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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 randomised trial, 890 patients received a new treatment and 870 received the control treatment. The table shows how many in each group had a side effect. Find the number needed to harm NNH: how many patients must be treated for one extra patient to have the side effect? Round up to a whole number of patients.
treatment: 387/890control: 252/870
NNH =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 50% for a disease. A test with sensitivity 74% and specificity 89% comes back negative. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 50%Se = 74%Sp = 89%negative
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 41, 30, 39, 52, 41.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 500 and standard deviation 60. What proportion P of values is below 410?
μ = 500σ = 60x = 410
P =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 40 phones, mean 145 h, known standard deviation 8 h. Model 2: 50 phones, mean 131 h, known standard deviation 14 h. Find the lower limit of the 95% confidence interval for μ₁ − μ₂.
The population standard deviation is 26. How large a sample n do you need so that a 95% confidence interval for the mean has a margin of error of at most 3?
σ = 26E = 395%
n =
7.Hypothesis tests & confidence intervalsMedium
A sample of 64 students has a mean score of 118.3 and a sample standard deviation of 15. Find the test statistic t for H₀: μ = 120.
μ₀ = 120x̄ = 118.3s = 15n = 64
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 50 of 200 patients on treatment A recover, and 44 of 120 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 50n₁ = 200x₂ = 44n₂ = 120
z =
3Solutions
1.Evidence-based practice statistics
Answer: NNH = 7
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.
Event rates
EER = 387/890 = 0.4348
CER = 252/870 = 0.2897
Number needed to harm
ARI = 0.4348 − 0.2897 = 0.1452
NNH = 1/0.1452 = 6.888 → 7
Answer
NNH = 7
About one in every 7 treated patients has the side effect because of the treatment.
2.Evidence-based practice statistics
Answer: p′ = 22.6 %
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 − 0.74)/0.89 = 0.2921
Pre-test odds
0.5/(1 − 0.5) = 1
Post-test odds
1·0.2921 = 0.2921
Post-test probability
p′ = 0.2921/(1 + 0.2921) = 0.2261 ≈ 22.6%
Answer
p′ = 22.6%
3.Descriptive statistics
Answer: s = 7.829
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: lower limit = 9.395 h
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.
Point estimate
x̄₁ − x̄₂ = 145 − 131 = 14
Standard error
SE = √(8²/40 + 14²/50) = 2.349
Margin of error
E = 1.96·2.349 = 4.605
Interval
14 ± 4.605 → [9.395, 18.6]
Answer
lower limit = 9.395 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: n = 289
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 = (1.96·26/3)² = 288.5
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 289
Answer
n = 289
7.Hypothesis tests & confidence intervals
Answer: t = −0.9067
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).
Standard error
SE = 15/√64 = 15/8 = 1.875
Test statistic
t = (118.3 − 120)/1.875 = −0.9067
Answer
t = −0.9067
t0.025; df = 63 = 1.998
|t| < 1.998, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: z = −2.218
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.
Sample proportions
p̂₁ = 50/200 = 0.25
p̂₂ = 44/120 = 0.3667
p̂ = 94/320 = 0.2938
Standard error under H₀
SE = √(0.2938·0.7063·(1/200 + 1/120)) = 0.05259
Test statistic
z = (0.25 − 0.3667)/0.05259 = −2.218
Answer
z = −2.218
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).