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Interpreting a Dosage-Recovery Regression: P-values, CIs, Causality

GoogleData Scientist · Onsite
MediumData ScientistOnsiteStatistics & Math

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Given a fitted dosage-vs-recovery-time regression, explain exactly what the reported p-value and confidence interval mean, what assumptions back that inference, how a confidence interval differs from a prediction interval, and why the result alone cannot establish causality. Tests statistical precision under pressure.

A hospital's analytics team ran an observational study: for 50 post-surgical patients, they regressed recovery time in days (y) on the dosage in mg of a new pain medication (x) that the attending physician chose for each patient, and reported:

y = 9.4 - 0.18 x        se(slope) = 0.07      residual sd = 3.1      n = 50
t = -2.57   p = 0.013   95% CI for slope: [-0.32, -0.04]

A clinician on the team wants to raise the standard dosage, reasoning "higher doses clearly speed recovery — the p-value is below 0.05." You've been asked to walk through what the numbers actually support before any dosage change is made.

Part 1: Model and test

Write the simple linear regression model and its parameters, state the null and alternative hypotheses behind the reported p-value, and show how the test statistic is built from the slope estimate and its standard error. Name the reference distribution and its degrees of freedom.

Part 2: Reading the numbers

Give a precise, one-sentence reading of p = 0.013, along with two common wrong readings of it. State what the 95% confidence interval means under repeated sampling, and what it does not claim about this particular interval. Explain how the two-sided test at α = 0.05 relates to whether the interval contains 0.

Part 3: Assumptions

List the assumptions the inference above depends on. For each, say what goes wrong if it fails in this specific setting, and how you'd check it.

Part 4: Mean response vs. a new patient

Someone asks: "If we give the next patient 50 mg, how many recovery days should we expect?" Contrast a confidence interval for the mean recovery time at a 50 mg dose with a prediction interval for one new patient's recovery time, and say which is wider and why.

Part 5: Causality

Explain why neither the small p-value nor the negative slope shows that giving more of the drug causes faster recovery. Name at least two concrete threats specific to how dosage was assigned here, and propose how you would obtain a causal estimate instead. Also explain how a tiny slope could still produce a tiny p-value.

No further computation is required — reason from the summary statistics given above.

Hints

Hint 1

A test statistic is always (estimate − hypothesized value) / standard error; with one predictor and an intercept, degrees of freedom are n − 2.

Hint 2

A p-value is a probability computed assuming the null and the model are both true — it is not the probability that the null is true, and it says nothing about effect size.

Hint 3

A prediction interval has to cover the spread of one new patient's own outcome on top of the uncertainty in the fitted line — that extra term never shrinks, no matter how large n gets.

Hint 4

Ask what decided the dosage each patient actually received, and whether that decision could itself be tangled up with how sick they were.

Follow-up questions

  • How would adding a covariate for surgery severity change how you interpret the dosage slope?
  • If recovery times for patients who shared the same surgeon were correlated, how would you adjust the inference?
  • How would you explain the same confidence interval result in Bayesian terms, and what extra ingredient would you need?
  • If the team had tested dosage against five different outcome measures and only reported this one, what's the statistical problem and how would you correct for it?