No abstract

Probit-based models relating a proportional response variable to a temporal explanatory variable, assuming that the times to response are normally distributed within the population, have been used in seed biology for describing the rate of loss of viability during seed ageing and the progress of germination over time in response to environmental signals (e.g. water, temperature). These models may be expressed as generalized linear models (GLMs) with a probit (cumulative normal distribution) link function, and, using GLM fitting procedures in current statistical software, parameters of these models are efficiently estimated while taking into account the binomial error distribution of the dependent variable. The fitted parameters can then be used to calculate the ‘traditional’ model parameters, such as the hydro- or hydrothermal time constant, the mean or median response of the seeds (e.g. mean time to death, median base water potential), and the standard deviation of the normal distribution of that response. Furthermore, through consideration of the deviance and residuals, performing model evaluation and modification can lead to improved understanding of the underlying physiological/ecological processes. However, fitting a binomial GLM is not appropriate for the cumulative count data often collected from germination studies, as successive observations are not independent, and time-to-event/survival analysis should be considered instead. This review discusses well-known probit-based models, providing advice on how to collect appropriate data and fit the models to those data, and gives an overview of alternative analysis approaches to improve understanding of the underlying mechanisms of seed dormancy and germination behaviour.

The objective of this study was to fit a hydrothermal germination model to germination data for a seedlot of radiata pine (Pinus radiata D. Don). Seeds were incubated for 50 d at constant temperatures and water potentials (T ¼ 12.5-32.58C, C ¼ 0 to 21.2 MPa). Most seeds completed germination within 50 d, but for low C and/or non-optimal temperatures (T , 17.58C, T . 258C) many seeds did not complete germination. In general, germination data conformed to the hydrothermal model. Departures from the model were encountered for slow-germinating seeds at suboptimal temperatures (T # 208C). To account for these departures, two alternative hydrothermal models were fitted with an additional term for an upwards shift in seed base water potential with increasing time to germination. The alternative models more correctly predicted germination time than the original model. Similarly, reduced percentage germination at supra-optimal temperatures (T . 208C) was explained by including a term in the hydrothermal model which shifted the base water potential of seeds upwards towards zero, which in turn reduced the predicted rate that hydrothermal time would be accumulated by seeds. The rate of this upwards shift in base water potential was dependent on time to complete germination and ambient water potential as well as supra-optimal temperature.

Tibial tunnel enlargement, consistent with that seen in humans after allograft anterior cruciate ligament reconstruction, did not appear to affect the ultimate incorporation of the allograft on a histologic level.