Zero-Inflated Models

Zero-inflated models decompose the data-generating process into a binary component and a count component. The binary component, typically a logistic regression, models the probability of belonging to the 'always-zero' group versus the 'potentially non-zero' group. The count component, often a Poisson or Negative Binomial regression, models the expected frequency conditional on being in the non-zero group. This separation allows the model to distinguish between a customer who will never transact and one who simply hasn't transacted yet during the observation window.

A defining feature is the explicit differentiation between two types of zeros:

• Structural zeros: Inherent absence of the event. In CLV contexts, this represents customers who have permanently churned or will never purchase a specific product category.
• Sampling zeros: Random realizations from the count distribution. A currently active customer may have zero transactions in a given month purely by chance. Standard Poisson models conflate these two sources, leading to underprediction of zeros and biased parameter estimates.

The ZIP model is the foundational formulation where the count component follows a Poisson distribution. The probability mass function is:

• P(Y=0) = π + (1-π) * e^(-λ)
• P(Y=k) = (1-π) * (λ^k * e^(-λ)) / k! for k > 0 Where π is the zero-inflation probability and λ is the Poisson rate parameter. Both π and λ can be modeled as functions of covariates using logit and log link functions respectively. ZIP is appropriate when the non-zero counts exhibit equi-dispersion.

The ZINB model extends ZIP by replacing the Poisson distribution with a Negative Binomial distribution to accommodate overdispersion in the count data. Overdispersion occurs when the variance of the non-zero counts exceeds the mean, a common phenomenon in transaction data where some customers exhibit highly variable purchase frequencies. The Negative Binomial introduces an additional dispersion parameter α that captures unobserved heterogeneity. When α approaches zero, ZINB reduces to the ZIP model.

Zero-inflated models are often compared to hurdle models, which also use a two-part structure but with a crucial conceptual difference:

• Hurdle models treat all zeros as arising from a single binary process. Once the hurdle is crossed, the count distribution is truncated at zero, generating only positive values.
• Zero-inflated models allow zeros to originate from both the binary process and the count process. Hurdle models are appropriate when zeros represent a decision threshold, while zero-inflated models are preferred when zeros have mixed origins.

The Vuong closeness test is a likelihood-ratio-based statistical test used to determine whether a zero-inflated model provides a significantly better fit than its non-inflated counterpart. The test evaluates the null hypothesis that both models are equally close to the true data-generating process against the alternative that one model is superior. A significant positive test statistic favors the zero-inflated specification. For nested comparisons between ZIP and ZINB, a likelihood ratio test on the dispersion parameter α is employed.

Zero-inflated models decompose CLV prediction into two distinct statistical processes. A binary logit or probit model first estimates the probability of a customer being a structural zero—permanently inactive with no future transactions. A count model, typically Poisson or Negative Binomial, then predicts the transaction frequency for the active segment. This separation prevents the inflation of zeros from biasing the rate parameter downward, which would systematically underestimate the value of truly active customers.

The ZIP model assumes that transaction counts for active customers follow a Poisson distribution with a constant rate parameter λ. It is defined by the mixture: P(Y=0) = π + (1-π)e^(-λ) and P(Y=y) = (1-π) * (λ^y * e^(-λ)) / y! for y>0. The parameter π represents the probability of being a structural zero. ZIP is most appropriate when the active customer base exhibits equidispersion—where the variance of transaction counts equals the mean—a condition rarely met in real-world purchasing data.

The ZINB model extends ZIP by adding a dispersion parameter α to accommodate overdispersion in transaction counts. The Negative Binomial component allows the variance to exceed the mean, capturing the heterogeneity in purchase rates among active customers. The probability mass function incorporates the same binary zero-inflation gate but replaces the Poisson kernel with a Negative Binomial kernel. ZINB is the default choice for CLV forecasting because individual customer transaction rates are inherently heterogeneous, leading to variance far exceeding the mean.

A critical distinction exists between hurdle and zero-inflated frameworks. Hurdle models treat all zeros as originating from a single process—a customer must first cross a 'hurdle' to become active, after which a truncated count distribution governs positive transactions. In contrast, zero-inflated models recognize two distinct sources of zeros: structural zeros from permanently inactive customers and sampling zeros from active customers who simply made no purchase during the observation window. For CLV, zero-inflated models are theoretically superior because they acknowledge that a currently inactive customer may still be alive in the relationship.

Both the zero-inflation and count components can be parameterized with customer-level covariates. The logit link for the zero-inflation probability π_i incorporates features like days since last purchase, email engagement, and return frequency: logit(π_i) = Z_i * γ. Simultaneously, the log link for the count rate λ_i incorporates features like average order value, product category diversity, and session frequency: log(λ_i) = X_i * β. This dual regression structure allows the model to learn that a long lapse in purchases increases the probability of being a structural zero while high historical spend increases the expected transaction rate if still active.

The Vuong closeness test provides a statistical framework for determining whether the zero-inflation specification is necessary. It compares the fit of a standard Poisson or Negative Binomial model against its zero-inflated counterpart. The test statistic follows a standard normal distribution, with significant positive values indicating the zero-inflated model is preferred. For CLV applications, the Vuong test should be applied to out-of-sample holdout data to avoid overfitting. A non-significant result suggests the simpler count model is sufficient, simplifying model maintenance and inference latency.