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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). A test has a sensitivity of 81% and a specificity of 76%. Find the positive likelihood ratio LR+. Round to two decimal places.
Se = 81%Sp = 76%
LR+ =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 80% and specificity 77%. The prevalence of the disease is 23%. How many positive results n₊ are expected in total?
N = 10 000p = 23%Se = 80%Sp = 77%
n₊ =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 26, 37, 48, 25, 32, 53.
n = 6
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 593.8.
μ = 500σ = 75x = 593.8
z =
5.Confidence intervalsMedium
A sample of 13 items has mean 81.5 and standard deviation s = 7. The population is roughly normal. Find the lower limit of the 90% confidence interval for the mean.
n = 13x̄ = 81.5s = 790%t(12) = 1.782
lower limit =
6.Confidence intervalsHard
The population standard deviation is 25. 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 = 25E = 299%
n =
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 100; the population standard deviation is known to be 24. A random sample of 64 observations has a mean of 102.64. Test H₀: μ = 100 against H₁: μ > 100. Find the p-value p.
μ₀ = 100σ = 24n = 64x̄ = 102.64H₁: μ > μ₀
p =
8.Hypothesis tests & confidence intervalsHard
In a trial, 90 of 300 patients on treatment A recover, and 70 of 200 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 90n₁ = 300x₂ = 70n₂ = 200
z =
3Solutions
1.Evidence-based practice statistics
Answer: LR+ = 3.38
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.
Positive likelihood ratio
LR+ = 0.81/(1 − 0.76) = 3.375 ≈ 3.38
Answer
LR+ = 3.38
2.Evidence-based practice statistics
Answer: n₊ = 3611
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
10 000·0.23 = 2300
10 000 − 2300 = 7700
All positive results
TP = 2300·0.8 = 1840
FP = 7700·0.23 = 1771
n₊ = 1840 + 1771 = 3611
Answer
n₊ = 3611
Of the 3611 positives only 1840 have the disease (PPV 51%).
3.Descriptive statistics
Answer: s = 11.55
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 = (593.8 − 500)/75 = 1.25
Answer
z = 1.25
5.Confidence intervals
Answer: lower limit = 78.04
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.
Critical value
df = n − 1 = 12, t = 1.782
Margin of error
E = 1.782·7/√13 = 3.46
Interval
81.5 ± 3.46 → [78.04, 84.96]
Answer
lower limit = 78.04
6.Confidence intervals
Answer: n = 1037
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 = (2.576·25/2)² = 1037
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 1037
Answer
n = 1037
7.Hypothesis tests & confidence intervals
Answer: p = 0.1894
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 = 24/√64 = 3
z = (102.64 − 100)/3 = 0.88
Tail area
Φ(0.88) = 0.8106
1 − Φ(0.88) = 0.1894
p-value
p = 0.1894
Answer
p = 0.1894
p > 0.05, so H₀ is not rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: z = −1.174
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 = 90/300 = 0.3
p̂₂ = 70/200 = 0.35
p̂ = 160/500 = 0.32
Standard error under H₀
SE = √(0.32·0.68·(1/300 + 1/200)) = 0.04258
Test statistic
z = (0.3 − 0.35)/0.04258 = −1.174
Answer
z = −1.174
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).