t50BluckCoward: Self-corrected t_50
In dmenne/d13cbreath: Reads and Evaluates Gastric Emptying 13C Breath Test Data

Description Usage Arguments Value Examples

Newton's method to solve the self-corrected BluckCoward equation for 1/2 to compute t_50. See also equation G(n,t) in "Bluck LJC, Jackson S, Vlasakakis G, Mander A (2011) Bayesian hierarchical methods to interpret the 13C-octanoic acid breath test for gastric emptying. Digestion 83_96-107, page 98.

1	t50BluckCoward(cf)

`cf`	named vector of coefficients; only `k` and `beta` are required Note that in this package, `k` is measured in 1/min (e.g. 0.01/min), usually it is quoted as 1/h (e.g. 0.6/h).

time where value is 1/2 of maximum, i.e. t_50 or t_1/2 in minutes; in the above paper, the parameter is called t_1/2(in).

# From table 3 and 4 in Bluck et al.; values for k and beta(NLS, Bayesian) are entered
# and checked against the tabulated values of t_{1/2(in)}.
# Most errors are small, but there are some outliers; errors in paper table?
# Parameters and Bluck results:
library(plyr) # for mutate
# Table 3 of Bluck et al.
cf3 = data.frame(
          method = rep(c("NLS","Bayesian")),
          group = rep(c("Lean","Obese"),each=2),
          k =    c(0.576,0.606,0.529,0.608),
          beta = c(5.24, 5.79, 5.95, 7.54),
          t12 =  c(3.67, 3.63, 4.23, 3.99),
          t12in = c(2.076,2.110,2.422,2.466),
          tlag = c(2.88,2.88,3.34,3.26),
          tlagin = c(1.632, 1.724,1.92,2.101)
)
cf3 = mutate(cf3,
          t50Maes = t50Maes(cf3),
          t50BluckCoward=t50BluckCoward(cf3),
          tLagMaes = tLagMaes(cf3),
          tLagBluckCoward = tLagBluckCoward(cf3),
          Err_t50Maes = round(100*(t50Maes-t12)/t12,2),
          Err_t50BluckCoward =
            round(100*(t50BluckCoward-t12in)/t12in,2),
          Err_LagMaes = round(100*(tLagMaes-tlag)/tlag,2),
          Err_LagBluckCoward =
            round(100*(tLagBluckCoward-tlagin)/tlagin,2)
)
cf3
# Table 4
# There are large differences for MJ3, both using the Bayesian (26%)
# and the NLS method (16%).  The other data are within the expected limits
cf4 = data.frame(
          method = rep(c("NLS","Bayesian"),each=3),
          group = rep(c("MJ1","MJ2","MJ3")),
          k = c(0.585,0.437,0.380,0.588,0.418,0.361),
          beta=c(4.35,4.08,4.44,4.49,4.30,4.29),
          t12 = c(3.39,4.25,4.82,3.40,4.61,5.09),
          t12in = c(1.77,2.16,2.19,1.81,2.34,2.43),
          tlag = c(2.56,3.17,3.39,2.58,3.40,3.62),
          tlagin = c(1.30,1.53,1.33,1.35,1.65,1.57)
)
cf4 = mutate(cf4,
          t50Maes = t50Maes(cf4),
          t50BluckCoward = t50BluckCoward(cf4),
          tLagMaes = tLagMaes(cf4),
          tLagBluckCoward = tLagBluckCoward(cf4),
          Err_t50Maes = unlist(round(100*(t50Maes-t12)/t12)),
          Err_t50BluckCoward =
            round(100*(t50BluckCoward-t12in)/t12in,2),
          Err_LagMaes = round(100*(tLagMaes-tlag)/tlag,2),
          Err_LagBluckCoward =
            round(100*(tLagBluckCoward-tlagin)/tlagin,2)
)
cf4

#' @seealso \code{\link{ExpBeta}}