By itself, statistical independence is too strong a condition to apply directly to most survey data. For example, for questions on the FSS in the CPS, the assumption of statistical independence asserts that the probability that an individual endorses or affirms any survey item is independent of whether or not they endorse any other item. On its face, this assumption seems too strong, since it would be expected that the endorsement of one food insecurity question would increase the probability of endorsing other food insecurity questions. However, a modified form of independence, conditional statistical independence, is a more useful idea and is described next.
Two variables, X1 and X2, are statistically independent conditionally given a third variable, Z, if their probabilities satisfy a conditional version of the product rule in equation (3), i.e.,
Equation (4) says that when Z is fixed at (or conditioned to be) a specific value, z, then X1 and X2 are statistically independent, using the product rule. It is possible for variables to be conditionally statistically independent given a third variable but not to be statistically independent themselves. In this circumstance, it is sometimes said that Z “explains” any association or dependence between X1 and X2, because, once the value of Z has been conditioned on or fixed, there is no more association left to explain. A coin tossing example of conditional independence arises if there are two unfair coins. For example, suppose that coin A is biased towards heads and produces heads with probability 2/3, while coin B is biased towards tails and produces heads with probability 1/3. Now the procedure is to pick one of the two coins out of a box at random and then toss it twice. If which coin is being tossed is known, then there is statistical independence between X1 and X2 as before. In this case Z is the coin being tossed, A or B. Conditioning on or knowing which coin was selected makes the results of the two tosses be independent. But if the coin being tossed is unknown, then X1 and X2 are statistically dependent. If, for example, the coin is pulled out of the box at random and tossed and X1 is heads, then it is more likely than not that the coin is A, and therefore X2 is more likely than not to be a heads as well.
One way to understand the role of conditional independence in latent variable models is in terms of measurement models. In this usage, the