The " Linear-by- Linear" the test is for ordinal (ordered) categories and assumes equal
and ordered intervals. The Linear-by-Linear Association test is a test for trends in
a larger-than- 2×2 table.
Suppose we investigate whether 78 employees' promotion (yes/no) is related to
their performance ranking in the previous year (1-4, 1=low), as follows:
Ranking 1: Not promoted 17, Promoted 2, Total 19.
Ranking 2: Not promoted 16, Promoted 4, Total 20.
Ranking 3: Not promoted 14, Promoted 6, Total 20.
Ranking 4: Not promoted 10, Promoted 9, Total 19.
SPSS shows a significant linear-by-linear association (p=.008) showing that
there is a significant association between the ranking and being promoted.
Some useful details of how these works are:
1. The test relates to the odds. Odds are used for their statistical properties and
are not quite the same as probabilities. For ranking 1, the odds of being
promoted are 2:17, as opposed to the probability which is 2:19.
2. Then, the test is on the odds ratios; e.g. if you move from rank 1 to rank 2, the
odds ratio is 4:16/2:17 = 0.250/0.118 = 2.12. (The null hypothesis is that the
odds ratio is 1, i.e. a change in ranks makes no difference to the odds.)
3. The procedure presumes that the odds ratios (in the population) are the same
for all steps (i.e. if moving from rank 1 to rank 2 doubles the odds of promotion,
moving from rank 2 to rank 3 would also double the odds of promotion). That is
why there is only 1 degree of freedom. (This assumption is known as " linearity
in the logit".)
4. The test is therefore conceptually the same (and gives a similar answer) to
doing logistic regression with just one covariate. (In logistic regression,
"covariate" means a variable like this one). In this case, the covariate would be
ranking, and the DV would be promotion decision.
By
Siddharth Singh
