Additive time-dependent hazard model with doubly truncated data

Cox regression with doubly truncated responses and time-dependent covariates: the impact of innovation on firm survival

The creation of new firms is an important incentive for the economic growth of a country, since it generates employment, it encourages the competition, and promotes innovation. In this work, we investigate the survival of Spanish firms which were created since 2001 and closed down between 2004 and 2012. The information was gathered from Technological Innovation Panel (PITEC), a survey with a focus the technological innovation in Spanish firms. In particular, a Cox regression model with time-dependent covariates was used in order to identify and quantify the determinants of the risk of exit for the firm. The selection bias due to the interval sampling for the firms was corrected by using methods for doubly truncated lifetimes. Interestingly, it is seen how the correction for the selection bias changes both the size and the statistical significance of the effects provided by standard Cox regression.

Truncating the exponential with a uniform distribution

For a sample of Exponentially distributed durations we aim at point estimation and a confidence interval for its parameter. A duration is only observed if it has ended within a certain time interval, determined by a Uniform distribution. Hence, the data is a truncated empirical process that we can approximate by a Poisson process when only a small portion of the sample is observed, as is the case for our applications. We derive the likelihood from standard arguments for point processes, acknowledging the size of the latent sample as the second parameter, and derive the maximum likelihood estimator for both. Consistency and asymptotic normality of the estimator for the Exponential parameter are derived from standard results on M-estimation. We compare the design with a simple random sample assumption for the observed durations. Theoretically, the derivative of the log-likelihood is less steep in the truncation-design for small parameter values, indicating a larger computational effort for root finding and a larger standard error. In applications from the social and economic sciences and in simulations, we indeed, find a moderately increased standard error when acknowledging truncation.

Left-censored dementia incidences in estimating cohort effects

We estimate the dementia incidence hazard in Germany for the birth cohorts 1900 until 1954 from a simple sample of Germany's largest health insurance company. Followed from 2004 to 2012, 36,000 uncensored dementia incidences are observed and further 200,000 right-censored insurants included. From a multiplicative hazard model we find a positive and linear trend in the dementia hazard over the cohorts. The main focus of the study is on 11,000 left-censored persons who have already suffered from the disease in 2004. After including the left-censored observations, the slope of the trend declines markedly due to Simpson's paradox, left-censored persons are imbalanced between the cohorts. When including left-censoring, the dementia hazard increases differently for different ages, we consider omitted covariates to be the reason. For the standard errors from large sample theory, left-censoring requires an adjustment to the conditional information matrix equality.