Scope of Inference

By the end of this lesson, you will be able to determine the appropriate scope of inference from a study's design, and correctly state whether conclusions can be generalized and whether causation can be claimed.

Two Key Questions About Any Study

After any statistical study, you should ask:

1. Can we generalize to a larger population? → Depends on whether the sample was randomly selected.

2. Can we establish a cause-and-effect relationship? → Depends on whether the study was a randomized experiment.

The 2×2 Framework

	Random Assignment?	No Random Assignment?
Random Sample?	Generalize to population; establish causation	Generalize to population; cannot establish causation
No Random Sample?	Cannot generalize; establish causation for this group	Cannot generalize; cannot establish causation

Generalization: Random Sampling

Random sampling means subjects were selected from the population using a probability mechanism (SRS, stratified, etc.).

→ Results can be generalized to the population from which the sample was drawn.

Without random sampling (convenience sample, volunteers): → Results apply only to the specific subjects studied. We cannot assume others would behave the same way.

Causation: Random Assignment

Random assignment means subjects were randomly assigned to treatment groups in an experiment.

→ We can conclude that the treatment caused the observed difference in the response.

Without random assignment (observational study): → We can only conclude association. Confounding variables could explain the relationship.

Common Scenarios

Survey with a random sample, no treatment:

• Observational study
• Can generalize results to the population (because of random sampling)
• Cannot claim causation (no random assignment)
• Correct language: "There is evidence of an association between X and Y in [population]"

Experiment with random assignment, volunteers recruited:

• Experiment