Beta Regression

Introduction

Beta Regression is tailored for modeling variables that take values in the open interval $(0, 1)$, making it ideal for proportions, rates, and other bounded continuous outcomes. This regression technique is particularly useful in fields like finance, biology, and social sciences, where outcomes are naturally constrained within a specific range.

Background and Theory

Beta Distribution

The Beta distribution is defined for values strictly between 0 and 1, described by two positive shape parameters, $\alpha$ and $\beta$. The probability density function (PDF) of the Beta distribution is:

where $B(\alpha, \beta)$ is the Beta function, and $y \in (0, 1)$.

Beta Regression Model

Beta regression models the dependent variable $Y$ that follows a Beta distribution. The mean $\mu$ of $Y$ is linked to a linear combination of explanatory variables $X$ through a link function, typically a logit link for $\mu$:

The variance of $Y$ is functionally related to $\mu$ and a dispersion parameter $\phi$, enhancing the model's flexibility to handle different levels of variability in the data.

Optimization Process

Maximum Likelihood Estimation (MLE)

MLE in Beta Regression involves estimating the parameters that maximize the likelihood of observing the given data under the assumed Beta distribution model. The likelihood function for parameters, given observations $(y_i, x_i)$, is the product of individual Beta PDFs for each observation.

Formulating the Likelihood Function

Given $N$ observations, the likelihood function $L(\alpha, \beta; y, X)$ can be expressed as:

In practice, $\alpha$ and $\beta$ are related to the explanatory variables and the parameters $\beta_0, \beta_1, \ldots, \beta_n$ through the link function and the dispersion parameter $\phi$.

Log-Likelihood and Optimization

The log-likelihood $\log L(\alpha, \beta; y, X)$ is often used for optimization:

The parameters are estimated by maximizing this log-likelihood through numerical optimization techniques, considering the constraints of $\alpha, \beta > 0$ and $y_i \in (0, 1)$.

Implementation

Parameters

• alpha: float, default = 1.0
  Shape parameter of gamma distribution
  
• beta: float, default = 1.0
  Scale parameter of gamma distribution
  
• learning_rate: float, default = 0.01
  Step size of the gradient descent update
  
• max_iter: int, default = 100
  Number of iteration
  
• l1_ratio: float, default = 0.5
  Balancing parameter of L1 and L2 in elastic-net regularization
  
• reg_strength: float, defualt = 0.01
  Regularization strength
  
• regularization: Literal['l1', 'l2', 'elastic-net'], default = None
  Regularization type
  

Applications

Beta Regression is widely applicable for:

• Finance: Modeling asset returns that are constrained between 0 and 1.
• Ecology: Analyzing rates and proportions, such as land cover percentages.
• Health Sciences: Studying rates of disease progression or recovery.

Strengths and Limitations

Strengths

• Flexibility: Can model outcomes that are ratios or proportions within a bounded interval.
• Dispersion Parameter: Accounts for variability in the data that is not captured by the mean alone.

Limitations

• Data Transformation: Requires that the dependent variable be strictly within the open interval $(0, 1)$, which may necessitate transformation of the data.
• Complexity: The estimation of parameters and interpretation of the model can be more complex compared to traditional linear regression.

Advanced Considerations

• Link Functions: Exploring different link functions for the mean and dispersion parameters can provide better model fit for specific datasets.
• Zero-One Inflation: For data including exact 0s and 1s, modifications to the Beta regression model or alternative approaches may be necessary.

Conclusion

Beta Regression offers a sophisticated framework for analyzing data constrained within the (0, 1) interval, leveraging the flexibility of the Beta distribution and the robustness of MLE for parameter estimation. Its application across various disciplines underscores its utility in modeling bounded continuous outcomes.

References

1. Ferrari, Silvia, and Francisco Cribari-Neto. "Beta Regression for Modelling Rates and Proportions." Journal of Applied Statistics, vol. 31, no. 7, 2004, pp. 799-815.
2. McCullagh, Peter, and John Nelder. "Generalized Linear Models." Chapman & Hall/CRC, 1989.