be in the range 0 to 31.
RESTRICTIONS
1. Predictor codes must be in the range 0 - 31 when more than one predictor is defined.
2. The total number of predictor codes, obtained by summing the number of codes for each predictor, must be less than or equal to the value assigned to the MAXC parameter. Thus, if there are two predictors, one with codes 0,1,2 and the other with codes 1,2,3,4, set MAXC to 7 or higher.
CONTROL STATEMENTS
Filter (optional)
Job Title (required if using a Runfile)
Options and Parameters
CRITERION=n
Tolerance (0.0-1.0) of the convergence test selected.
Default: CRITERION=.005.
DELETE=(MD1,MD2)
MD1 Delete all cases where any independent variable equals its first missing-data code.
MD2 Delete all cases where any independent variable equals its second missing-data code.
DEPV=variable number The dependent variable.
MAXC=n
The maximum total number of predictor codes for all predictors .
Default: MAXC=99.
MAXI=n
The maximum number of iterations.
Default: 25 iterations.
PRINT=(DICT|CODES,TABLES,TRACE)
DICT Print the input dictionary.
CODES Print the input dictionary and category labels.
TABLES Print pair-wise cross-tabulations of independent variables.
TRACE Print the coefficients from all iterations.
RECODE=n Use RECODE n, previously entered via the RECODE command.
RESIDUALS Write RECODE control statements for computing predicted and residual values to RESIDUAL file. The predicted value variable number will be R10000 and the residual value variable number will be R10001.
TEST=%MEAN|CUTOFF|%RATIO.
The convergence test desired. If not specified, MCA iterates until the maximum number of iterations (MAXI) is exceeded. (see CRITERION)
%MEAN Test whether the change in all coefficients from one iteration to the next is below a specified fraction of the grand mean.
CUTOFF Test whether the change in all coefficients from one iteration to the next is less than a specified value.
%RATIO Test whether the change is less than a specified fraction of the ratio of the standard deviation of the dependent variable to its mean.
VARS=variable numbers
The list of independent variables. One-way analysis of variance is performed if
only one variable is specified.
WT=n Use variable n as a weight variable
REFERENCES
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.
EXAMPLES
Example 1: Predicting income (V268) from occupation, marital status, and education.
File assignments: dictin=scf.dic datain=scf.dat
Filter include v37=1
Job Title PREDICTING INCOME
Options and parameters: print=(dict) depv=v268 V=v251,v30,v32 del=(md1,md2) test=%mean
*** MCA -- MULTIPLE CLASSIFICATION ANALYSIS ***
PREDICTING INCOME
Number of variables: 4
The data are not weighted
For the independent variables, cases with MD1 or MD2 values will be deleted
The iteration maximum is 25
The convergence test is %MEAN
The tolerance factor is .00500
INPUT DICTIONARY:
VNUM NAME TYPE LOC WID NDEC MD1 MD2 REFNO
V30 MARITAL STATUS I 9 2 0 9 30
V32 EDUC OF HEAD I 11 2 0 9 32
V37 RACE I 13 2 0 9 37
V251 OCCUPATION B I 25 2 0 251
V268 TOTAL FAMILY INC I 27 4 0 268
0 cases deleted due to missing data on the dependent variable.
0 cases deleted due to missing data on the independent variables.
0 cases deleted due to predictor codes outside the range 0 to 31.
299 cases were used in the analysis.
RESULTS BASED ON ITERATION 6
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
UNADJUSTED
NO OF SUM OF CLASS DEVIATION FROM
CLASS CASES WEIGHTS % MEAN GRAND MEAN COEFFICIENT
0 68 68 22.7 4592.206 -5936.115 -4256.094
1 30 30 10.0 16396.07 5867.746 1165.547
2 22 22 7.4 19716.09 9187.770 7577.927
3 14 14 4.7 15615.71 5087.393 3987.124
4 22 22 7.4 9988.636 -539.6847 547.4017
5 42 42 14.0 12596.05 2067.727 1663.999
6 36 36 12.0 10407.06 -121.2655 461.7471
7 36 36 12.0 7910.333 -2617.988 -1574.841
8 21 21 7.0 11960.00 1431.679 1774.740
9 8 8 2.7 4009.000 -6519.321 -5901.890
STANDARD
CLASS ADJUSTED MEAN DEVIATION
0 6272.228 4161.586
1 11693.87 9158.358
2 18106.25 6896.417
3 14515.45 11944.88
4 11075.72 5269.902
5 12192.32 5372.033
6 10990.07 4254.318
7 8953.480 5063.992
8 12303.06 6163.097
9 4626.431 2196.427
ETA-SQUARE = .380238 BETA-SQUARE .195452
ETA = .616634 BETA .442099
ETA-SQUARE (ADJ) = .360938
ETA (ADJ) = .600781
UNADJUSTED DEVIATION SS = .646484E+10
ADJUSTED DEVIATION SS = .332309E+10
PREDICTOR V30 MARITAL STATUS
UNADJUSTED
NO OF SUM OF CLASS DEVIATION FROM
CLASS CASES WEIGHTS % MEAN GRAND MEAN COEFFICIENT
1 221 221 73.9 12449.90 1921.575 1123.470
2 17 17 5.7 7115.882 -3412.439 -2828.932
3 41 41 13.7 3732.463 -6795.858 -2956.380
4 16 16 5.4 5748.750 -4779.571 -4603.841
5 4
4 1.3
7640.000
-2888.321 -1330.495
STANDARD
CLASS ADJUSTED MEAN DEVIATION
1 11651.79 7563.060
2 7699.389 4465.809
3 7571.941 2752.520
4 5924.480 4340.339
5 9197.826 8306.206
ETA-SQUARE = .194470 BETA-SQUARE .658475E-01
ETA = .440988 BETA .256608
ETA-SQUARE (ADJ) = .183511
ETA (ADJ) = .428382
UNADJUSTED DEVIATION SS = .330640E+10
ADJUSTED DEVIATION SS = .111955E+10
PREDICTOR SUMMARY STATISTICS
PREDICTOR V32 EDUC OF HEAD
UNADJUSTED
NO OF SUM OF CLASS DEVIATION FROM
CLASS CASES WEIGHTS % MEAN GRAND MEAN COEFFICIENT
1 16 16 5.4 5973.375 -4554.946 -564.7311
2 71 71 23.7 6579.493 -3948.828 -2085.182
3 44 44 14.7 11013.86 485.5426 397.8526
4 70 70 23.4 10257.70 -270.6211 -789.0604
5 37 37 12.4 11210.03 681.7060 -1273.955
6 30 30 10.0 14161.87 3633.546 2836.744
7 17 17 5.7 16022.71 5494.385 3034.737
8 14 14 4.7 19327.71 8799.393 7518.277
STANDARD
CLASS ADJUSTED MEAN DEVIATION
1 9963.590 6006.004
2 8443.139 4868.404
3 10926.17 8730.284
4 9739.261 6009.121
5 9254.365 5760.727
6 13365.06 7470.542
7 13563.06 6769.267
8 18046.60 12470.24
ETA-SQUARE = .203802 BETA-SQUARE .949135E-01
ETA = .451445 BETA .308080
ETA-SQUARE (ADJ) = .184650
ETA (ADJ) = .429709
UNADJUSTED DEVIATION SS = .346507E+10
ADJUSTED DEVIATION SS = .161373E+10
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
*** MULTIPLE R (ADJUSTED) = .67430 MULTIPLE R-SQUARED (ADJUSTED) = .45468
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
*** MULTIPLE R (ADJUSTED) = .67430 MULTIPLE R-SQUARED (ADJUSTED) = .45468