21  hdDRS with a binary outcome

Similar to hdPS, there are seven steps in High-dimensional disease risk score (hdDRS). However, the sixth step is different. In the sixth step, we fit a propensity score model in the hdPS analysis, while we fit a disease risk score model in the hdDRS analysis. Moreover in the seventh step, estimating the treatment/exposure effect with the matched or inverse probability weighted data is a popular choice in the hdPS analysis, while adjusting for deciles of the disease risk score is a popular choice in the hdDRS analysis.

#	hdPS	hdDRS
Step 1	Proxy sources	Proxy sources
Step 2	Empirical	Empirical
Step 3	Recurrence	Recurrence
Step 4	Prioritize	Prioritize
Step 5	Covariates	Covariates
Step 6	Propensity score	Disease risk score
Step 7	Association	Association

hdPS vs hdDRS

• Utilizing these empirically identified proxies with investigator-specified measured confounders, the hdPS and hdDRS are proposed to reduce residual confounding bias.
• Both methods have seven steps.
• The sixth step is different: in the hdPS analysis, we we model the exposure (propensity score model), while in the hdDRS analysis, we model the outcome (disease risk score model).

(Kumamaru et al. 2016; Hossain 2025)

21.1 Step 0: Analytic data

We will use the same NHANES data for this hdDRS demonstration as we did for the hdPS with a binary exposure and a binary outcome demonstration.

dim(hdps.data)
#> [1] 3839  119

21.2 Step 6: Disease risk score (DRS)

Hansen (2008) shows that the DRS has a balancing property, called prognostic balance. Individuals sharing a similar DRS value can be regarded as having the same risk/prognosis for the outcome. In contrast, the propensity score has a covariate balance property.

(Hansen 2008)

21.2.1 Create DRS formula

With a binary outcome, there are two approaches to estimate the disease risk score (DRS):

• Fitting DRS model on unexposed individuals: Covariates include the investigator-specified and empirical covariates
• Fitting DRS model on the full cohort: Covariates include the exposure, investigator-specified and empirical covariates.

In this example, we will focus on fitting the DRS model on unexposed individuals:

hdps.data$outcome <- as.numeric(I(hdps.data$diabetes=='Yes'))
proxy.list.sel <- names(out3$autoselected_covariate_df[,-1])
proxyform <- paste0(proxy.list.sel, collapse = "+")
covform <- paste0(investigator.specified.covariates, collapse = " + ")

rhsformula <- paste0(c(covform, proxyform), collapse = "+")
drs.formula <- as.formula(paste0("outcome", "~", rhsformula))
drs.formula
#> outcome ~ age.cat + sex + education + race + marital + income + 
#>     born + year + diabetes.family.history + medical.access + 
#>     smoking + diet.healthy + physical.activity + sleep + uric.acid + 
#>     protein.total + bilirubin.total + phosphorus + sodium + potassium + 
#>     globulin + calcium.total + systolicBP + diastolicBP + high.cholesterol + 
#>     rec_dx_I10_once + rec_dx_R73_once + rec_dx_I10_frequent + 
#>     rec_dx_R60_once + rec_dx_E78_once + rec_dx_M79_once + rec_dx_I51_once + 
#>     rec_dx_M10_once + rec_dx_I50_once + rec_dx_K21_once + rec_dx_D75_once + 
#>     rec_dx_Z79_once + rec_dx_F41_once + rec_dx_M1A_once + rec_dx_E87_once + 
#>     rec_dx_R12_once + rec_dx_R51_once + rec_dx_J45_once + rec_dx_I50_frequent + 
#>     rec_dx_L70_once + rec_dx_M25_once + rec_dx_I63_once + rec_dx_R39_once + 
#>     rec_dx_N28_once + rec_dx_K25_once + rec_dx_F90_once + rec_dx_B00_once + 
#>     rec_dx_J42_once + rec_dx_R41_once + rec_dx_I20_once + rec_dx_M54_once + 
#>     rec_dx_J44_once + rec_dx_K08_once + rec_dx_I21_once + rec_dx_F32_once + 
#>     rec_dx_J30_once + rec_dx_F43_once + rec_dx_R06_once + rec_dx_I48_once + 
#>     rec_dx_R32_once + rec_dx_R42_once + rec_dx_N92_once + rec_dx_N95_once + 
#>     rec_dx_M19_once + rec_dx_E07_once + rec_dx_R25_once + rec_dx_G43_once + 
#>     rec_dx_R52_once + rec_dx_M81_once + rec_dx_T78_once + rec_dx_G47_once + 
#>     rec_dx_R11_once + rec_dx_B35_once + rec_dx_M06_once + rec_dx_E04_once + 
#>     rec_dx_H40_frequent + rec_dx_T14_once + rec_dx_C50_once + 
#>     rec_dx_H40_once + rec_dx_N32_once + rec_dx_A49_once + rec_dx_J45_frequent + 
#>     rec_dx_N39_once + rec_dx_R09_once + rec_dx_N94_once + rec_dx_F39_once + 
#>     rec_dx_E03_once + rec_dx_K59_once + rec_dx_F31_once + rec_dx_L40_once + 
#>     rec_dx_M06_frequent + rec_dx_M13_once + rec_dx_I48_frequent + 
#>     rec_dx_G25_once + rec_dx_F31_frequent + rec_dx_M62_once + 
#>     rec_dx_K92_once + rec_dx_R10_once + rec_dx_G40_once + rec_dx_G40_frequent + 
#>     rec_dx_J44_frequent + rec_dx_K04_once + rec_dx_I49_once + 
#>     rec_dx_R00_once + rec_dx_R07_once + rec_dx_H04_once + rec_dx_K27_once + 
#>     rec_dx_R05_once + rec_dx_K30_once + rec_dx_N40_once + rec_dx_R35_once

21.2.2 Unexposed cohort

# Unexposed cohort
dat.unexposed <- subset(hdps.data, obese == "No")

21.2.3 Fit DRS model

Here we fit a logistic regression model on the unexposed individuals. Similar to the hdPS example, we are adding only the main effects in the non-transformed form.

fit.drs <- glm(drs.formula, data = dat.unexposed, family = binomial)

21.2.4 Obtain DRS

Now we can obtain the DRS for the full cohort and check the summary. Unlike in hdPS, balance checking is difficult in hdDRS analysis. There are some proposed matching and weighting techniques with hdDRS where balance checking is possible. These techniques are developed/proposed with only investigator-specified covariates. In this demonstration, we adjust our outcome model for deciles of the DRS, which is a common practice in the hdDRS literature.

(Wyss et al. 2015; Nguyen et al. 2024)

hdps.data$drs <- predict(fit.drs, type = "response", newdata = hdps.data)

# Sumamry
summary(hdps.data$drs)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.00000 0.01255 0.06931 0.19362 0.26619 0.99998

# Summary by exposure status
tapply(hdps.data$drs, hdps.data$obese, summary)
#> $No
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.00000 0.01281 0.06271 0.17814 0.23135 0.99989 
#> 
#> $Yes
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.00000 0.01186 0.07603 0.21491 0.31354 0.99998

21.3 Step 7: Association

21.3.1 Adjtsing for deciles of DRS

# Deciles of DRS
hdps.data$drs.decile <- as.factor(dplyr::ntile(hdps.data$drs, 10))

# Outcome analysis
fit.hddrs <- glm(outcome ~ obese + drs.decile, 
                 data = hdps.data,
                 family = binomial)

publish(fit.hddrs, pvalue.method = "robust", confint.method = "robust", 
        print = F)$regressionTable[1:2,]

21.3.2 Adjtsing for deciles of DRS and covariates

Another popular approach is to adjust for deciles of the DRS as well as for the investigator-specified covariates:

# Outcome analysis
fit.hddrs1 <- glm(outcome ~ obese + drs.decile + age.cat + sex + education + 
                   race + marital + income + born + year + 
                   diabetes.family.history + medical.access + smoking + 
                   diet.healthy + physical.activity + sleep + uric.acid + 
                   protein.total + bilirubin.total + phosphorus + sodium + 
                   potassium + globulin + calcium.total + systolicBP + 
                   diastolicBP + high.cholesterol, data = hdps.data,
                 family = binomial)

publish(fit.hddrs1, pvalue.method = "robust", confint.method = "robust", 
        print = F)$regressionTable[1:2,]

Hansen, Ben B. 2008. “The Prognostic Analogue of the Propensity Score.” Biometrika 95 (2): 481–88.

Hossain, Md Belal. 2025. “Chapter 2: High-Dimensional Disease Risk Score for Dealing with Residual Confounding in Estimating Treatment Effects with a Survival Outcome.” In. Harnessing the power of causal inference; predictive analytics for survival outcomes with health administrative data: applications to tuberculosis research.

Kumamaru, Hiraku, Joshua J Gagne, Robert J Glynn, Soko Setoguchi, and Sebastian Schneeweiss. 2016. “Comparison of High-Dimensional Confounder Summary Scores in Comparative Studies of Newly Marketed Medications.” Journal of Clinical Epidemiology 76: 200–208.

Nguyen, Tri-Long, Thomas PA Debray, Bora Youn, Gabrielle Simoneau, and Gary S Collins. 2024. “Confounder Adjustment Using the Disease Risk Score: A Proposal for Weighting Methods.” American Journal of Epidemiology 193 (2): 377–88.

Wyss, Richard, Alan R Ellis, M Alan Brookhart, Michele Jonsson Funk, Cynthia J Girman, Ross J Simpson Jr, and Til Stürmer. 2015. “Matching on the Disease Risk Score in Comparative Effectiveness Research of New Treatments.” Pharmacoepidemiology and Drug Safety 24 (9): 951–61.