Heterogeneity & Demographic Analysis

Introduction

Heterogeneity analysis is a way to explore how the results of a model can vary depending on the characteristics of individuals in a population, and demographic analysis estimates the average values of a model over an entire population.

In practice these two analyses naturally complement each other: heterogeneity analysis runs the model on multiple sets of parameters (reflecting different characteristics found in the target population), and demographic analysis combines the results.

For this example we will use the result from the assessment of a new total hip replacement previously described in vignette("d-non-homogeneous", "heemod").

Population characteristics

The characteristics of the population are input from a table, with one column per parameter and one row per individual. Those may be for example the characteristics of the indiviuals included in the original trial data.

For this example we will use the characteristics of 100 individuals, with varying sex and age, specified in the data frame tab_indiv:

tab_indiv

## # A tibble: 100 × 2
##      age   sex
##    <dbl> <int>
##  1    49     1
##  2    57     0
##  3    65     1
##  4    39     0
##  5    72     0
##  6    58     1
##  7    60     1
##  8    55     0
##  9    56     1
## 10    47     0
## # ℹ 90 more rows

library(ggplot2)
ggplot(tab_indiv, aes(x = age)) +
  geom_histogram(binwidth = 2)

Running the analysis

res_mod, the result we obtained from run_model() in the Time-varying Markov models vignette, can be passed to update() to update the model with the new data and perform the heterogeneity analysis.

res_h <- update(res_mod, newdata = tab_indiv)

## No weights specified in update, using equal weights.

## Updating strategy 'standard'...

## Updating strategy 'np1'...

Interpreting results

The summary() method reports summary statistics for cost, effect and ICER, as well as the result from the combined model.

summary(res_h)

## An analysis re-run on 100 parameter sets.
## 
## * Unweighted analysis.
## 
## * Values distribution:
## 
##                                  Min.      1st Qu.     Median        Mean
## standard - Cost          438.70535048  605.0062810 621.589242 683.2237991
## standard - Effect          5.05860925   24.4991251  27.780658  26.3457276
## standard - Cost Diff.               -            -          -           -
## standard - Effect Diff.             -            -          -           -
## standard - Icer                     -            -          -           -
## np1 - Cost               590.76054210  635.5509751 640.058850 657.8358957
## np1 - Effect               5.07524179   24.8264025  27.975477  26.6017928
## np1 - Cost Diff.        -159.96283707  -99.5031416  18.469607 -25.3879034
## np1 - Effect Diff.         0.01159912    0.1948185   0.220806   0.2560652
## np1 - Icer              -351.98058303 -304.0330575  83.646307 198.7461029
##                             3rd Qu.         Max.
## standard - Cost         786.6690449 8.711621e+02
## standard - Effect        29.0596426 3.152925e+01
## standard - Cost Diff.             -            -
## standard - Effect Diff.           -            -
## standard - Icer                   -            -
## np1 - Cost              687.1659033 7.111993e+02
## np1 - Effect             29.2683350 3.176519e+01
## np1 - Cost Diff.         30.5446941 1.520552e+02
## np1 - Effect Diff.        0.3272774 4.544649e-01
## np1 - Icer              156.7853582 1.310920e+04
## 
## * Combined result:
## 
## 2 strategies run for 60 cycles.
## 
## Initial state counts:
## 
## PrimaryTHR = 1000L
## SuccessP = 0L
## RevisionTHR = 0L
## SuccessR = 0L
## Death = 0L
## 
## Counting method: 'beginning'.
## 
## Values:
## 
##           utility     cost
## standard 26345.73 683223.8
## np1      26601.79 657835.9
## 
## Efficiency frontier:
## 
## np1
## 
## Differences:
## 
##     Cost Diff. Effect Diff.      ICER     Ref.
## np1   -25.3879    0.2560652 -99.14626 standard

The variation of cost or effect can then be plotted.

plot(res_h, result = "effect", binwidth = 5)

plot(res_h, result = "cost", binwidth = 50)

plot(res_h, result = "icer", type = "difference",
     binwidth = 500)

plot(res_h, result = "effect", type = "difference",
     binwidth = .1)

plot(res_h, result = "cost", type = "difference",
     binwidth = 30)

The results from the combined model can be plotted similarly to the results from run_model().

plot(res_h, type = "counts")

Weighted results

Weights can be used in the analysis by including an optional column .weights in the new data to specify the respective weights of each strata in the target population.

tab_indiv_w

## # A tibble: 100 × 3
##      age   sex .weights
##    <dbl> <int>    <dbl>
##  1    57     1   0.0102
##  2    61     0   0.495 
##  3    71     1   0.666 
##  4    49     1   0.936 
##  5    49     1   0.387 
##  6    79     1   0.233 
##  7    65     0   0.895 
##  8    61     1   0.686 
##  9    55     0   0.701 
## 10    60     1   0.533 
## # ℹ 90 more rows

res_w <- update(res_mod, newdata = tab_indiv_w)

## Updating strategy 'standard'...

## Updating strategy 'np1'...

res_w

## An analysis re-run on 100 parameter sets.
## 
## * Weights distribution:
## 
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.003172 0.251257 0.541292 0.510826 0.753942 0.992491 
## 
## Total weight: 51.08255
## 
## * Values distribution:
## 
##                                  Min.      1st Qu.      Median        Mean
## standard - Cost          500.08967163  605.0062810 684.5319673 694.8918701
## standard - Effect         10.06345874   24.4991251  25.9857701  25.5972990
## standard - Cost Diff.               -            -           -           -
## standard - Effect Diff.             -            -           -           -
## standard - Icer                     -            -           -           -
## np1 - Cost               607.16692250  635.5509751 658.3345507 661.1555528
## np1 - Effect              10.13073146   24.8264025  26.1614223  25.8570922
## np1 - Cost Diff.        -159.96283707  -99.5031416 -12.4380622 -33.7363174
## np1 - Effect Diff.         0.05767389    0.1948185   0.2324224   0.2597931
## np1 - Icer              -351.98058303 -304.0330575 -90.1547672 -15.5133867
##                             3rd Qu.         Max.
## standard - Cost         786.6690449  871.1621236
## standard - Effect        28.5543952   31.6837747
## standard - Cost Diff.             -            -
## standard - Effect Diff.           -            -
## standard - Icer                   -            -
## np1 - Cost              687.1659033  711.1992865
## np1 - Effect             28.9688773   31.9214350
## np1 - Cost Diff.         30.5446941  107.0772509
## np1 - Effect Diff.        0.3272774    0.4544649
## np1 - Icer              156.7853582 1856.5985016
## 
## * Combined result:
## 
## 2 strategies run for 60 cycles.
## 
## Initial state counts:
## 
## PrimaryTHR = 1000L
## SuccessP = 0L
## RevisionTHR = 0L
## SuccessR = 0L
## Death = 0L
## 
## Counting method: 'beginning'.
## 
## Values:
## 
##           utility     cost
## standard 25597.30 694891.9
## np1      25857.09 661155.6
## 
## Efficiency frontier:
## 
## np1
## 
## Differences:
## 
##     Cost Diff. Effect Diff.      ICER     Ref.
## np1  -33.73632    0.2597931 -129.8584 standard