Lecture 6: Inference & Sampling Methods

Flashcards reviewing statistical inference, sampling methods, mathematical descriptors, estimator evaluations, inverse probability weighting, and population divergence concepts.

What is statistical inference?

Statistical inference is the attempt to reach a conclusion concerning the population from observations of the sample.

What is the distinction between a population parameter and a sample statistic?

A population parameter is a numerical descriptor of a population (such as population mean $\mu$ or population variance $\sigma^2$), whereas a sample statistic is a numerical descriptor calculated from a sample (such as sample mean $\bar{x}$ or sample variance $s^2$).

What are the mathematical formulas for population variance and sample variance?

Population variance is $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$, while sample variance is $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$.

What are the two primary goals of sampling?

The two primary goals are representativeness (ensuring the sample is a miniature replica of the population) and precision (securing statistical precision to answer scientific questions with less sampling error).

How is sampling error defined?

Sampling error is the difference between an estimated statistic calculated from a sample and the true parameter in the population.

What defines Simple Random Sampling (SRS)?

Simple Random Sampling is a sampling strategy where every individual in the population has an equal chance of selection, and every possible sample of size $n$ has an equal probability of being selected.

Under what condition can sampling without replacement be treated like sampling with replacement?

Sampling without replacement can be treated like sampling with replacement when the sample size $n$ is less than or equal to $5\%$ of the population size $N$ ($n \le 0.05N$).

How is the selection interval $k$ calculated in systematic sampling, and what warning is associated with this strategy?

The selection interval is calculated as $k = \frac{N}{n}$. The main warning is that hidden patterns within the sampling list can introduce severe bias.

How do Stratified Random Sampling and Cluster Sampling differ in how groups are handled?

In Stratified Random Sampling, the population is divided into mutually exclusive groups (strata) based on a characteristic and individuals are randomly selected from each group. In Cluster Sampling, the population is divided into naturally occurring groups (clusters) and whole groups are randomly selected.

What is Complex Multistage Sampling, and what are its potential stages?

Complex Multistage Sampling is a step-by-step sampling process from larger groups to smaller groups. Its stages can include Primary (e.g., states), Secondary (e.g., counties), Tertiary (e.g., census tracts), and Subsequent (e.g., individuals).

What is the objective of oversampling?

The objective of oversampling is to intentionally sample a higher proportion of individuals from a specific subgroup than naturally occurs in the population to ensure a sufficient sample size for analysis.

How do Design-Based Inference and Model-Based Inference differ regarding randomness?

In Design-Based Inference, population values are fixed and randomness comes solely from the sampling design. In Model-Based Inference, population values are treated as random variables from a superpopulation, and randomness comes from both the theoretical model and the sampling design.

What is the inclusion indicator $Z_i$, and what is its probability of selection under Simple Random Sampling (SRS)?

The inclusion indicator $Z_i$ equals $1$ if person $i$ is selected and $0$ if not. Under SRS, the selection probability is $P(Z_i = 1) = \frac{n}{N}$.

What are the two main criteria for evaluating an estimator $T$ for a parameter $\theta$?

The two criteria are unbiasedness ($E[T] = \theta$), which measures accuracy, and variance ($Var(T)$), which measures precision across repeated samples.

What is the Finite Population Correction (FPC) formula, and how does it affect the variance of the sample mean $Var(\bar{x})$?

The Finite Population Correction is $\text{FPC} = \frac{N - n}{N - 1}$. Under SRS without replacement, the variance of the sample mean is $Var(\bar{x}) = \left(\frac{\sigma^2}{n}\right) \times \text{FPC}$.

In Inverse Probability Weighting (IPW), how is sampling weight $w_i$ calculated from selection probability $\pi_i$?

The sampling weight is the inverse of the selection probability, defined as $w_i = \frac{1}{\pi_i}$, representing how many population members each sampled person represents.

What are the selection probability $\pi_i$ and sampling weight $w_i$ for each individual under Simple Random Sampling (SRS)?

Under SRS, every individual has an identical selection probability $\pi_i = \frac{n}{N}$ and an identical sampling weight $w_i = \frac{N}{n}$.

What is the formula for the Horvitz-Thompson Estimator of the population mean $\mu$?

The Horvitz-Thompson Estimator is $\hat{\mu} = \frac{1}{N} \sum_{i=1}^{N} \frac{Z_i x_i}{\pi_i}$.

What is the impact of extremely large weights in Inverse Probability Weighting (IPW), and how does weight truncation resolve it?

Extremely large weights increase estimator variance and elevate the influence of individual outliers. Weight truncation caps extreme weights at a ceiling value, introducing slight estimator bias to achieve substantial variance reduction.

What are three broader applications of Inverse Probability Weighting (IPW)?

The three broader applications are Survey Sampling (adjusting for non-response and complex designs), Missing Data Analysis (weighting observed cases by inverse probability of observation), and Causal Inference (propensity score weighting to adjust for confounding).

How is the Sampled Population defined compared to the Target Population?

The Sampled Population is the population from which the sample is actually drawn, whereas the Target Population is the population to which conclusions are ultimately intended to apply.

What are the primary arguments FOR allowing divergence between the sampled population and target population?

The primary arguments FOR divergence are practical feasibility, lower study cost, participant accessibility, and operational constraints (geographic, logistical, or eligibility limitations).

What are the primary arguments AGAINST allowing divergence between the sampled population and target population?

The primary arguments AGAINST divergence are reduced representativeness, potential bias from underrepresenting key subgroups, systematic differences in treatment effects or outcomes, and reduced external validity.

According to the inference pathway model, how do Statistical Inference and Scientific/Medical Inference differ in what they connect?

Statistical Inference connects the Sample to the Sampled Population via probability mechanisms, while Scientific or Medical Inference connects the Sampled Population to the Target Population via biological mechanisms and common or medical sense.