Bishop, Fienberg, and Holland (1975) present data from D. Friedlander that shows the distribution of women in England and Wales according to their marital status in 1957. One year later, new official marginal estimates were announced. The problem is to adjust the entries in the 1957 table so as to fit the new marginals while retaining the interaction between cells. This problem can arise when you have internal cells that are known from sampling a population and then get margins based on a complete census.

When you want to adjust an observed table of cell frequencies to a new set of margins, you must set the initab parameter to be the table of observed values. The new marginals are specified through the table argument. The particular cell values for table are not important, since only the marginals are used (the proportionality between cells is determined by initab).

There are two easy ways to create a table that contains given margins. Recall that a table of independent variables has an expected cell value . Thus you could form a table with these cell entries. Another possibility is to use a “greedy algorithm” to assign as many of the marginals as possible to the first cell, then assign as many of the remaining marginals as possible to the second cell, and so on until all of the marginals have been distributed. Both of these approaches are encapsulated into modules in the following program:

/* Return a table such that cell (i,j) has value
   (sum of row i)(sum of col j)/(sum of all cells) */
start GetIndepTableFromMargins( bottom, side );
   if bottom[+] ^= side[+] then do;
      print "Marginal totals are not equal";
      abort;
   end;
   table = side*bottom/side[+];
   return (table);
finish;

/* Use a "greedy" algorithm to create a table whose
   marginal totals match given marginal totals.
   Margin1 is the vector of frequencies totaled down
           each column. Margin1 means that
           Variable 1 has NOT been summed over.
   Margin2 is the vector of frequencies totaled across
           each row. Margin2 means that Variable 2
           has NOT been summed over.
   After calling, use SHAPE to change the shape of
           the returned argument. */
start GetGreedyTableFromMargins( Margin1, Margin2 );
   /* copy arguments so they are not corrupted */
   m1 = colvec(Margin1); /* colvec is in IMLMLIB */
   m2 = colvec(Margin2);
   if m1[+] ^= m2[+] then do;
      print "Marginal totals are not equal";
      abort;
   end;
   dim1 = nrow(m1);
   dim2 = nrow(m2);
   table = j(1,dim1*dim2,0);
   /* give as much to cell (1,1) as possible,
      then as much as remains to cell (1,2), etc,
      until all the margins have been distributed */
   idx = 1;
   do i2 = 1 to dim2;
      do i1 = 1 to dim1;
         t = min(m1[i1],m2[i2]);
         table[idx] = t;
         idx = idx + 1;
         m1[i1] = m1[i1]-t;
         m2[i2] = m2[i2]-t;
      end;
   end;
   return (table);
finish;

Mod = {0.01 15}; /* tighten stopping criterion */

Columns = {"  Single" " Married" "Widow/Divorced"};
Rows    = {"15 - 19" "20 - 24" "25 - 29" "30 - 34"
           "35 - 39" "40 - 44" "45 - 49" "50 Or Over"};

/* Marital status has 3 levels. Age has 8 levels */
Dim = {3 8};

/* Use known distribution for start-up values */
IniTab = { 1306  83    0 ,
           619  765    3 ,
           263 1194    9 ,
           173 1372   28 ,
           171 1393   51 ,
           159 1372   81 ,
           208 1350  108 ,
          1116 4100 2329 };

/* New marginal totals for age by marital status */
NewMarital = { 3988 11702 2634 };
NewAge = {1412,1402,1450,1541,1681,1532,1662,7644};

/* Create any table with these marginals */
Table = GetGreedyTableFromMargins(NewMarital, NewAge);
Table = shape(Table, nrow(IniTab), ncol(IniTab));

/* Consider all main effects */
Config = {1 2};

call ipf(Fit, Status, Dim, Table, Config, IniTab, Mod);

if Status[1] = 0 then
   print "Known Distribution (1957)",
      IniTab [colname=Columns rowname=Rows format=8.0],,
      "Adjusted Estimates of Distribution (1958)",
      Fit [colname=Columns rowname=Rows format=8.2];
else
   print "IPF did not converge in "
         (Status[3]) " iterations";

The results of this program are shown in Figure 24.170. The same results are obtained if the table parameter is formed by using the “independent algorithm.”

A similar technique can be used to standardize data from raw counts into percentages. For example, consider data from a 1836 vote in the U.S. House of Representatives on a resolution that the House should adopt a policy of tabling all petitions for the abolition of slavery. Attitudes toward abolition were different among slaveholding states that would later secede from the Union (“the South”), slaveholding states that refused to secede (“the Border States”), and nonslaveholding states (“the North”).

The raw votes for the resolution are defined in the following statements. The data are hard to interpret because the margins are not homogeneous.

/*         Yea Abstain Nay */
IniTab = {  61    12    60,   /* North */
            17     6     1,   /* Border */
            39    22     7 }; /* South */

Standardizing the data by specifying homogeneous margins reveals interactions and symmetry that were not apparent in the raw data. Suppose the margins are specified as follows:

NewVotes   = {100 100 100};
NewSection = {100,100,100};

In this case, the program for marital status by age can be easily rewritten to adjust the votes into a standardized form. The resulting output is shown in Figure 24.171:

The “greedy algorithm” presented in the Marital-Status-By-Age example can be extended in a natural way to the case where you have one-way marginals and want to form an -dimensional table. For example, a three-dimensional “greedy algorithm” would allocate the vector table as table=j(dim1*dim2*dim3,1,0); and have three nested loops as indicated in the following statements. Afterwards, the table parameter can be reshaped by using the SHAPE function.

do i3 = 1 to dim3;
   do i2 = 1 to dim2;
      do i1 = 1 to dim1;
         t = min(m1[i1],m2[i2],m3[i3]);
         table[idx] = t;
         idx = idx + 1;
         m1[i1] = m1[i1]-t;
         m2[i2] = m2[i2]-t;
         m3[i3] = m3[i3]-t;
      end;
   end;
end;

The idea of the “greedy algorithm” can be extended to marginals that are not one-way. For example, the following three-dimensional table is similar to one that appears in Christensen (1997) based on data from M. Rosenberg. The table presents data on a person’s self-esteem for people classified according to their religion and their father’s educational level.

Since the father’s education level is nested across columns, it is Variable 1 with levels that correspond to not finishing high school, graduating from high school, attending college, graduating from college, and attending graduate courses. The variable that varies the quickest across rows is Self-Esteem, so Self-Esteem is Variable 2 with values “High” and “Low.” The Religion variable is Variable 3 with values “Catholic,” “Jewish,” and “Protestant.”

The following program encodes this table by using the MARG call to compute a two-way marginal table by summing over the third variable, and a one-way marginal by summing over the first two variables. Then a new table (NewTable) is created by applying the greedy algorithm to the two marginals. Finally, the marginals of NewTable are computed and compared with those of table.

dim={5 2 3};
table={
/* Father's Education:
           NotHSGrad HSGrad Col ColGrad PostCol
          Self-
   Relig  Esteem                           */
/* Cath-   Hi */ 575   388  100    77   51,
/* olic    Lo */ 267   153   40    37   19,

/* Jew-    Hi */ 117   102   67    87   62,
/*  ish    Lo */  48    35   18    12   13,

/* Prot-   Hi */ 359   233  109   197   90,
/* estant  Lo */ 159   173   47    82   32
        };

config = { 1 3,
           2 0 };
call marg(locmar, marginal, dim, table, config);
print locmar, marginal, table;

/* Examine marginals: The name indicates the
   variable(s) that are NOT summed over.
   The locmar variable tells where to index
   into the marginal variable. */
Var12_Marg = marginal[1:(locmar[2]-1)];
Var12_Marg = shape(Var12_Marg,dim[2],dim[1]);
Var3_Marg  = marginal[locMar[2]:ncol(marginal)];

NewTable = j(nrow(table),ncol(table),0);
/* give as much to cell (1,1,1) as possible,
   then as much as remains to cell (1,1,2), etc,
   until all the margins have been distributed. */
idx = 1;
do i3 = 1 to dim[3];        /* over Var3 */
   do i2 = 1 to dim[2];     /* over Var2 */
      do i1 = 1 to dim[1];  /* over Var1 */
         /* Note Var12_Marg has Var1 varying across
            the columns */
         t = min(Var12_Marg[i2,i1],Var3_Marg[i3]);
         NewTable[idx] = t;
         idx = idx + 1;
         Var12_Marg[i2,i1] = Var12_Marg[i2,i1]-t;
         Var3_Marg[i3] = Var3_Marg[i3]-t;
      end;
   end;
end;

call marg(locmar, NewMarginal, dim, table, config);
maxDiff = abs(marginal-NewMarginal)[<>];
if maxDiff=0 then
   print "Marginals are unchanged";
print NewTable;

Korff, Taback, and Beard (1952) reported statistics related to the outbreak of food poisoning at a company picnic. A total of 304 people at the picnic were surveyed to determine who had eaten either of two suspect foods: potato salad and crabmeat. The predictor variables are whether the individual ate potato salad (Variable 1: “Yes” or “No”) and whether the person ate crabmeat (Variable 2: “Yes” or “No”). The response variable is whether the person was ill (Variable 3: “Ill” or “Not Ill”). The order of the variables is determined by the dim and table arguments to the IPF subroutine. The variables are nested across columns, then across rows.

The following program defines the variables and observations, and then fits three separate models. How well each model fits the data is determined by computing a Pearson chi-square statistic , where the sum is over all cells, stands for the observed cell count, and stands for the fitted estimate. Other statistics, such as the likelihood-ratio chi-square statistic , could also be used.

The program first fits a model that excludes the three-way interaction. The model fits well, so you can conclude that an association between illness and potato salad does not depend on whether an individual ate crabmeat. The next model excludes the interaction between potato salad and illness. This model is rejected with a large chi-square value, so the data support an association between potato salad and illness. The last model excludes the interaction between the crabmeat and the illness. This model fits moderately well.

/* Compute a chi-square score for a table of observed
   values, given a table of expected values. Compare
   this score to a chi-square value with given degrees
   of freedom at 95% confidence level. */
start ChiSqTest( obs, model, degFreedom );
   diff = (obs - model)##2 / model;
   chiSq = diff[+];
   chiSqCutoff = cinv(0.95, degFreedom);
   print chiSq chiSqCutoff;
   if chiSq > chiSqCutoff then
      print "Reject hypothesis";
   else
      print "No evidence to reject hypothesis";
finish;

dim={2 2 2};

/* Crab meat:   Y E S        N O
   Potato:   Yes   No      Yes  No   */
table={      120    4      22   0,   /* Ill */
              80   31      24  23 }; /* Not Ill */

crabmeat = "          C R A B      N O  C R A B";
potato  = {"YesPot" "NoPot" "YesPot" "NoPot"};
illness = {"Ill", "Not Ill"};

hypoth = "Hypothesis: no three-factor interaction";
config={1  1  2,
        2  3  3};
call ipf(fit,status,dim,table,config);

print hypoth, "Fitted Model:",
      fit[label=crabmeat colname=potato
          rowname=illness format=6.2];
run ChiSqTest(table, fit, 1); /* 1 deg of freedom */

/* Test for interaction between Var 3 (Illness) and
   Var 1 (Potato Salad) */
hypoth = "Hypothesis: no Illness-Potato Interaction";
config={1 2,
        2 3};
call ipf(fit,status,dim,table,config);

print hypoth, "Fitted Model:",
      fit[label=crabmeat colname=potato
          rowname=illness format=6.2];
run ChiSqTest(table, fit, 2); /* 2 deg of freedom */

/* Test for interaction between Var 3 (Illness) and
   Var 2 (Crab meat) */
hypoth = "Hypothesis: no Illness-Crab Interaction";
config={1 1,
        2 3};
call ipf(fit,status,dim,table,config);

print hypoth, "Fitted Model:",
      fit[label=crabmeat colname=potato
          rowname=illness format=6.2];
run ChiSqTest(table, fit, 2); /* 2 deg of freedom */