

   FFiitt LLoogg--LLiinneeaarr MMooddeellss bbyy IItteerraattiivvee PPrrooppoorrttiioonnaall SSccaalliinngg

        loglm(formula, data=sys.parent(), subset, na.action, ...)

   AArrgguummeennttss::

    formula: A linear model formula specifying the log-linear
             model.

             If the left-hand side is empty, the `data' argu-
             ment is required and must be a (complete) array of
             frequencies.  In this case the variables on the
             right-hand side may be the names of the `dimnames'
             attribute of the frequency array, or may be the
             positive integers: 1, 2, 3, ...{} used as alterna-
             tive names for the 1st, 2nd, 3rd, ...{} dimension
             (classifying factor).  If the left-hand side is
             not empty it specifies a vector of frequencies.
             In this case the data argument, if present, must
             be a data frame from which the left-hand side vec-
             tor and the classifying factors on the right-hand
             side are (preferentially) obtained.  The usual
             abbreviation of a `.' to stand for "all other
             variables in the data frame" is allowed.  Any non-
             factors on the right-hand side of the formula are
             coerced to factor.

       data: Numeric array or data frame.  In the first case it
             specifies the array of frequencies; in then second
             it provides the data frame from which the vari-
             ables occurring in the formula are preferentially
             obtained in the usual way.

             This argument may be the result of a call to
             `crosstabs'.

     subset: Specifies a subset of the rows in the data frame
             to be used.  The default is to take all rows.

   na.action: Specifies a method for handling missing observa-
             tions.  The default is to fail if missing values
             are present.

   keep.frequencies: If `TRUE' specifies that the (possibly
             constructed) array of frequencies is to be
             retained as part of the fitted model object.  The
             default action is to use the same value as that
             used for `fit'.

        ...: May supply other arguments to the function
             `loglin'.

   DDeessccrriippttiioonn::

        This function provides a front-end to the standard
        function, `loglin', to allow log-linear models to be
        specified and fitted in a manner similar to that of
        other fitting functions, such as `glm'.

   DDeettaaiillss::

        If the left-hand side of the formula is empty the
        `data' argument supplies the frequency array and the
        right-hand side of the formula is used to construct the
        list of fixed faces as required by `loglin'.  Struc-
        tural zeros may be specified by giving a `start' argu-
        ment with those entries set to zero, as described in
        the help information for `loglin'.

        If the left-hand side is not empty, all variables on
        the right-hand side are regarded as classifying factors
        and an array of frequencies is constructed.  If some
        cells in the complete array are not specified they are
        treated as structural zeros.  The right-hand side of
        the formula is again used to construct the list of
        faces on which the observed and fitted totals must
        agree, as required by `loglin'.  Hence terms such as
        `a:b', `a*b' and `a/b' are all equivalent.

   VVaalluuee::

        An object of class `loglm' conveying the results of the
        fitted log-linear model.  Methods exist for the generic
        functions `print', `summary', `deviance', `fitted',
        `coef', `resid', `anova' and `update', which perform
        the expected tasks.  Only log-likelihood ratio tests
        are allowed using `anova'.

        The deviance is simply an alternative name for the log-
        likelihood ratio statistic for testing the current
        model within a saturated model, in accordance with
        standard usage in generalized linear models.

   WWAARRNNIINNGG::

        If structural zeros are present, the calculation of
        degrees of freedom may not be correct.  `loglin' itself
        takes no action to allow for structural zeros.  `loglm'
        deducts one degree of freedom for each structural zero,
        but cannot make allowance for gains in error degrees of
        freedom due to loss of dimension in the model space.
        (This would require checking the rank of the model
        matrix, but since iterative proportional scaling meth-
        ods are developed largely to avoid constructing the
        model matrix explicitly, the computation is at least
        difficult.)

        When structural zeros (or zero fitted values) are pre-
        sent the estimated coefficients will not be available
        due to infinite estimates.  The deviances will normally
        continue to be correct, though.

   SSeeee AAllssoo::

        `loglin'

   EExxaammpplleess::

        ### Not usable in R

        # The data frames  Cars93, minn38 and quine are available
        # in the MASS library.

        # Case 1: frequencies specified as an array.
        > sapply(minn38, function(x) length(levels(x)))
         hs phs fol sex f
          3   4   7   2 0
        > minn38a <- array(0, c(3,4,7,2), lapply(minn38[, -5], levels))
        > minn38a[data.matrix(minn38[,-5])] <- minn38$f
        > fm <- loglm(~1 + 2 + 3 + 4, minn38a)  # numerals as names.
        > deviance(fm)
        [1] 3711.9
        > fm1 <- update(fm, .~.^2)
        > fm2 <- update(fm, .~.^3, print = T)
        5 iterations: deviation 0.0750732
        > anova(fm, fm1, fm2)
        LR tests for hierarchical log-linear models

        Model 1:
          ~  1 + 2 + 3 + 4
        Model 2:
         .  ~  1 + 2 + 3 + 4 + 1:2 + 1:3 + 1:4 + 2:3 + 2:4 + 3:4
        Model 3:
         .  ~  1 + 2 + 3 + 4 + 1:2 + 1:3 + 1:4 + 2:3 + 2:4 + 3:4 +
             1:2:3 + 1:2:4 + 1:3:4 + 2:3:4

                  Deviance  df Delta(Dev) Delta(df) P(> Delta(Dev)
          Model 1 3711.915 155
          Model 2  220.043 108   3491.873        47        0.00000
          Model 3   47.745  36    172.298        72        0.00000
        Saturated    0.000   0     47.745        36        0.09114

        # Case 1. An array generated with crosstabs.

        > loglm(~Type + Origin, crosstabs(~Type + Origin, Cars93))
        Call:
        loglm(formula =  ~ Type + Origin, data = crosstabs( ~ Type +
             Origin, Cars93))

        Statistics:
                            X^2 df  P(> X^2)
        Likelihood Ratio 18.362  5 0.0025255
                 Pearson 14.080  5 0.0151101

        # Case 2.  Frequencies given as a vector in a data frame
        > names(quine)
        [1] "Eth"  "Sex"  "Age"  "Lrn"  "Days"
        > fm <- loglm(Days ~ .^2, quine)
        > gm <- glm(Days ~ .^2, poisson, quine)  # check glm.
        > c(deviance(fm), deviance(gm))          # deviances agree
        [1] 1368.7 1368.7
        > c(fm$df, gm$df)                        # resid df do not!
        [1] 127 128
        # The loglm residual degrees of freedom is wrong because of
        # a non-detectable redundancy in the model matrix.

