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When preparing data for use with Covariance-based Structural Equation Modeling (CB-SEM), what are the different effects of replacing missing data with mean or median? When is one of them more appropriate than the other? Is any of them generally recommended?

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The general effect of replacing missing values with means or medians is to give you wrong results and neither is generally recommended. Better methods are things like Multiple Imputation or the EM algorythm (or both) to estimate the covariance matrix and take into account the uncertainty due to the missing information. But before using either of those methods you need to understand the science and your data enough to decide on why the information is missing and decide if you are willing to assume that the missing data is missing completely at random, missing at random, or missing not at random.

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