Calculating Bias For Trimmed Mean Linear Models:

tm_bias calculates the bias and the bias-adjusted estimate for a trimmed means analysis (tm) of a given dataset, for a user-specified trimming fraction and dropout spread. tm_bias calculates, under assumption of normally distributed outcomes, the bias components resulting from violation of the location shift assumption and violation of the strong MNAR assumption.

Usage

tm_bias(
  formula,
  GR,
  trF,
  side = c("LOW", "HIGH"),
  spread_TG = "max_bias",
  spread_CG = "max_bias",
  data
)

Arguments

formula: an object of class formula, specifying the model, of the form outcome ~ terms, where terms must include the binary treatment variable, with additional variables optional.
GR: a string denoting the name of the binary treatment variable. This function assumes the lowest value to be the comparator/reference group
trF: a number between 0 and 1, specifying the trimming fraction: the proportion of the data that is trimmed away for each treatment group. trF should be equal to or greater than the largest observed dropout proportion. If left unspecified, a default trimming fraction of 0.5 is assumed.
side: specifies if higher value trimming ("HIGH") or lower value trimming ("LOW") should be performed.
spread_TG: a number between 0 and 1, specifying the dropout spread for the treatment group. spread_TG should be equal to or greater than the observed dropout proportion. If left unspecified, the worst-case scenario is assumed, in which dropout is located on the side of the distribution opposite from the one that is being trimmed (spread_TG="max_bias").
spread_CG: a number between 0 and 1, specifying the dropout spread for the comparator group. spread_CG should be equal to or greater than the observed dropout proportion. If left unspecified, the worst-case scenario is assumed, in which dropout is located on the side of the distribution opposite from the one that is being trimmed (spread_CG="max_bias").
data: a data frame containing the variables in the model. data should contain at least the following: a numeric outcome variable and a binary treatment variable (numeric, character or factor).

Value

tm_bias returns an object of class tm_bias.

An object of class "tm_bias" is a list containing the following components:

call: the matched call
bias_components: an array of bias components, including location shift assumption bias (LS), Strong MNAR bias in the treatment group (TG) and the comparator group (CG)
total_bias: the sum of all bias components
TM_estimate: the trimmed means estimate of the treatment effect
bias_adj_TM_estimate: the bias adjusted trimmed means estimate
analysis_details: the user-specified trimming fraction, trimming side, and dropout spread in the treatment (TG) and comparator groups (CG)
observed_TG_SD: observed standard deviation of the treatment group (TG) outcome
observed_CG_SD: observed standard deviation of the comparator group (CG) outcome
inferred_TG_SD: inferred full sample standard deviation of the treatment group (TG) outcome
max_bias_CG: an array of bias components, total bias, the bias adjusted estimate, and inferred full sample group standard deviations, calculated under the assumption of worst-case scenario dropout, with dropout in the comparator group (CG) on the opposite side of the distribution from the one that is being trimmed
max_bias_TG: an array of bias components, total bias, the bias adjusted estimate, and inferred full sample group standard deviations, calculated under the assumption of worst-case scenario dropout, with dropout in the treatment group (TG) on the opposite side of the distribution from the one that is being trimmed

Details

The trimmed means estimate is subject to two assumptions: the strong MNAR assumption requires that all dropouts (unobserved outcome values) are located in the fraction of the distribution that is trimmed away; the location shift assumption requires the group variances of the full sample to be equal. The bias resulting from the violation of either assumption can be calculated under assumption of normally distributed outcomes.

Obtaining the strong MNAR assumption bias requires an additional assumption about the distribution of the dropouts: it is assumed that the dropouts are spread homogeneously across the specified dropout spread. For example, under lower value trimming (side="LOW"), and a treatment group dropout spread of 0.6 (spread_TG=0.6), any value in the bottom 60% of the treatment group outcome distribution is equally likely to be missing.

The specified dropout spread for a given treatment group has implications for the unobserved full sample variance that is inferred from the observed data. For example, for an observed dropout of 0.4 and an assumed dropout spread of 0.5, the inferred full sample variance will be larger than for an assumed dropout spread of e.g., 0.8.

In addition to calculating the bias for a user-specified dropout spread, tm_bias also calculates the maximal bias. For example, for lower value trimming (side="LOW"), the worst-case scenario would involve lower value dropout in the treatment group (TG) and higher value dropout in the comparator group (CG), and vice versa. Bias components are calculated for both scenarios. If the dropout spread (spread_TG, spread_CG) is left unspecified for either treatment group, the function will return only these quantities.

Examples

test_dat <- as.data.frame(cbind(c(rep(0, 500), rep(1, 500)),
  c(sort(rnorm(500, 0, 1)), sort(rnorm(500, 1, 1.5)))))
colnames(test_dat) <- c("TR", "Y")
test_dat$Y[which(test_dat$TR == 0)[1:150]] <- NA
test_dat$Y[which(test_dat$TR == 1)[sample(seq(1, 400), 200, replace = FALSE)]] <- NA
tm_bias_obj <- tm_bias(formula = Y ~ TR, "TR", trF = 0.5,
                       side = "LOW", spread_TG = 0.4,
                       spread_CG = 0.6, data = test_dat)
print(tm_bias_obj)
#> 
#> Call:
#> tm_bias(formula = Y ~ TR, GR = "TR", trF = 0.5, side = "LOW", 
#>     spread_TG = 0.4, spread_CG = 0.6, data = test_dat)
#> 
#> Analysis details:
#>                                              
#>  50% trimming,                               
#>  under assumption of lower value dropout,    
#>  with 40% dropout spread in the TG group (1),
#>  and 60% dropout spread in the CG group (0)  
#> 
#> Bias components:
#>   Bias type                 Bias   
#>   LS                        1.36138
#>   Strong MNAR TG group (1)  0.00000
#>   Strong MNAR CG group (0)  0.01751
#> 
#> Total bias:
#>       
#>  1.379
#> 
#> TM estimate:
#>        
#>  0.8824
#> 
#> Bias adjusted TM estimate:
#>         
#>  -0.4965
#> 
#> Observed SDs:
#>                                           
#>  Observed TG group ( 1) SD for 40% dropout
#>  1.56                                     
#>                                           
#>  Observed CG group ( 0) SD for 30% dropout
#>  0.6928                                   
#> 
#> Inferred SDs:
#>                                                                                            
#>  Inferred TG group ( 1) full sample SD for 40% dropout in the lower 40% of the distribution
#>  2.401                                                                                     
#>                                                                                            
#>  Inferred CG group ( 0) full sample SD for 30% dropout in the lower 60% of the distribution
#>  0.695                                                                                     
#> 
#> Bias under maximal violation of the strong MNAR assumption in the CG group (0)                                
#> Strong MNAR bias          0.6579
#> LS bias                   1.1285
#> Total bias                1.7864
#> Bias adjusted estimate   -0.9040
#> Inferred SD TG group (1)  2.4012
#> Inferred SD CG group (0)  0.9869
#> 
#> Bias under maximal violation of the strong MNAR assumption in the TG group (1)                                
#> Strong MNAR bias         -2.4404
#> LS bias                   1.1285
#> Total bias               -1.3119
#> Bias adjusted estimate    2.1943
#> Inferred SD TG group (1)  2.4012
#> Inferred SD CG group (0)  0.9869