ABSTRACT
The first assumption is concerned with independence of the observations. We discuss this first as it is applicable broadly to inferential procedures, although testing for independence may be approached in different ways given different analytic procedures. As you may recall, there is a connection between sampling method and independence. Simple random sampling is defined as the process of selecting sample observations from a population so that each observation has an equal and independent probability of being selected. If the sampling process is truly random, then (a) each observation in the population has an equal chance of being included in the sample, and (b) each observation selected into the sample is independent of (or not affected by) every other selection. Independence implies that each observation is selected without regard to any other observation sampled. We also would fail to have equal and independent probability of selection if the sampling procedure employed was something other than a simple random sample-because it is only with a simple random sample that we have met the conditions (a) and (b) presented earlier. (Although there are statistical means to correct for nonsimple random samples, such as weighting and variance correction methods, they are beyond the scope of this textbook.) This concept of independence is an important assumption. If we have independence, then generalizations from the sample back to the population can be made (you may remember this as external validity which was likely introduced in your research methods course). Violations of this assumption can detrimentally impact standard error values and thus any resulting hypothesis tests. In particular, even small violations of this assumption can result in a quite dramatically increased actual alpha level as compared to nominal alpha level (Barcikowski, 1981;
Scariano & Davenport, 1987). At minimum, when there is dependence, tests should be conducted at a decreased alpha level (e.g., .01), given that the actual alpha will likely be higher. Lack of independence affects the estimated standard errors of the model. For serious violations, transformations or other estimation procedures can be considered.
