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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, 370 patients received a new treatment and 850 received the control treatment. The table shows how many in each group had the outcome (an unwanted event such as a stroke). Find the number needed to treat NNT to prevent one event. Round up to a whole number of patients.
treatment: 108/370control: 332/850
NNT =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 10% for a disease. A test with sensitivity 96% and specificity 73% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 10%Se = 96%Sp = 73%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 23, 17, 28, 14, 57, 28, 10.
n = 7
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 50 and standard deviation 12. Find the z-score z of a score of 38.
μ = 50σ = 12x = 38
z =
5.Confidence intervalsMedium
The blood pressure of 11 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 3.5 and standard deviation 7.8. Find the lower limit of the 90% confidence interval for the mean difference.
n = 11d̄ = 3.5sd = 7.890%t(10) = 1.812
lower limit = mmHg
6.Confidence intervalsHard
The population standard deviation is 15. 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 5?
σ = 15E = 599%
n =
7.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 80; the population standard deviation is known to be 20. A random sample of 25 observations has a mean of 71.76. Test H₀: μ = 80 against H₁: μ ≠ 80. Find the p-value p.
μ₀ = 80σ = 20n = 25x̄ = 71.76H₁: μ ≠ μ₀
p =
8.Hypothesis tests & confidence intervalsHard
A one-sided z test of H₀: μ = 50 against H₁: μ > 50 is done at α = 0.10 with a sample of 36; the standard deviation is known to be 15. In reality the mean is 53.5. Find the power 1 − β of the test in percent.
μ₀ = 50μ = 53.5σ = 15n = 36α = 0.10
1 − β = %
3Solutions
1.Evidence-based practice statistics
Answer: NNT = 11
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 = 108/370 = 0.2919
CER = 332/850 = 0.3906
Number needed to treat
ARR = 0.3906 − 0.2919 = 0.0987
NNT = 1/0.0987 = 10.13 → 11
Answer
NNT = 11
Treating 11 patients prevents about one event.
2.Evidence-based practice statistics
Answer: p′ = 28.3 %
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.
Likelihood ratio
LR+ = 0.96/(1 − 0.73) = 3.556
Pre-test odds
0.1/(1 − 0.1) = 0.1111
Post-test odds
0.1111·3.556 = 0.3951
Post-test probability
p′ = 0.3951/(1 + 0.3951) = 0.2832 ≈ 28.3%
Answer
p′ = 28.3%
3.Descriptive statistics
Answer: s = 15.57
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 = (38 − 50)/12 = −1
Answer
z = −1
5.Confidence intervals
Answer: lower limit = −0.7614 mmHg
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.
Critical value
df = 10, t = 1.812
Margin of error
E = 1.812·7.8/√11 = 4.261
Interval
3.5 ± 4.261 → [−0.7614, 7.761]
Answer
lower limit = −0.7614 mmHg
6.Confidence intervals
Answer: n = 60
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 = (2.576·15/5)² = 59.72
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 60
Answer
n = 60
7.Hypothesis tests & confidence intervals
Answer: p = 0.0394
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 = 20/√25 = 4
z = (71.76 − 80)/4 = −2.06
Tail area
Φ(2.06) = 0.9803
1 − Φ(2.06) = 0.0197
p-value
p = 2·0.0197 = 0.0394
Answer
p = 0.0394
p ≤ 0.05, so H₀ is rejected at the 5% level.
8.Hypothesis tests & confidence intervals
Answer: 1 − β = 54.7 %
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).