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1Formulas and methods
Evidence-based practice statistics
Sensitivity and specificity
Se = a/(a + c), Sp = d/(b + d)
Predictive values
PPV = a/(a + b), NPV = d/(c + d)
Accuracy
(a + d)/n
Likelihood ratios
LR+ = Se/(1 − Sp), LR− = (1 − Se)/Sp
PPV at prevalence p
Se·p / (Se·p + (1 − Sp)(1 − p))
Odds
odds = p/(1 − p); p = odds/(1 + odds); post-test odds = pre-test odds × LR
Risk measures
RR = EER/CER, ARR = CER − EER, RRR = ARR/CER
Number needed to treat
NNT = 1/ARR, rounded up; NNH = 1/(EER − CER)
Odds ratio
OR = ad/(bc)
Incidence rate
new cases / person-time
Method
Draw the 2 × 2 table and label a, b, c and d before you calculate.
Read sensitivity and specificity down the columns, predictive values along the rows.
For another prevalence, imagine 10 000 people and fill the table with Se, Sp and the prevalence.
For a likelihood ratio, convert the probability to odds, multiply, and convert back.
For a trial, find EER and CER first; then RR = EER/CER, ARR = CER − EER and NNT = 1/ARR rounded up.
Descriptive statistics
Mean
x̄ = Σx/n
Sample standard deviation
s = √(Σ(x − x̄)²/(n − 1))
Median
middle value of the sorted data
z-score
z = (x − μ)/σ
Normal distribution
68% within ±1σ, 95% within ±2σ
Confidence intervals
σ known
x̄ ± z·σ/√n
σ unknown
x̄ ± t·s/√n, df = n − 1
Critical z
90%: 1.645, 95%: 1.96, 99%: 2.576
Sample size
n = (z·σ/E)², rounded up
Method
Decide between z (population σ known) and t (only the sample s, with n − 1 degrees of freedom).
Find the critical value for the confidence level.
Margin of error: E = critical value × standard deviation/√n.
Interval: x̄ ± E.
Hypothesis tests & confidence intervals
z-test for a mean
z = (x̄ − μ0)/(σ/√n)
t-test for a mean
t = (x̄ − μ0)/(s/√n), df = n − 1
Two-sided at 5%
reject H0 if |z| > 1.96
Confidence interval
x̄ ± z·σ/√n
Method
State H₀ and H₁ and decide whether the test is one- or two-sided.
Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
Compare with the critical value, or find the p-value.
Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.
2Practice problems
1.Evidence-based practice statisticsMedium
Practice exercise (made-up values, not clinical guidance). In a cohort, groups of people were followed for different lengths of time (see the table: number of people and years of follow-up for each). During follow-up, 33 new cases occurred. Find the incidence rate IR per 1000 person-years. Round to one decimal place.
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 25% for a disease. A test with sensitivity 90% and specificity 87% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 25%Se = 90%Sp = 87%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 20, 32, 27, 17, 29, 38.
n = 6
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 8. What proportion P of values is below 52?
μ = 60σ = 8x = 52
P =
5.Confidence intervalsMedium
The blood pressure of 6 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean −3.5 and standard deviation 7.3. Find the upper limit of the 90% confidence interval for the mean difference.
n = 6d̄ = −3.5sd = 7.390%t(5) = 2.015
upper limit = mmHg
6.Confidence intervalsHard
Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 99% confidence interval for the proportion has a margin of error of at most 0.05?
p = 0.5E = 0.0599%
n =
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 50; the population standard deviation is known to be 6. A random sample of 144 observations has a mean of 51.04. Test H₀: μ = 50 against H₁: μ ≠ 50. Find the p-value p.
μ₀ = 50σ = 6n = 144x̄ = 51.04H₁: μ ≠ μ₀
p =
8.Hypothesis tests & confidence intervalsHard
How large must a sample be so that the 90% confidence interval for a mean has a margin of error of at most 1.5? The standard deviation is 20. Find the smallest sample size n.
σ = 20E = 1.590%
n =
3Solutions
1.Evidence-based practice statistics
Answer: IR = 17.6 per 1000 person-years
Likelihood ratios and risk measures: LR+ = sensitivity/(1 − specificity), LR− = (1 − sensitivity)/specificity. In a trial, EER and CER are the event rates with and without treatment: RR = EER/CER, ARR = CER − EER, RRR = ARR/CER and NNT = 1/ARR, rounded up. In a case-control study only the odds ratio OR = ad/(bc) can be used. An incidence rate = new cases/person-time.
Person-years
PY = 140·2.5 + 260·2 + 500·2 = 1870
Incidence rate
IR = 33/1870·1000 = 17.6 per 1000 person-years
Answer
IR = 17.6 per 1000 person-years
2.Evidence-based practice statistics
Answer: p′ = 69.8 %
Predictive values depend on prevalence: PPV = Se·p/(Se·p + (1 − Sp)(1 − p)) and NPV = Sp(1 − p)/(Sp(1 − p) + (1 − Se)p). With likelihood ratios: pre-test odds = p/(1 − p), post-test odds = pre-test odds × LR, probability = odds/(1 + odds). For a population of N, true positives = N·p·Se and false positives = N·(1 − p)·(1 − Sp). NNT = 1/(CER × RRR), rounded up.
Likelihood ratio
LR+ = 0.9/(1 − 0.87) = 6.923
Pre-test odds
0.25/(1 − 0.25) = 0.3333
Post-test odds
0.3333·6.923 = 2.308
Post-test probability
p′ = 2.308/(1 + 2.308) = 0.6977 ≈ 69.8%
Answer
p′ = 69.8%
3.Descriptive statistics
Answer: s = 7.731
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Standardise
z = −1
Standard normal table
Φ(−1) = 0.1587
P = 0.1587
Answer
P = 0.1587
5.Confidence intervals
Answer: upper limit = 2.505 mmHg
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
Critical value
df = 5, t = 2.015
Margin of error
E = 2.015·7.3/√6 = 6.005
Interval
−3.5 ± 6.005 → [−9.505, 2.505]
Answer
upper limit = 2.505 mmHg
6.Confidence intervals
Answer: n = 664
Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.
Solve for n
n = 2.576²·0.5·0.5/0.05² = 663.6
Round up
n = 664
Answer
n = 664
7.Hypothesis tests & confidence intervals
Answer: p = 0.03753
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.
Test statistic
SE = σ/√n = 6/√144 = 0.5
z = (51.04 − 50)/0.5 = 2.08
Tail area
Φ(2.08) = 0.9812
1 − Φ(2.08) = 0.0188
p-value
p = 2·0.0188 = 0.03753
Answer
p = 0.03753
p ≤ 0.05, so H₀ is rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: n = 482
Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).