# Tutorial: Identify the optimal D subtype solution

## Introduction

Sometimes subtypes will not be pre-defined, but rather it will be of interest to identify subtypes that differ maximally with respect to risk heterogeneity based on some number of disease markers.

This tutorial will introduce you to the optimal_kmeans_d() function, which will run $$k$$-means clustering, using the kmeans() function, on disease marker data and identify the subtype solution that maximizes $$D$$. When $$k$$-means clustering is run with multiple random starts, it will return a variety of class solutions, as the algorithm typically reaches a local rather than global maxima. Then for each candidate class solution, $$D$$ can be computed and the solution that maximizes $$D$$, a measure of etiologic heterogeneity, can be identified. This function is currently for use with case-control data only. See Tutorial: Estimate the extent of etiologic heterogeneity for details on the calculation of the $$D$$ value.

## Example

This tutorial will make use of a simulated example dataset named subtype_data. This simulated dataset contains 1200 case subjects and 800 control subjects. There are 30 continuous disease markers available on the cases. There are two continuous risk factors and one binary risk factor available on all subjects.

Say for example that interest was in identifying the optimally etiologically heterogeneous 3-subtype solution based on all 30 markers y1-y30 available in subtype_data, using all three risk factors x1, x2 and x3. We use the default of 100 random starts of the $$k$$-means clustering algorithm, and set a seed so that results would be reproduced if we were to run this code again on a later date. Note that this function is currently a bit slow to run due to the fitting of numerous models, so please be patient.

library(riskclustr)

res <- optimal_kmeans_d(
markers = c(paste0("y", seq(1:30))),
M = 3,
factors = list("x1", "x2", "x3"),
case = "case",
data = subtype_data,
nstart = 100,
seed = 81110224)

The function returns a list with one element for the optimal $$D$$ value and one element that is the original data frame with a column added for the optimal $$D$$ class solution.

First let’s look at a tabulation of optimal_d_label from the optimal_d_data object to see how each case was classified into a subtype:

table(res[["optimal_d_data"]]\$optimal_d_label)
#>
#>   0   1   2   3
#> 800 301 300 599

These subtype labels could now be used as the outcome in a polytomous logistic regression model fit with eh_test_subtype() to obtain measures of risk factor association as well as heterogeneity $$p$$-values. See Tutorial: test for etiologic heterogeneity in a case-control study for details.

Next we can see the $$D$$ value that goes long with the optimal subtype solution:

res[["optimal_d"]]
#> [1] 0.3385876

We see that this optimal 3-subtype solution based on clustering the 30 disease markers results in $$D=0.339$$.