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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 study, exposed people contributed 4000 person-years with 48 new cases, and unexposed people 2500 person-years with 15 new cases. Find the incidence rate ratio IRR (exposed versus unexposed). Round to two decimal places.
exposed: 48/4000 PYnot exposed: 15/2500 PY
IRR =
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 94%. The prevalence of the disease is 28%. How many people with the disease are missed by the test (false negatives, FN)?
N = 10 000p = 28%Se = 71%Sp = 94%
FN =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 30, 10, 12, 16, 22, 29.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 170 and standard deviation 5. Find the z-score z of a score of 164.
μ = 170σ = 5x = 164
z =
5.Confidence intervalsMedium
A sample of 15 items has mean 33 and standard deviation s = 12.5. The population is roughly normal. Find the lower limit of the 95% confidence interval for the mean.
n = 15x̄ = 33s = 12.595%t(14) = 2.145
lower limit =
6.Confidence intervalsHard
The population standard deviation is 40. 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 0.5?
σ = 40E = 0.595%
n =
7.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 6 students, mean score 31, standard deviation 3. Group 2: 8 students, mean score 29.1, standard deviation 4.5. 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 = 6x̄₁ = 31s₁ = 3n₂ = 8x̄₂ = 29.1s₂ = 4.5
t =
8.Hypothesis tests & confidence intervalsHard
A one-sided z test of H₀: μ = 500 against H₁: μ > 500 is done at α = 0.05 with a sample of 64; the standard deviation is known to be 16. In reality the mean is 503.4. Find the power 1 − β of the test in percent.
μ₀ = 500μ = 503.4σ = 16n = 64α = 0.05
1 − β = %
3Solutions
1.Evidence-based practice statistics
Answer: IRR = 2
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.
Incidence rates
exposed: 48/4000 = 12 per 1000 person-years
not exposed: 15/2500 = 6 per 1000 person-years
Rate ratio
IRR = 12/6 = 2
Answer
IRR = 2
2.Evidence-based practice statistics
Answer: FN = 812
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
False negatives
FN = 2800·0.29 = 812
Answer
FN = 812
3.Descriptive statistics
Answer: s = 8.542
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 = (164 − 170)/5 = −1.2
Answer
z = −1.2
5.Confidence intervals
Answer: lower limit = 26.08
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 = 14, t = 2.145
Margin of error
E = 2.145·12.5/√15 = 6.923
Interval
33 ± 6.923 → [26.08, 39.92]
Answer
lower limit = 26.08
6.Confidence intervals
Answer: n = 24587
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 = (1.96·40/0.5)² = 24586
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 24587
Answer
n = 24587
7.Hypothesis tests & confidence intervals
Answer: t = 0.8918
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 = (5·3² + 7·4.5²)/12 = 15.56
sp = 3.945
Standard error
SE = 3.945·√(1/6 + 1/8) = 2.131
Test statistic
t = (31 − 29.1)/2.131 = 0.8918
Answer
t = 0.8918
With df = 12, the two-sided 5% critical value is 2.179; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: 1 − β = 52.19 %
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).