Human postmortem interval estimation from vitreous potassium: an analysis of original data from six different studies

The investigation of the postmortem interval (PMI) by determining potassium levels in the vitreous humor (KV) has been a subject of forensic pathology research for more than a quarter of a century. The numerous studies to date have yielded a variety of linear or piecewise-linear relationships between KV and PMI, i.e., different estimated intercepts and slopes of regression line(s) as well as different reliabilities of these estimates. This lack of agreement is due in part to the variable numbers of cases reported from study to study, differing observed ranges of KV and PMI, and the unaccommodated effects of factors on potassium concentration, including age of subject, amount of urea nitrogen, ambient temperature, and presence of illness. Original data from six of these studies, for a total of 790 cases, are reanalyzed together. The relationship between KV and PMI is not completely linear, and the residual variability of KV as a function of PMI is not constant. Thus, two main assumptions of the simple linear model, linearity and constant variance, are not supported by the data. It is clearly problematic to report statistical summaries such as the slope of an estimated regression line and the reliability of that estimate based on a model with faulty assumptions. Yet even after rescaling the data in an attempt to achieve linearity in the KV-PMI relationship and to stabilize residual variation, the relationship continues to be non-linear and its variability unstable. A new approach is developed for modeling KV and PMI that accommodates non-linearities and changing residual variability. A local regression model, specifically a loess smooth curve, is fitted separately to the data from each of the six studies. The loess smooth curve adapts locally to the changing and possibly non-linear relationship between KV and PMI across their observed ranges. The data from all six studies are then combined to yield a single loess curve with 95% confidence bands. The estimated loess curve and confidence bands are used in an inverse prediction method to construct low, middle and high PMI estimates at given values of KV. The reliability of estimated PMI decreases as KV increases. Although the confidence bands surrounding the overall curve widen in the extreme high end due to there being fewer available data in that region, PMI estimates are more precise over the entire range of KV and PMI than those obtained from any single study alone. A cross-validation procedure provides an independent check of the predictive performance of the method.