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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 1200 person-years with 65 new cases, and unexposed people 4700 person-years with 127 new cases. Find the incidence rate ratio IRR (exposed versus unexposed). Round to two decimal places.
exposed: 65/1200 PYnot exposed: 127/4700 PY
IRR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A screening test has a sensitivity of 88% and a specificity of 86%. It is used in a population where the prevalence is 3%. Find the positive predictive value PPV as a percentage. Round to one decimal place.
Se = 88%Sp = 86%p = 3%
PPV = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 20, 34, 22, 45, 33, 10.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 500 and standard deviation 25. Find the z-score z of a score of 470.
μ = 500σ = 25x = 470
z =
5.Confidence intervalsMedium
A 90% confidence interval for a mean, based on 36 observations and a known σ, has a margin of error of 1. What is the margin of error E₂ of a 99% interval based on 324 observations?
90%n₁ = 36E₁ = 199%n₂ = 324
E₂ =
6.Confidence intervalsHard
Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 90% confidence interval for the proportion has a margin of error of at most 0.02?
p = 0.5E = 0.0290%
n =
7.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 5 students, mean score 36, standard deviation 6. Group 2: 26 students, mean score 42.1, standard deviation 7.5. 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 = 5x̄₁ = 36s₁ = 6n₂ = 26x̄₂ = 42.1s₂ = 7.5
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 163 of 250 patients on treatment A recover, and 64 of 80 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 163n₁ = 250x₂ = 64n₂ = 80
z =
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: 65/1200 = 54.17 per 1000 person-years
not exposed: 127/4700 = 27.02 per 1000 person-years
Rate ratio
IRR = 54.17/27.02 = 2.005 ≈ 2
Answer
IRR = 2
2.Evidence-based practice statistics
Answer: PPV = 16.3 %
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.
True and false positives (per person)
Se·p = 0.88·0.03 = 0.0264
(1 − Sp)·(1 − p) = 0.14·0.97 = 0.1358
Positive predictive value
PPV = 0.0264/(0.0264 + 0.1358) = 0.1628 ≈ 16.3%
Answer
PPV = 16.3%
The lower the prevalence, the lower the PPV of the same test.
3.Descriptive statistics
Answer: s = 12.42
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 = (470 − 500)/25 = −1.2
Answer
z = −1.2
5.Confidence intervals
Answer: E₂ = 0.522
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.
Ratio of margins
E₂/E₁ = (2.576/1.645)·√(36/324) = 0.522
New margin
E₂ = 1·0.522 = 0.522
Answer
E₂ = 0.522
6.Confidence intervals
Answer: n = 1692
Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.
Solve for n
n = 1.645²·0.5·0.5/0.02² = 1691
Round up
n = 1692
Answer
n = 1692
7.Hypothesis tests & confidence intervals
Answer: t = −1.709
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 = (4·6² + 25·7.5²)/29 = 53.46
sp = 7.311
Standard error
SE = 7.311·√(1/5 + 1/26) = 3.57
Test statistic
t = (36 − 42.1)/3.57 = −1.709
Answer
t = −1.709
With df = 29, the two-sided 5% critical value is 2.045; |t| is smaller, so H₀ is not rejected.
8.Hypothesis tests & confidence intervals
Answer: z = −2.487
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 = 163/250 = 0.652
p̂₂ = 64/80 = 0.8
p̂ = 227/330 = 0.6879
Standard error under H₀
SE = √(0.6879·0.3121·(1/250 + 1/80)) = 0.05952
Test statistic
z = (0.652 − 0.8)/0.05952 = −2.487
Answer
z = −2.487
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).