Statistics

Dose-Response Curves and Biomarker Diagnostic Power

The other day, one of my friends and colleagues (I’ll refer to him as “Dr. A”) asked me if I knew anything about assessing biomarker diagnostic power. He went on to describe his clinical problem, which I’ll try to recount here …

The Delta Method

Here, we’ll look at various applications of the Delta Method, especially in the context of variance stabilizing transformations, along with looking at the confidence intervals of estimates. The Delta Method is used as a way to approximate the …

Mahalanobis Distances of Brain Connectivity

For one of the projects I’m working on, I have an array of multivariate data relating to brain connectivity patterns. Briefly, each brain is represented as a surface mesh, which we represent as a graph $G = (V,E)$, where $V$ is a set of $n$ …

Convergence In Probability

I’m going over Chapter 5 in Casella and Berger’s (CB) “Statistical Inference”, specifically Section 5.5: Convergence Concepts, and wanted to document the topic of convergence in probability with some plots demonstrating the …

Overview of Poisson-Multinomial Relationship

In this post, I’m going to briefly cover the relationship between the Poisson distribution and the Multinomial distribution. Let’s say that we have a set of independent, Poisson-distributed random variables $Y_{1}, Y_{2}… Y_{k}$ …

Multivariate Normal Distribution

In this post, I’ll be covering the basics of Multivariate Normal Distributions, with special emphasis on deriving the conditional and marginal distributions. Given a random variable under the usual Gauss-Markov assumptions, with $y_{i} \sim …

Blockwise Matrix Inversion

I’m taking a Statistics course on the theory of linear models, which covers Gauss-Markov models and various extensions of them. Sometimes, when dealing with partitioned matrices, and commonly Multivariate Normal Distributions, we’ll often …