Biostatistics for the MCAT

Biostatistics Fundamentals

Part 1 of 4 — Descriptive Statistics & Data Distributions

Types of Data

Measures of Central Tendency

Example: A drug trial shows patient recovery times: 5, 6, 7, 8, 100 days.

• Mean = 25.2 days (affected by outlier)
• Median = 7 days (better representation)

Measures of Spread

• A sample variance divides by $n - 1$ instead of $N$ (Bessel's correction), which slightly enlarges it for small samples.

68-95-99.7 Rule (Normal Distribution):

• 68% of data within 1 SD of mean
• 95% within 2 SD
• 99.7% within 3 SD

Descriptive Statistics 🎯

Key Takeaways — Part 1

• Central Tendency: Use median for skewed data; mean for symmetric distributions
• Spread: SD is most useful on MCAT; interpret via 68-95-99.7 rule
• Outliers: Robust stats (median, IQR) better than mean ± SD when outliers present
• Log scales: Many biomedical values are log-normally distributed (viral loads, enzyme concentrations—use log-transform)

• Low-yield extras: a sample variance divides by n − 1 (Bessel's correction) rather than N

Worked Examples — Descriptive Statistics

Data: 4, 5, 5, 6, 40

1. Mean = 60/5 = 12.
2. Median = 5.
3. Outlier (40) inflates the mean.

Best central tendency: median.

Mean = 70, SD = 5. Estimate the range containing about 95% of values.

1. 95% is roughly mean ± 2 SD.
2. 70 ± 10 gives 60 to 80.

Approximate 95% interval: 60 to 80.

Two test forms have the same mean score (80). Form A has SD 3; Form B has SD 12.

1. Same mean means same average performance.
2. Lower SD means scores cluster more tightly.
3. Higher SD means performance is more variable.

Conclusion: Form A is more consistent across students.

Biostatistics Fundamentals

Part 2 of 4 — Hypothesis Testing & p-values

Hypothesis Types

One-tailed vs Two-tailed:

• One-tailed: Predicts direction (Drug lowers BP) → all of α sits in one tail
• Two-tailed: No direction (Drug changes BP) → α is split between the two tails (0.025 each at α = 0.05)

Type I & II Errors

Critical: α = 0.05 means a 5% chance of a Type I error when $H_{0}$ is true, i.e., a 5% chance of rejecting a true null (standard MCAT threshold)

p-value Interpretation

• p < 0.05 → Reject $H_{0}$ (statistically significant)
• p ≥ 0.05 → Fail to reject $H_{0}$ (not significant)

Example: A study finds p = 0.03 for a new antibiotic efficacy.

• Interpretation: 3% probability of obtaining results at least as extreme as those observed, assuming the null hypothesis (no real antibiotic effect) is true
• Conclusion: Reject $H_{0}$; the antibiotic likely has real efficacy

Power & Sample Size

• Power = 1 − β (ability to detect a true effect)
• Larger sample size → More power
• Higher power = better study (typically aim for 80%+ power)

Hypothesis Testing 🎯

Key Takeaways — Part 2

• p-value = Probability of observing data (or more extreme) if $H_{0}$ is true
• α=0.05 = 5% threshold; p < 0.05 → reject $H_{0}$
• Type I error (α): False positive; Type II error (β): False negative
• Power (1−β): Increases with larger sample size; typical goal is 80%+
• MCAT Tip: p < 0.05 = statistically significant; always check the p-value first

Worked Examples — Hypothesis Testing & p-values

Study result: p = 0.03, α = 0.05.

1. Compare p to α.
2. Since 0.03 < 0.05, reject $H_{0}$.

Conclusion: statistically significant finding.

A test concludes a drug works, but in reality it does not.

1. Rejected $H_{0}$ when $H_{0}$ was true.
2. This is a false positive.

Error type: Type I error (α).

Small trial p = 0.07; larger trial on same effect p = 0.02.

1. Larger n reduces standard error.
2. Smaller uncertainty improves ability to detect true effects.
3. Power increases, reducing Type II error risk.

Takeaway: bigger sample, higher power.

Biostatistics Fundamentals

Part 3 of 4 — Confidence Intervals & Effect Size

Confidence Intervals (CI)

A CI gives a range where the true parameter likely lies (unlike a single p-value).

$\text{95\% CI} = \bar{x} \pm 1.96 \times SE, \qquad SE = \frac{SD}{\sqrt{n}}$

(1.96 is often rounded to 2, so a 95% CI is roughly the estimate ± 2 SE.)

Interpretation: "We are 95% confident the true population mean falls within this range."

The null value is 0 for a difference (mean difference, change in BP) and 1 for a ratio (odds ratio, relative risk).

Example: Study finds mean blood pressure reduction of 10 mmHg (95% CI: 5–15 mmHg).

• Interpretation: Likely true reduction is between 5–15 mmHg
• Since CI doesn't include 0, the effect is significant

Effect Size

Effect size quantifies magnitude of difference (independent of sample size).

• Rough benchmarks for Cohen's d: 0.2 (small), 0.5 (medium) and 0.8 (large).

Example: A very large trial compares two antacids head to head:

• Drug A: Mean relief = 7.0 hours
• Drug B: Mean relief = 6.9 hours
• p = 0.02, so the difference is "significant," but the effect size is trivial (0.1 hour, about 6 minutes)

Confidence Intervals & Effect Size 🎯

Key Takeaways — Part 3

• 95% CI: Range where true population parameter likely falls
• Narrow CI = Better precision (larger N); CI includes the null value (0 for differences, 1 for ratios such as OR/RR) = Not significant
• Effect Size: Magnitude of difference (Cohen's d, OR, r); independent of sample size
• p-value vs Effect Size: p-value answers "Is there an effect?" (yes/no). Effect size answers "How big?"
• MCAT Tip: Always check both—significant p-value ≠ meaningful effect; large CI suggests underpowered study

• Low-yield extras: Cohen's d benchmarks (0.2 small, 0.5 medium, 0.8 large)

Worked Examples — Confidence Intervals & Effect Size

Treatment effect = 4 units, 95% CI: 1 to 7.

1. Interval does not include 0.
2. Effect is statistically significant.
3. True effect is plausibly between 1 and 7 units.

Conclusion: significant positive effect with moderate precision.

Study A CI: 10 to 30. Study B CI: 18 to 22.

1. Study B has a much narrower CI.
2. Narrower CI means lower uncertainty in estimate.

More precise estimate: Study B.

A huge sample finds p < 0.001 for a score increase of 0.2 points.

1. p-value says the effect is unlikely due to chance.
2. Effect size is tiny.
3. Statistical significance does not guarantee clinical relevance.

Takeaway: evaluate both p-value and effect size.

Biostatistics Fundamentals

Part 4 of 4 — Correlation, Causation, Study Design, Diagnostic Tests & Risk

Correlation Coefficient (r)

Measures strength and direction of linear relationship between two variables.

r = -1 → Perfect negative correlation
r = 0 → No correlation
r = +1 → Perfect positive correlation

Critical: r close to ±1 does NOT prove causation!

Correlation ≠ Causation

Three mechanisms for correlation:

1. Causation: X → Y (aspirin → reduced heart attack risk)
2. Reverse Causation: Y → X (depression is linked to poor physical health, but the poor health may be causing the depression rather than the reverse)
3. Confounding Variable: Z → both X and Y (smoking → both yellow teeth AND lung cancer)

Example: Ice cream sales correlate with drowning deaths.

• Confounder: Summer heat drives both (neither causes the other)

Study Design & Confounders

Cross-sectional, case-control and cohort studies are all observational: the researcher only records exposure. Only the RCT assigns it.

RCT Gold Standard: Randomization balances known and unknown confounders across groups.

Observational Study: 
  Does statin use → lower cholesterol?
  (Confounded by: diet, exercise, genetics)

RCT Gold Standard:
  Randomize patients to statin vs placebo
  (Randomization balances confounders)

Diagnostic Tests: The 2×2 Table

Every screening or diagnostic question on the MCAT can be laid out in one 2×2 table. Rows are the test result; columns are the true disease status.

Shortcut: sensitivity and specificity read down the columns (start from disease status); PPV and NPV read across the rows (start from the test result).

Mnemonics:

• SnNout: a highly Sensitive test, when Negative, rules out disease (few false negatives). Good for screening.
• SpPin: a highly Specific test, when Positive, rules in disease (few false positives). Good for confirming.

Key reasoning:

• Sensitivity and specificity are properties of the test: they do not change with prevalence.
• PPV and NPV depend on prevalence. Higher prevalence → higher PPV, lower NPV. Lower prevalence → lower PPV, higher NPV. This is why screening a low-prevalence population produces many false positives.
• Moving the cutoff trades one for the other. Lowering the threshold for a "positive" result catches more true cases (sensitivity ↑, fewer false negatives) but flags more healthy people (specificity ↓, more false positives). Raising the threshold does the reverse.

Risk Measures

For risk questions, rows are exposure and columns are outcome:

• RR or OR = 1 → no association (the null value for a ratio, as in Part 3). Greater than 1 → exposure linked to more of the outcome; less than 1 → exposure is protective.
• When the disease is rare, the OR closely approximates the RR.

Treatment effect measures (RCTs):

• Absolute risk reduction (ARR) = risk in control group − risk in treated group
• Relative risk reduction (RRR) = ARR ÷ risk in control group (equivalently, 1 − RR)
• Number needed to treat (NNT) = $\frac{1}{\text{ARR}}$: how many patients must be treated to prevent one bad outcome

Example: Heart attacks occur in 10% of placebo patients and 6% of drug patients.

• RR = 0.06 ÷ 0.10 = 0.6 (drug patients have 60% of the placebo risk)
• ARR = 10% − 6% = 4% = 0.04, so RRR = 0.04 ÷ 0.10 = 40%
• NNT = 1 ÷ 0.04 = 25 patients treated to prevent one heart attack

Correlation, Causation, Diagnostic Tests & Risk 🎯

Key Takeaways — Part 4

• Correlation (r): Measures linear relationship; high |r| ≠ causation
• Confounding: Third variable influences both exposure and outcome
• Study Hierarchy (for causation inference): Cross-sectional < Case-Control < Cohort < RCT
• RCT Gold Standard: Randomization balances known and unknown confounders
• Temporal Relationship: Exposure must precede outcome for causation (cohort better than cross-sectional)
• 2×2 table (test rows, disease columns): Sensitivity = a/(a+c), Specificity = d/(b+d), PPV = a/(a+b), NPV = d/(c+d)
• SnNout / SpPin: a negative result on a highly sensitive test rules disease out; a positive result on a highly specific test rules it in
• Prevalence: Sensitivity and specificity are fixed properties of the test; higher prevalence → higher PPV, lower NPV
• Cutoff trade-off: Lowering the threshold for "positive" raises sensitivity and lowers specificity (and vice versa)
• Risk measures: Relative risk = [a/(a+b)] ÷ [c/(c+d)] for cohort studies; odds ratio = ad/bc for case-control studies; RR or OR = 1 means no association
• ARR and NNT: ARR = control risk − treated risk; NNT = 1/ARR
• MCAT Tip: Always ask "Could a confounder explain this correlation?" and "What study design would prove causation?"

Worked Examples — Correlation, Causation & Design

Data show coffee intake correlates with heart disease.

1. Correlation indicates association only.
2. Smoking could confound both coffee intake and disease risk.
3. Need stronger design (e.g., randomized intervention) for causal inference.

Conclusion: association present, causation unproven.

Given cross-sectional, case-control, cohort, and RCT:

1. RCT is strongest due to randomization.
2. Cohort is next because exposure precedes outcome.
3. Case-control and cross-sectional are more confounded.

Strongest to weakest: RCT > cohort > case-control > cross-sectional.

Study records current depression and current sleep quality at one time point.

1. Exposure and outcome measured simultaneously.
2. Cannot determine which came first.
3. Reverse causation remains possible.

Takeaway: temporal ambiguity weakens causal claims.

A screening test has 90% sensitivity and 80% specificity. It is given to 1,000 people, and disease prevalence is 10%.

1. Diseased = 10% × 1,000 = 100; disease-free = 900.
2. True positives = 90% × 100 = 90; false negatives = 100 − 90 = 10.
3. True negatives = 80% × 900 = 720; false positives = 900 − 720 = 180.
4. Fill the 2×2 table:

1. PPV = a/(a+b) = 90/270 ≈ 33%: only one in three positives truly has the disease.
2. NPV = d/(c+d) = 720/730 ≈ 98.6%: a negative result is very reassuring.

Same test, prevalence 50% (500 diseased, 500 disease-free): true positives = 450, false positives = 100, false negatives = 50, true negatives = 400. PPV = 450/550 ≈ 82% and NPV = 400/450 ≈ 89%.

Takeaway: sensitivity and specificity did not change, but raising prevalence raised PPV and lowered NPV.