Summary of relationships between exchangeability, biasing paths and bias

Definitions and conceptualizations of confounding and selection bias have evolved over the past several decades. An important advance occurred with development of the concept of exchangeability. For example, if exchangeability holds, risks of disease in an unexposed group can be compared with risks in an exposed group to estimate causal effects. Another advance occurred with the use of causal graphs to summarize causal relationships and facilitate identification of causal patterns that likely indicate bias, including confounding and selection bias. While closely related, exchangeability is defined in the counterfactual-model framework and confounding paths in the causal-graph framework. Moreover, the precise relationships between these concepts have not been fully described. Here, we summarize definitions and current views of these concepts. We show how bias, exchangeability and biasing paths interrelate and provide justification for key results. For example, we show that absence of a biasing path implies exchangeability but that the reverse implication need not hold without an additional assumption, such as faithfulness. The close links shown are expected. However confounding, selection bias and exchangeability are basic concepts, so comprehensive summarization and definitive demonstration of links between them is important. Thus, this work facilitates and adds to our understanding of these important biases.