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Fakultät Statistik

P3: IncTimeDose-Splines

Two-dimensional spline regression to model incubation time-dose curves

In project P3 dose-response relationships of ultraviolet B radiation (UVB radiation) and long-wave UVA radiation of natural sunlight are investigated with the help of two-dimensional spline regression.

Ultraviolet B radiation (290-315 nm) is a complete carcinogen and contributes significantly to the development of skin cancer in humans. Under physiological conditions, human skin is exposed to a mixture of UVB and long-wave UVA (315-400 nm) of natural sunlight (IARC, 2012) and the ratio between UVB and UVA varies throughout the day (Kollias et al., 2011). We could show that UVB and UVA radiation interact with each other: The addition of UVA radiation to UVB radiation weakens the skin-damaging effect of UVB (Sondenheimer et al., 2018).

In this project, the dose-response relationships involved will be examined more closely. In this case, the different dosages are varied UVB-to-UVA ratios. We use already existing RNAseq data sets of skin cells. Fibroblasts and keratinocytes were irradiated either with only UVB or UVA, or with a combination of both. The RNA of the cells was isolated at two different time points after irradiation (incubation time). RNA-sequencing was performed with an Illumina HiSeq3000. We aim to investigate the dose-response relationships of two incubation time points in a joint analysis. Additional data will be acquired for further incubation times and different physiologically relevant dosages. These will be chosen to best support the statistical analysis.

Currently, these data sets, which are essentially two-dimensional, are usually analysed using multiple pairwise comparisons, separately for each incubation time. A statistically more sophisticated approach would be to model the dose-response relationships as one-dimensional parametric functions for each incubation time. This project aims to go beyond this and tackles two drawbacks of this approach.

1. Parametric regression has low estimation variance, but high bias due to inflexible and possibly misspecified parametric functions describing the dose-response relationship. Semiparametric spline regression is more flexible, and penalised splines (P-splines) yield a variancebias or smoothness-goodness of fit compromise (Eilers and Marx, 1996). It has been demonstrated that shape constraints such as monotonicity, unimodality, or convexity/concavity can further improve estimation. When using the B-spline basis, these shape constraints can be incorporated by constrained estimation of the spline coefficients (Köllmann et al., 2014) or by the use of asymmetric penalty terms (Eilers, 2005). The effect of UVA irradiation might as well increase monotonically or first increase and then decrease (unimodality) with higher dosages.

2. When modelling univariate dose-response relationships separately for each incubation time, the information shared across the incubation times is not taken into account. However, it could be critical to identify certain features of the relationship. In this project, we aim to consider data of all incubation times at once by modelling a two-dimensional surface with the help of tensor product B-splines. As for the one-dimensional case, a compromise between smoothness and goodness of fit can be achieved using (two-dimensional) penalty terms, and shape constraints can be incorporated using asymmetric penalties on the spline coefficients (Bollaerts et al., 2006).

Looking at only one incubation time, it is also of interest to identify the minimum UVA dose, at which the combination irradiation shows a significantly different effect compared to UVB irradiation alone. Thus, this project seeks to also answer the question, if the use of spline regression enables the identification of this minimum effective UVA dose, for example with the help of bootstrap tests or confidence intervals.


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  • Eilers PHC, Marx BD (1996). Flexible smoothing with B-splines and penalties. Statistical Science 11(2), 89-121, doi: 10.1214/ss/1038425655.
  • Eilers PHC (2005). Unimodal smoothing. J Chemometrics 19, 317-328, doi: 10.1002/cem.935.
  • IARC (2012). A review of human carcinogens. Part D: Radiation. Lyon, France: International Agency for Research on Cancer (IARC monographs on the evaluation of carcinogenic risks to humans, v. 100D).
  • Kollias N, Ruvolo E Jr, Sayre RM (2011). The value of the ratio of UVA to UVB in sunlight. Photochem Photobiol 87(6), 1474-1475, doi: 10.1111/j.1751-1097.2011.00980.x.
  • Köllmann C, Bornkamp B, Ickstadt K (2014). Unimodal regression using BernsteinSchoenberg-splines and penalties. Biometrics 70, 783-793, doi: 10.1111/biom.12193.
  • Sondenheimer K, Neumann C, Majora M, Rapp A, Greinert R, Boukamp P, Krutmann J (2018). 1155 Analysis of the interaction of different wavelengths present in natural sunlight. J Investigative Dermatology 138(5), 196, Supplement, doi: 10.1016/j.jid.2018.03.1169.