Exploring the equivalence of two common mixture models for duration data

The beta-geometric (BG) distribution and the Pareto distribution of the second kind (P(II)) are two basic models for duration-time data that share some underlying characteristics (i.e., continuous mixtures of memoryless distributions), but differ in two important respects: first, the BG is the natural model to use when the event of interest occurs in discrete time, while the P(II) is the right choice for a continuous-time setting. Second, the underlying mixing distributions (the beta and gamma for the BG and P(II), respectively, are very different—and often believed to be non-comparable with each other. Despite these and other key differences, the two models are strikingly similar in terms of their fit and predictive performance as well as their parameter estimates. We explore this equivalence, both empirically and analytically, and discuss the implications from both a substantive and methodological standpoint

"How to project customer retention" revisited: the role of duration dependence

Cohort-level retention rates typically increase over time, and the beta-geometric (BG) distribution has proven to be a robust model for capturing and projecting these patterns into the future. According to this model, the phenomenon of increasing cohort-level retention rates is purely due to cross-sectional heterogeneity; an individual customer’s propensity to churn does not change over time. In this paper we present the beta-discrete-Weibull (BdW) distribution as an extension to the BG model, one that allows individual-level churn probabilities to increase or decrease over time. In addition to capturing the phenomenon of increasing cohort-level retention rates, this new model can also accommodate situations in which there is an initial dip in retention rates before they increase (i.e., a U-shaped cohort-level retention curve). A key finding is that even when aggregate retention rates are monotonically increasing, the individual-level churn probabilities are unlikely to be declining over time, as conventional wisdom would suggest. We carefully explore these connections between heterogeneity, duration dependence, and the shape of the retention curve, and draw some managerially relevant conclusions, e.g., that accounting for cross-sectional heterogeneity is more important than accounting for any individual-level dynamics in churn propensities

Valuing Subscription-Based Businesses Using Publicly Disclosed Customer Data

The growth of subscription-based commerce has seen a change in the types of data firms report to external shareholders. More than ever before, companies are discussing and disclosing data on the number of customers acquired and lost, customer lifetime value, and more. This has fueled an increasing interest in linking the value of a firm’s customers to the overall value of the firm, with the term customer-based corporate valuation being using to describe such efforts. While a number of researchers in the fields of marketing and accounting have explored this idea, their underlying models of customer acquisition and retention do not adequately reflect the empirical realities associated with these behaviors, and the associated valuation models do not live up to the standards of finance professionals. We develop a framework for valuing subscription-based firms that addresses both issues, and apply it to data from Dish Network and Sirius XM Holdings

Some Customers Would Rather Leave Without Saying Goodbye

We investigate the increasingly common business setting in which companies face the possibility of both observed and unobserved customer attrition (i.e., "overt" and "silent" churn) in the same pool of customers. This is the case for many online-based services where customers have the choice to stop interacting with the firm either by formally terminating the relationship (e.g., cancelling their account) or by simply ignoring all communications coming from the firm. The standard contractual versus noncontractual categorization of customer-firm relationships does not apply in such hybrid settings, which means the standard models for analyzing customer attrition do not apply. We propose a hidden Markov model (HMM)-based framework to capture silent and overt churn. We apply our modeling framework to two different contexts - a daily deals website and a performing arts organization. In contrast to previous studies that have not separated the two types of churn, we find that overt churners in these hybrid settings tend to interact more, rather than less, with the firm prior to churning. That is, in settings where both types of churn are present, a high level of activity - such as customers actively opening emails received from the firm - is not necessarily a good indicator of future engagement; rather it is associated with higher risk of overt churn. We also identify a large number of "silent churners" in both empirical applications - customers who disengage with the company very early on, rarely exhibit any type of activity, and almost never churn overtly. Furthermore, we show how the two types of churners respond very differently to the firm's communications, implying that a common retention strategy for proactive churn management is not appropriate in these hybrid settings

The ePetri dish, an on-chip cell imaging platform based on subpixel perspective sweeping microscopy (SPSM)

We report a chip-scale lensless wide-field-of-view microscopy imaging technique, subpixel perspective sweeping microscopy, which can render microscopy images of growing or confluent cell cultures autonomously. We demonstrate that this technology can be used to build smart Petri dish platforms, termed ePetri, for cell culture experiments. This technique leverages the recent broad and cheap availability of high performance image sensor chips to provide a low-cost and automated microscopy solution. Unlike the two major classes of lensless microscopy methods, optofluidic microscopy and digital in-line holography microscopy, this new approach is fully capable of working with cell cultures or any samples in which cells may be contiguously connected. With our prototype, we demonstrate the ability to image samples of area 6 mm × 4 mm at 660-nm resolution. As a further demonstration, we showed that the method can be applied to image color stained cell culture sample and to image and track cell culture growth directly within an incubator. Finally, we showed that this method can track embryonic stem cell differentiations over the entire sensor surface. Smart Petri dish based on this technology can significantly streamline and improve cell culture experiments by cutting down on human labor and contamination risks

Determination of Deoxynivalenol in the Urine of Pregnant Women in the UK

Deoxynivalenol (DON) is one of the most commonly occurring trichothecenes, produced mainly by Fusarium graminearum. Little is known about the effect of DON exposure or the levels of DON exposure that occur during pregnancy. The project aimed to provide data on levels of total DON and de-epoxi Deoxynivalenol (DOM-1) in pregnant human urine samples analysed by liquid chromatography-mass spectrometry (LC-MS). Morning urine samples were collected over two consecutive days from 42 volunteers and associated food consumption was recorded for the 24 h prior to the sample. Spearman’s rho non-parametric test for correlation was used to assess the data. Levels of DON did not differ significantly between day 1 (mean 29.7 ng/mL urine or 40.1 ng DON/mg creatinine) and day 2 (mean 28.7 ng/mL urine or 38.8 ng DON/mg creatinine ng/mL/day) urine samples. The only significant positive correlation was found between total ng DON/mg creatinine and parity (rho = 0.307, n = 42, p < 0.005 two-tailed) and total ng DON/mg creatinine with baked goods on day 1 (rho = 0.532, n = 42, p < 0.0005 two-tailed). This study provides data on the DON levels in pregnancy in this suburban population and reassurance that those levels are within acceptable limits

Reference Distorted Prices

I show that when consumers (mis)perceive prices relative to reference prices, budgets turn out to be soft, prices tend to be lower and the average quality of goods sold decreases. These observations provide explanations for decentralized purchase decisions, for people being happy with a purchase even when they have paid their evaluation, and for why trade might affect high quality local firms 'unfairly'

Scaling and Formulary cross sections for ion-atom impact ionization

The values of ion-atom ionization cross sections are frequently needed for many applications that utilize the propagation of fast ions through matter. When experimental data and theoretical calculations are not available, approximate formulas are frequently used. This paper briefly summarizes the most important theoretical results and approaches to cross section calculations in order to place the discussion in historical perspective and offer a concise introduction to the topic. Based on experimental data and theoretical predictions, a new fit for ionization cross sections is proposed. The range of validity and accuracy of several frequently used approximations (classical trajectory, the Born approximation, and so forth) are discussed using, as examples, the ionization cross sections of hydrogen and helium atoms by various fully stripped ions.Comment: 46 pages, 8 figure