Interpretation and Basic Statistical Concepts

age of classmates: what type of data?

ratio data

mean (sensitive to outliers), median (robust), mode (most common).

central tendency

standard deviation (SD), interquartile range (IQR), range. use these to describe your sample before you start inferring things about the population.

spread

all possible values and how often they occur.

distribution

what you actually observed in your sample (your histogram).

empirical distribution

the mathematical model you assume (e.g., normal, binomial, Poisson) to do inference.

statistical (theoretical) distribution

symmetric bell; many biological means approximate this (e.g., average SBP across many samples via CLT).

normal distribution

number of "successes" in n trials (e.g., number with cough among 20 patients on ACE inhibitors).

binomial distribution

counts of rare events over time/space (e.g., infusion pump errors per 1,000 patient-days).

Poisson distribution

even if individual measurements are skewed, the distribution of sample means becomes ~normal as n grows (usually n≥30 is decent). this is why we can build CIs and run t-tests on means without demanding perfect normality.

Central Limit Theorem

single best guess from your sample. mean difference, proportion, risk ratio, etc.

point estimate

range that likely captures the true parameter.

interval estimate

for _______________ (means): if a 95% CI includes 0, not statistically significant at α=0.05.

differences

for ______________ (RR, OR, HR): if the 95% CI includes 1, not significant.

ratios

select all that apply:

what are the hypothesis testing mechanics?

1. state hypotheses

2. choose α (usually 0.05)

3. pick test that matches data type/assumptions

4. compute test statistic & p-value

5. decision: p

false positive (you said "works" when it doesn't).

type I (α) error

false negative (you missed a real effect).

type II (β) error

probability to detect a real effect; increases with bigger n, larger effect, lower variance, higher α.

power (1−β) error

approximately normal or large n:

1 group vs a value → one-sample t-test

2 independent groups → independent t-test

pre/post same patients → paired t-test

≥3 groups → ANOVA (post-hoc tests if significant)

means

chi-square or Fisher's exact (small counts)

proportions/categorical

two independent groups → Mann-Whitney U

paired → Wilcoxon signed-rank

≥3 groups → Kruskal-Wallis

skewed/ordinal

Pearson r (linear, normal-ish) vs Spearman ρ (monotonic, rank-based)

association of two continuous variables

log-rank test, Cox regression

time-to-event

linear (continuous outcome), logistic (binary outcome), Poisson/negative binomial (counts)

regression

unlikely due to chance under H₀ (p

statistically significant

big enough to matter to patients (meets or exceeds a minimal clinically important difference, MCID).

clinically significant

(bell curve): where means live. this family also gives us the z and t tools.

normal

yes/no outcomes counted over n trials (e.g., cough on ACE-I: yes/no across 100 patients).

binomial

counts in categories; tests independence/association in contingency tables ("ACE-I use" × "cough: yes/no").

chi-square (χ²)

ratio of two variances; powers ANOVA and compares model fits (think: "do these groups have different means?" via variance ratios).

F distribution

means →

normal/t

proportions & counts →

binomial/χ²

comparing spreads or ≥3 means →

F

for continuous data (interval/ratio) that cluster around a center. symmetric around the mean; mean = median = mode. fully described by μ (mu, the mean) and σ (sigma, the standard deviation). change μ → the whole curve slides left/right. change σ → the curve gets skinny/tall (small σ) or wide/flat (large σ).

normal distribution

standardize any value to the standard normal (mean 0, SD 1) using a:

z-score

tells you how many SDs away from the mean your observation sits.

z-score

~68% within ±1 SD, _____________ , ~99.7% within ±3 SD.

~95% within ±2 SD

if σ is ____________ and you're estimating it with sample s, and the sample is small, your test statistic follows t, not z (heavier tails, df = n−1).

unknown

no matter the original μ and σ, you compute z=x−μ/σ and now you can use the _____________ to get probabilities.

standard normal table (Φ)

when comparing means using sample SD (unknown σ).

1 mean vs a value (one-sample)

2 independent means (independent)

paired pre/post (paired)

t distribution

goodness-of-fit (do observed counts match expected?) independence in a contingency table (drug class × adverse event).

χ²

select all that apply:

what are the four ingredients for binomial distribution?

1. each trial has two outcomes

2. independent trials

3. constant probability of success = p across trials

4. fixed n trials

this is for categorical outcomes with two options: yes/no, responder/non-responder, cough/no cough.

you repeat the trial n times (each patient is a "flip").

each trial has the same chance of "success" p.

we care about how many successes we get out of n.

binomial

1 group vs a number →

one-sample t

proportions / yes-no counts (AE happened? responder?)

chi-square

bell; mean=median=mode; use z for "how far."

normal

bell with chunky tails; use for means with small n/unknown σ.

t

counts of yes/no across n; proportions live here.

binomial

compare proportions between groups.

chi-square/Fisher

compare 3+ means.

ANOVA/F

select all that apply:

what is the 95% CI test hack?

1. differences (mean or percent): CI crosses 0 → not significant

2. ratios (RR/OR/HR): CI crosses 1 → not significant

new GLP-1 shows 48% responders in n=50.

binomial

compare mean A1c change in 2 independent groups (n=25 each). test?

independent t-test

same patients before/after SGLT2 added. test?

paired t-test

3 antihypertensives, compare mean SBP across the three. test?

ANOVA/F

lives in categorical land. powers the chi-square test: "are two categorical variables associated?" (e.g., ACE-I vs. ARB × cough yes/no). always skewed right; values are ≥ 0. bigger χ² → bigger mismatch between observed vs expected counts.

chi-square (χ²) distribution

ratio of two variances. powers ANOVA ("are any of ≥3 group means different?") and a bunch of regression model comparisons. also right-skewed; ≥ 0. says "at least one group mean isn't like the others."

F distribution

for counts of rare events in a fixed space/time (not yes/no per person, but events per exposure). pharmacy vibes: med errors per 1,000 patient-days, CLABSI per 10,000 line-days, code blues per month. mean = variance = λ (lambda). when events get common, ______________ ~ normal.

Poisson distribution

which is a normal distribution?

Z distribution.

the engine behind most tests

if you take lots of samples and compute their means, those means line up into a bell curve even when the raw data are ugly. the center of that bell is the true population mean (μ). the spread of that bell is the standard error (SE): x-bar = lambda/square root of n

unknown σ → estimate with s

bigger n shrinks SE → narrower bell → tighter confidence intervals → more power.

central limit theorem

you measure a sample and make your best, math-honest statement about the population you care about.

statistical inference

select all that apply:

what are the two complementary tools in statistical inference?

1.estimation

2. hypothesis testing

tells you size + precision

estimation

tells you whether the data are strong enough to reject "no effect."

hypothesis testing

everyone you care about (e.g., all adults with T2DM in Houston).

population

the people you actually measured (e.g., the 220 T2DM patients in your health system this year).

sample

mean μ, SD σ, proportion π

population parameters

mean x̄, SD s, proportion p

sample statistics

you use x̄, s, p to say something about ________________

all of the above

you calculate:

point estimates: one number (x̄, p, mean difference, odds ratio, etc.).

confidence intervals (CIs): a range of plausible values for the true effect.

estimation

select all that apply:

why do CIs matter more than p-values?

1. precision

2. clinical relevance

select all that apply:

what controls CI width?

1. variability

2. sample size

3. confidence level

(s or σ): messier data → wider CI

variability

(n): bigger n → smaller SE = s/√n → tighter CI

sample size

99% CI > 95% CI (wider because more cautious)

confidence level

(means, absolute risk): if the CI crosses 0, not statistically significant at 0.05.

differences

(RR, OR, HR): if the CI crosses 1, not significant.

ratios

estimate ± critical value × standard error (SE)

confidence interval

the number you computed (mean, proportion, mean difference, risk ratio, etc.).

estimate

how "strict" the interval is:

90% → 1.645 (z)

95% → 1.96 (z)

99% → 2.576 (z)

small samples for means → use t* with df (heavier tails).

critical value

"how wobbly is my estimate?" (shrinks with bigger n).

standard error

no effect, no difference.

null (H₀)

there is an effect/difference (two-sided by default).

alternative (Hₐ)

the test is designed to attack H₀ only. at the end you either: reject H₀, or ________________ (you don't "accept" it—you just didn't get enough receipts).

fail to reject H₀

if H₀ were actually true, this is the probability of seeing data this extreme (or more) just by chance.

p-value

using α = 0.05 unless pre-specified otherwise: p ≤ α →

reject H₀ (statistically significant)

using α = 0.05 unless pre-specified otherwise: p > α →

fail to reject H₀ (not statistically significant)

you are conducting a study and set an α level of 0.05. your analysis produces a p-value of 0.02. which of the following conclusions should you make?

reject H₀; you reached statistical significance

increases when the effect is bigger, variability is lower, n is larger, or α is higher.

power

select all that apply:

how do we run a hypothesis test?

1. state H₀/Hₐ

2. choose α

3. compute test statistic → p-value

4. decision + effect size + CI

5. write it cleanly

asks "different?" (up or down). safer default; protects from surprises.

non-directional tests

only if a decrease (or increase) is the only meaningful direction and you pre-registered it. otherwise reviewers will side-eye you.

directional tests

tells you the strength of evidence against H₀.

p

tells you how big the effect could realistically be (precision).

CI

question: "are they different in either direction?" null (H₀): μ₁ − μ₂ = 0Alt (Hₐ): μ₁ − μ₂ ≠ 0 (non-directional)

CI cheat code: 95% CI for (μ₁ − μ₂) does not include 0 → significant difference.

example: mean SBP drop: ARB − ACEI = −4 mmHg (95% CI −7 to −1) → different.

two-sided (non-directional)

question: "are the two within a pre-set 'clinically unimportant' window ± Δ?" null (H₀): |μ₁ − μ₂| ≥ Δ (they differ too much) alt (Hₐ): |μ₁ − μ₂| < Δ (they are equivalent)

how it's done: TOST (Two One-Sided Tests) or the CI rule. CI cheat code (α=0.05): the 90% CI for (μ₁ − μ₂) must lie entirely inside (−Δ, +Δ). (why 90%? because you split α across two one-sided tests.)

example (bioequivalence flavor): generic vs brand AUC ratio. Δ on the log-ratio scale corresponds to the famous 80-125% bounds: the 90% CI for the ratio must sit entirely within 0.80 to 1.25.

testing for equivalence

question: "is treatment 1 better (or worse) than treatment 2? "if "higher is better": H₀: μ₁ − μ₂ ≤ 0Hₐ: μ₁ − μ₂ > 0

CI cheat code (α=0.05): the 95% CI lower bound for (μ₁ − μ₂) is > 0 (for "better"). example: A1c reduction (positive numbers = larger drop better). Drug X − control = +0.4% (95% CI +0.1 to +0.7) → superior.

use one-sided only if the undesired direction truly doesn't matter and you pre-registered it.

testing for superiority

question: "is the new option no worse than control by more than Δ?" pick the direction carefully. suppose higher is better (e.g., A1c drop): H₀: μ₁ − μ₂ ≤ −Δ (new is worse by ≥Δ) Hₐ: μ₁ − μ₂ > −Δ (new is not worse than Δ → non-inferior)

CI cheat code (α=0.05): the 95% CI lower bound for (μ₁ − μ₂) is > −Δ. (equivalently, many protocols use a 97.5% one-sided CI.)

example: new oral GLP-1 vs injectable: Δ = 0.4% A1c. observed difference = −0.10% (95% CI −0.30 to +0.10). lower bound −0.30 > −0.40 → non-inferior. not superior (CI crosses 0).

testing for non-inferiority

____________ in a difference test ≠ equivalence or non-inferiority. you need the right design and the right CI rule.

not significant

select all that apply:

what is the 5-step "traditional" workflow?

1. translate the question

2. choose the test

3. set α (0.05 typical)

4. compute test/CI

5. conclude