,
Alina Malyutina [ctb] | Title: | Dose-Response Data Analysis |
|---|---|
| Description: | Fit logistic functions to observed dose-response continuous data and evaluate goodness-of-fit measures. See Malyutina A., Tang J., and Pessia A. (2023) <doi:10.18637/jss.v106.i04>. |
| Authors: | Alberto Pessia [aut, cre] ,
Alina Malyutina [ctb] |
| Maintainer: | Alberto Pessia <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 2.0.5 |
| Built: | 2025-02-16 05:01:47 UTC |
| Source: | https://github.com/albertopessia/drda |
Use the Newton's with a trust-region method to fit non-linear growth curves to observed data.
drda( formula, data, subset, weights, na.action, mean_function = "logistic4", lower_bound = NULL, upper_bound = NULL, start = NULL, max_iter = 1000 )drda( formula, data, subset, weights, na.action, mean_function = "logistic4", lower_bound = NULL, upper_bound = NULL, start = NULL, max_iter = 1000 )
formula |
an object of class |
data |
an optional data frame, list or environment (or object coercible
by |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
weights |
an optional vector of weights to be used in the fitting
process. If provided, weighted least squares is used with weights |
na.action |
a function which indicates what should happen when the data
contain |
mean_function |
the model to be fitted. See |
lower_bound |
numeric vector with the minimum admissible values of the
parameters. Use |
upper_bound |
numeric vector with the maximum admissible values of the
parameters. Use |
start |
starting values for the parameters. |
max_iter |
maximum number of iterations in the optimization algorithm. |
The 5-parameter logistic function can be selected by choosing
mean_function = "logistic5" or mean_function = "l5". The function is
defined here as
alpha + delta / (1 + nu * exp(-eta * (x - phi)))^(1 / nu)
where eta > 0 and nu > 0. When delta is positive (negative) the curve
is monotonically increasing (decreasing).
Parameter alpha is the value of the function when x -> -Inf.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
The 4-parameter logistic function is the default model of drda. It can be
explicitly selected by choosing mean_function = "logistic4" or
mean_function = "l4". The function is obtained by setting nu = 1 in the
generalized logistic function, that is
alpha + delta / (1 + exp(-eta * (x - phi)))
where eta > 0. When delta is positive (negative) the curve is
monotonically increasing (decreasing).
Parameter alpha is the value of the function when x -> -Inf.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi represents the x value at which the curve is equal to its
mid-point, i.e. f(phi; alpha, delta, eta, phi) = alpha + delta / 2.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
The 2-parameter logistic function can be selected by choosing
mean_function = "logistic2" or mean_function = "l2". For a monotonically
increasing curve set nu = 1, alpha = 0, and delta = 1:
1 / (1 + exp(-eta * (x - phi)))
For a monotonically decreasing curve set nu = 1, alpha = 1, and
delta = -1:
1 - 1 / (1 + exp(-eta * (x - phi)))
where eta > 0. The lower bound of the curve is zero while the upper bound
of the curve is one.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi represents the x value at which the curve is equal to its
mid-point, i.e. f(phi; eta, phi) = 1 / 2.
The Gompertz function is the limit for nu -> 0 of the 5-parameter logistic
function. It can be selected by choosing mean_function = "gompertz" or
mean_function = "gz". The function is defined in this package as
alpha + delta * exp(-exp(-eta * (x - phi)))
where eta > 0.
Parameter alpha is the value of the function when x -> -Inf.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi sets the displacement along the x-axis.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
The mid-point of the function, that is alpha + delta / 2, is achieved at
x = phi - log(log(2)) / eta.
The 5-parameter log-logistic function is selected by setting
mean_function = "loglogistic5" or mean_function = "ll5". The function is
defined here as
alpha + delta * (x^eta / (x^eta + nu * phi^eta))^(1 / nu)
where x >= 0, eta > 0, phi > 0, and nu > 0. When delta is
positive (negative) the curve is monotonically increasing (decreasing). The
function is defined only for positive values of the predictor variable x.
Parameter alpha is the value of the function at x = 0.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
The 4-parameter log-logistic function is selected by setting
mean_function = "loglogistic4" or mean_function = "ll4". The function is
obtained by setting nu = 1 in the generalized log-logistic function, that
is
alpha + delta * x^eta / (x^eta + phi^eta)
where x >= 0 and eta > 0. When delta is positive (negative) the curve
is monotonically increasing (decreasing). The function is defined only for
positive values of the predictor variable x.
Parameter alpha is the value of the function at x = 0.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi represents the x value at which the curve is equal to its
mid-point, i.e. f(phi; alpha, delta, eta, phi) = alpha + delta / 2.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
The 2-parameter log-logistic function is selected by setting
mean_function = "loglogistic2" or mean_function = "ll2". For a
monotonically increasing curve set nu = 1, alpha = 0, and delta = 1:
x^eta / (x^eta + phi^eta)
For a monotonically decreasing curve set nu = 1, alpha = 1, and
delta = -1:
1 - x^eta / (x^eta + phi^eta)
where x >= 0, eta > 0, and phi > 0. The lower bound of the curve is
zero while the upper bound of the curve is one.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi represents the x value at which the curve is equal to its
mid-point, i.e. f(phi; eta, phi) = 1 / 2.
The log-Gompertz function is the limit for nu -> 0 of the 5-parameter
log-logistic function. It can be selected by choosing
mean_function = "loggompertz" or mean_function = "lgz". The function is
defined in this package as
alpha + delta * exp(-(phi / x)^eta)
where x > 0, eta > 0, and phi > 0. Note that the limit for x -> 0 is
alpha. When delta is positive (negative) the curve is monotonically
increasing (decreasing). The function is defined only for positive values of
the predictor variable x.
Parameter alpha is the value of the function at x = 0.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi sets the displacement along the x-axis.
The value of the function when x -> Inf is alpha + delta. In
dose-response studies delta can be interpreted as the maximum theoretical
achievable effect.
It is possible to search for the maximum likelihood estimates within pre-specified interval regions.
Note: Hypothesis testing is not available for constrained estimates because asymptotic approximations might not be valid.
An object of class drda and model_fit, where model is the
chosen mean function. It is a list containing the following components:
boolean value assessing if the optimization algorithm converged or not.
total number of iterations performed by the optimization algorithm
boolean value set to TRUE if optimization was
constrained.
boolean vector indicating which parameters were estimated from the data.
maximum likelihood estimates of the model parameters.
minimum value (found) of the residual sum of squares.
residual degrees of freedom.
fitted mean values.
residuals, that is response minus fitted values.
(only for weighted fits) the specified weights.
model that was used for fitting.
effective sample size.
corrected maximum likelihood estimate of the standard deviation.
maximum value (found) of the log-likelihood function.
observed Fisher information matrix evaluated at the maximum likelihood estimator.
approximate variance-covariance matrix of the model parameters.
the matched call.
the terms object used.
the model frame used.
(where relevant) information returned by
model.frame on the special handling of NAs.
# by default `drda` uses a 4-parameter logistic function for model fitting fit_l4 <- drda(response ~ log_dose, data = voropm2) # get a general overview of the results summary(fit_l4) # compare the model against a flat horizontal line and the full model anova(fit_l4) # 5-parameter logistic curve appears to be a better model fit_l5 <- drda(response ~ log_dose, data = voropm2, mean_function = "l5") plot(fit_l4, fit_l5) # fit a 2-parameter logistic function fit_l2 <- drda(response ~ log_dose, data = voropm2, mean_function = "l2") # compare our models anova(fit_l2, fit_l4) # use log-logistic functions when utilizing doses (instead of log-doses) # here we show the use of other arguments as well fit_ll5 <- drda( response ~ dose, weights = weight, data = voropm2, mean_function = "loglogistic5", lower_bound = c(0.5, -1.5, 0, -Inf, 0.25), upper_bound = c(1.5, 0.5, 5, Inf, 3), start = c(1, -1, 3, 100, 1), max_iter = 10000 ) # note that the maximum likelihood estimate is outside the region of # optimization: not only the variance-covariance matrix is now singular but # asymptotic assumptions do not hold anymore.# by default `drda` uses a 4-parameter logistic function for model fitting fit_l4 <- drda(response ~ log_dose, data = voropm2) # get a general overview of the results summary(fit_l4) # compare the model against a flat horizontal line and the full model anova(fit_l4) # 5-parameter logistic curve appears to be a better model fit_l5 <- drda(response ~ log_dose, data = voropm2, mean_function = "l5") plot(fit_l4, fit_l5) # fit a 2-parameter logistic function fit_l2 <- drda(response ~ log_dose, data = voropm2, mean_function = "l2") # compare our models anova(fit_l2, fit_l4) # use log-logistic functions when utilizing doses (instead of log-doses) # here we show the use of other arguments as well fit_ll5 <- drda( response ~ dose, weights = weight, data = voropm2, mean_function = "loglogistic5", lower_bound = c(0.5, -1.5, 0, -Inf, 0.25), upper_bound = c(1.5, 0.5, 5, Inf, 3), start = c(1, -1, 3, 100, 1), max_iter = 10000 ) # note that the maximum likelihood estimate is outside the region of # optimization: not only the variance-covariance matrix is now singular but # asymptotic assumptions do not hold anymore.
Estimate effective doses, that is the x values for which f(x) = y.
effective_dose(object, y, type, level)effective_dose(object, y, type, level)
object |
fit object as returned by |
y |
numeric vector with response levels (default 0.5). |
type |
character string with either "relative" (default) or "absolute". |
level |
level of confidence intervals (default 0.95). |
Given a fitted model f(x; theta) we seek the values x at which the
function is equal to the specified response values.
If responses are given on a relative scale (type = "relative"), then y is
expected to range in the interval (0, 1).
If responses are given on an absolute scale (type = "absolute"), then y
is free to vary on the whole real line. Note, however, that the solution
does not exist when y is not in the image of the function.
Numeric matrix with the effective doses and the corresponding
confidence intervals. Each row is associated with each value of y.
drda_fit <- drda(response ~ log_dose, data = voropm2) effective_dose(drda_fit) # relative values are given on the (0, 1) range effective_dose(drda_fit, y = c(0.2, 0.8)) # explicitly say when we are using actual response values effective_dose(drda_fit, y = c(0.2, 0.8), type = "absolute") # use a different confidence level effective_dose(drda_fit, y = 0.6, level = 0.8)drda_fit <- drda(response ~ log_dose, data = voropm2) effective_dose(drda_fit) # relative values are given on the (0, 1) range effective_dose(drda_fit, y = c(0.2, 0.8)) # explicitly say when we are using actual response values effective_dose(drda_fit, y = c(0.2, 0.8), type = "absolute") # use a different confidence level effective_dose(drda_fit, y = 0.6, level = 0.8)
Evaluate at a particular set of parameters the Gompertz function.
gompertz_fn(x, theta)gompertz_fn(x, theta)
x |
numeric vector at which the Gompertz function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The Gompertz function f(x; theta) is defined here as
g(x; theta) = exp(-exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi), alpha is the value of the
function when x -> -Inf, delta is the (signed) height of the curve,
eta > 0 is the steepness of the curve or growth rate, and phi is related
with the value of function at x = 0.
When delta < 0 the curve is monotonically decreasing while it is
monotonically increasing for delta > 0.
Numeric vector of the same length of x with the values of the
Gompertz function.
Evaluate at a particular set of parameters the gradient and Hessian of the Gompertz function.
gompertz_gradient(x, theta) gompertz_hessian(x, theta) gompertz_gradient_hessian(x, theta)gompertz_gradient(x, theta) gompertz_hessian(x, theta) gompertz_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The Gompertz function f(x; theta) is defined here as
g(x; theta) = exp(-exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. When delta is
positive (negative) the curve is monotonically increasing (decreasing).
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the Gompertz function.
gompertz_gradient_2(x, theta) gompertz_hessian_2(x, theta) gompertz_gradient_hessian_2(x, theta)gompertz_gradient_2(x, theta) gompertz_hessian_2(x, theta) gompertz_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The Gompertz function f(x; theta) is defined here as
g(x; theta) = exp(-exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. When delta is
positive (negative) the curve is monotonically increasing (decreasing).
This set of functions use a different parameterization from
link[drda]{gompertz_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the log-Gompertz function.
loggompertz_fn(x, theta)loggompertz_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The log-Gompertz function f(x; theta) is defined here as
f(x; theta) = alpha + delta exp(-(phi / x)^eta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. By convention we set
f(0; theta) = lim_{x -> 0} f(x; theta) = alpha.
Numeric vector of the same length of x with the values of the
log-logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the log-Gompertz function.
loggompertz_gradient(x, theta) loggompertz_hessian(x, theta) loggompertz_gradient_hessian(x, theta)loggompertz_gradient(x, theta) loggompertz_hessian(x, theta) loggompertz_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The log-Gompertz function f(x; theta) is defined here as
f(x; theta) = alpha + delta exp(-(phi / x)^eta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. By convention we set
f(0; theta) = lim_{x -> 0} f(x; theta) = alpha.
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the log-Gompertz function.
loggompertz_gradient_2(x, theta) loggompertz_hessian_2(x, theta) loggompertz_gradient_hessian_2(x, theta)loggompertz_gradient_2(x, theta) loggompertz_hessian_2(x, theta) loggompertz_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The log-Gompertz function f(x; theta) is defined here as
f(x; theta) = alpha + delta exp(-(phi / x)^eta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. By convention we set
f(0; theta) = lim_{x -> 0} f(x; theta) = alpha.
This set of functions use a different parameterization from
link[drda]{loggompertz_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta) and phi2 = log(phi).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 2-parameter logistic function.
logistic2_fn(x, theta)logistic2_fn(x, theta)
x |
numeric vector at which the logistic function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 2-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. Only eta and phi
are free to vary (therefore the name) while vector c(alpha, delta) is
constrained to be either c(0, 1) (monotonically increasing curve) or
c(1, -1) (monotonically decreasing curve).
This function allows values other than {0, 1, -1} for alpha and delta
but will coerce them to their proper constraints.
Numeric vector of the same length of x with the values of the
logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 2-parameter logistic function.
logistic2_gradient(x, theta, delta) logistic2_hessian(x, theta, delta) logistic2_gradient_hessian(x, theta, delta)logistic2_gradient(x, theta, delta) logistic2_hessian(x, theta, delta) logistic2_gradient_hessian(x, theta, delta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the two parameters in the form
|
delta |
value of delta parameter (either 1 or -1). |
The 2-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. Only eta and phi
are free to vary (therefore the name) while vector c(alpha, delta) is
constrained to be either c(0, 1) (monotonically increasing curve) or
c(1, -1) (monotonically decreasing curve).
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 2-parameter logistic function.
logistic2_gradient_2(x, theta, delta) logistic2_hessian_2(x, theta, delta) logistic2_gradient_hessian_2(x, theta, delta)logistic2_gradient_2(x, theta, delta) logistic2_hessian_2(x, theta, delta) logistic2_gradient_hessian_2(x, theta, delta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the two parameters in the form
|
delta |
value of delta parameter (either 1 or -1). |
The 2-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. Only eta and phi
are free to vary (therefore the name) while vector c(alpha, delta) is
constrained to be either c(0, 1) (monotonically increasing curve) or
c(1, -1) (monotonically decreasing curve).
This set of functions use a different parameterization from
link[drda]{logistic2_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 4-parameter logistic function.
logistic4_fn(x, theta)logistic4_fn(x, theta)
x |
numeric vector at which the logistic function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi), alpha is the value of the
function when x -> -Inf, delta is the (signed) height of the curve,
eta > 0 is the steepness of the curve or growth rate (also known as the
Hill coefficient), and phi is the value of x at which the curve is equal
to its mid-point.
When delta < 0 the curve is monotonically decreasing while it is
monotonically increasing for delta > 0.
The mid-point f(phi; theta) is equal to alpha + delta / 2 while the value
of the function for x -> Inf is alpha + delta.
Numeric vector of the same length of x with the values of the
logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 4-parameter logistic function.
logistic4_gradient(x, theta) logistic4_hessian(x, theta) logistic4_gradient_hessian(x, theta)logistic4_gradient(x, theta) logistic4_hessian(x, theta) logistic4_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. When delta is
positive (negative) the curve is monotonically increasing (decreasing).
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 4-parameter logistic function.
logistic4_gradient_2(x, theta) logistic4_hessian_2(x, theta) logistic4_gradient_hessian_2(x, theta)logistic4_gradient_2(x, theta) logistic4_hessian_2(x, theta) logistic4_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + exp(-eta * (x - phi)))
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi) and eta > 0. When delta is
positive (negative) the curve is monotonically increasing (decreasing).
This set of functions use a different parameterization from
link[drda]{logistic4_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 5-parameter logistic function.
logistic5_fn(x, theta)logistic5_fn(x, theta)
x |
numeric vector at which the logistic function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu), eta > 0, and nu > 0.
When delta is positive (negative) the curve is monotonically increasing
(decreasing). When x -> -Inf the value of the function is alpha while the
value of the function for x -> Inf is alpha + delta .
Parameter alpha is the value of the function when x -> -Inf.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
Numeric vector of the same length of x with the values of the
logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 5-parameter logistic function.
logistic5_gradient(x, theta) logistic5_hessian(x, theta) logistic5_gradient_hessian(x, theta)logistic5_gradient(x, theta) logistic5_hessian(x, theta) logistic5_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu), eta > 0, and nu > 0. When
delta is positive (negative) the curve is monotonically increasing
(decreasing).
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 5-parameter logistic function.
logistic5_gradient_2(x, theta) logistic5_hessian_2(x, theta) logistic5_gradient_hessian_2(x, theta)logistic5_gradient_2(x, theta) logistic5_hessian_2(x, theta) logistic5_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (1 + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu), eta > 0, and nu > 0. When
delta is positive (negative) the curve is monotonically increasing
(decreasing).
This set of functions use a different parameterization from
link[drda]{logistic5_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta) and nu2 = log(nu).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 6-parameter logistic function.
logistic6_fn(x, theta)logistic6_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (xi + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu, xi), eta > 0, nu > 0, and
xi > 0. When delta is positive (negative) the curve is monotonically
increasing (decreasing).
Parameter alpha is the value of the function when x -> -Inf.
Parameter delta affects the value of the function when x -> Inf.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
Parameter xi affects the value of the function when x -> Inf.
Note: The 6-parameter logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Numeric vector of the same length of x with the values of the
logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 6-parameter logistic function.
logistic6_gradient(x, theta) logistic6_hessian(x, theta) logistic6_gradient_hessian(x, theta)logistic6_gradient(x, theta) logistic6_hessian(x, theta) logistic6_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (xi + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu, xi), eta > 0, nu > 0, and
xi > 0. When delta is positive (negative) the curve is monotonically
increasing (decreasing).
Note: The 6-parameter logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 6-parameter logistic function.
logistic6_gradient_2(x, theta) logistic6_hessian_2(x, theta) logistic6_gradient_hessian_2(x, theta)logistic6_gradient_2(x, theta) logistic6_hessian_2(x, theta) logistic6_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter logistic function f(x; theta) is defined here as
g(x; theta) = 1 / (xi + nu * exp(-eta * (x - phi)))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where theta = c(alpha, delta, eta, phi, nu, xi), eta > 0, nu > 0, and
xi > 0. When delta is positive (negative) the curve is monotonically
increasing (decreasing).
This set of functions use a different parameterization from
link[drda]{logistic6_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta), nu2 = log(nu), and xi2 = log(xi).
Note that argument theta is on the original scale and not on the log scale.
Note: The 6-parameter logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 2-parameter log-logistic function.
loglogistic2_fn(x, theta)loglogistic2_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 2-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. Only eta and phi are free to vary (therefore the name) while
vector c(alpha, delta) is constrained to be either c(0, 1) (monotonically
increasing curve) or c(1, -1) (monotonically decreasing curve).
This function allows values other than {0, 1, -1} for alpha and delta
but will coerce them to their proper constraints.
Numeric vector of the same length of x with the values of the
log-logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 2-parameter log-logistic function.
loglogistic2_gradient(x, theta, delta) loglogistic2_hessian(x, theta, delta) loglogistic2_gradient_hessian(x, theta, delta)loglogistic2_gradient(x, theta, delta) loglogistic2_hessian(x, theta, delta) loglogistic2_gradient_hessian(x, theta, delta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the two parameters in the form
|
delta |
value of delta parameter (either 1 or -1). |
The 2-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. Only eta and phi are free to vary (therefore the name), while
c(alpha, delta) are constrained to be either c(0, 1) (monotonically
increasing curve) or c(1, -1) (monotonically decreasing curve).
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 2-parameter log-logistic function.
loglogistic2_gradient_2(x, theta, delta) loglogistic2_hessian_2(x, theta, delta) loglogistic2_gradient_hessian_2(x, theta, delta)loglogistic2_gradient_2(x, theta, delta) loglogistic2_hessian_2(x, theta, delta) loglogistic2_gradient_hessian_2(x, theta, delta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the two parameters in the form
|
delta |
value of delta parameter (either 1 or -1). |
The 2-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0. Only eta and phi are free to vary (therefore the name), while
c(alpha, delta) are constrained to be either c(0, 1) (monotonically
increasing curve) or c(1, -1) (monotonically decreasing curve).
This set of functions use a different parameterization from
link[drda]{loglogistic2_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta) and phi2 = log(phi).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 4-parameter log-logistic function.
loglogistic4_fn(x, theta)loglogistic4_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0.
Numeric vector of the same length of x with the values of the
log-logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 4-parameter log-logistic function.
loglogistic4_gradient(x, theta) loglogistic4_hessian(x, theta) loglogistic4_gradient_hessian(x, theta)loglogistic4_gradient(x, theta) loglogistic4_hessian(x, theta) loglogistic4_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi), eta > 0, and
phi > 0.
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 4-parameter log-logistic function.
loglogistic4_gradient_2(x, theta) loglogistic4_hessian_2(x, theta) loglogistic4_gradient_hessian_2(x, theta)loglogistic4_gradient_2(x, theta) loglogistic4_hessian_2(x, theta) loglogistic4_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the four parameters in the form
|
The 4-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = x^eta / (x^eta + phi^eta)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu), eta > 0, and
phi > 0.
This set of functions use a different parameterization from
link[drda]{loglogistic4_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta) and phi2 = log(phi).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 5-parameter log-logistic function.
loglogistic5_fn(x, theta)loglogistic5_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu), eta > 0,
phi > 0, and nu > 0.
Parameter alpha is the value of the function when x = 0.
Parameter delta is the (signed) height of the curve.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
Numeric vector of the same length of x with the values of the
log-logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 5-parameter log-logistic function.
loglogistic5_gradient(x, theta) loglogistic5_hessian(x, theta) loglogistic5_gradient_hessian(x, theta)loglogistic5_gradient(x, theta) loglogistic5_hessian(x, theta) loglogistic5_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu), eta > 0,
phi > 0, and nu > 0.
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 5-parameter log-logistic function.
loglogistic5_gradient_2(x, theta) loglogistic5_hessian_2(x, theta) loglogistic5_gradient_hessian_2(x, theta)loglogistic5_gradient_2(x, theta) loglogistic5_hessian_2(x, theta) loglogistic5_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the five parameters in the form
|
The 5-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu), eta > 0,
phi > 0, and nu > 0.
This set of functions use a different parameterization from
link[drda]{loglogistic5_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta), phi2 = log(phi), and nu2 = log(nu).
Note that argument theta is on the original scale and not on the log scale.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate at a particular set of parameters the 6-parameter log-logistic function.
loglogistic6_fn(x, theta)loglogistic6_fn(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (xi * x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu, xi), eta > 0,
phi > 0, nu > 0, and xi > 0. When delta is positive (negative) the
curve is monotonically increasing (decreasing).
Parameter alpha is the value of the function when x = 0.
Parameter delta affects the value of the function when x -> Inf.
Parameter eta represents the steepness (growth rate) of the curve.
Parameter phi is related to the mid-value of the function.
Parameter nu affects near which asymptote maximum growth occurs.
Parameter xi affects the value of the function when x -> Inf.
Note: The 6-parameter log-logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Numeric vector of the same length of x with the values of the
log-logistic function.
Evaluate at a particular set of parameters the gradient and Hessian of the 6-parameter log-logistic function.
loglogistic6_gradient(x, theta) loglogistic6_hessian(x, theta) loglogistic6_gradient_hessian(x, theta)loglogistic6_gradient(x, theta) loglogistic6_hessian(x, theta) loglogistic6_gradient_hessian(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (xi * x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu, xi), eta > 0,
phi > 0, nu > 0, and xi > 0. When delta is positive (negative) the
curve is monotonically increasing (decreasing).
Note: The 6-parameter log-logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Gradient or Hessian evaluated at the specified point.
Evaluate at a particular set of parameters the gradient and Hessian of the 6-parameter log-logistic function.
loglogistic6_gradient_2(x, theta) loglogistic6_hessian_2(x, theta) loglogistic6_gradient_hessian_2(x, theta)loglogistic6_gradient_2(x, theta) loglogistic6_hessian_2(x, theta) loglogistic6_gradient_hessian_2(x, theta)
x |
numeric vector at which the function is to be evaluated. |
theta |
numeric vector with the six parameters in the form
|
The 6-parameter log-logistic function f(x; theta) is defined here as
g(x; theta) = (x^eta / (xi * x^eta + nu * phi^eta))^(1 / nu)
f(x; theta) = alpha + delta g(x; theta)
where x >= 0, theta = c(alpha, delta, eta, phi, nu, xi), eta > 0,
phi > 0, nu > 0, and xi > 0.
This set of functions use a different parameterization from
link[drda]{loglogistic6_gradient}. To avoid the non-negative
constraints of parameters, the gradient and Hessian computed here are for
the function with eta2 = log(eta), phi2 = log(phi), nu2 = log(nu), and
xi2 = log(xi).
Note that argument theta is on the original scale and not on the log scale.
Note: The 6-parameter log-logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.
Gradient or Hessian of the alternative parameterization evaluated at the specified point.
Evaluate the normalized area above the curve (NAAC).
naac(object, xlim, ylim)naac(object, xlim, ylim)
object |
fit object as returned by |
xlim |
numeric vector of length 2 with the lower and upped bound of the
integration interval. Default is |
ylim |
numeric vector of length 2 with the lower and upped bound of the
allowed function values. Default is |
The area under the curve (AUC) is the integral of the chosen model
y(x; theta) with respect to x.
In real applications the response variable is usually contained within a
known interval. For example, if our response represents relative viability
against a control compound, the curve is expected to be between 0 and 1.
Let ylim = c(yl, yu) represent the admissible range of our function
y(x; theta), that is yl is its lower bound and yu its upper bound.
Let xlim = c(xl, xu) represent the admissible range of the predictor
variable x. For example, when x represent the dose, the boundaries
are the minimum and maximum doses we can administer.
To make the AUC value comparable between different compounds and/or studies,
this function sets a hard constraint on both the x variable and the
function y. The intervals can always be changed if needed.
The integral calculated by this function is of the piece-wise function
f(x; theta) defined as
f(x; theta) = yl, if y(x; theta) < yl
f(x; theta) = y(x; theta), if yl <= y(x; theta) <= yu
f(x; theta) = yu, if y(x; theta) > yu
The AUC is finally normalized by its maximum possible value, that is the
area of the rectangle with width xu - xl and height yu - yl.
The normalized area above the curve (NAAC) is simply NAAC = 1 - NAUC.
Numeric value representing the normalized area above the curve.
nauc for the Normalized Area Under the Curve
(NAUC).
drda_fit <- drda(response ~ log_dose, data = voropm2) naac(drda_fit) naac(drda_fit, xlim = c(6, 8), ylim = c(0.2, 0.5))drda_fit <- drda(response ~ log_dose, data = voropm2) naac(drda_fit) naac(drda_fit, xlim = c(6, 8), ylim = c(0.2, 0.5))
Evaluate the normalized area under the curve (NAUC).
nauc(object, xlim, ylim)nauc(object, xlim, ylim)
object |
fit object as returned by |
xlim |
numeric vector of length 2 with the lower and upped bound of the
integration interval. Default is |
ylim |
numeric vector of length 2 with the lower and upped bound of the
allowed function values. Default is |
The area under the curve (AUC) is the integral of the chosen model
y(x; theta) with respect to x.
In real applications the response variable is usually contained within a
known interval. For example, if our response represents relative viability
against a control compound, the curve is expected to be between 0 and 1.
Let ylim = c(yl, yu) represent the admissible range of our function
y(x; theta), that is yl is its lower bound and yu its upper bound.
Let xlim = c(xl, xu) represent the admissible range of the predictor
variable x. For example, when x represent the dose, the boundaries
are the minimum and maximum doses we can administer.
To make the AUC value comparable between different compounds and/or studies,
this function sets a hard constraint on both the x variable and the
function y. The intervals can always be changed if needed.
The integral calculated by this function is of the piece-wise function
f(x; theta) defined as
f(x; theta) = yl, if y(x; theta) < yl
f(x; theta) = y(x; theta), if yl <= y(x; theta) <= yu
f(x; theta) = yu, if y(x; theta) > yu
The AUC is finally normalized by its maximum possible value, that is the
area of the rectangle with width xu - xl and height yu - yl.
Numeric value representing the normalized area under the curve.
naac for the Normalized Area Above the Curve
(NAAC).
drda_fit <- drda(response ~ log_dose, data = voropm2) nauc(drda_fit) nauc(drda_fit, xlim = c(6, 8), ylim = c(0.2, 0.5))drda_fit <- drda(response ~ log_dose, data = voropm2) nauc(drda_fit) nauc(drda_fit, xlim = c(6, 8), ylim = c(0.2, 0.5))
Plot maximum likelihood curves fitted with drda.
## S3 method for class 'drda' plot(x, ...)## S3 method for class 'drda' plot(x, ...)
x |
|
... |
other |
This function provides a scatter plot of the observed data, overlaid with the maximum likelihood curve fit. If multiple fit objects from the same family of models are given, they will all be placed in the same plot.
Accepted plotting arguments are:
character string with the base used for printing the values on
the x axis. Accepted values are 10 for base 10, 2 for base 2, e
for base e, or n (default) for no log-scale printing.
curve color(s). By default, up to 9 color-blind friendly colors are provided.
axis labels.
the range of x and y values with sensible defaults.
level of confidence intervals (default is 0.95). Set to zero or a negative value to disable confidence intervals.
if TRUE (default) shows guidelines associated with the
curve mid-point.
if TRUE (default) shows in the plot the data points used
for fitting.
if TRUE (default) shows the legend.
character string with custom legend position. See
link[graphics]{legend} for possible keywords.
custom labels for the legend model names.
If TRUE (default) a figure is plotted, otherwise the function
returns a list with values to create the figure manually.
If argument show is TRUE, no return value. If argument show is
FALSE, a list with all plotting data.
A dataset containing dose-response data of drug Vorinostat tested ex-vivo on the OPM-2 cell-line.
voropm2voropm2
A data frame with 45 rows and 4 variables:
viability measures normalized using positive and negative controls
drug concentrations (nM) used for testing
natural logarithm of variable dose
random weights included only for package demonstration