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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 1100 person-years with 14 new cases, and unexposed people 6300 person-years with 32 new cases. Find the incidence rate ratio IRR (exposed versus unexposed). Round to two decimal places.
exposed: 14/1100 PYnot exposed: 32/6300 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 72% and specificity 87%. The prevalence of the disease is 9%. How many false-positive results FP are expected?
N = 10 000p = 9%Se = 72%Sp = 87%
FP =
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 37, 18, 26, 35, 39.
n = 5
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 100 and standard deviation 5. Find the z-score z of a score of 92.5.
μ = 100σ = 5x = 92.5
z =
5.Confidence intervalsMedium
A sample of 15 items has mean 21 and standard deviation s = 12.5. The population is roughly normal. Find the upper limit of the 99% confidence interval for the mean.
n = 15x̄ = 21s = 12.599%t(14) = 2.977
upper limit =
6.Confidence intervalsHard
The population standard deviation is 23. 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 1.5?
σ = 23E = 1.599%
n =
7.Hypothesis tests & confidence intervalsMedium
A sample of 16 students has a mean score of 101.4 and a sample standard deviation of 8. Find the test statistic t for H₀: μ = 100.
μ₀ = 100x̄ = 101.4s = 8n = 16
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 56 of 80 patients on treatment A recover, and 71 of 120 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 56n₁ = 80x₂ = 71n₂ = 120
z =
3Solutions
1.Evidence-based practice statistics
Answer: IRR = 2.51
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: 14/1100 = 12.73 per 1000 person-years
not exposed: 32/6300 = 5.079 per 1000 person-years
Rate ratio
IRR = 12.73/5.079 = 2.506 ≈ 2.51
Answer
IRR = 2.51
2.Evidence-based practice statistics
Answer: FP = 1183
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
10 000·0.09 = 900
10 000 − 900 = 9100
False positives
FP = 9100·0.13 = 1183
Answer
FP = 1183
Of the 1831 positives only 648 have the disease (PPV 35.4%).
3.Descriptive statistics
Answer: s = 8.803
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.
Mean
x̄ = 155/5 = 31
Squared deviations
Σ(x − x̄)² = 6² + (−13)² + (−5)² + 4² + 8² = 310
Standard deviation
s = √(310/4) = 8.803
Answer
s = 8.803
4.Descriptive statistics
Answer: z = −1.5
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 = (92.5 − 100)/5 = −1.5
Answer
z = −1.5
5.Confidence intervals
Answer: upper limit = 30.61
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 = 2.977
Margin of error
E = 2.977·12.5/√15 = 9.608
Interval
21 ± 9.608 → [11.39, 30.61]
Answer
upper limit = 30.61
6.Confidence intervals
Answer: n = 1561
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 = (2.576·23/1.5)² = 1560
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 1561
Answer
n = 1561
7.Hypothesis tests & confidence intervals
Answer: t = 0.7
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).
Standard error
SE = 8/√16 = 8/4 = 2
Test statistic
t = (101.4 − 100)/2 = 0.7
Answer
t = 0.7
t0.025; df = 15 = 2.131
|t| < 2.131, so H₀ is not rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: z = 1.559
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 = 56/80 = 0.7
p̂₂ = 71/120 = 0.5917
p̂ = 127/200 = 0.635
Standard error under H₀
SE = √(0.635·0.365·(1/80 + 1/120)) = 0.06949
Test statistic
z = (0.7 − 0.5917)/0.06949 = 1.559
Answer
z = 1.559
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).