# Sensitivity analysis for Ad exchange

I’ve created a sensitivity analysis function using Sovan Lek who developed the approach in the mid-1990s. The ‘Lek-profile method’ is described briefly in Lek et al. 19962 and in more detail in Gevrey M, Dimopoulos I, Lek S. 2003. Review and comparison of methods to study the contribution of variables in artificial neural network models. Ecological Modelling. 160:249-264.

In fact, the profile method can be extended to any statistical model and is not specific to neural networks, although it is one of few methods used to evaluate the latter. For any statistical model where multiple response variables are related to multiple explanatory variables, we choose one response and one explanatory variable. We obtain predictions of the response variable across the range of values for the given explanatory variable. All other explanatory variables are held constant at a given set of respective values (e.g., minimum, 20th percentile, maximum). The final product is a set of response curves for one response variable across the range of values for one explanatory variable, while holding all other explanatory variables constant. This is implemented in R by creating a matrix of values for explanatory variables where the number of rows is the number of observations and the number of columns is the number of explanatory variables. All explanatory variables are held at their mean (or other constant value) while the variable of interest is sequenced from its minimum to maximum value across the range of observations. This matrix (actually a data frame) is then used to predict values of the response variable from a fitted model object. This is repeated for different variables.

I’ll illustrate the function using Ad exchange optimisation data, a problem I’m currently working for a costumer, using the R package NNET. The response variable is the eCPM – effective cost per mile. The relationships between the variables are determined a set of parameters CPC, CTR, etc). The explanatory variables are partially correlated and taken from a multivariate normal distribution.