Fit a four-parameter hill function (El-Kassaby et al. 2008) to cumulative germination count data and compute the associated parameters.
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.countsis considered as partial and ifFALSE, it is considered as cumulative. Default isTRUE.- 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. SeedDetails.- umax
The maximum germination percentage value for computing uniformity. Default is
90. SeedDetails.- tries
The number of tries to be attempted to fit the curve. Default is 3.
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
NaNif 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 beNaNif 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% andumax% 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.
Details
The cumulative germination count data of a seed lot can be modelled to fit a four-parameter hill function defined as follows (El-Kassaby et al. 2008) .
\[y = y_{0}+\frac{ax^{b}}{c^{b}+x^{b}}\]
Where, \(y\) is the cumulative germination percentage at time \(x\), \(y_{0}\) is the intercept on the y axis, \(a\) is the asymptote, or maximum cumulative germination percentage, which is equivalent to germination capacity, \(b\) is a mathematical parameter controlling the shape and steepness of the germination curve (the larger the \(b\) parameter, the steeper the rise toward the asymptote \(a\), and the shorter the time between germination onset and maximum germination), and \(c\) is the "half-maximal activation level" which represents the time required for 50% of viable seeds to germinate (\(c\) is equivalent to the germination speed).
In FourPHFfit, this model has be reparameterized by substituting
\(b\) with \(e^{\beta}\) to constraint \(b\) to positive
values only.
\[y = y_{0}+\frac{ax^{e^{\beta}}}{c^{e^{\beta}}+x^{e^{\beta}}}\]
Where, \(b = e^{\beta}\) and \(\beta = \log_{e}(b)\).
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
(Chau 2023)
.
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 (\(lag\)) can be computed as follows.
\[lag = b\sqrt{\frac{-y_{0}c^{b}}{a + y_{0}}}\]
The value \(D_{lag-50}\) is defined as the duration between the time at germination onset (lag) and that at 50% germination (\(c\)).
The time interval between the percentages of viable seeds specified in the
arguments umin and umin to germinate is computed as uniformity
(\(U_{t_{max}-t_{min}}\)).
\[U_{t_{max}-t_{min}} = t_{max} - t_{min}\]
The partial derivative of the four-parameter hill function gives the instantaneous rate of germination (\(s\)) as follows.
\[s = \frac{\partial y}{\partial x} = \frac{abc^{b}x^{b-1}}{(c^{b}+x^{b})^{2}}\]
From this function for instantaneous rate of germination, the time at maximum germination rate (\(TMGR\)) can be estimated as follows.
\[TMGR = b \sqrt{\frac{c^{b}(b-1)}{b+1}}\]
TMGR represents the point in time when the instantaneous rate of germination starts to decline.
The area under the curve (\(AUC\)) 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 (\(MGT\)) and the skewness of the germination curve is computed as the ratio of \(MGT\) and the time for 50% of viable seeds to germinate (\(t_{50}\)).
\[Skewness = \frac{MGT}{t_{50}}\]
If final germination percentage is less than 10%, a warning is given, as the results may not be informative.
References
Chau J (2023).
“gslnls: GSL Nonlinear Least-Squares Fitting.”
El-Kassaby YA, Moss I, Kolotelo D, Stoehr M (2008).
“Seed germination: Mathematical representation and parameters extraction.”
Forest Science, 54(2), 220–227.
Galassi M (ed.) (2009).
GNU Scientific Library Reference Manual: For GSL Version 1.12, 3. ed edition.
Network Theory, Bristol.
ISBN 978-0-9546120-7-8.
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)
#> $data
#> gp csgp intervals
#> 1 0 0 1
#> 2 0 0 2
#> 3 0 0 3
#> 4 0 0 4
#> 5 8 8 5
#> 6 34 42 6
#> 7 20 62 7
#> 8 14 76 8
#> 9 2 78 9
#> 10 0 78 10
#> 11 2 80 11
#> 12 0 80 12
#> 13 0 80 13
#> 14 0 80 14
#>
#> $Parameters
#> term estimate std.error statistic p.value
#> 1 bta 2.290709 0.05602634 40.88628 2.965932e-14
#> 2 c 6.034954 0.03872162 155.85488 3.270089e-21
#>
#> $Fit
#> sigma isConv finTol logLik AIC BIC deviance df.residual
#> 1 1.61522 TRUE 6.039613e-14 -25.49868 56.99736 58.91453 31.30723 12
#> nobs
#> 1 14
#>
#> $a
#> [1] 80
#>
#> $b
#> [1] 9.881937
#>
#> $c
#> [1] 6.034954
#>
#> $y0
#> [1] 0
#>
#> $lag
#> [1] 0
#>
#> $Dlag50
#> [1] 6.034954
#>
#> $t50.total
#> [1] 6.355121
#>
#> $txp.total
#> 10 60
#> 4.956264 6.744598
#>
#> $t50.Germinated
#> [1] 6.034954
#>
#> $txp.Germinated
#> 10 60
#> 4.831807 6.287724
#>
#> $Uniformity
#> 90 10 uniformity
#> 7.537690 4.831807 2.705882
#>
#> $TMGR
#> [1] 5.912194
#>
#> $AUC
#> [1] 1108.976
#>
#> $MGT
#> [1] 6.632252
#>
#> $Skewness
#> [1] 1.098973
#>
#> $msg
#> [1] "#1. success "
#>
#> $isConv
#> [1] TRUE
#>
#> $model
#> Nonlinear regression model
#> model: csgp ~ FourPHF_fixa_fixy0(x = intervals, a = max(csgp), bta, c)
#> data: data
#> bta c
#> 2.291 6.035
#> residual sum-of-squares: 31.31
#>
#> Algorithm: multifit/levenberg-marquardt, (scaling: levenberg, solver: qr)
#>
#> Number of iterations to convergence: 9
#> Achieved convergence tolerance: 6.04e-14
#>
#> attr(,"class")
#> [1] "FourPHFfit" "list"
# From cumulative germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = y, intervals = int, total.seeds = 50, tmax = 20,
partial = FALSE)
#> $data
#> gp csgp intervals
#> 1 0 0 1
#> 2 0 0 2
#> 3 0 0 3
#> 4 0 0 4
#> 5 8 8 5
#> 6 34 42 6
#> 7 20 62 7
#> 8 14 76 8
#> 9 2 78 9
#> 10 0 78 10
#> 11 2 80 11
#> 12 0 80 12
#> 13 0 80 13
#> 14 0 80 14
#>
#> $Parameters
#> term estimate std.error statistic p.value
#> 1 bta 2.290709 0.05602634 40.88628 2.965932e-14
#> 2 c 6.034954 0.03872162 155.85488 3.270089e-21
#>
#> $Fit
#> sigma isConv finTol logLik AIC BIC deviance df.residual
#> 1 1.61522 TRUE 6.039613e-14 -25.49868 56.99736 58.91453 31.30723 12
#> nobs
#> 1 14
#>
#> $a
#> [1] 80
#>
#> $b
#> [1] 9.881937
#>
#> $c
#> [1] 6.034954
#>
#> $y0
#> [1] 0
#>
#> $lag
#> [1] 0
#>
#> $Dlag50
#> [1] 6.034954
#>
#> $t50.total
#> [1] 6.355121
#>
#> $txp.total
#> 10 60
#> 4.956264 6.744598
#>
#> $t50.Germinated
#> [1] 6.034954
#>
#> $txp.Germinated
#> 10 60
#> 4.831807 6.287724
#>
#> $Uniformity
#> 90 10 uniformity
#> 7.537690 4.831807 2.705882
#>
#> $TMGR
#> [1] 5.912194
#>
#> $AUC
#> [1] 1108.976
#>
#> $MGT
#> [1] 6.632252
#>
#> $Skewness
#> [1] 1.098973
#>
#> $msg
#> [1] "#1. success "
#>
#> $isConv
#> [1] TRUE
#>
#> $model
#> Nonlinear regression model
#> model: csgp ~ FourPHF_fixa_fixy0(x = intervals, a = max(csgp), bta, c)
#> data: data
#> bta c
#> 2.291 6.035
#> residual sum-of-squares: 31.31
#>
#> Algorithm: multifit/levenberg-marquardt, (scaling: levenberg, solver: qr)
#>
#> Number of iterations to convergence: 9
#> Achieved convergence tolerance: 6.04e-14
#>
#> attr(,"class")
#> [1] "FourPHFfit" "list"