lefttruncated

Lefttruncated, in statistics and data analysis, refers to left truncation or delayed entry in which observations are only included if their value exceeds a left-hand threshold a. Formally, a random variable X is left-truncated at a if only X > a (or X ≥ a) values are observed. The observed distribution is the conditional distribution of X given X > a; its density is f(x | x > a) = f(x) / (1 - F(a)) for x > a, where f is the original density and F is the cumulative distribution function. This truncation often arises when individuals enter a study after time zero, so cases are observed only if they have survived or remained at risk up to the entry time. It is distinct from left-censoring, where the exact value below a threshold is unknown but known to lie below it.

Left truncation affects inference because naive analyses that ignore the truncation can be biased or inconsistent. It is commonly encountered in survival analysis, reliability engineering, and econometrics. For example, in medical follow-up studies, patients enrolled at different times may only contribute data if they survive to enrollment; in reliability data, components that fail before testing begins are not observed.

Estimation uses truncated likelihoods or models that account for delayed entry. The likelihood for a sample {x_i} left-truncated at a is the product of f(x_i) divided by (1 - F(a)). In regression contexts, methods such as Cox models with delayed entry or parametric/likelihood-based approaches for truncated distributions are applied. Related concepts include right truncation, interval truncation, and left-censoring. Identifying the truncation threshold and correctly modeling the conditional distribution are essential for valid inference.