In the multiplicative loglinear model the product of all parameters belonging to one effect (e.g., t^XjXj ) is 1. Thus, their values are situated around 1, with no upper bound and a lower bound of zero. A value of 3 represents the same deviation as a value of 0.33, albeit in the opposite direction. To facilitate the intuitive understanding of these values, the natural logarithm (in) is usually applied to make the values more comparable. Working with the In turns the product into an additive combination, which gives the model its name:
p = ln(t!), V = In«), V. = ln(x;), and Xf = In (if)
The parameters of the additive loglinear model are symmetrically situated around zero with no negative or positive limit value. Consequently, the equally strong multiplicative parameters of 3 and 0.33 become In (3) = 1.10 and In (0.33) = -1.10.
Products of multiplicative parameters correspond to sums of additive parameters, and ratios correspond to differences. The model in Equations (1) and (2) is called a saturated model because it implies no constraints on the data; its estimated parameters can be found in Table 17.4. Hence, the model-implied cell frequencies always equal the observed cell frequencies. To generally identify loglinear models the parameters have to be constrained. Usually, the product of the multiplicative parameters has to equal 1 for each effect and, consequently, the sum of parameters belonging to one effect of the additive parameterization has to equal zero.
As stated earlier, multiplicative parameters of one-variable effects indicate the ratio to which the geometric mean of the frequencies pertaining to the three cells of this category differs from the overall geometric mean. For example, the geometric mean of the second category of Rater A (dyslexic) is 0.69 times as large as the overall geometric mean; the geometric mean of the third category of Rater A (normal) is 1.42 times larger than the overall geometric mean. This means that A categorized fewer students as dyslexic than normal. Two-variable effects denote the ratio to which the expected frequency of a particular cell differs from the expectation on the basis of the lower-order effects. For example, the frequency of the cell dyslexic by A and dyslexic by B is 6.11 times as large as expected on the basis of the one-variable effects. The parameters belonging to the same symptoms rated by different raters are all larger than
1 = 2.48, = 6.11,and = 5.75) showing that the ratings are related to each other and that both ratings converge to a certain degree.
Expected cell frequencies depend on the product of the overall effect, the one-variable effects, and the two-variable effect. For example, the expected cell frequency (e22), which is the combination of the ratings dyslexic by A and dyslexic by B, can be computed as
The product of the one- and two-variable parameters indicates that the expected frequency of this particular cell is 2.32 times larger than the overall geometric mean. The expected cell frequency of 15.05 equals the observed cell frequency (15) except for rounding errors.
The T parameters can additionally be used to compare expected frequencies. One-variable effects
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