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When we speak of likelihood, we mean the probability of some sample data or set of observations conditional on some hypothesized parameter or set of parameters (Everitt, 2002). Conditional probability statements such as p(D/H₀) can very generally be considered simple examples of likelihoods, where typically the set of parameters, in this case, may be simply μ and σ². A likelihood function is the likelihood of a parameter given data (see Fox, 2016).

When we speak of maximum‐likelihood estimation, we mean the process of maximizing a likelihood subject to certain parameter conditions. As a simple example, suppose we obtain 8 heads on 10 flips of a presumably fair coin. Our null hypothesis was that the coin is fair, meaning that the probability of heads is p(H) = 0.5. However, our actual obtained result of 8 heads on 10 flips would suggest the true probability of heads to be closer to p(H) = 0.8. Thus, we ask the question:

Which value of θmakes the observed result most likely?

If we only had two choices of θ to select from, 0.5 and 0.8, our answer would have to be 0.8, since this value of the parameter θ makes the sample result of 8 heads out of 10 flips most likely. That is the essence of how maximum‐likelihood estimation works (see Hays, 1994, for a similar example). ML is the most common method of estimating parameters in many models, including factor analysis, path analysis, and structural equation models to be discussed later in the book. There are very good reasons why mathematical statisticians generally approve of maximum likelihood. We summarize some of their most favorable properties.

Firstly, ML estimators are asymptotically unbiased, which means that bias essentially vanishes as sample size increases without bound (Bollen, 1989). Secondly, ML estimators are consistent and asymptotically efficient, the latter meaning that the estimator has a small asymptotic variance relative to many other estimators. Thirdly, ML estimators are asymptotically normally distributed, meaning that as sample size grows, the estimator takes on a normal distribution. Finally, ML estimators possess the invariance property (see Casella and Berger, 2002, for details).