Within
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Local independence is the underlying assumption of
latent variable model
A latent variable model is a statistical model that relates a set of observable variables (also called ''manifest variables'' or ''indicators'') to a set of latent variables.
It is assumed that the responses on the indicators or manifest variable ...
s.
The observed items are
conditionally independent
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabil ...
of each other given an individual score on the
latent variable
In statistics, latent variables (from Latin: present participle of ''lateo'', “lie hidden”) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or me ...
(s). 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 (1968). Suppose that a sample of 1000 people was asked whether they read journals A and B. Their responses were as follows:
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.
See also
*
Conditional independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabil ...
References
*Lazarsfeld, P.F., and Henry, N.W. (1968) ''Latent Structure analysis.'' Boston: Houghton Mill.
Further reading
*{{cite journal , author=Henning, G. , title=Meanings and implications of the principle of local independence , journal=Language Testing , volume=6 , issue=1 , pages=95–108 , year=1989 , doi=10.1177/026553228900600108
External links
''Local independence'' by Jeroen K. Vermunt & Jay Magidson
Econometric modeling
Independence (probability theory)
Latent variable models