# Statistical model validation

Short description: Evaluating whether a chosen statistical model is appropriate or not

In statistics, model validation is the task of evaluating whether a chosen statistical model is appropriate or not. Oftentimes in statistical inference, inferences from models that appear to fit their data may be flukes, resulting in a misunderstanding by researchers of the actual relevance of their model. To combat this, model validation is used to test whether a statistical model can hold up to permutations in the data. This topic is not to be confused with the closely related task of model selection, the process of discriminating between multiple candidate models: model validation does not concern so much the conceptual design of models as it tests only the consistency between a chosen model and its stated outputs.

There are many ways to validate a model. Residual plots plot the difference between the actual data and the model's predictions: correlations in the residual plots may indicate a flaw in the model. Cross validation is a method of model validation that iteratively refits the model, each time leaving out just a small sample and comparing whether the samples left out are predicted by the model: there are many kinds of cross validation. Predictive simulation is used to compare simulated data to actual data. External validation involves fitting the model to new data. Akaike information criterion estimates the quality of a model.

## Overview

Model validation comes in many forms and the specific method of model validation a researcher uses is often a constraint of their research design. To emphasize, what this means is that there is no one-size-fits-all method to validating a model. For example, if a researcher is operating with a very limited set of data, but data they have strong prior assumptions about, they may consider validating the fit of their model by using a Bayesian framework and testing the fit of their model using various prior distributions. However, if a researcher has a lot of data and is testing multiple nested models, these conditions may lend themselves toward cross validation and possibly a leave one out test. These are two abstract examples and any actual model validation will have to consider far more intricacies than describes here but these example illustrate that model validation methods are always going to be circumstantial.

In general, models can be validated using existing data or with new data, and both methods are discussed more in the following subsections, and a note of caution is provided, too.

### Validation with Existing Data

Validation based on existing data involves analyzing the goodness of fit of the model or analyzing whether the residuals seem to be random (i.e. residual diagnostics). This method involves using analyses of the models closeness to the data and trying to understand how well the model predicts its own data. One example of this method is in Figure 1, which shows a polynomial function fit to some data. We see that the polynomial function does not conform well to the data, which appears linear, and might invalidate this polynomial model.

Commonly, statistical models on existing data are validated using a validation set, which may also be referred to as a holdout set. A validation set is a set of data points that the user leaves out when fitting a statistical model. After the statistical model is fitted, the validation set is used as a measure of the model's error. If the model fits well on the initial data but has a large error on the validation set, this is a sign of overfitting, as seen in Figure 1.

Figure 1.  Data (black dots), which was generated via the straight line and some added noise, is perfectly fitted by a curvy polynomial.

### Validation with New Data

If new data becomes available, an existing model can be validated by assessing whether the new data is predicted by the old model. If the new data is not predicted by the old model, then the model might not be valid for the researcher's goals.

With this in mind, a modern approach is to validate a neural network is to test its performance on domain-shifted data. This ascertains if the model learned domain-invariant features.[1]

### A Note of Caution

A model can be validated only relative to some application area.[2][3] A model that is valid for one application might be invalid for some other applications. As an example, consider the curve in Figure 1: if the application only used inputs from the interval [0, 2], then the curve might well be an acceptable model.

## Methods for validating

When doing a validation, there are three notable causes of potential difficulty, according to the Encyclopedia of Statistical Sciences.[4] The three causes are these: lack of data; lack of control of the input variables; uncertainty about the underlying probability distributions and correlations. The usual methods for dealing with difficulties in validation include the following: checking the assumptions made in constructing the model; examining the available data and related model outputs; applying expert judgment.[2] Note that expert judgment commonly requires expertise in the application area.[2]

Expert judgment can sometimes be used to assess the validity of a prediction without obtaining real data: e.g. for the curve in Figure 1, an expert might well be able to assess that a substantial extrapolation will be invalid. Additionally, expert judgment can be used in Turing-type tests, where experts are presented with both real data and related model outputs and then asked to distinguish between the two.[5]

For some classes of statistical models, specialized methods of performing validation are available. As an example, if the statistical model was obtained via a regression, then specialized analyses for regression model validation exist and are generally employed.

### Residual diagnostics

Residual diagnostics comprise analyses of the residuals to determine whether the residuals seem to be effectively random. Such analyses typically requires estimates of the probability distributions for the residuals. Estimates of the residuals' distributions can often be obtained by repeatedly running the model, i.e. by using repeated stochastic simulations (employing a pseudorandom number generator for random variables in the model).

If the statistical model was obtained via a regression, then regression-residual diagnostics exist and may be used; such diagnostics have been well studied.

### Cross validation

Cross validation is a method of sampling that involves leaving some parts of the data out of the fitting process and then seeing whether those data that are left out are close or far away from where the model predicts they would be. What that means practically is that cross validation techniques fit the model many, many times with a portion of the data and compares each model fit to the portion it did not use. If the models very rarely describe the data that they were not trained on, then the model is probably wrong.

## References

1. Feng, Cheng; Zhong, Chaoliang; Wang, Jie; Zhang, Ying; Sun, Jun; Yokota, Yasuto (July 2022). "Learning Unforgotten Domain-Invariant Representations for Online Unsupervised Domain Adaptation". Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization. pp. 2958–2965. doi:10.24963/ijcai.2022/410. ISBN 978-1-956792-00-3.
2. National Research Council (2012), "Chapter 5: Model validation and prediction", Assessing the Reliability of Complex Models: Mathematical and statistical foundations of verification, validation, and uncertainty quantification, Washington, DC: National Academies Press, pp. 52–85, doi:10.17226/13395, ISBN 978-0-309-25634-6 .
3. Batzel, J. J.; Bachar, M.; Kappel, F., eds. (2013), "Chapter 1: Merging mathematical and physiological knowledge", Mathematical Modeling and Validation in Physiology, Springer, pp. 3–19, doi:10.1007/978-3-642-32882-4_1 .
4. Deaton, M. L. (2006), Kotz, S., ed., Simulation models, validation of, Wiley .
5. "Statistical validation", Ecological Modelling 68 (1–2): 21–32, 1993, doi:10.1016/0304-3800(93)90105-2 .