Self-starting Gompertz models

Creates initial coefficient estimates for selfStart wrappers around gompertz() and gompertz_left(), for use with stats::nls(). Both wrappers use the same 4-parameter (A, B, xmid, slope) interface.

Usage

SSgompertz(t, A, B, xmid, slope)

SSgompertz_left(t, A, B, xmid, slope)

SSgompertz_left(t, A, B, xmid, slope)

Arguments

t: A numeric vector of the predictor variable (time).
A: A numeric parameter for the starting asymptote of the response variable.
B: A numeric parameter for the ending asymptote of the response variable.
xmid: A numeric parameter for the time at the inflection point (the steepest point) of the curve, in units of the predictor variable t.
slope: A numeric parameter for the response rate dx/dt at the inflection xmid.

Value

A numeric vector of predicted values the same length as the predictor variable t.

Details

Model formulas

Right-Gompertz: x ~ SSgompertz(t, A, B, xmid, slope)
Left-Gompertz: x ~ SSgompertz_left(t, A, B, xmid, slope)

Used by analyse_kinetics() with method = "sigmoidal" and shape = "gompertz" or "gompertz_left". Starting estimates locate the inflection from a smoothed first derivative. SSgompertz() masks stats::SSgompertz().

Fixing parameters

Any parameter may be held constant by writing a value in place of its name in the formula, e.g. x ~ SSgompertz(t, A = 0, B, xmid, slope) fixes the starting asymptote at A = 0. Fixed parameters are excluded from estimation and are not returned by stats::coef().

Examples

## create a Gompertz curve with random noise
set.seed(15)
t <- 1:60
x <- gompertz(t, A = 10, B = 100, xmid = 30, slope = 4) +
    rnorm(length(t), 0, 2)
data <- data.frame(t, x)

model <- nls(x ~ SSgompertz(t, A, B, xmid, slope), data = data)
summary(model)
#> 
#> Formula: x ~ SSgompertz(t, A, B, xmid, slope)
#> 
#> Parameters:
#>        Estimate Std. Error t value Pr(>|t|)    
#> A      10.23048    0.43131   23.72   <2e-16 ***
#> B     100.77389    0.84867  118.74   <2e-16 ***
#> xmid   29.92865    0.17117  174.85   <2e-16 ***
#> slope   4.01621    0.09135   43.96   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 1.811 on 56 degrees of freedom
#> 
#> Number of iterations to convergence: 4 
#> Achieved convergence tolerance: 1.076e-06
#> 

## fix the starting asymptote `A` at a known value
model_fixed <- nls(x ~ SSgompertz(t, A = 10, B, xmid, slope), data = data)
summary(model_fixed)
#> 
#> Formula: x ~ SSgompertz(t, A = 10, B, xmid, slope)
#> 
#> Parameters:
#>        Estimate Std. Error t value Pr(>|t|)    
#> B     100.85346    0.83300  121.07   <2e-16 ***
#> xmid   29.88864    0.15332  194.94   <2e-16 ***
#> slope   4.00873    0.08903   45.03   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 1.8 on 57 degrees of freedom
#> 
#> Number of iterations to convergence: 3 
#> Achieved convergence tolerance: 2.247e-06
#> 

## left-Gompertz
set.seed(16)
x2 <- gompertz_left(t, A = 10, B = 100, xmid = 30, slope = 4) +
    rnorm(length(t), 0, 2)
data2 <- data.frame(t, x = x2)

model_left <- nls(x ~ SSgompertz_left(t, A, B, xmid, slope), data = data2)
summary(model_left)
#> 
#> Formula: x ~ SSgompertz_left(t, A, B, xmid, slope)
#> 
#> Parameters:
#>       Estimate Std. Error t value Pr(>|t|)    
#> A       9.8781     1.0370   9.525 2.58e-13 ***
#> B     100.6108     0.4727 212.839  < 2e-16 ***
#> xmid   30.0446     0.1987 151.192  < 2e-16 ***
#> slope   3.9205     0.1002  39.127  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 2.021 on 56 degrees of freedom
#> 
#> Number of iterations to convergence: 6 
#> Achieved convergence tolerance: 1.273e-06
#>