Local independence
Encyclopedia
Local independence is the underlying assumption of latent variable model
s.
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.
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.
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.
Latent variable model
A latent variable model is a statistical model that relates a set of variables to a set of latent variables.It is assumed that 1) the responses on the indicators or manifest variables are the result of...
s.
The observed items are conditionally independent
Conditional independence
In probability theory, two events R and B are conditionally independent given a third event Y precisely if the occurrence or non-occurrence of R and the occurrence or non-occurrence of B are independent events in their conditional probability distribution given Y...
of each other given an individual score on the latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...
(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 | Low education | Read A | Did not read A | Total | |
Read B | 240 | 60 | 300 | Read B | 20 | 80 | 100 | |
Did not read B | 160 | 40 | 200 | Did not read B | 80 | 320 | 400 | |
Total | 400 | 100 | 500 | 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.
Further reading
- Henning, G. (1989) Meanings and implications of the principle of local independence. Language Testing, Vol. 6 (1), 95–108 DOI: 10.1177/026553228900600108