Two-component error model — sigma

@@ -149,40 +88,36 @@ dependence of the measured value $y$:" /> dependence of the measured value $y$:

sigma_twocomp(y, sigma_low, rsd_high)

- -

Arguments

- - - - - - - - - - - - - - -

y	The magnitude of the observed value
sigma_low	The asymptotic minimum of the standard deviation for low -observed values
rsd_high	The coefficient describing the increase of the standard -deviation with the magnitude of the observed value

- -

Value

sigma_twocomp(y, sigma_low, rsd_high)

Arguments

y: The magnitude of the observed value
sigma_low: The asymptotic minimum of the standard deviation for low +observed values
rsd_high: The coefficient describing the increase of the standard +deviation with the magnitude of the observed value

Value

The standard deviation of the response variable.

Details

- +

Details

$$\sigma = \sqrt{ \sigma_{low}^2 + y^2 * {rsd}_{high}^2}$$ sigma = sqrt(sigma_low^2 + y^2 * rsd_high^2)

This is the error model used for example by Werner et al. (1978). The model proposed by Rocke and Lorenzato (1995) can be written in this form as well, but assumes approximate lognormal distribution of errors for high values of y.

References

- +

References

Werner, Mario, Brooks, Samuel H., and Knott, Lancaster B. (1978) Additive, Multiplicative, and Mixed Analytical Errors. Clinical Chemistry 24(11), 1895-1898.

@@ -190,61 +125,62 @@ Additive, Multiplicative, and Mixed Analytical Errors. Clinical Chemistry measurement error in analytical chemistry. Technometrics 37(2), 176-184.

Ranke J and Meinecke S (2019) Error Models for the Kinetic Evaluation of Chemical Degradation Data. Environments 6(12) 124 -doi: 10.3390/environments6120124 +doi: 10.3390/environments6120124 .

Examples

times <- c(0, 1, 3, 7, 14, 28, 60, 90, 120)
-d_pred <- data.frame(time = times, parent = 100 * exp(- 0.03 * times))
-set.seed(123456)
-d_syn <- add_err(d_pred, function(y) sigma_twocomp(y, 1, 0.07),
-  reps = 2, n = 1)[[1]]
-f_nls <- nls(value ~ SSasymp(time, 0, parent_0, lrc), data = d_syn,
- start = list(parent_0 = 100, lrc = -3))
-library(nlme)
-f_gnls <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
-  data = d_syn, na.action = na.omit,
-  start = list(parent_0 = 100, lrc = -3))
-if (length(findFunction("varConstProp")) > 0) {
-  f_gnls_tc <- update(f_gnls, weights = varConstProp())
-  f_gnls_tc_sf <- update(f_gnls_tc, control = list(sigma = 1))
-}
-f_mkin <- mkinfit("SFO", d_syn, error_model = "const", quiet = TRUE)
-f_mkin_tc <- mkinfit("SFO", d_syn, error_model = "tc", quiet = TRUE)
-plot_res(f_mkin_tc, standardized = TRUE)
-
AIC(f_nls, f_gnls, f_gnls_tc, f_gnls_tc_sf, f_mkin, f_mkin_tc)
-
#>              df      AIC
-#> f_nls         3 114.4817
-#> f_gnls        3 114.4817
-#> f_gnls_tc     5 103.6447
-#> f_gnls_tc_sf  4 101.6447
-#> f_mkin        3 114.4817
-#> f_mkin_tc     4 101.6446

Examples

times <- c(0, 1, 3, 7, 14, 28, 60, 90, 120)
+d_pred <- data.frame(time = times, parent = 100 * exp(- 0.03 * times))
+set.seed(123456)
+d_syn <- add_err(d_pred, function(y) sigma_twocomp(y, 1, 0.07),
+  reps = 2, n = 1)[[1]]
+f_nls <- nls(value ~ SSasymp(time, 0, parent_0, lrc), data = d_syn,
+ start = list(parent_0 = 100, lrc = -3))
+library(nlme)
+f_gnls <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
+  data = d_syn, na.action = na.omit,
+  start = list(parent_0 = 100, lrc = -3))
+if (length(findFunction("varConstProp")) > 0) {
+  f_gnls_tc <- update(f_gnls, weights = varConstProp())
+  f_gnls_tc_sf <- update(f_gnls_tc, control = list(sigma = 1))
+}
+f_mkin <- mkinfit("SFO", d_syn, error_model = "const", quiet = TRUE)
+f_mkin_tc <- mkinfit("SFO", d_syn, error_model = "tc", quiet = TRUE)
+plot_res(f_mkin_tc, standardized = TRUE)
+
+AIC(f_nls, f_gnls, f_gnls_tc, f_gnls_tc_sf, f_mkin, f_mkin_tc)
+#>              df      AIC
+#> f_nls         3 114.4817
+#> f_gnls        3 114.4817
+#> f_gnls_tc     5 103.6447
+#> f_gnls_tc_sf  4 101.6447
+#> f_mkin        3 114.4817
+#> f_mkin_tc     4 101.6446
+