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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, 38 new cases occurred. Find the incidence rate IR per 1000 person-years. Round to one decimal place.
340 × 2.5 years160 × 1.5 years38 new cases
IR = per 1000 person-years
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 60% for a disease. A test with sensitivity 89% and specificity 71% comes back negative. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 60%Se = 89%Sp = 71%negative
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 52, 50, 20, 26, 24, 19, 29.
n = 7
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 8. What proportion P of values is below 48?
μ = 60σ = 8x = 48
P =
5.Confidence intervalsMedium
A 95% confidence interval for a mean, based on 50 observations and a known σ, has a margin of error of 4.9. What is the margin of error E₂ if the same study uses 100 observations instead?
95%n₁ = 50E₁ = 4.9n₂ = 100
E₂ =
6.Confidence intervalsHard
The population standard deviation is 5. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 2?
σ = 5E = 299%
n =
7.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 9 students, mean score 36.5, standard deviation 7.5. Group 2: 8 students, mean score 39.2, standard deviation 6.5. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).
n₁ = 9x̄₁ = 36.5s₁ = 7.5n₂ = 8x̄₂ = 39.2s₂ = 6.5
t =
8.Hypothesis tests & confidence intervalsHard
A sample of 64 measurements has mean 120.4. The population standard deviation is 5. Find the lower limit L of the 90% confidence interval for the mean.
x̄ = 120.4σ = 5n = 6490%
L =
3Solutions
1.Evidence-based practice statistics
Answer: IR = 34.9 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 = 340·2.5 + 160·1.5 = 1090
Incidence rate
IR = 38/1090·1000 = 34.9 per 1000 person-years
Answer
IR = 34.9 per 1000 person-years
2.Evidence-based practice statistics
Answer: p′ = 18.9 %
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− = (1 − 0.89)/0.71 = 0.1549
Pre-test odds
0.6/(1 − 0.6) = 1.5
Post-test odds
1.5·0.1549 = 0.2324
Post-test probability
p′ = 0.2324/(1 + 0.2324) = 0.1886 ≈ 18.9%
Answer
p′ = 18.9%
3.Descriptive statistics
Answer: s = 13.81
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 = −1.5
Standard normal table
Φ(−1.5) = 0.06681
P = 0.06681
Answer
P = 0.06681
5.Confidence intervals
Answer: E₂ = 3.465
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
Ratio of margins
E₂/E₁ = (1.96/1.96)·√(50/100) = 0.7071
New margin
E₂ = 4.9·0.7071 = 3.465
Answer
E₂ = 3.465
6.Confidence intervals
Answer: n = 42
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
Solve for n
n = (2.576·5/2)² = 41.47
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 42
Answer
n = 42
7.Hypothesis tests & confidence intervals
Answer: t = −0.7881
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.
Pooled variance
sp² = (8·7.5² + 7·6.5²)/15 = 49.72
sp = 7.051
Standard error
SE = 7.051·√(1/9 + 1/8) = 3.426
Test statistic
t = (36.5 − 39.2)/3.426 = −0.7881
Answer
t = −0.7881
With df = 15, the two-sided 5% critical value is 2.131; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: L = 119.4
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).