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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 new test was compared with the reference standard in a study (see the 2 × 2 table). Find the positive likelihood ratio LR+. Round to two decimal places.
a = 204b = 400c = 136d = 980
LR+ =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). Without a drug, 23% of patients have a certain side effect during a year. The drug raises this risk with a relative risk of RR = 2.27. Find the number needed to harm NNH: how many patients must take the drug for a year for one extra patient to have the side effect? Round up to a whole number of patients.
control 23%RR = 2.27
NNH =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 57, 54, 31, 34, 17, 48.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 500 and standard deviation 60. Find the z-score z of a score of 428.
μ = 500σ = 60x = 428
z =
5.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 100 phones, mean 145.5 h, known standard deviation 8 h. Model 2: 100 phones, mean 130.5 h, known standard deviation 15 h. Find the lower limit of the 99% confidence interval for μ₁ − μ₂.
Two fertilisers are tested. Plot group 1: 12 plots, mean yield 53 kg, standard deviation 4.5 kg. Plot group 2: 5 plots, mean 45 kg, standard deviation 7.5 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the margin of error E of the 90% confidence interval for μ₁ − μ₂.
The mean of a population is claimed to be 120; the population standard deviation is known to be 25. A random sample of 100 observations has a mean of 116.15. Test H₀: μ = 120 against H₁: μ < 120. Find the p-value p.
μ₀ = 120σ = 25n = 100x̄ = 116.15H₁: μ < μ₀
p =
8.Hypothesis tests & confidence intervalsHard
A one-sided z test of H₀: μ = 500 against H₁: μ > 500 is done at α = 0.10 with a sample of 100; the standard deviation is known to be 30. In reality the mean is 503.3. Find the probability of a Type II error β in percent.
μ₀ = 500μ = 503.3σ = 30n = 100α = 0.10
β = %
3Solutions
1.Evidence-based practice statistics
Answer: LR+ = 2.07
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.
Sensitivity and specificity
Se = 204/340 = 0.6
Sp = 980/1380 = 0.7101
Positive likelihood ratio
LR+ = 0.6/(1 − 0.7101) = 2.07
Answer
LR+ = 2.07
2.Evidence-based practice statistics
Answer: NNH = 4
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.
Risk with the drug
0.23·2.27 = 0.5221
Absolute risk increase
ARI = 0.5221 − 0.23 = 0.2921
Number needed to harm
NNH = 1/0.2921 = 3.423 → 4
Answer
NNH = 4
3.Descriptive statistics
Answer: s = 15.46
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 = (428 − 500)/60 = −1.2
Answer
z = −1.2
5.Confidence intervals
Answer: lower limit = 10.62 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 = 145.5 − 130.5 = 15
Standard error
SE = √(8²/100 + 15²/100) = 1.7
Margin of error
E = 2.576·1.7 = 4.379
Interval
15 ± 4.379 → [10.62, 19.38]
Answer
lower limit = 10.62 h
The interval does not contain 0, so the means differ significantly.
6.Confidence intervals
Answer: E = 5.098 kg
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.
Pooled standard deviation
sp² = (11·4.5² + 4·7.5²)/15 = 29.85
sp = 5.464
Margin of error
E = 1.753·5.464·√(1/12 + 1/5) = 5.098
Answer
E = 5.098 kg
7.Hypothesis tests & confidence intervals
Answer: p = 0.06178
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 = 25/√100 = 2.5
z = (116.15 − 120)/2.5 = −1.54
Tail area
Φ(1.54) = 0.9382
1 − Φ(1.54) = 0.0618
p-value
p = 0.06178
Answer
p = 0.06178
p > 0.05, so H₀ is not rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: β = 57.22 %
Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).