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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, 130 patients received a new treatment and 540 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: 55/130control: 113/540
NNH =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 71% and specificity 95%. The prevalence of the disease is 28%. How many true-positive results TP are expected?
N = 10 000p = 28%Se = 71%Sp = 95%
TP =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 48, 40, 24, 32, 16.
n = 5
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 500 and standard deviation 75. Find the z-score z of a score of 650.
μ = 500σ = 75x = 650
z =
5.Confidence intervalsMedium
A 99% confidence interval for a mean, based on 100 observations and a known σ, has a margin of error of 5.8. What is the margin of error E₂ if the same study uses 25 observations instead?
99%n₁ = 100E₁ = 5.8n₂ = 25
E₂ =
6.Confidence intervalsHard
The population standard deviation is 29. How large a sample n do you need so that a 90% confidence interval for the mean has a margin of error of at most 0.5?
σ = 29E = 0.590%
n =
7.Hypothesis tests & confidence intervalsMedium
Two production lines make the same part. Line 1: a sample of 30 parts has mean length 75 mm, and the line's known standard deviation is 17 mm. Line 2: a sample of 50 parts has mean 71.2 mm, known standard deviation 10 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
In a trial, 150 of 250 patients on treatment A recover, and 42 of 80 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 150n₁ = 250x₂ = 42n₂ = 80
z =
3Solutions
1.Evidence-based practice statistics
Answer: NNH = 5
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 = 55/130 = 0.4231
CER = 113/540 = 0.2093
Number needed to harm
ARI = 0.4231 − 0.2093 = 0.2138
NNH = 1/0.2138 = 4.677 → 5
Answer
NNH = 5
About one in every 5 treated patients has the side effect because of the treatment.
2.Evidence-based practice statistics
Answer: TP = 1988
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.
With and without the disease
10 000·0.28 = 2800
10 000 − 2800 = 7200
True positives
TP = 2800·0.71 = 1988
Answer
TP = 1988
3.Descriptive statistics
Answer: s = 12.65
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.
Mean
x̄ = 160/5 = 32
Squared deviations
Σ(x − x̄)² = 16² + 8² + (−8)² + 0² + (−16)² = 640
Standard deviation
s = √(640/4) = 12.65
Answer
s = 12.65
4.Descriptive statistics
Answer: z = 2
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 = (650 − 500)/75 = 2
Answer
z = 2
5.Confidence intervals
Answer: E₂ = 11.6
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 = (2.576/2.576)·√(100/25) = 2
New margin
E₂ = 5.8·2 = 11.6
Answer
E₂ = 11.6
6.Confidence intervals
Answer: n = 9104
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 = (1.645·29/0.5)² = 9103
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 9104
Answer
n = 9104
7.Hypothesis tests & confidence intervals
Answer: z = 1.114
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.
Standard error of the difference
SE = √(17²/30 + 10²/50) = √11.63 = 3.411
Test statistic
z = (75 − 71.2)/3.411 = 1.114
Answer
z = 1.114
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: z = 1.184
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 = 150/250 = 0.6
p̂₂ = 42/80 = 0.525
p̂ = 192/330 = 0.5818
Standard error under H₀
SE = √(0.5818·0.4182·(1/250 + 1/80)) = 0.06336
Test statistic
z = (0.6 − 0.525)/0.06336 = 1.184
Answer
z = 1.184
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).