Is "linear-by-linear association" in SPSS another name for the chi-squared test for trend? If not, what is it?
It is. You might have a look at IBM's support page for SPSS, where it is stated in a technote on the Chi² test: 'The Crosstabs procedure includes the Mantel-Haenszel test of trend among its chi-square test statistics. ... The MH test for trend will be printed in the "Chi-Square Tests" table and labelled "Linear-by-Linear Association".'
As a previous reply mentioned, yes it is and the technical description is at SPSS's support page: https://www-304.ibm.com/support/docview.wss?uid=swg21477269
This is a useful statistic for those who understand it. 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 this 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.