FourPHFfit: Fit four-parameter hill function

In germinationmetrics: Seed Germination Indices and Curve Fitting

Fit four-parameter hill function

Description

Fit a four-parameter hill function \insertCiteel-kassaby_seed_2008germinationmetrics to cumulative germination count data and compute the associated parameters. \loadmathjax

Usage

FourPHFfit(
  germ.counts,
  intervals,
  total.seeds,
  partial = TRUE,
  fix.y0 = TRUE,
  fix.a = TRUE,
  tmax,
  xp = c(10, 60),
  umin = 10,
  umax = 90,
  tries = 3
)

Arguments

germ.counts	Germination counts at each time interval. Can be partial or cumulative as specified in the argument partial.
intervals	The time intervals.
total.seeds	Total number of seeds.
partial	logical. If TRUE, germ.counts is considered as partial and if FALSE, it is considered as cumulative. Default is TRUE.
fix.y0	Force the intercept of the y axis through 0.
fix.a	Fix a as the actual maximum germination percentage at the end of the experiment.
tmax	The time up to which AUC is to be computed.
xp	Germination percentage value(s) for which the corresponding time is to be computed as a numeric vector. Default is c(10, 60).
umin	The minimum germination percentage value for computing uniformity. Default is 10. Seed Details.
umax	The maximum germination percentage value for computing uniformity. Default is 90. Seed Details.
tries	The number of tries to be attempted to fit the curve. Default is 3.

Details

The cumulative germination count data of a seed lot can be modelled to fit a four-parameter hill function defined as follows \insertCiteel-kassaby_seed_2008germinationmetrics.

\mjsdeqn

y = y_0+\fracax^bc^b+x^b

Where, \mjseqny is the cumulative germination percentage at time \mjseqnx, \mjseqny_0 is the intercept on the y axis, \mjseqna is the asymptote, or maximum cumulative germination percentage, which is equivalent to germination capacity, \mjseqnb is a mathematical parameter controlling the shape and steepness of the germination curve (the larger the \mjseqnb parameter, the steeper the rise toward the asymptote \mjseqna, and the shorter the time between germination onset and maximum germination), and \mjseqnc is the "half-maximal activation level" which represents the time required for 50% of viable seeds to germinate (\mjseqnc is equivalent to the germination speed).

In FourPHFfit, this model has be reparameterized by substituting \mjseqnb with \mjseqne^\beta to constraint \mjseqnb to positive values only.

\mjsdeqn

y = y_0+\fracax^e^\betac^e^\beta+x^e^\beta

Where, \mjseqnb = e^\beta and \mjseqn\beta = \log_e(b).

The curve fitting is performed with nonlinear gslnls package, a R interface to the least-squares optimization with the GNU Scientific Library (GSL) with the Levenberg-Marquardt algorithm \insertCitegalassi_gnu_2009,chau_gslnls_2023germinationmetrics.

Once this function is fitted to the curve, FourPHFfit computes the time to 50% germination of total seeds (t50.total) or viable seeds (t50.Germinated). Similarly the time at any percentage of germination (in terms of both total and viable seeds) as specified in argument xp can be computed.

The time at germination onset (\mjseqnlag) can be computed as follows.

\mjsdeqn

lag = b\sqrt\frac-y_0c^ba + y_0

The value \mjseqnD_lag-50 is defined as the duration between the time at germination onset (lag) and that at 50% germination (\mjseqnc).

The time interval between the percentages of viable seeds specified in the arguments umin and umin to germinate is computed as uniformity (\mjseqnU_t_max-t_min).

\mjsdeqn

U_t_max-t_min = t_max - t_min

The partial derivative of the four-parameter hill function gives the instantaneous rate of germination (\mjseqns) as follows.

\mjsdeqn

s = \frac\partial y\partial x = \fracabc^bx^b-1(c^b+x^b)^2

From this function for instantaneous rate of germination, the time at maximum germination rate (\mjseqnTMGR) can be estimated as follows.

\mjsdeqn

TMGR = b \sqrt\fracc^b(b-1)b+1

TMGR represents the point in time when the instantaneous rate of germination starts to decline.

The area under the curve (\mjseqnAUC) is obined by integration of the fitted curve between time 0 and time specified in the argument 'tmax'.

Integration of the fitted curve gives the value of mean germination time (\mjseqnMGT) and the skewness of the germination curve is computed as the ratio of \mjseqnMGT and the time for 50% of viable seeds to germinate (\mjseqnt_50).

\mjsdeqn

Skewness = \fracMGTt_50

If final germination percentage is less than 10%, a warning is given, as the results may not be informative.

Value

A list with the following components:

data	A data frame with the data used for computing the model.
Parameters	A data frame of parameter estimates, standard errors and p value.
Fit	A one-row data frame with estimates of model fitness such as log likelyhoods, Akaike Information Criterion, Bayesian Information Criterion, deviance and residual degrees of freedom.
a	The asymptote or the maximum cumulative germination percentage.
b	The mathematical parameter controlling the shape and steepness of the germination curve.
c	The half-maximal activation level.
y0	The intercept on the y axis.
lag	Time at germination onset.
Dlag50	duration between the time at germination onset (lag) and that at 50% germination.
t50.total	Time required for 50% of total seeds to germinate. Will be NaN if more than 50% of total seeds do not germinate.
txp.total	Time required for x% (as specified in argument xp) of total seeds to germinate. Will be NaN if more than x% of total seeds do not germinate.
t50.Germinated	Time required for 50% of viable/germinated seeds to germinate.
txp.Germinated	Time required for x% (as specified in argument xp) of viable/germinated seeds to germinate.
Uniformity	Time interval between umin% and umax% of viable seeds to germinate.
TMGR	Time at maximum germination rate.
AUC	The estimate of area under the curve.
MGT	Mean germination time.
Skewness	Skewness of mean germination time.
msg	The message from gsl_nls.
isConv	Logical value indicating whether convergence was achieved.
model	The raw fitted model output as a list of class gsl_nls.

References

\insertAllCited

Examples

x <- c(0, 0, 0, 0, 4, 17, 10, 7, 1, 0, 1, 0, 0, 0)
y <- c(0, 0, 0, 0, 4, 21, 31, 38, 39, 39, 40, 40, 40, 40)
int <- 1:length(x)
total.seeds = 50

# From partial germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = x, intervals = int, total.seeds = 50, tmax = 20)

# From cumulative germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = y, intervals = int, total.seeds = 50, tmax = 20,
           partial = FALSE)

germinationmetrics documentation built on Aug. 19, 2023, 1:07 a.m.