The g-formula is a foundational tool for identifying causal effects in observational data. This tool is based on the law of iterated expectation, a key mathematical identity in statistics. However, the notation with which the law of iterated expectation and the g-formula is expressed can be opaque to those with little background in statistics. We provide a primer introducing the law of iterated expectation, the integration notation used to express it, and its role for causal effect identification via the g-formula. Under the assumptions of causal consistency, positivity, and conditional exchangeability, the law of iterated expectation can be rewritten as a causal standardization formula (the g-formula) in two nonparametrically equivalent forms: a non-iterative conditional expectation (NICE) form involving a single weighted average of conditional outcome means, and an iterative conditional expectation (ICE) form involving nested expectations. We illustrate both forms using three progressively complex numerical examples: a time-fixed example with a single binary confounder, a time-fixed example with discrete and continuous confounders, and a time-varying example with two timepoints. We provide clarity on what the law of iterated expectation is, how it is related to the g-formula, and how to gain intuition of its mathematical formulations in actual data examples that can be generalized to a range of settings.

Many real-world networks are incomplete, making link prediction a fundamental challenge in network science. To train parameters and evaluate algorithms, observed links are usually divided into three subsets, namely training, validation, and probe sets. This division implicitly involves two sampling processes: first-stage sampling yields the probe set and second-stage sampling obtains the variation set. To date, our understanding of how these two sampling processes affect algorithm performance remains quite limited. To address this issue, we propose a sampling scheme called $β$-sampling, where the sampling probability of a link is proportional to the product of the degrees of its two endpoints raised to the power of $β$. Experiments on 45 real-world networks reveal that the structural characteristics of missing links, as simulated via varying probe sets, substantially impact prediction accuracy. When missing links tend to connect high-degree nodes, such links can be predicted accurately with ease. Furthermore, even with a fixed probe set, second-stage sampling still exerts a significant influence on prediction accuracy. Notably, the optimal second-stage sampling strategy differs from \textit{random sampling} (which randomly selects links to form the validation set) and \textit{consistent sampling} (which guarantees that links in the validation and probe sets share identical structural characteristics).

A utility function has been proposed that values more lives that are lost than those that are saved. I do not dispute the ethical motivation behind this kind of asymmetry. However, I show with a simple example that the scope of applicability of such a decision criterion is considerably more limited than it may first appear.

As data science emerges as a distinct academic discipline, introductory data science (IDS) courses play a key role in shaping students foundational understanding. Often taught by instructors without formal training in data science or pedagogy, these courses present a unique and globally relevant context for examining pedagogical content knowledge (PCK). Drawing on semi-structured interviews with 14 IDS instructors and their course syllabi, this study explores how IDS instructors describe and make sense of their teaching practices, which are analyzed through the lens of PCK. The findings highlight key components of PCK about IDS and offer insights into supporting instructor development. This work contributes to expanding the scope of PCK research into new interdisciplinary domains and ongoing global efforts to build capacity in data science education. It could serve as a starting point for developing a PCK framework specific to IDS.