Direct Standardization

Introduction

Consider a set of observations \((x_i,y_i)\) drawn non-uniformly from an unknown distribution. We know the expected value of the columns of \(X\), denoted by \(b \in {\mathbf R}^n\), and want to estimate the true distribution of \(y\). This situation may arise, for instance, if we wish to analyze the health of a population based on a sample skewed toward young males, knowing the average population-level sex, age, etc. The empirical distribution that places equal probability \(1/m\) on each \(y_i\) is not a good estimate.

So, we must determine the weights \(w \in {\mathbf R}^m\) of a weighted empirical distribution, \(y = y_i\) with probability \(w_i\), which rectifies the skewness of the sample (Fleiss, Levin, and Paik 2003, 19.5). We can pose this problem as

\[ \begin{array}{ll} \underset{w}{\mbox{maximize}} & \sum_{i=1}^m -w_i\log w_i \\ \mbox{subject to} & w \geq 0, \quad \sum_{i=1}^m w_i = 1,\quad X^Tw = b. \end{array} \]

Our objective is the total entropy, which is concave on \({\mathbf R}_+^m\), and our constraints ensure \(w\) is a probability distribution that implies our known expectations on \(X\).

To illustrate this method, we generate \(m = 1000\) data points \(x_{i,1} \sim \mbox{Bernoulli}(0.5)\), \(x_{i,2} \sim \mbox{Uniform}(10,60)\), and \(y_i \sim N(5x_{i,1} + 0.1x_{i,2},1)\). Then we construct a skewed sample of \(m = 100\) points that overrepresent small values of \(y_i\), thus biasing its distribution downwards. This can be seen in Figure , where the sample probability distribution peaks around \(y = 2.0\), and its cumulative distribution is shifted left from the population’s curve. Using direct standardization, we estimate \(w_i\) and reweight our sample; the new empirical distribution cleaves much closer to the true distribution shown in red.

In the CVXR code below, we import data from the package and solve for \(w\).

## Import problem data
data(dspop)   # Population
data(dssamp)  # Skewed sample

ypop <- dspop[,1]
Xpop <- dspop[,-1]
y <- dssamp[,1]
X <- dssamp[,-1]
m <- nrow(X)

## Given population mean of features
b <- as.matrix(apply(Xpop, 2, mean))

## Construct the direct standardization problem
w <- Variable(m)
objective <- sum(entr(w))
constraints <- list(w >= 0, sum(w) == 1, t(X) %*% w == b)
prob <- Problem(Maximize(objective), constraints)

## Solve for the distribution weights
result <- solve(prob)
weights <- result$getValue(w)
result$value
## [1] 4.223305

We can plot the density functions using linear approximations for the range of \(y\).

## Plot probability density functions
dens1 <- density(ypop)
dens2 <- density(y)
dens3 <- density(y, weights = weights)
yrange <- seq(-3, 15, 0.01)
d <- data.frame(x = yrange,
                True = approx(x = dens1$x, y = dens1$y, xout = yrange)$y,
                Sample = approx(x = dens2$x, y = dens2$y, xout = yrange)$y,
                Weighted = approx(x = dens3$x, y = dens3$y, xout = yrange)$y)
plot.data <- gather(data = d, key = "Type", value = "Estimate", True, Sample, Weighted,
                    factor_key = TRUE)
ggplot(plot.data) +
    geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
    theme(legend.position = "top")
## Warning: Removed 300 row(s) containing missing values (geom_path).
Probability distribution functions population, skewed sample and reweighted sample

Figure 1: Probability distribution functions population, skewed sample and reweighted sample

Followed by the cumulative distribution function.

## Return the cumulative distribution function
get_cdf <- function(data, probs, color = 'k') {
    if(missing(probs))
        probs <- rep(1.0/length(data), length(data))
    distro <- cbind(data, probs)
    dsort <- distro[order(distro[,1]),]
    ecdf <- base::cumsum(dsort[,2])
    cbind(dsort[,1], ecdf)
}

## Plot cumulative distribution functions
d1 <- data.frame("True", get_cdf(ypop))
d2 <- data.frame("Sample", get_cdf(y))
d3 <- data.frame("Weighted", get_cdf(y, weights))

names(d1) <- names(d2) <- names(d3) <- c("Type", "x", "Estimate")
plot.data <- rbind(d1, d2, d3)

ggplot(plot.data) +
    geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
    theme(legend.position = "top")

Session Info

sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin19.5.0 (64-bit)
## Running under: macOS Catalina 10.15.7
## 
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.10_1/lib/libopenblasp-r0.3.10.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices datasets  utils     methods   base     
## 
## other attached packages:
## [1] tidyr_1.1.0   ggplot2_3.3.2 CVXR_1.0-9   
## 
## loaded via a namespace (and not attached):
##  [1] tidyselect_1.1.0 xfun_0.15        slam_0.1-47      purrr_0.3.4     
##  [5] lattice_0.20-41  Rmosek_9.2.3     colorspace_1.4-1 vctrs_0.3.2     
##  [9] generics_0.0.2   htmltools_0.5.0  yaml_2.2.1       gmp_0.6-0       
## [13] rlang_0.4.7      pillar_1.4.6     glue_1.4.1       Rmpfr_0.8-1     
## [17] withr_2.2.0      Rcplex_0.3-3     bit64_0.9-7      lifecycle_0.2.0 
## [21] stringr_1.4.0    munsell_0.5.0    blogdown_0.19    gtable_0.3.0    
## [25] gurobi_9.0.3.1   codetools_0.2-16 evaluate_0.14    labeling_0.3    
## [29] knitr_1.28       cccp_0.2-4       highr_0.8        Rcpp_1.0.5      
## [33] scales_1.1.1     farver_2.0.3     bit_1.1-15.2     digest_0.6.25   
## [37] stringi_1.4.6    bookdown_0.19    dplyr_1.0.0      grid_4.0.2      
## [41] Rglpk_0.6-4      ECOSolveR_0.5.3  tools_4.0.2      magrittr_1.5    
## [45] tibble_3.0.3     crayon_1.3.4     pkgconfig_2.0.3  ellipsis_0.3.1  
## [49] rcbc_0.1.0.9001  Matrix_1.2-18    assertthat_0.2.1 rmarkdown_2.3   
## [53] R6_2.4.1         compiler_4.0.2

Source

R Markdown

References

Fleiss, J. L., B. Levin, and M. C. Paik. 2003. Statistical Methods for Rates and Proportions. Wiley-Interscience.