Granger causality analysis has become very popular in neuroscience in recent years. I will briefly describe here what it is and how it works:

Granger causality is a metric that can assess the amount of causal influence one thing has upon another. For example, the quality of the banana harvest will have a causal effect upon the price of bananas. There may be several causal variables. For example, the quality of banana harvest in India, the exchange rate between India and England, and the price of fuel for transport ships will all affect the price of a banana in England. The causal variables are called independent variables and the effect variable (price of banana in England) is called the dependant variable. Granger causality analysis will allow one to assess how much causal influence a particular causal variable has on the effect variable.

In order to perform Granger causal analysis one must first perform two regression analyses. Regression is a statistical method that allows one to model a system by assessing the relationship between independent variables and the dependant variable. Basically one has a set of sample data output from the system (e.g. values for harvest quality, fuel price, the banana price in England etc) and an equation one wishes to use to model the system. In that equation are several variable parameters that describe a particular relationship between the independent variables as well as between the independent variables and the dependant variable. Computational techniques are used to find the best values to set the parameters to so that the model fits the sample data. In the graph below the scattered dots are different samples for which the independent variable x and dependant variable y are plotted. The line running through them is the plot of a model equation for which a parameter has been fitted so that the line runs though the sample in a way that models the system well.

Simple linear regresion model fitted to sample data

Once we have a model the independent variables can be varied thus allowing one to see and predict the effect of changes on the system.

In order to perform Granger causality analysis one first builds a regression model which makes use of all the independent variables and then assesses how good the model is at making predictions. Next one builds another regression model the same as the first but this time with one of the independent causal variables removed from the model. The latter model is then tested for how good it is at making predictions. Now that one has a value for the prediction quality of each of the models one can use the discrepancy between the two values as a metric for how much causal influence the removed causal variable has.

Anil Seth from Sussex University has developed a MatLab toolbox for Granger causality which is reviewed here, and available here.