Loglinear modeling is a method for detecting associations among multiple categorical variables
and is the component of the modified LISREL approach that performs structural analysis.(84 85, ) Using maximum likelihood estimation, the cell frequencies are estimated based on the specified model. The lambdas (λ), or effect parameters, are then determined as part of the loglinear
modeling. A main effect parameter indicates the effect that an individual variable has on the cell frequencies. An interaction effect parameter indicates the presence of an interaction, or association, between two or more variables.(86)
The expected frequencies are used to assess the goodness of fit of the loglinear model by a comparison to the observed frequencies. Either the Pearson chi square statistic (χ2) or the Likelihood Ratio chi square statistic (L2) can be used to assess the fit, although L2 is the preferred statistic. The Likelihood Ratio statistic has additive properties and can be partitioned for testing conditional independence.(87)
There are two versions of loglinear modeling. In the asymmetric version, also known as a
logit analysis, a response variable is assumed or chosen, and the effects of the explanatory
variables and their associations on the response variable are determined. Specifically, the log of the odds of the expected frequencies of the response variable is modeled in terms of the variables and their associations. In the symmetric version, a response variable is not assumed or chosen. Rather, patterns of mutual association among the categorical variables are explored. In a symmetric loglinear model, the log of the expected cell frequency is modeled in terms f the variables and their associations.(88 89, )
3.5.1 Associations in Loglinear Models
The reason for the use of loglinear modeling in this research is to assess the associations among the hazmat variables for development of an accurate network-based model. There are various types of associations among categorical variables that can be determined or measured using loglinear modeling. For example, one can test for either a marginal or partial association between two variables. A marginal association between two variables is determined by summing or collapsing over all other variables in the model. The other variables are in essence
ignored, and the association in the two-way table is assessed exclusively. A partial association between two variables is an association after removing the effects of other variables. Based on this, it is a conservative test. Partial association is related to the concept of conditional
independence. If variables X and Y are conditionally independent given a third variable Z, then a
partial association does not exist between X and Y given Z. This is depicted in the figure below.
X
Z Y
X
Z Y
Figure 3: Conditional Independence of X and Y.
One will notice the absence of an arrow, or arc, from X to Y, indicating conditional independence, or lack of a direct association.(90) The establishment of conditional independence between two variables simplifies an influence diagram or Bayesian network by reducing the number of needed arcs.(91) If there are no associations among variables X, Y, and Z, then the model of mutual independence holds.
3.5.2 Testing Significance of Associations
Marginal and partial associations between variables are determined based on differences in the
L2 statistics of the pertinent loglinear models. This L2 difference is known as a component and also follows the chi square distribution.(92) For example, suppose one wishes to test the significance of a marginal association between variables X and Y in a three-way table for X, Y,
and Z. The statisticfor the model of mutual independence among X, Y, and Z ( ) is compared to the statistic for the model that that additionally contains an interaction term for X and Y ( ), thereby obtaining the component L
2 0 L 2 i L 2. Specifically, . 2 2 0 2 i L L L = −
In addition, the difference in their degrees of freedom is also calculated, as shown below. .
0 dfi
df
df = −
If the component L2 is large relative to its component degrees of freedom (df), then the association between X and Y is significant.(93) In this test of marginal association between X and
Y, the variable Z was ignored, or summed over.(94)
The previous test and loglinear models in general are represented using a conventional notation. For example, for variables X, Y, and Z, the model of mutual independence is represented as follows:
[X] [Y] [Z].
A test of marginal association between X and Y is indicated by a comparison of the above model with the following model, which additionally contains an interaction term for X and Y:
[X] [Y] [Z] [XY].
A test of partial association can also be represented using the conventional notation. To test for a partial association between X and Y in the presence of Z, the following model containing all two- way associations except [XY]:
[X] [Y] [Z] [XZ] [YZ],
is compared to the model additionally containing an association term for X and Y [X] [Y] [Z] [XZ] [YZ] [XY].
If the component L2 is large enough, then a partial association between X and Y exists.(95) Note that the effect of Z was removed by the inclusion of all two-way associations involving Z in the former model.
3.5.3 Assessing Associations in Large, Sparse Tables
Significance testing is problematic in the case of a large, sparse contingency table as well as a large sample size. However, a large, sparse table is often the type of table that investigators work with.(96) Such tables contain many variables or categories and thus many cells with zeros or small cell counts less than five, despite a large sample size. This is problematic for the use of chi-square statistics, such as L2. These statistics are suspect under conditions of sparseness because L2does not follow the chi square distribution in this case.(97) This is also known as Cochran’s Rule.(98)
However, although significance testing is suspect with sparse tables, the existence of associations between variables can still be determined using the effects, or lambda (λ), parameters.(99 100, ) A lambda parameter indicates the strength of an effect, or its importance in explaining any deviation from a flat distribution of the cases among the categories, or cells. Thus, there is a lambda parameter for each combination of the categories of the variables, and a lambda parameter can be positive or negative, with the sign indicating the direction of influence of the effect. For example, in a symmetric loglinear model, a lambda with a positive sign indicates that the effect is responsible for a relative increase in the number of cases in the cell.(101) For an asymmetric model, which assumes a response variable, a lambda with a positive sign indicates increased odds that the response variable equals a given value. In other words, there are a larger proportion of cases associated with the particular value of the response variable.(102 103, ) The lambda parameters are much more robust than a chi-square or standardized
lambda test. The lambdas are also insensitive to sample size if the sample size is not small.(104) In general, two variables are considered not directly associated if the maximum lambda for the variable pair is less than 0.20 in absolute value. Hence, X and Y are not associated if max|λij| < 0.20, where i and j represent any two categories of X and Y, respectively.(105 106, )