Evaluating model performance without validation data

Question

I've found myself in a situation where I have been given a model that has been developed from what I believe to be the "this seems right" approach. The model in question is very complex but it has no associated error margins or tests of correctness. I.e., there is nothing supporting this model to be good.

To verify the model I've generated 100,000 uniformly distributed data points (the model inputs are all categorical) that I ran through the model (to get the categorical output). I've then started to plot each input variable vs the output variable density. I expect a good model to have plots with varying density for each input value, like this one:

(Note, A,B,C,D,E denotes the different values of a input variable and 1,2,3,4 the different values of the output variable.)

What I found however is that on almost all the input variables I get plots with constant densities for each input value, like this:

(Note, A,B,C,D,E denotes the different values of a input variable and 1,2,3,4 the different values of the output variable.)

My question is simply: am I right to say that variables with densities like the ones plotted in the second graph are not contributing to the model? That they do not have any predictive power? Or am I missing something key here in evaluating this model?

Thanks a bunch for any answers and/or comments regarding my little situation!

P.S. Any hints/tricks on how to further evaluate a model without any validation data would be greatly appreciated!

Answer 1

Given you have no validation data the best you can do is to look at the sensitivity of the model to input factors and their interactions. I'd recommend having a look at global sensitivity analysis. Depending on how long it takes you to run your model, you could be interested in variance-based techniques.

I. M. Sobol’ : Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55:271–280, 2001

You can have a look at this question for a presentation of other sensitivity analysis questions.

Answer 2

The simple answer is that if varying an input variable to your model yields no variation in the model's output, then those variables are most likely not contributing to the model. Note that there could be complex interactions in the model, such that variation in an input variable only makes a difference under some conditions (e.g., when another variable is set to a specific value) and that you simply didn't test the model under those conditions. However, it is not entirely clear what your situation is (as others have pointed out), so there may be more going on here.

Answer 3

I am not sure that

I expect a good model to have plots with varying density for each input value

is generally true.

I assume the model was given to you in the form of a $f(\vec{x})$. I suppose that you have some restrictions / constraints on $\vec{x}$ that is $\vec{x} \in X$ where $X$ is the set of all $\vec{x}$'s that match the constraint.

In this case you could sample N $(\vec{x}_1, \ldots \vec{x}_N)$ from $X$, like what you mentioned, then compute $y^*_i = f(\vec{x_i}) $ this will give you $(y^*_1 \ldots y^*_N)$ sequence. You could then check if the sequence $(y^*_1 \ldots y^*_N)$ is a random or not. For instance if $y$ is binary you could calculate the probability that the sequence $(y^*_1 \ldots y^*_N)$ is drawn from a Bernoulli distribution.