Tuesday, March 25, 2014

Can Graphs Cure Cancer?

Judea Pearl
In my previous post, I introduced the idea of a "graph".  Graphs, are a mathematic representation of relationships involving nodes and directed edges.  Graphs are commonly used in a number of branches of science and mathematics.  Economists, like myself, are familiar with graphs from many many many courses of game theory we took in grad school.  However, it was only recently I became aware of the use of graphs in statistics.  I was an immediate convert.

The foremost proponent of the use of graph theory in statistics is the artificial intelligence researcher Judea Pearl.  Pearl argues that graphs provide a simple, clear and coherent representation of many statistical models.

Consider we are interested in the causal relationship between two observable variables (X) and (Y).  We may be interested in the causal effect of a particular chemotherapy (X) on patient survival (Y).  In Figure 1, the causal relationship between X and Y is represented by the directed edge (arrow) from X to Y.  If our data always looked like this, science would be pretty dang easy.  We see some change in chemo treatment and then the resulting change in patient survival and we know how chemo affects survival.

But no.  Science is not that easy.
Unconfounded Graph

Usually, there is some other variable (U) that is some characteristic of the patient that we don't observe.  This may be some genetic factor that we are unable to measure.  This unobserved characteristic also determines survival.  Importantly, this unobserved characteristic may interact with the chemo treatment.  For example, in colon cancer the drug Cetuximab (Erbitux) is only indicated for treatment of KRAS wild-type metastatic patients.  Imagine if we didn't know this genetic marker, we may see some patients do well on Cetuximab and other patients not so well.  

Worse, we may aggregate across patients who are KRAS wild-type and patients who are KRAS mutated and conclude Cetuximab is much less effective than it really is.

Graphs may not directly cure cancer, but I believe they will help statisticians and lay people better understand what the data says about potential cancer treatments.

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