Isolating bias in association indices

Association indices have been a mainstay of social behaviour analysis for decades. However, researchers have long recognized that these indices can be biased under certain conditions. In this paper, I develop a process map of the steps necessary to transform social behaviour into estimates of association rates. This helps to distinguish the subject population's behaviour from the researcher's data collection protocol. By doing this, we can isolate the sources of bias. I also show that bias in association indices is often a function of the true association rate. This means that while bias does not affect the ordering of associations, it can impact analysis in unpredictable ways. Performing network analysis with biased association indices can lead researchers to arrive at different conclusions than if they had used unbiased estimators. To simplify the mathematical task of deriving unbiased estimators, I introduce three properties of maximum likelihood estimators that allow one to treat association data as output from a multinomial distribution, then use the functional invariance property of maximum likelihood estimators to solve for estimators. I apply these properties to a selection of common data collection protocols to show that there is no single association index that is appropriate for all cases. Instead, each of the commonly used indices is unbiased under appropriate conditions. Furthermore, when it is possible that some of the individuals are not identified, I introduce some new unbiased estimators. I close with a discussion of nontraditional techniques of collecting data that provide an opportunity to increase the number of outputs from the data collection process. These techniques may ultimately make it possible to specify association behaviour more carefully by allowing for more parameters in the data generation process.