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Local independence
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<h2 id="example">Example</h2> <p>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: </p> <table><tbody><tr><td></td><td>Read A</td><td>Did not read A</td><td>Total</td></tr><tr><td>Read B</td><td>260</td><td>140</td><td>400</td></tr><tr><td>Did not read B</td><td>240</td><td>360</td><td>600</td></tr><tr><td>Total</td><td>500</td><td>500</td><td>1000</td></tr></tbody></table> <p>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%=260/500) than non-readers of A (28%=140/500). 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. </p><p>If the analysis is extended to also look at the education level of these people, the following tables are found. </p> <table><tbody><tr><td><table><tbody><tr><td>High education</td><td>Read A</td><td>Did not read A</td><td>Total</td></tr><tr><td>Read B</td><td>240</td><td>60</td><td>300</td></tr><tr><td>Did not read B</td><td>160</td><td>40</td><td>200</td></tr><tr><td>Total</td><td>400</td><td>100</td><td>500</td></tr></tbody></table></td><td><table><tbody><tr><td>Low education</td><td>Read A</td><td>Did not read A</td><td>Total</td></tr><tr><td>Read B</td><td>20</td><td>80</td><td>100</td></tr><tr><td>Did not read B</td><td>80</td><td>320</td><td>400</td></tr><tr><td>Total</td><td>100</td><td>400</td><td>500</td></tr></tbody></table></td></tr></tbody></table> <p>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. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Conditional_independence/udUXFaMw">Conditional independence</a></li></ul> <ul><li>Lazarsfeld, P.F., and Henry, N.W. (1968) <i>Latent Structure analysis.</i> Boston: Houghton Mill.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Henning, G. (1989). "Meanings and implications of the principle of local independence". <i>Language Testing</i>. 6 (1): 95–108. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1177%2F026553228900600108">10.1177/026553228900600108</a>.</li></ul> <h2 id="external-links">External links</h2> <a href="http://www.statisticalinnovations.com/articles/Locindep.pdf"><i>Local independence</i> by Jeroen K. Vermunt & Jay Magidson</a> <a href="http://www.rasch.org/rmt/rmt53b.htm">Local Independence and Latent Class Analysis</a>