3-Way Contingency Table

Log-Linear Analysis for an AxBxC Contingency Table

Log-linear analysis is a version of chi-square analysis in which the relevant values are calculated by way of weighted natural logarithms. The first advantage of this procedure is that it is easier to program in the case of a complex 3-way contingency table, since it allows all chi-square values to be derived through simple addition and subtraction of various combinations of the weighted logarithms. The second advantage is that the chi-square values thus derived are linear, which allows for more complex analyses not readily available through the conventional chi-square computational procedure. When a chi-square value is calculated by the log- linear method, it is typically designated as G² as an indication of its computational origin. Since G² is distributed approximately as chi-square, its associated probability under the null hypothesis can be estimated through reference to the appropriate sampling distribution of chi-square, as defined by its degrees of freedom. Values of G² will usually be quite close to the corresponding values of chi-square that would be calculated using the conventional procedure.

For a 3-way contingency table containing up to 5 rows (A), 5 columns (B), and 5 layers (C), the present page will calculate the following seven values of G²:_Q

ABC	Representing the 3-way interaction between A, B, and C.
AB	Representing the 2-way interactions for AB, AC, and BC, respectively. These are the same measures that would be obtained from an AB table collapsed across the levels of C, an AC table collapsed across the levels of B, and a BC table collapsed across the levels of A.
AC
BC
AB(C)	Representing the 2-way interactions for each pair of variables, AB, AC, and BC, when the effects of the third variable are removed from the picture. Thus, AB(C) represents the AB interaction when the AC and BC interactions are removed. It is the same measure that would be obtained by constructing a separate AB table for each level of C, calculating a separate G² measure for each, and then summing the results.
AC(B)
BC(A)

To begin, enter the numbers of rows, columns, and layers in the designated places, then click the «Setup» button and enter your data into the appropriate cells of the data-entry matrices. After all data have been entered, click one of the «Calculate» buttons.

Data Entry

	C1
	B1	B2	B3	B4	B5
A1
A2
A3
A4
A5

	C2
	B1	B2	B3	B4	B5
A1
A2
A3
A4
A5

	C3
	B1	B2	B3	B4	B5
A1
A2
A3
A4
A5

	C4
	B1	B2	B3	B4	B5
A1
A2
A3
A4
A5

	C5
	B1	B2	B3	B4	B5
A1
A2
A3
A4
A5

Results:

Source	G²	df	P
ABC
AB
AC
BC
AB(C)
AC(B)
BC(A)

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