Population Stability Index

One way to measure 'shifts' in the proportion of observations within subgroups is by the Population Stability Index (PSI). When a sample population is classified into various subgroups, one might want to establish whether those subgroups are stable with respect to a base population. The PSI is quantified by the formula:

 PSI= \sum_{i=1}^K(X_i-Y_i)*\ln(X_i/Y_i)

where X_i and Y_i refer to the proportion of observations belonging to the i^{th} subgroup in the sample and base population, respectively. Naturally, K is the number of subgroups.

A general rule-of-thumb is when PSI \geq 0.25, a significant change in the sample population has occurred. When 0.25 \geq PSI \geq 0.10, slight changes have occurred. Below 0.10 generally indicates stability.

In R, the PSI can be calculated using the follow function:

#######################: Population Stability Index

#index=unique identifier of accounts (e.g. positive integer)
#segment= segment/group number (e.g. positive integer)
#subgroup=population subgroup (e.g. quarters, years)
#data=dataset containing index, segment, and subgroup columns 

#matrix of PSI (subgroup x segments)

#if no index, the function will create an index
#if no segment, function will run without segments
#if no subgroup, function will not run
#if no data, function will not run

PSI_function <- function(index, segment, subgroup, data){
 #no subgroup or data? exit function
 if (missing(subgroup) || missing(data)) {return(NULL)}
 #no index? create index
 if (missing(index)) {
 data[,index] <- 1:nrow(data)}

 #no segments?
 if (missing(segment)){
 form <- as.formula(paste(index, "~", subgroup, sep=""))
 X <- aggregate(form, data=data, length)[,2]/nrow(data)
 Y <- X[[length(X)]]
 PSI <- as.matrix((X-Y)*log(X/Y))
 rownames(PSI) <- sort(unique(data[,subgroup]))
 if (!missing(segment)){
 PSI <- matrix(0, nrow=length(unique(data[,subgroup])), ncol=length(unique(data[,segment])))
 for (i in 1:length(unique(data[,segment]))){
 form <- as.formula(paste(index, "~", subgroup, sep=""))
 data_subset <- data[data[,segment]==sort(unique(data[,segment]))[i],]
 X <- aggregate(form, data=data_subset, length)[,2]/nrow(data_subset)
 Y <- X[[length(X)]]
 PSI[,i] <- (X-Y)*log(X/Y)
 colnames(PSI) <- sort(unique(data[,segment]))
 rownames(PSI) <- sort(unique(data[,subgroup]))


#PSI_function(index="N", subgroup="Obs_Date", segment="Seg", data=SSL_data)

Note: If a model is developed using a sample population from some specified time period, it may be worthwhile to measure the PSI at every observation point spanning this period (e.g. yearly, quarterly, monthly, etc.). A dramatic shift in the sample population, as evidence by the PSI, may warrant investigation of the model. This is because the model, that is used to make inferences about subgroups of the development sample population, may no longer be representative of the base population required to make inferences for. This does not mean the model is not longer applicable for making inferences about a base population, it merely indicates shifts in the proportion of subgroup observations has occurred.
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