Diagnostic Accuracy and Agreement
Estimate the number of diseased and nondiseased participants needed to obtain a specified confidence bound for a positive or negative diagnostic likelihood ratio. The calculation uses the log-transformed likelihood-ratio confidence interval described by Simel, Samsa & Matchar (1991).
The positive diagnostic likelihood ratio is LR+ = Se / (1 − Sp), while the negative diagnostic likelihood ratio is LR− = (1 − Se) / Sp. Likelihood ratios combine sensitivity and specificity into a single measure of how strongly a test result changes the odds of disease.
Simel, Samsa & Matchar (1991) treat the likelihood ratio as a ratio estimator and use its natural logarithm because the log likelihood ratio is approximately normally distributed. For probabilities p1 and p2 in the diseased and nondiseased groups, respectively, the approximate standard error is
For LR+, p1 = Se and p2 = 1 − Sp. For LR−, p1 = 1 − Se and p2 = Sp.
The calculator specifies a confidence bound that should be excluded from the confidence interval. With a control-to-diseased ratio r = n₂/n₁, the required number of diseased participants is obtained by solving the log-scale confidence-bound equation:
For LR+, B is a lower confidence bound, so the calculation requires B < LR+. For LR−, B is an upper confidence bound, so the calculation requires B > LR− and the denominator is equivalently [log(B) − log(LR−)]². The total sample size is n₁ + n₂, with n₂ = r n₁.
Simel, Samsa & Matchar give an example with expected sensitivity 0.80, specificity 0.73, equal numbers of diseased and nondiseased participants, and an expected LR+ of 2.96. The investigators require the lower 95% confidence bound for LR+ to exceed 2.0. Solving their equation gives approximately 73.4 participants in each group, which is rounded up to 74 diseased and 74 nondiseased participants, for a total of 148.
This is a precision/confidence-bound calculation rather than a conventional null-hypothesis power calculation. It plans the study so that the specified likelihood-ratio confidence bound is separated from the anticipated likelihood ratio under the assumed sensitivity and specificity.
Simel, D. L., Samsa, G. P., & Matchar, D. B. (1991). Likelihood ratios with confidence: Sample size estimation for diagnostic test studies. Journal of Clinical Epidemiology, 44(8), 763–770. doi:10.1016/0895-4356(91)90128-V.
Hajian-Tilaki, K. (2014). Sample size estimation in diagnostic test studies of biomedical informatics. Journal of Biomedical Informatics, 48, 193–204. doi:10.1016/j.jbi.2014.02.013. Section 6.4 discusses sample-size estimation for likelihood ratios.