salbmCombine: Sensitivity Analysis for Binary Missing Data

Description

Combines main and bootstrap results.

Usage

salbmCombine(x0, Samps=NULL, div=c(NA,NA))

Arguments

main results from original data.

Samps

list of sample results

div

low and high alpha points

Value

a list with combined results

Details

Combines data and produces bootstrap summaries including confidence intervals.

Four types of confidence intervals are calculated after combining the results. Fixing the sensitivity parameter, let $E$ be an estimate for a treatment group and for $b=1,2,\ldots,B$ let $E_b$ be the corresponding estimate for bootstrap $b$. Let $SE$ be a standard error estimate for $E$ and $SE_b$ be a standard error estimate for $E_b$. Let $q( \{x_i\}, v)$ be the vth quantile for the collection $\{x_i\}$.

1. Let $q_l = q( \{E_b\}, 0.025 )$ and $q_u = q( \{E_b\}, 0.975 )$. Then a 95% confdence interval for $E$ is given by $( q_l, q_u )$. These are refered to as c("lb1","ub1") or type 1 confidence intervals.

2. Let $q_s = q( \{\mid E_b - E\mid\}, 0.95 )$. Then a 95% confdence interval for $E$ is given by $( E - qs, E + qs )$. These are refered to as c("lb2","ub2") or type 2 confidence intervals.

3. Let $t_b = ( E_b - E ) / SE_b$, eqnq_l = q( {t_b}, 0.025 ) and $q_u = q( \{t_b\}, 0.975 )$. Let $M$ be the mean of $E_b$. Then a 95% confdence interval for eqnE is given by $( M - q_u SE, M - q_l SE )$ These are refered to as c("lb3","ub3") or type 3 confidence intervals.

4. Let $t_b = ( E_b - E ) / SE_b$, and $q_s = q( \{\mid t_b \mid\}, 0.95 )$. Let $M$ be the mean of $E_b$. Then a 95% confdence interval for E is given by $( M - q_s SE, M + q_s SE )$ These are refered to as c("lb4","ub4") or type 4 confidence intervals.

Computing SE

Let D be a dataset with n rows and T a fixed timepoint. Three standard deviations are computed

$SD_0$ the standard deviation of the data in D at timepoint T when 0 is substituted for missing values in D,

$SD_1$ the standard deviation of the data in D at timepoint T when 1 is substituted for missing values in D, and

$SD_m$ the standard deviation of the data in D at timepoint T when mean value at time T is substituted for missing values at time T.

A low, mid and high values of alpha are choosen and denoted by $a_l$, $a_0$, and $a_u$ respectively. In salbm $a_0 = 0$. Then a standard error is computed as:

$$ SE = \left\{\Large\begin{array}{ll} \frac{( \alpha - a_l ) SD_0 + ( a_0 - \alpha ) SD_m }{ ( a_0 - a_l ) \sqrt{n} }&\normalsize \mbox{\hspace{0.2in} when } \normalsize \alpha < 0 \\[0.15in]

\frac{( a_u - \alpha ) SD_1 + ( \alpha - a_0 ) SD_m }{ ( a_u - a_0 ) \sqrt{n} }&\normalsize \mbox{\hspace{0.2in} when } \normalsize \alpha \ge 0 \end{array} \right. $$

Examples

Run this code

# NOT RUN {
  data(trt1)
  data(trt2)
  data <- list( trt1 = trt1, trt2 = trt2 )

  ## main run
  x0 <- salbm( data = data , K = 6, ntree = 250, 
               seeds = c(22,18), seeds2 = c(-2,-3),
               alphas = -5:5, NBootstraps=0 )

  ## add Bootstraps
  samp1 <- addSamples(obj=x0, seeds=c(99,12), 
               seeds2 = c(-45,-80), bBS=1,
               NBootstraps=250)
  ## add more Bootstraps
  samp2 <- addSamples(obj=x0, seeds=c(9,2), 
               seeds2 = c(-54,-8), bBS=251,
               NBootstraps=250)

  ## Together
  R <- salbmCombine(x0=x0, Samps=list(samp1,samp2))
# }