Advanced Topics in Statistical Data Science

 

Propensity Scores When Treatment Isn’t Binary

causal inference
statistical inference
You know how to work with propensity scores when treatment is binary. Treatment is on or off, you model the probability of being treated given covariates, and then you…
Jul 1, 2026

Two reliability diagrams side by side. The left panel shows a well-calibrated model whose curve lies on the 45-degree diagonal. The right panel shows an over-confident model whose curve lies above the diagonal for low predicted probabilities and below it for high predicted probabilities.

Probability Calibration in ML

machine learning
statistical inference
When are the outputs of predict_proba actual probabilities? If the model says \(0.9\), should I assume that among all the cases it scored \(0.9\), roughly \(90\%\) were…
Jul 1, 2026
 

Kronecker Products in Econometrics

parametric models
statistical inference
Open any econometrics text past the introductory chapters and you will eventually hit a \(\otimes\) — the Kronecker product — usually bolted onto a covariance matrix and…
Jun 25, 2026
 

Working with Weak Instruments

causal inference
statistical inference
hypothesis testing
Weak instruments are not merely unhelpful; they can waste your time and your nerves. With a weak IV in hand, you can spend months trying to “fix” it, only for your paper or…
Jun 24, 2026
 

Always-Valid \(p\)-Values and Online Multiple Testing

multiple testing
hypothesis testing
statistical inference
The classical \(p\)-value is built for a world that modern data science has largely abandoned. That world has two defining features. First, you collect a fixed sample…
Jun 24, 2026
 

Linear Models and \(p > n\): What Can You Do?

parametric models
statistical inference
machine learning
It seems intuitive that you cannot fit a regression with more predictors than observations. With \(p\) parameters and only \(n < p\) data points, the reasoning goes, you…
Jun 22, 2026
 

A Simple Estimation Recipe for Targeted Learning

causal inference
machine learning
Targeted learning is extremely popular in public health and epidemiology, yet it has never really made its way into the economics and econometrics world. It was introduced…
Jun 12, 2026
 

The Role of Instrumental Variables in Randomized Experiments

causal inference
randomized experiments
Instrumental variables (IV) are econometricians’ pride. They underpin a great many scientific findings, and even the Nobel committee in Stockholm seems fond of them, judging…
Jun 11, 2026
 

Lasso and Its Theoretical Guarantees

machine learning
variable selection
statistical inference
Every data scientist has heard of the Lasso, and I have written plenty about it here as well — its many flavors, its oracle property, and its use in estimating heterogeneous…
Jun 11, 2026
 

Second-Generation \(p\)-Values: A Worthy Idea That Did Not Catch On

hypothesis testing
statistical inference
Hardly a data science interview goes by without the person in power bringing up the definition of the infamous \(p\)-value. Normally, chaos ensues. For good or ill, \(p\)-val…
Jun 11, 2026

Q-Q plot of iris sepal length

6 Underrated Plot Types

correlation
statistical inference
Most data science workflows rely on a familiar trio of plots: histograms, scatterplots, and boxplots. They are useful, but they leave a lot of structure hidden in the data.
Apr 6, 2026
 

The Many Flavors of Principal Component Analysis

machine learning
flavors
Principal component analysis (PCA) is one of those methods that everyone learns early and then quietly keeps using for years. The appeal is obvious: take a high-dimensional…
Apr 5, 2026
 

Brief Overview of Treatment Effect Bounds

causal inference
In applied causal work, the real problem is often not estimation but identification. Attrition, imperfect take-up, endogenous selection, and missing outcomes can all make…
Apr 2, 2026
 

What OLS Estimates in Causal Inference

causal inference
parametric models
OLS is still the default causal estimator in a surprising amount of applied work. That is often understandable. Regression is simple, transparent, and often a reasonable…
Apr 1, 2026
 

The Many Flavors of Lasso

machine learning
flavors
The Lasso (Least Absolute Shrinkage and Selection Operator), introduced by Tibshirani in 1996, has become one of the go-to tools for variable selection and shrinkage in…
Mar 14, 2026
 

The Oracle Property: What It Promises (and What It Doesn’t)

machine learning
variable selection
In high-dimensional regression, we sometimes hear that a variable selection method possesses the oracle property. The phrase sounds impressive: it suggests that an estimator…
Mar 13, 2026
 

Why Some Confidence Intervals Are Not Symmetric

statistical inference
hypothesis testing
Most of us were trained to think of a \(95\%\) confidence interval as
Mar 10, 2026
 

OLS with Fixed vs Random \(X\): What Actually Changes?

parametric models
statistical inference
In regression courses, you will eventually hear the phrase: “OLS works whether \(X\) is fixed or random.” That statement is correct, but dangerously compressed.
Mar 8, 2026
 

Logistic Regression in Randomized Trials?

causal inference
parametric models
Randomized controlled trials (RCTs) are the gold standard for causal inference. Random assignment guarantees that treatment is independent of potential outcomes. As a…
Feb 17, 2026
 

Randomization Inference: A Gentle Introduction

causal inference
statistical inference
Randomization inference offers a refreshing alternative to traditional parametric inference, providing exact control over Type I error rates without relying on large-sample…
Feb 12, 2026
 

Generalized Additive Models: What You Need to Know

semiparametric models
machine learning
Generalized Additive Models (GAMs) are one of the most powerful and flexible tools in a data scientist’s toolbox for modeling complex, nonlinear relationships between…
Feb 12, 2026
 

Understanding Correlated Random Effects Models

causal inference
For decades, panel data analysis has largely revolved around a familiar dichotomy: fixed effects (FE) versus random effects (RE). More recently, generalized fixed effects…
Feb 11, 2026
 

The Many Flavors of Propensity Score Methods In Causal Inference

causal inference
flavors
Introduced by Rosenbaum and Rubin in 1983, the propensity score, the probability of receiving treatment given observed covariates, has become the workhorse for handling…
Jan 22, 2026
 

The Wilcoxon-Mann-Whitney Test is Not a Test of Medians

statistical inference
hypothesis testing
Nonparametric tests like the Wilcoxon-Mann-Whitney (WMW) are among the most popular alternatives to the \(t\)- and \(z\)-tests in settings where normality assumptions break…
Jan 20, 2026
 

Unconditional Quantile Regression and Treatment Effects

causal inference
Quantile regression has become a widely used tool in econometrics and statistics, thanks to its ability to model the entire distribution of an outcome variable rather than…
Dec 21, 2025
 

The Many Flavors of Matching for Causal Inference

causal inference
flavors
If you’ve worked on causal inference with observational data, you’ve likely faced the fundamental challenge: the treated and control groups often look very different.…
May 27, 2025
 

The Many Flavors of Variable Selection

variable selection
machine learning
flavors
If you’ve ever worked with high-dimensional data, you’ve likely faced a familiar challenge: too many variables. Some features are pure noise, others are redundant or…
May 26, 2025
 

The Many Flavors of Bootstrap

bootstrap
statistical inference
flavors
At its heart, the bootstrap poses a simple yet powerful question: “What if we could resample from our existing data, treating it as a stand-in for the population?” By doing…
May 26, 2025
 

The Secret Life of Correlation: Myths and Thirteen Views

statistical inference
correlation
Statistical correlation has long captivated me—it’s probably the topic I’ve written about most on this blog. What makes it so compelling is the combination of theoretical…
May 24, 2025
 

The Kolmogorov–Smirnov Test as a Goodness-of-fit

statistical inference
hypothesis testing
The Kolmogorov–Smirnov (KS) test is a staple in the statistical toolbox for checking how well data fit a hypothesized distribution. It comes in both a one-sample and a…
May 5, 2025
 

Jackknife vs. Bootstrap: A Tale of Two Resamplers

bootstrap
statistical inference
If you’ve ever dived into resampling methods, you’ve likely come across the jackknife and the bootstrap. They both aim to help us estimate uncertainty or bias without…
May 4, 2025
 

The Roles of Covariates in Randomized Experiments

randomized experiments
causal inference
Properly implemented randomized experiments (such as randomized controlled trials (RCTs) and A/B tests) guarantee unbiased estimates of the causal effect of a treatment \(T\)…
May 2, 2025
 

Causal Inference with Residualized Regressions

causal inference
statistical inference
The Frisch-Waugh-Lovell (FWL) theorem offers an elegant alternative to standard multivariate linear regression when estimating causal effects. Instead of running a full…
May 1, 2025
 

Causal vs. Predictive Modeling: Subtle, but Crucial Differences

causal inference
machine learning
It’s one of the most common mix-ups I see among data scientists—especially those coming from a machine learning background: confusing causal modeling with predictive…
Apr 30, 2025
 

The Two Types of Weights in Causal Inference

weights
causal inference
Causal inference fundamentally seeks to answer: What is the effect of a treatment or intervention? The challenge lies in ensuring that the comparison groups—treated versus…
Feb 28, 2025

Binscatter: A New Visual Tool for Data Analysis

correlation
In the realm of data visualization, the classical scatter plot has long been a staple for exploring bivariate relationships. However, as datasets grow larger and more…
Feb 9, 2025
 

Filling in Missing Data with MCMC

statistical inference
Every dataset inevitably contains missing or incomplete values. Practitioners then face the dilemma of how to address these missing observations. A common approach, though…
Jan 31, 2025
 

The Limits of Semiparametric Models: The Efficiency Bound

statistical inference
semiparametric models
The efficiency bound is a cornerstone of the academic literature on semiparametric models, and it’s easy to see why. This bound quantifies the potential loss in efficiency…
Jan 22, 2025

The Limits of Nonparametric Models

statistical inference
nonparametric models
Nonparametric statistics offers a powerful toolkit for data analysis when the underlying data-generating process is too complex or unknown to be captured by parametric…
Jan 22, 2025
 

The Limits of Parametric Models: The Cramér-Rao Bound

statistical inference
parametric models
Obtaining the lowest possible variance is a primary goal for anyone working with statistical models. Efficiency (or precision), as is the jargon, is a cornerstone of…
Jan 12, 2025

The Three Classes of Statistical Models

parametric models
semiparametric models
nonparametric models
Statistical modeling is among the most exciting elements of working with data. When mentoring junior data scientists, I never fail to see the spark in their eyes when our…
Jan 12, 2025
 

The Delta Method: Simplifying Confidence Intervals for Complex Estimators

statistical inference
You’ve likely encountered this scenario: you’ve calculated an estimate for a particular parameter, and now you require a confidence interval. Seems straightforward, doesn’t…
Jan 10, 2025
 

Stein’s Paradox: A Simple Illustration

statistical inference
paradox
In the realm of statistics, few findings are as counterintuitive and fascinating as Stein’s paradox. It defies our common sense about estimation and provides a glimpse into…
Jan 10, 2025

Mutual Information: What, Why, How, and When

correlation
When exploring dependencies between variables, the data scientist’s toolbox often relies on correlation measures to reveal relationships and potential patterns. But what if…
Jan 2, 2025

Generating Variables with Predefined Correlation

correlation
Suppose you are working on a project where the relationship between two variables is influenced by an unobserved confounder, and you want to simulate data that reflects this…
Dec 20, 2024
 

Stratified Sampling with Continuous Variables

randomized experiments
causal inference
Stratified sampling is a foundational technique in survey design, ensuring that observations capture key characteristics of a population. By dividing the data into distinct…
Dec 18, 2024
 

Column-Sampling Bootstrap?

bootstrap
statistical inference
The bootstrap is a versatile resampling technique traditionally focused on rows. Let’s add a twist to the plain vanilla bootstrap. Imagine you have a wide dataset—many…
Dec 16, 2024
 

The Bootstrap and its Limitations

bootstrap
statistical inference
The bootstrap is a powerful resampling technique used to estimate the sampling distribution of a statistic. By repeatedly drawing observations with replacement from the…
Dec 16, 2024

Simpson’s Paradox: A Simple Illustration

paradox
causal inference
Simpson’s paradox is one of the most counterintuitive phenomena in data analysis. It describes situations where a trend observed within groups disappears—or even…
Dec 6, 2024
 

Causation without Correlation

causal inference
correlation
While most people understand that correlation doesn’t imply causation, it might surprise many to learn that causation doesn’t always result in correlation. In the absence of…
Nov 21, 2024
 

The Many Flavors of Gradient Boosting

machine learning
flavors
Gradient boosting has emerged as one of the most powerful techniques for predictive modeling. In its simplest form, we can think of gradient boosting like having a team of…
Nov 6, 2024

Bayesian Analysis of Randomized Experiments: A Modern Approach

randomized experiments
Imagine you’re a data scientist evaluating an A/B test of a new recommendation algorithm. The results show a modest but promising 0.5% lift in conversion rate—up from \(8\%\)…
Oct 29, 2024
 

Weights in Statistical Analyses

weights
statistical inference
Weights in statistical analyses offer a way to assign varying importance to observations in a dataset. Although powerful, they can be quite confusing due to the various…
Sep 18, 2024
 

Causality without Experiments, Unconfoundedness, or Instruments

causal inference
Causality is central to many practical data-related questions. Conventional methods for isolating causal relationships rely on experimentation, assume unconfoundedness, or…
Aug 12, 2024
 

FOCI: A New Variable Selection Method

variable selection
machine learning
In our data-abundant world, we often have access to tens, hundreds, or even thousands of variables. Most of these features are usually irrelevant or redundant, leading to…
Jun 11, 2024
 

Nonlinear Correlations and Chatterjee’s Coefficient

correlation
flavors
Much of data science is concerned with learning about the relationships between different variables. The most basic tool to quantify relationship strength is the correlation…
Apr 12, 2024
 

A Brief Introduction to Conformal Inference

machine learning
statistical inference
Traditional confidence intervals estimate the range in which a population parameter, such as a mean or regression coefficient, is likely to fall with a specified level of…
Dec 20, 2023
 

Using Conformal Inference for Variable Importance in Machine Learning

machine learning
statistical inference
variable selection
Many machine learning (ML) methods operate as opaque systems, generating predictions when given a dataset as input. Identifying which variables have the greatest impact on…
Dec 20, 2023
 

Recent Developments in False Discovery Rate

multiple testing
statistical inference
A while back I wrote an article summarizing various approaches to correcting for multiple hypothesis testing. The dominant framework, False Discovery Rate (FDR), controls…
Oct 27, 2023
 

ML-Based Regression Adjustments in Randomized Experiments

machine learning
randomized experiments
Randomized experiments are the gold standard when interested in measuring causal relationships with data. In settings with small treatment effects or underpowered designs, a…
Aug 1, 2023
 

The Alphabet of Learners for Heterogeneous Treatment Effects

machine learning
randomized experiments
heterogeneous treatment effects
flavors
Numerous tales illustrate the inadequacy of the average to capture meaningful quantities. Statisticians love these. In my favorite one the protagonist places her head in a…
Jul 28, 2023
 

Lasso for Heterogeneous Treatment Effects Estimation

heterogeneous treatment effects
causal inference
variable selection
Lasso is one of my favorite machine learning algorithms. It is so simple, elegant, and powerful. My feelings aside, Lasso indeed has a lot to offer. While, admittedly, it is…
Jun 30, 2023
 

An Overview of Machine Learning Methods in Causal Inference

machine learning
causal inference
flavors
The most exciting trend in causal inference over the last decade has been the infusion of machine learning (ML) techniques. Supervised machine learning is designed to find…
Apr 30, 2023
 

The Variance of Propensity Score Matching Estimators

statistical inference
causal inference
Propensity score matching (PSM) is among the most popular methods for estimating causal effects with observational data. It lends its fame to both its power and simplicity.…
Mar 30, 2023
 

Correlation is a Cosine

correlation
statistical inference
You might have come across the statement, “correlation is a cosine,” but never taken the time to explore its precise meaning. It certainly sounds intriguing—how can the…
Feb 9, 2023
 

Correlation is Not (Always) Transitive

correlation
statistical inference
At first, I found this really puzzling. \(X\) is correlated (Pearson) with \(Y\), and \(Y\) is correlated with \(Z\). Does this mean \(X\) is necessarily correlated with \(Z\)…
Dec 22, 2022
 

Lord’s Paradox: A Simple Illustration

correlation
paradox
Lord’s paradox presents a fascinating challenge in causal inference and statistics. It highlights how different statistical methods applied to the same data can lead to…
Dec 18, 2022
 

Hypothesis Testing in Linear Machine Learning Models

statistical inference
machine learning
variable selection
Machine learning models are an indispensable part of data science. They are incredibly good at what they are designed for – making excellent predictions. They fall short in…
Nov 6, 2022

Number of Statistically Significant Variables

The Many Flavors of Multiple Testing Adjustments

multiple testing
statistical inference
flavors
The abundance of data around us is a major factor making the data science field so attractive. It enables all kinds of impactful, interesting, or fun analyses. I admit this…
Oct 22, 2022
 

Hypothesis Testing with Population Data

hypothesis testing
statistical inference
Classical statistical theory is built on the idea of working with a sample of data from a given population of interest. Our software packages compute confidence intervals to…
Sep 23, 2022
 

Overlapping Confidence Intervals and Statistical (In)Significance

statistical inference
hypothesis testing
This is a mistake I’ve made myself—more times than I’d like to admit. Even seasoned professors and expert data scientists sometimes fall into the same trap.
Aug 12, 2022
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