Relative Mortality Metric (RMM) Calculation

Calculates the Relative Mortality Metric (RMM) from Napoli et al. (2017) based on patient survival probabilities (Ps) and actual outcomes. The function groups patients into bins based on their survival probability scores (Ps) and computes a weighted mortality metric along with confidence intervals. For more information on the methods used in this function, see as well Schroeder et al. (2019), and Kassar et al. (2016).

The Relative Mortality Metric (RMM) quantifies the performance of a center in comparison to the anticipated mortality based on the TRISS national benchmark. The RMM measures the difference between observed and expected mortality, with a range from -1 to 1.

An RMM of 0 indicates that the observed mortality aligns with the expected national benchmark across all acuity levels.
An RMM greater than 0 indicates better-than-expected performance, where the center is outperforming the national benchmark.
An RMM less than 0 indicates under-performance, where the center’s observed mortality is higher than the expected benchmark.

This metric helps assess how a center's mortality compares to the national standards, guiding quality improvement efforts. rmm() utilizes bootstrap sampling to calculate the confidence intervals via the standard error method.

Usage

rmm(
  data,
  Ps_col,
  outcome_col,
  group_vars = NULL,
  n_samples = 100,
  Divisor1 = 5,
  Divisor2 = 5,
  Threshold_1 = 0.9,
  Threshold_2 = 0.99,
  bootstrap_ci = TRUE,
  pivot = FALSE,
  seed = NULL
)

Arguments

data: A data frame or tibble containing the data.
Ps_col: The name of the column containing the survival probabilities (Ps). Should be numeric on a scale from 0 to 1.
outcome_col: The name of the column containing the outcome data. It should be binary, with values indicating patient survival. A value of 1 should represent "alive" (survived), while 0 should represent "dead" (did not survive). TRUE/FALSE are accepted as well. Ensure the column contains only these possible values.
group_vars: Optional character vector specifying grouping variables for stratified analysis. If NULL, the calculation is performed on the entire dataset.
n_samples: A numeric value indicating the number of bootstrap samples to take from the data source.
Divisor1: A divisor used for binning the survival probabilities (default is 5).
Divisor2: A second divisor used for binning the survival probabilities (default is 5).
Threshold_1: The first threshold for dividing the survival probabilities (default is 0.9).
Threshold_2: The second threshold for dividing the survival probabilities (default is 0.99).
bootstrap_ci: A logical indicating whether to return the relative mortality metric estimate and 95% confidence intervals using bootstrap sampling. Default is TRUE.
pivot: A logical indicating whether to return the results in a long format (pivot = TRUE) or wide format (pivot = FALSE, default). Use with caution in tandem with group_vars if the grouping variable is of a different class than rmm()'s outputs, such as factor or character grouping variables.
seed: Optional numeric value to set a random seed for reproducibility. If NULL (default), no seed is set.

Value

A tibble containing the Relative Mortality Metric (RMM) and related statistics:

population_RMM_LL: The lower bound of the 95% confidence interval for the population RMM.
population_RMM: The final calculated Relative Mortality Metric for the population existing in data.
population_RMM_UL: The upper bound of the 95% confidence interval for the population RMM.
population_CI: The confidence interval width for the population RMM.
bootstrap_RMM_LL: The lower bound of the 95% confidence interval for the bootstrap RMM. (optional, if bootstrap_ci = TRUE)
bootstrap_RMM: The average RMM value calculated for the bootstrap sample. (optional, if bootstrap_ci = TRUE)
bootstrap_RMM_UL: The upper bound of the 95% confidence interval for the bootstrap RMM. (optional, if bootstrap_ci = TRUE)
bootstrap_CI: The width of the 95% confidence interval for the bootstrap RMM. (optional, if bootstrap_ci = TRUE)
If pivot = TRUE, the results will be in long format with two columns: stat and value, where each row corresponds to one of the calculated statistics.
If pivot = FALSE (default), the results will be returned in wide format, with each statistic as a separate column.

Details

Like other statistical computing functions, rmm() is happiest without missing data. It is best to pass complete probability of survival and mortality outcome data to the function for optimal performance. With smaller datasets, this is especially helpful. However, rmm() will throw a warning about missing values, if any exist in Ps_col and/or outcome_col.

rmm() assumes Ps_col contains probabilities derived from real-world inputs for the Trauma Injury Severity Score (TRISS) model. Synthetic or low-variability data (especially with small sample sizes) may not reflect the distribution of TRISS-derived survival probabilities. This can result in unstable estimates or function failure due to insufficient dispersion. With small sample sizes, it may be important to use smaller values with the divisor arguments and adjust the thresholds (based on the distribution of the Ps_col values) to create bins that better accommodate the data.

Due to the use of bootstrap sampling within the function, users should consider setting the random number seed within rmm() for reproducibility.

Note

This function will produce the most reliable and interpretable results when using a dataset that has one row per patient, with each column being a feature.

By default, rmm() derives bin cut points from the full dataset’s distribution. This ensures comparability across groups when group_vars is used. To tailor results to a specific group (e.g., a single hospital), filter the dataset to that subgroup before calling rmm(). The function will then compute bins and related statistics using only that subset’s Ps_col distribution. When group_vars is used, and ff a group lacks observations within one or more bins, rm_bin_summary() will compute statistics only for the bins that contain data. Bins with no observations are excluded from the summary for that group.

References

Kassar, O.M., Eklund, E.A., Barnhardt, W.F., Napoli, N.J., Barnes, L.E., Young, J.S. (2016). Trauma survival margin analysis: A dissection of trauma center performance through initial lactate. The American Surgeon, 82(7), 649-653. doi:10.1177/000313481608200733

Napoli, N. J., Barnhardt, W., Kotoriy, M. E., Young, J. S., & Barnes, L. E. (2017). Relative mortality analysis: A new tool to evaluate clinical performance in trauma centers. IISE Transactions on Healthcare Systems Engineering, 7(3), 181–191. doi:10.1080/24725579.2017.1325948

Schroeder, P. H., Napoli, N. J., Barnhardt, W. F., Barnes, L. E., & Young, J. S. (2018). Relative mortality analysis of the “golden hour”: A comprehensive acuity stratification approach to address disagreement in current literature. Prehospital Emergency Care, 23(2), 254–262. doi:10.1080/10903127.2018.1489021

Author

Nicolas Foss, Ed.D, MS, original implementation in MATLAB by Nicholas J. Napoli, Ph.D., MS

Examples

# Generate example data
set.seed(123)

# Parameters
# Total number of patients
n_patients <- 5000

# Arbitrary group labels
groups <- sample(x = LETTERS[1:2], size = n_patients, replace = TRUE)

# Trauma types
trauma_type_values <- sample(
  x = c("Blunt", "Penetrating"),
  size = n_patients,
  replace = TRUE
)

# RTS values
rts_values <- sample(
  x = seq(from = 0, to = 7.8408, by = 0.005),
  size = n_patients,
  replace = TRUE
)

# patient ages
ages <- sample(
  x = seq(from = 0, to = 100, by = 1),
  size = n_patients,
  replace = TRUE
)

# ISS scores
iss_scores <- sample(
  x = seq(from = 0, to = 75, by = 1),
  size = n_patients,
  replace = TRUE
)

# Generate survival probabilities (Ps)
Ps <- traumar::probability_of_survival(
  trauma_type = trauma_type_values,
  age = ages,
  rts = rts_values,
  iss = iss_scores
)

# Simulate survival outcomes based on Ps
survival_outcomes <- rbinom(n_patients, size = 1, prob = Ps)

# Create data frame
data <- data.frame(Ps = Ps, survival = survival_outcomes, groups = groups) |>
  dplyr::mutate(death = dplyr::if_else(survival == 1, 0, 1))

# Example usage of the `rmm` function
rmm(data = data, Ps_col = Ps,
    outcome_col = survival,
    Divisor1 = 4,
    Divisor2 = 4,
    n_samples = 10
    )
#> # A tibble: 1 × 8
#>   population_RMM_LL population_RMM population_RMM_UL population_CI
#>               <dbl>          <dbl>             <dbl>         <dbl>
#> 1            -0.106        0.00247             0.111         0.108
#> # ℹ 4 more variables: bootstrap_RMM_LL <dbl>, bootstrap_RMM <dbl>,
#> #   bootstrap_RMM_UL <dbl>, bootstrap_CI <dbl>

# pivot!
rmm(data = data, Ps_col = Ps,
    outcome_col = survival,
    Divisor1 = 4,
    Divisor2 = 4,
    n_samples = 10,
    pivot = TRUE
    )
#> # A tibble: 8 × 2
#>   stat                 value
#>   <chr>                <dbl>
#> 1 population_RMM_LL -0.106  
#> 2 population_RMM     0.00247
#> 3 population_RMM_UL  0.111  
#> 4 population_CI      0.108  
#> 5 bootstrap_RMM_LL  -0.0145 
#> 6 bootstrap_RMM     -0.00544
#> 7 bootstrap_RMM_UL   0.00366
#> 8 bootstrap_CI       0.00910

# Create example grouping variable (e.g., hospital)
hospital <- sample(c("Hospital A", "Hospital B"), n_patients, replace = TRUE)

# Create data frame
data <- data.frame(
  Ps = Ps,
  survival = survival_outcomes,
  hospital = hospital
) |>
  dplyr::mutate(death = dplyr::if_else(survival == 1, 0, 1))

# Example usage of the `rmm` function with grouping by hospital
rmm(
  data = data,
  Ps_col = Ps,
  outcome_col = survival,
  group_vars = "hospital",
  Divisor1 = 4,
  Divisor2 = 4,
  n_samples = 10
)
#> # A tibble: 2 × 9
#>   hospital   population_RMM_LL population_RMM population_RMM_UL population_CI
#>   <chr>                  <dbl>          <dbl>             <dbl>         <dbl>
#> 1 Hospital A            -0.131         0.0211             0.173         0.152
#> 2 Hospital B            -0.169        -0.0149             0.139         0.154
#> # ℹ 4 more variables: bootstrap_RMM_LL <dbl>, bootstrap_RMM <dbl>,
#> #   bootstrap_RMM_UL <dbl>, bootstrap_CI <dbl>

# Pivoted output for easier visualization
rmm(
  data = data,
  Ps_col = Ps,
  outcome_col = survival,
  group_vars = "hospital",
  Divisor1 = 4,
  Divisor2 = 4,
  n_samples = 10,
  pivot = TRUE
)
#> # A tibble: 16 × 3
#>    hospital   stat                   value
#>    <chr>      <chr>                  <dbl>
#>  1 Hospital A population_RMM_LL -0.131    
#>  2 Hospital A population_RMM     0.0211   
#>  3 Hospital A population_RMM_UL  0.173    
#>  4 Hospital A population_CI      0.152    
#>  5 Hospital A bootstrap_RMM_LL   0.00784  
#>  6 Hospital A bootstrap_RMM      0.0202   
#>  7 Hospital A bootstrap_RMM_UL   0.0326   
#>  8 Hospital A bootstrap_CI       0.0124   
#>  9 Hospital B population_RMM_LL -0.169    
#> 10 Hospital B population_RMM    -0.0149   
#> 11 Hospital B population_RMM_UL  0.139    
#> 12 Hospital B population_CI      0.154    
#> 13 Hospital B bootstrap_RMM_LL  -0.0231   
#> 14 Hospital B bootstrap_RMM     -0.0115   
#> 15 Hospital B bootstrap_RMM_UL   0.0000725
#> 16 Hospital B bootstrap_CI       0.0116