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(s). This means that the latent variable explains why the observed items are related to another. This can be explained by the following example.

Example

Local independence can be explained by an example of Lazarsfeld and Henry (1968). Suppose that a sample of 1000 people asked whether they read journals A and B. Their responses were as follows:

Read A Did not read A Total
Read B 260 140 400
Did not read B 240 360 600
Total 500 500 1000

One can easily see that the two variables (reading A and reading B) are strongly related, and thus dependent on each other. Readers of A tend to read B more often (52%) than non-readers of A (28%). If reading A and B were independent, then the formula P(A&B) = P(A)×P(B) 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 education Read A Did not read A Total
Read B 240 60 300
Did not read B 160 40 200
Total 400 100 500
Low education Read A Did not read A Total
Read B 20 80 100
Did not read B 80 320 400
Total 100 400 500

Again, if reading A and B were independent, then P(A&B) = P(A)×P(B) 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.

See also

References

Further reading

Local independence by Jeroen K. Vermunt & Jay Magidson
Local Independence and Latent Class Analysis
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