SEARCH -- SEARCHING FOR STRUCTURE

SEARCH is a binary segmentation procedure used to develop a predictive model for a dependent variable. It searches among a set of predictor variables for those predictors which most increase the researcher's ability to account for the variance or distribution of a depen­dent vari­able. The question, "what dichotomous split on which single predictor variable will give us a maximum improve­ment in our ability to predict values of the dependent variable?," embedded in an iterative scheme, is the basis for the algorithm used in this command.

SEARCH divides the sample, through a series of binary splits, into a mutually exclusive series of subgroups. After each split, every observation is a member of exactly one of these subgroups.  They are chosen so that, at each step in the procedure, the split into the two new sub­groups accounts for more of the variance or distribution (reduces the predictive error more) than a split into any other pair of sub­groups. The predictor variables may be ordinally or nominally scaled. The dependent variable may be continuous or categorical.

Research questions are often of the type "What is the effect of X on Y?" But the answer requires answering a larger question "What set of variables and their combinations seems to affect Y?" With SEARCH a variable X that seems to have an overall effect may have its apparent influence disappear after a few splits, with the final groups, while varying greatly as to their levels of Y, showing no effect of X. The implication is that, given other things, X does not really affect Y.

Conversely, while X may seem to have no overall effect on Y, after splitting the sample into groups that take account of other powerful factors, there may be some groups in which X has a substantial effect. Think of economists' notion of the actor at the margin. A motivating factor might affect those not constrained or compelled by other forces. Those who, other things considered, have a 40-60 percent probability of acting, might show substantial response to some motivator. Or a group with very high or very low likelihood of acting might be discouraged or encouraged by some motivator. But if X has no effect on any of the subgroups generated by Search, one has pretty good evidence that it does not matter, even in an interactive way.

SEARCH makes a sequence of binary divisions of a dataset in such a way that each split maximally reduces the error variance or increases the information (chi-square or rank correlation). It finds the best split on each predictor and takes the best of the best.

The process stops when additional splits are not likely to improve predictions to a fresh sample or to the population, i.e., when the null probability from that split rises above some selected level (e.g., .05, .025, .01 or .005). Of course, having tried several possibilities for each of several predictors, the null probability is clearly understated. Alternative stopping rules can be used in any combination: minimum group size, maximum number of splits, minimum reduction in explained variance relative to the original total, or maximum null probability

Splitting creteria:

There can be four splitting criteria, based on the dependent variable type:

  Means

  Regressions

  Classifications

  Ranks

The splitting criterion in each case is the reduction in ignorance (error variance, etc.) or increase in information. Terms like classification and regression trees should be replaced by binary segmentation or unrestricted analysis of variance components, or searching for structure. With rich bodies of data, many non-linearity’s and non-additivity possible, and many competing theories, the usual restrictions and assumptions that one is testing a single model are not appropriate. What does remain, however, is a systematic, pre-stated searching strategy that is reproducible, not a free ransacking.

Means. For means the splitting criterion is the reduction in error variance, that is, the sum of squares around the mean, using two subgroup means instead of one parent group mean.

Regressions. For regressions (y=a+bx) the splitting criterion is the reduction in error variance from using two regressions rather than one.

Classifications (Chi option). For classifications (categorical dependent variable), the splitting criterion is the likelihood-ratio chi-square for dividing the parent group into two subgroups.

Ranks (Tau option). For rankings (ordered dependent variable), the splitting criterion is Kendall's tau-b, a rank correlation measure.

The major components of output:

The analysis of variance or distribution on final groups (except for “analysis=tau”)

The split summary

The final group summary

Summary table of best splits for each predictor for each group (except for “analysis= tau”)

The predictor summary table. You may request the first group (PRINT=FIRST), the final groups (PRINT=FINAL), or all groups (PRINT=TABLE). The tables are printed in reverse group order, i.e., last group first and first group last.

Group Tree Structure

A structure table with entries for each group, numbered in order and indented, so that one can easily see the pedigree of each final group and its detail.  

References:

Agresti, Alan (1996), Introduction to Categorical Data Analysis, New York: John Wiley & Sons, Inc.
Dunn, Olive Jean, and Virginia A. Clark (1974), Applied Statistics: Analysis of Variance and Regression, New York: Holt, Rinehart and Winston.
Chow, G. (1960), "Test of Equality between Sets of Coefficients in Two Linear Regressions," Econometrica, 29:591-605.
Gibbons, Jean Dickinson (1997), Nonparametric Methods for Quantitative Analysis, 3rd edition, Syracuse: American Sciences Press.
Hays, William (1988), Statistics, 4th edition, New York: Holt, Rinehart, & Winston.
Klem, Laura (1974), "Formulas and Statistical References," in Osiris III, Volume 5, Ann Arbor: Institute for Social Research.
Sonquist, J. A., E. L. Baker and J. N. Morgan (1974), Searching for Structure, revised edition, Ann Arbor: Institute for Social Research, The University of Michigan.

 

Example:   Investigates income (V268)

 

 

ANALYSIS TYPE: MEANS    

 

Dependent variable: 268 Income

 

Predictor variables: 32 37 251 30

 

The number of cases is     326

 

The partitioning ends with 9 final groups

 

The variation explained is 38.2 percent

 

One-way Analysis of Final Groups

 

  Source       Variation          DF

 

  Explained  .701177E+10           8

  Error      .113438E+11         317

  Total      .183555E+11         325

 

Split Summary Table

 

Group 1, N=326

  Mean(Y)=10451.0, Var(Y)=.564786E+08, Variation=.183555E+11

  Split on V37: RACE, Var expl=.216040E+08, Significance=.544344

  Into Group 2, Codes 1

   And Group 3, Codes 0,2-9

 

Group 2, N=299

  Mean(Y)=10528.3, Var(Y)=.570540E+08, Variation=.170021E+11

  Split on V30: MARITAL STATUS, Var expl=.312812E+10, Significance=0.000100

  Into Group 4, Codes 1

   And Group 5, Codes 2-5

 

Group 4, N=221

  Mean(Y)=12449.9, Var(Y)=.571999E+08, Variation=.125840E+11

  Split on V32: EDUC OF HEAD, Var expl=.173944E+10, Significance=0.000100

  Into Group 6, Codes 1-5

   And Group 7, Codes 6-8

 

Group 6, N=171

  Mean(Y)=10932.9, Var(Y)=.430128E+08, Variation=.731217E+10

  Split on V251: OCCUPATION B, Var expl=.140900E+10, Significance=0.000100

  Into Group 8, Codes 0

   And Group 9, Codes 1-9

 

Group 9, N=142

  Mean(Y)=12230.1, Var(Y)=.402303E+08, Variation=.567247E+10

  Split on V251: OCCUPATION B, Var expl=.423362E+09, Significance=0.001380

  Into Group 10, Codes 1-3

   And Group 11, Codes 4-9

 

Group 11, N=115

  Mean(Y)=11393.4, Var(Y)=.249652E+08, Variation=.284603E+10

  Split on V251: OCCUPATION B, Var expl=.495146E+08, Significance=.156284

  Into Group 12, Codes 4-6

   And Group 13, Codes 7-9

 

Group 12, N=69

  Mean(Y)=11929.2, Var(Y)=.212965E+08, Variation=.144816E+10

  Split on V32: EDUC OF HEAD, Var expl=.571610E+08, Significance=0.097853

  Into Group 14, Codes 1-3

   And Group 15, Codes 4,5

 

Group 5, N=78

  Mean(Y)=5083.86, Var(Y)=.167531E+08, Variation=.128999E+10

  Split on V251: OCCUPATION B, Var expl=.183562E+09, Significance=0.000992

  Into Group 16, Codes 0

   And Group 17, Codes 1,2,4-9

 

Final Group Summary Table

 

Group 3, N=27

  Mean(Y)=9594.30, Var(Y)=.512249E+08, Variation=.133185E+10

 

Group 7, N=50

  Mean(Y)=17638.2, Var(Y)=.720890E+08, Variation=.353236E+10

 

Group 8, N=29

  Mean(Y)=4580.97, Var(Y)=.823915E+07, Variation=.230696E+09

 

Group 10, N=27

  Mean(Y)=15793.6, Var(Y)=.924261E+08, Variation=.240308E+10

 

Group 13, N=46

  Mean(Y)=10589.8, Var(Y)=.299634E+08, Variation=.134835E+10

 

Group 14, N=28

  Mean(Y)=13030.6, Var(Y)=.309307E+08, Variation=.835128E+09

 

Group 15, N=41

  Mean(Y)=11177.0, Var(Y)=.138968E+08, Variation=.555873E+09

 

Group 16, N=35

  Mean(Y)=3383.49, Var(Y)=.515942E+07, Variation=.175420E+09

 

Group 17, N=43

  Mean(Y)=6467.88, Var(Y)=.221668E+08, Variation=.931006E+09

 

Percent Total Variation Explained by Best Split for Each Group (*=Final Groups)

 

          1      2     3*      4      5      6     7*     8*      9    10*

V32   12.00  11.90   0.00   9.48   0.86   3.62   0.00   0.00   0.68   0.00

V37    0.12   0.00   0.00   0.00   0.00   0.00   0.00   0.00   0.00   0.00

V251  18.12  16.90   0.00   9.14   1.00   7.68   0.00   0.00   2.31   0.00

V30   17.92  17.04   0.00   0.00   0.00   0.00   0.00   0.00   0.00   0.00

 

Percent Total Variation Explained by Best Split for Each Group (*=Final Groups) - continued

 

         11     12    13*    14*    15*    16*    17*

V32    0.16   0.31   0.00   0.00   0.00   0.00   0.00

V37    0.00   0.00   0.00   0.00   0.00   0.00   0.00

V251   0.27   0.01   0.00   0.00   0.00   0.00   0.00

V30    0.00   0.00   0.00   0.00   0.00   0.00   0.00

 

Group TREE Structure

 

Group 1: All Cases

  N=326, Mean(Y)=10451.0

  Group 2 V37: RACE, Codes 1

    N=299, Mean(Y)=10528.3

    Group 4 V30: MARITAL STATUS, Codes 1

      N=221, Mean(Y)=12449.9

      Group 6 V32: EDUC OF HEAD, Codes 1-5

        N=171, Mean(Y)=10932.9

        Group 8 V251: OCCUPATION B, Codes 0

          N=29, Mean(Y)=4580.97

        Group 9 V251: OCCUPATION B, Codes 1-9

          N=142, Mean(Y)=12230.1

          Group 10 V251: OCCUPATION B, Codes 1-3

            N=27, Mean(Y)=15793.6

          Group 11 V251: OCCUPATION B, Codes 4-9

            N=115, Mean(Y)=11393.4

            Group 12 V251: OCCUPATION B, Codes 4-6

              N=69, Mean(Y)=11929.2

              Group 14 V32: EDUC OF HEAD, Codes 1-3

                N=28, Mean(Y)=13030.6

              Group 15 V32: EDUC OF HEAD, Codes 4,5

                N=41, Mean(Y)=11177.0

            Group 13 V251: OCCUPATION B, Codes 7-9

              N=46, Mean(Y)=10589.8

      Group 7 V32: EDUC OF HEAD, Codes 6-8

        N=50, Mean(Y)=17638.2

    Group 5 V30: MARITAL STATUS, Codes 2-5

      N=78, Mean(Y)=5083.86

      Group 16 V251: OCCUPATION B, Codes 0

        N=35, Mean(Y)=3383.49

      Group 17 V251: OCCUPATION B, Codes 1,2,4-9

        N=43, Mean(Y)=6467.88

  Group 3 V37: RACE, Codes 0,2-9

    N=27, Mean(Y)=9594.30