Bowker Test of Symmetry

Bowker's test of symmetry checks whether a square table of paired categorical ratings is symmetric, that is whether changes from category i to j are as frequent as changes from j to i.

Bowker 대칭성 검정은 같은 대상을 두 번 분류한 정방 분할표에서 i에서 j로 바뀐 빈도와 j에서 i로 바뀐 빈도가 같은지 검정합니다.

The same patients were classified into three categories before and after treatment; are shifts in one direction more common than in the other?

같은 환자를 치료 전후에 세 범주로 분류했는데, 한쪽 방향으로 바뀐 경우가 더 많은가요?

Bowker's test compares each pair of off-diagonal cells, summing (n_ij − n_ji)² / (n_ij + n_ji) over all pairs of categories, and refers the total to a chi-square distribution with k(k − 1)/2 degrees of freedom. With two categories it reduces to McNemar's test.

When to use Bowker Test of Symmetry

• The same subjects are classified twice into the same set of three or more categories.
• Two raters classify the same items and you want to know whether disagreements run in one direction.
• You want a McNemar-type test beyond two categories.

When to use something else

• You only want to compare the marginal distributions; use the Stuart-Maxwell test.
• The two classifications come from different, independent samples; use a chi-square test of independence.

Assumptions and requirements

• Both classifications use the same categories.
• Pairs are independent of each other.
• Each pair of off-diagonal cells has a reasonable total count (for example 10 or more).

What the analysis reports

• Chi-square statistic, degrees of freedom and p value
• Contribution of each pair of categories
• Number of discordant pairs and the proportion on the diagonal
• Stacked bar chart of the paired table

How to report it

Bowker's test indicated that the [k]×[k] table was not symmetric, χ²([df], N = [value]) = [value], p = [value].

Related methods

• McNemar's Test
• Stuart-Maxwell Marginal Homogeneity Test
• Cohen's Kappa

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