Identifying the Sufficient Set in a Directed Acyclic Graph: The Clock-and-Grid Approach

Background and Purpose: Social work researchers conducting causal analyses of risk factors and interventions can use directed acyclic graphs (DAGs) as a visual aid to represent assumptions about how the hypothesized causal effect is affected by confounders not under the researcher’s control. These confounders represent “backdoor paths” that terminate at the exposure, and carry non-causal association. These backdoor paths must be accounted for, e.g., through regression adjustment, to estimate an unbiased causal effect of the exposure. A set of confounders that, when conditioned on, gives us a causal effect is called a “sufficient set”. DAGs often consist of numerous confounders and backdoor paths, and it can be challenging to (1) ensure that all of the backdoor paths are identified and (2) extract from these backdoor paths one or more sufficient sets. A structured and thorough approach is needed. This is the second step, of three, in using DAGs in social work research.

Methods: In this oral presentation we discuss the “Clock and Grid” (C&G) method, a graphical approach to identifying the inventory of backdoor paths and then subsequently reducing this inventory to one or more sufficient sets. The Clock approach minimizes the risk of overlooking backdoor paths by using a counterclockwise motion to exhaustively follow all paths into and out of each variable, starting from the exposure and ending at the outcome. This counterclockwise motion around each variable is repeated until it returns to the starting point. We enter the inventory of paths into a table, with one row for each path. The Grid of variables is constructed from the inventory, listing each variable in its own column. A structured approach, of circling variables and crossing out rows in which these variables appear, identifies variables that must be included in the sufficient set. When complete, all sufficient sets will be identified.

Results: We demonstrate several examples to which we have applied the C&G approach. These include published DAGs in the didactic literature and in social work research. C&G correctly inventories the backdoor paths and identifies the sufficient sets that these authors found. We then applied C&G to examples from our own work, helping us to identify the set of variables that must be conditioned. In many cases, easy-to-miss paths, overlooked using an ad hoc approach, were found.

Conclusions and Implications: Social work researchers are beginning to rely on DAGs as a tool for causal inference. They can aid in measuring or identifying appropriate variables and choosing a method that might best eliminate backdoor paths. They can also be used, as shown by the C&G approach, to identify covariates for conditioning in regression. This has the potential to encourage the field to undertake more and improved causal research, in both research design and analyses of secondary data. Identifying the backdoor paths and sufficient sets can be challenging when confronted with complex DAGs. The C&G approach described here includes a thorough, ordered approach to identifying paths that is intended to minimize errors and extract the sufficient sets from the inventory.