Local independence


Local independence is the underlying assumption of latent variable models.
The observed items are conditionally independent of each other given an individual score on the latent variable. This means that the latent variable explains why the observed items are related to one another. This can be explained by the following example.

Example

Local independence can be explained by an example of Lazarsfeld and Henry. Suppose that a sample of 1000 people was asked whether they read journals A and B. Their responses were as follows:
Read ADid not read ATotal
Read B260140400
Did not read B240360600
Total5005001000

One can easily see that the two variables are strongly related, and thus dependent on each other. Readers of A tend to read B more often than non-readers of A. If reading A and B were independent, then the formula P = P×P would hold. But 260/1000 isn't 400/1000 × 500/1000. Thus, reading A and B are statistically dependent on each other.
If the analysis is extended to also look at the education level of these people, the following tables are found.
High educationRead ADid not read ATotal
Read B24060300
Did not read B16040200
Total400100500

Low educationRead ADid not read ATotal
Read B2080100
Did not read B80320400
Total100400500

Again, if reading A and B were independent, then P = P×P would hold separately for each education level. And, in fact, 240/500 = 300/500×400/500 and 20/500 = 100/500×100/500. Thus if a separation is made between people with high and low education backgrounds,
there is no dependence between readership of the two journals. That is, reading A and B are independent once educational level is taken into consideration. The educational level 'explains' the difference in reading of A and B. If educational level is never actually observed or known, it may still appear as a latent variable in the model.