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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). A new test was compared with the reference standard in a study (see the 2 × 2 table). Find the negative likelihood ratio LR−. Round to two decimal places.
a = 194b = 59c = 76d = 121
LR− =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 40% for a disease. A test with sensitivity 88% and specificity 72% comes back positive. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 40%Se = 88%Sp = 72%positive
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 50, 10, 60, 56, 40, 17.
n = 6
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 20. What proportion P of values is below 44?
μ = 60σ = 20x = 44
P =
5.Confidence intervalsMedium
The blood pressure of 9 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 6.6 and standard deviation 7.2. Find the margin of error E of the 99% confidence interval for the mean difference.
n = 9d̄ = 6.6sd = 7.299%t(8) = 3.355
E = mmHg
6.Confidence intervalsHard
The diameters of 25 machined shafts have a sample standard deviation of 5.1 µm. Assuming the diameters are normally distributed, find the upper limit σ of the 95% confidence interval for the population standard deviation. With df = 24, 2.5% of the χ² distribution lies below 12.401 and 2.5% lies above 39.364.
n = 25s = 5.1 µmχ² = 12.401, 39.364
σ = µm
7.Hypothesis tests & confidence intervalsMedium
16 employees take a typing test before and after a training course. For each person the difference d = after − before (words per minute) is recorded. The mean difference is −8.55 and the standard deviation of the differences is 11.8. Find the paired test statistic t for H₀: μd = 0.
n = 16d̄ = −8.55sd = 11.8
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 56 of 80 patients on treatment A recover, and 50 of 80 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 56n₁ = 80x₂ = 50n₂ = 80
z =
3Solutions
1.Evidence-based practice statistics
Answer: LR− = 0.42
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.
Sensitivity and specificity
Se = 194/270 = 0.7185
Sp = 121/180 = 0.6722
Negative likelihood ratio
LR− = (1 − 0.7185)/0.6722 = 0.4187 ≈ 0.42
Answer
LR− = 0.42
2.Evidence-based practice statistics
Answer: p′ = 67.7 %
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+ = 0.88/(1 − 0.72) = 3.143
Pre-test odds
0.4/(1 − 0.4) = 0.6667
Post-test odds
0.6667·3.143 = 2.095
Post-test probability
p′ = 2.095/(1 + 2.095) = 0.6769 ≈ 67.7%
Answer
p′ = 67.7%
3.Descriptive statistics
Answer: s = 20.87
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 = −0.8
Standard normal table
Φ(−0.8) = 0.2119
P = 0.2119
Answer
P = 0.2119
5.Confidence intervals
Answer: E = 8.052 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 = 8, t = 3.355
Margin of error
E = 3.355·7.2/√9 = 8.052
Answer
E = 8.052 mmHg
6.Confidence intervals
Answer: σ = 7.095 µm
Interval for a variance: (n − 1)s²/σ² has a χ² distribution with n − 1 degrees of freedom. The 95% interval for σ² runs from (n − 1)s² divided by the upper χ² value (2.5% above it) to (n − 1)s² divided by the lower χ² value (2.5% below it); the interval for σ is the square root of both limits. It is not symmetric around s.
Sum of squares
(n − 1)s² = 24·5.1² = 624.2
Limits for σ²
624.2/39.364 = 15.86
624.2/12.401 = 50.34
Limit for σ
σ = √50.34 = 7.095 µm
Answer
σ = 7.095 µm
95% interval for σ: [3.982, 7.095] µm.
7.Hypothesis tests & confidence intervals
Answer: t = −2.898
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 of the mean difference
SE = 11.8/√16 = 2.95
Test statistic
t = −8.55/2.95 = −2.898
Answer
t = −2.898
With df = 15, the two-sided 5% critical value is 2.131; |t| is larger, so H₀ is rejected.
8.Hypothesis tests & confidence intervals
Answer: z = 1.003
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 = 50/80 = 0.625
p̂ = 106/160 = 0.6625
Standard error under H₀
SE = √(0.6625·0.3375·(1/80 + 1/80)) = 0.07477
Test statistic
z = (0.7 − 0.625)/0.07477 = 1.003
Answer
z = 1.003
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).