Advanced Modeling
What is advanced modeling?
Advanced modeling includes statistical methods that go beyond traditional linear models such as simple regression, multiple regression, ANOVA, and others. These methods help analysts answer complex questions, work with non-normal data, account for complicated data structures, and uncover patterns that simple models cannot capture.
Advanced models are useful when your data involve categorical outcomes, repeated measurements, an overparameterized model (fewer observations than the model supports), time-dependent observations, complex hierarchical structures, functional curves, or specialized signals such as spectra and chromatograms.
When should you use advanced modeling?
Consider advanced modeling when:
- The response variable is binary, categorical, a count, or highly skewed.
- You have many potential predictors and want to reduce overfitting.
- Data are grouped, nested, or collected repeatedly from the same experimental units.
- Observations are collected over time.
Each observation is a curve, profile, spectrum, or signal.
Common advanced modeling methods
Generalized linear models (GLMs)
Generalized linear models extend linear models to handle responses that do not follow a normal distribution. GLMs allow you to model binary outcomes, counts, rates, proportions, and other types of data by combining a linear predictor with an appropriate distribution and link function. Logistic regression and Poisson regression are common examples of GLMs.
Example: Predicting whether a customer will purchase a product (yes/no) or modeling the number of defects produced in a manufacturing process.
Penalized regression
Penalized regression methods fit linear models by adding a penalty to the error function and are useful in the following situations:
- You want to perform automatic model term selection on a large set of predictors.
- Your predictors are correlated; that is, there is collinearity among the predictors.
- You don’t have enough data to estimate parameters with good precision, or at all (n < p).
Common approaches include ridge regression, LASSO, and elastic net methods.
Example: Identifying the most important process variables from hundreds of sensor measurements.
Mixed models
Mixed models extend linear models by incorporating both fixed effects and random effects. They are useful when observations are grouped or correlated, such as measurements collected from multiple batches, operators, sites, fields, or equipment. They are also useful for studies with repeated measurements. Mixed models provide more accurate estimates by accounting for variability introduced by these groups.
Example: Comparing treatments while accounting for differences among manufacturing lots or experimental fields.
Generalized linear mixed models (GLMMs)
GLMMs combine the strengths of GLMs and mixed models. They allow non-normal responses while also accounting for grouped or repeated observations.
Example: Modeling the defects produced by a manufacturing process over different pieces of equipment, gauges, and processing conditions.
Nonlinear models
Nonlinear regression models are used as an alternative to linear models. The model is a nonlinear function of its parameters. You specify the nonlinear model, then estimate the parameters using nonlinear least squares regression.
Example: Fitting growth and decay models, chemical kinetics models, or peak shape models.
Time series models
Time series methods analyze data collected sequentially over time. These models help identify trends, seasonal patterns, cycles, and other temporal relationships while providing forecasts of future outcomes.
Example: Forecasting product demand, energy consumption, or process performance over time.
Functional data analysis
Functional data analysis treats each observation as a continuous curve or function rather than as a collection of separate measurements. This approach is often used for spectra, sensor profiles, growth curves, and other continuous signals. These models can be also used for peak-finding and capturing other descriptive measures of the curves.
The types of models used to describe the curves are basis function models, including spline, Fourier, and wavelet basis functions. These models extract the shape components of the curves which can then be used in subsequent analyses like DOE, predictive modeling, or process control, among others.
Example: Comparing process sensor profiles to identify meaningful differences among materials, batches, and operating conditions.
Choosing an advanced model
An appropriate modeling approach depends on your analytical goals and the structure of your data.
- Which type of response are you modeling? For non-normal responses, consider GLMs.
- Are the levels of your predictor variables fixed or random? For models with random factors, consider mixed models and GLMMs.
- Do you want to perform model term selection, do you have correlated predictors, or do you have too little data to fit your model? If so, consider penalized regression.
- Do you want to explain seasonal or other temporal signals in the data? If so, consider time series models.
- Do your data consist of continuous curves rather than single observations? If so, consider functional data analysis.
Understanding these questions helps narrow the range of appropriate modeling methods and improves both model performance and interpretability.