Signature
post_1 = pi_1 * f_1 / (pi_1 * f_1 + pi_2 * f_2)
| Inputs | Definition | Unit |
|---|---|---|
pi_1 | Estimated share of the population in class 1 | proportion |
f_1 | Value of class 1's fitted distribution at the unit's observed data | probability or density |
pi_2 | Estimated share of the population in class 2, equal to 1 minus pi_1 with two classes | proportion |
f_2 | Value of class 2's fitted distribution at the unit's observed data, on the same scale as f_1 | probability or density |
post_1 | Probability that the unit belongs to class 1 given its data | probability |
|---|
Function
Grouping patients or other units by dissimilarity in cluster analysis
Maps the clustering variables measured on each unit, such as a patient's counts of emergency and outpatient care, to a set of groups found in the data without an outcome variable. Combinatorial methods such as k-means assign each unit to one group by minimising the dissimilarity within groups; mixture models such as latent class analysis give each unit a probability of belonging to each class. The notation follows the Cluster Analysis article, whose six-patient example is used throughout.
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Implementations
Excel
Posterior class probability from named shares and likelihoods
With ShareOne, DensOne, ShareTwo and DensTwo named, the formula returns the posterior probability of class 1, held in PostOne. For K classes in ranges Shares and Dens, =INDEX(Shares,1)*INDEX(Dens,1)/SUMPRODUCT(Shares,Dens) gives class 1.
=ShareOne*DensOne/(ShareOne*DensOne+ShareTwo*DensTwo)
Assumptions
Class shares and class distributions taken from the fitted mixture model
pi_k and f_k come from a mixture model fitted by maximum likelihood or Bayesian methods, with the shares adding to 1; the posterior inherits their estimation uncertainty.
Each unit belongs to exactly one latent class
The classes are mutually exclusive and exhaustive, so the posterior probabilities for a unit add to 1.
Worked examples
Patient with 3 emergency attendances and a high-use class of 20 per cent
If attendances follow a Poisson distribution with mean 5 in a high-use class holding 20 per cent of patients and mean 1 in a low-use class, 3 attendances have likelihoods of about 0.1404 and 0.0613; the posterior probability of the high-use class is about 0.364 (computed here for illustration).
pi_1 = 0.2; f_1 = 0.140374; pi_2 = 0.8; f_2 = 0.061313; post_1 = 0.364
Patient with 6 emergency attendances in the same two-class model
With 6 attendances the likelihoods are about 0.1462 and 0.0005, so the posterior probability of the high-use class is about 0.9862 (computed here for illustration).
pi_1 = 0.2; f_1 = 0.146223; pi_2 = 0.8; f_2 = 0.000511; post_1 = 0.9862
Patient with 1 emergency attendance in the same two-class model
With 1 attendance the likelihoods are about 0.0337 and 0.3679, giving a high-use probability of about 0.0224 (computed here for illustration).
pi_1 = 0.2; f_1 = 0.03369; pi_2 = 0.8; f_2 = 0.367879; post_1 = 0.0224
Common errors
Treating the most probable class as certain
The patient with 3 attendances is allocated to the low-use class with probability of only about 0.64; carrying modal classes into a model as if known hides the allocation uncertainty, which adds parameter uncertainty to any class-specific analysis.
Reading class shares as individual membership probabilities
The 20 per cent share is the prior; a patient's own probability depends on the data and ranges from about 0.02 to 0.99 across the three examples.
Sources
Posterior probability that an observation belongs to a mixture component
Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd ed. New York: Springer; 2009. Section 14.3.4: mixture models describe each cluster by a component density and are fitted by maximum likelihood or corresponding Bayesian approaches; section 6.8, equation 6.33, p. 215: the mixture model provides an estimate of the probability that observation i belongs to component m, the component share times its density at x_i divided by the sum over components; section 14.3.7, p. 511: with Gaussian components of shared scalar variance EM is a soft version of k-means, and as the variance tends to 0 the probabilities become 0 and 1 and the two methods coincide.
Individuals assigned to the cluster with the maximum posterior probability
Yan S, Kwan YH, Tan CS, Thumboo J, Low LL. A systematic review of the clinical application of data-driven population segmentation analysis. BMC Medical Research Methodology. 2018;18(1):121. doi:10.1186/s12874-018-0584-9. Methods and Discussion: parametric methods such as latent class analysis assign an individual to a cluster with, for example, the maximum posterior probability of membership, and latent class analysis has goodness-of-fit measures to help determine the statistically optimal number of segments; latent class methods were the most common segmentation approach among the 216 included studies (96).
Canonical Identity
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