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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, groups of people were followed for different lengths of time (see the table: number of people and years of follow-up for each). During follow-up, 57 new cases occurred. Find the incidence rate IR per 1000 person-years. Round to one decimal place.
Practice exercise (made-up values, not clinical guidance). 10 000 people are screened with a test of sensitivity 89% and specificity 90%. 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 = 89%Sp = 90%
FN =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 22, 53, 36, 24, 48, 38.
n = 6
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 5. What proportion P of values is above 67.5?
μ = 60σ = 5x = 67.5
P =
5.Confidence intervalsMedium
A sample of 11 items has mean 65.5 and standard deviation s = 4. The population is roughly normal. Find the margin of error E of a 90% confidence interval for the mean.
n = 11x̄ = 65.5s = 490%t(10) = 1.812
E =
6.Confidence intervalsHard
The population standard deviation is 35. 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 0.5?
σ = 35E = 0.599%
n =
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 120; the population standard deviation is known to be 4. A random sample of 64 observations has a mean of 119.12. Test H₀: μ = 120 against H₁: μ < 120. Find the p-value p.
μ₀ = 120σ = 4n = 64x̄ = 119.12H₁: μ < μ₀
p =
8.Hypothesis tests & confidence intervalsHard
A sample of 25 measurements has mean 75.3. The population standard deviation is 4. Find the upper limit U of the 95% confidence interval for the mean.
x̄ = 75.3σ = 4n = 2595%
U =
3Solutions
1.Evidence-based practice statistics
Answer: IR = 50.2 per 1000 person-years
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.
Person-years
PY = 200·4 + 90·3 + 130·0.5 = 1135
Incidence rate
IR = 57/1135·1000 = 50.2 per 1000 person-years
Answer
IR = 50.2 per 1000 person-years
2.Evidence-based practice statistics
Answer: FN = 308
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 = 2800·0.11 = 308
Answer
FN = 308
3.Descriptive statistics
Answer: s = 12.43
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) = 0.9332
P = 1 − 0.9332 = 0.06681
Answer
P = 0.06681
5.Confidence intervals
Answer: E = 2.185
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 = 10, t = 1.812
Margin of error
E = 1.812·4/√11 = 2.185
Answer
E = 2.185
6.Confidence intervals
Answer: n = 32516
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 = (2.576·35/0.5)² = 32515
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 32516
Answer
n = 32516
7.Hypothesis tests & confidence intervals
Answer: p = 0.0392
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 = 4/√64 = 0.5
z = (119.12 − 120)/0.5 = −1.76
Tail area
Φ(1.76) = 0.9608
1 − Φ(1.76) = 0.0392
p-value
p = 0.0392
Answer
p = 0.0392
p ≤ 0.05, so H₀ is rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: U = 76.87
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).