Linear-by-Linear Association Test

The linear-by-linear association test checks for an ordinal trend in an R x C table using M^2 = (N - 1) r^2, where r is the correlation between assigned row and column scores.

선형-선형 연관 검정은 순서형 행·열 범주에 점수를 주고 그 상관 r 로 M² = (N − 1) r² 를 계산해 R × C 표의 순서 추세를 검정하는 방법입니다.

Both my variables are ordered categories; is there a trend where higher levels of one go with higher levels of the other?

두 변수가 모두 순서형 범주인데, 한쪽 수준이 높을수록 다른 쪽도 높아지는 경향이 있는지 검정할 수 있나요?

Assign scores to the ordered row and column categories, compute their Pearson correlation r over the table and test M^2 = (N - 1) r^2 against a chi-square distribution with one degree of freedom. Because it targets a single linear trend, it has more power than the general chi-square test when such a trend exists.

When to use Linear-by-Linear Association Test

• Both variables are ordinal and you expect a monotone association.
• The table is sparse in some cells, so a one-degree-of-freedom test is preferable.
• You want the trend test reported by SPSS Crosstabs as linear-by-linear association.

When to use something else

• Either variable is nominal; use the chi-square test of independence instead.
• The outcome is binary and the groups are ordered with a focus on proportions; the Cochran-Armitage trend test is the usual choice.

Assumptions and requirements

• Independent observations, each counted once (or with frequency weights).
• Category order and score spacing are meaningful.
• Large enough N for the chi-square(1) approximation.

What the analysis reports

• M^2 with df = 1 and p-value
• Score correlation r and signed z with a one-sided p
• Pearson chi-square for comparison
• The contingency table with the scores used

How to report it

The linear-by-linear association test indicated an ordinal trend, M^2(1, N = [value]) = [value], p = [value], r = [value].

Related methods

• Chi-Square Test of Independence
• Cochran-Armitage Trend Test
• Spearman Rank Correlation

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