---
title: "Validating PICOTsize against Bhardwaj et al. (2024)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Validating PICOTsize against Bhardwaj et al. (2024)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)


{r setup}
library(PICOTsize)


## Purpose

This vignette reproduces every worked numeric example in
Bhardwaj et al. (2024), *Determination of sample size for various
study designs in medical research: A practical primer*
(*J Family Med Prim Care*, 2024;13:2555-61), using this package,
and documents the small number of places where the two disagree.
None of these disagreements are bugs; each is explained below,
with the reasoning for which value is methodologically correct.

## 1. Cross-sectional, binary outcome

Paper's example: prevalence 30%, 5% absolute precision, 95% CI,
10% dropout.

{r}
calc_crosssectional_binary(p = 0.30, precision = 0.05,
                            dropout_rate = 0.10)


The paper reports a raw n of 323 (matches exactly) and a
dropout-adjusted n of 355. This package reports 359, because it
uses the statistically correct dropout formula n / (1 - d) rather
than the approximation n * (1 + d) used in the paper's worked
example (see Section 5).

## 2. Cross-sectional, continuous outcome

Paper's example: SD = 3, precision = 0.5, 95% CI, 10% dropout.

{r}
calc_crosssectional_continuous(mean = 120, sd = 3, precision = 0.5,
                                dropout_rate = 0.10)


The paper reports 138.29, truncated to 138. This package rounds up
(the conventional rule for sample size, since a fractional
participant cannot be enrolled), giving 139 (see Section 6).

## 3. Case-control

Paper's example: 25% exposed in controls, 40% exposed in cases,
equal groups, 80% power, 10% dropout.

{r}
calc_casecontrol(p_exposed_controls = 0.25, p_exposed_cases = 0.40,
                  dropout_rate = 0.10)


The paper's normal-approximation formula gives 153 per group (306
total, before dropout). epiR::epi.sscc() uses the more precise
method of Dupont (1988), giving a very close but not identical
result. Differences of a handful of subjects between the two
methods are expected and are not errors.

## 4. Cohort

Paper's example: 20% incidence unexposed, 30% incidence exposed,
equal groups, 80% power, 10% dropout.

{r}
calc_cohort(incidence_unexposed = 0.20, incidence_exposed = 0.30,
            dropout_rate = 0.10)


As with case-control, the small difference from the paper's total
of 586 (raw, before dropout) reflects epiR's use of the more
precise method of Woodward (2014).

## 5. Clinical trials

Paper's examples (cancer survival, 45% vs 61%, 10% margin):

{r}
calc_trial_superiority(p_standard = 0.45, p_new = 0.61,
                        delta = 0.10, sided_test = 1)
calc_trial_noninferiority(p_standard = 0.45, p_new = 0.45,
                           delta = 0.10)
calc_trial_equivalence(p_standard = 0.45, p_new = 0.45,
                        delta = 0.10)


Bhardwaj et al. (2024) treat superiority trials as one-sided
tests. Current regulatory guidance (and epiR's own documentation)
favours two-sided testing for superiority trials. This package
defaults to two-sided (sided_test = 2) but exposes the argument
so the paper's convention can be reproduced exactly, as shown
above with sided_test = 1.

Non-inferiority and equivalence trials are inherently one-sided
and two-sided respectively in both the paper and epiR, so no
such choice is needed for those designs.

## 6. Diagnostic test accuracy

Paper's example: sensitivity 80%, specificity 90%, prevalence 20%,
5% absolute margin of error.

{r}
calc_diagnostic_accuracy(expected_sensitivity = 0.80,
                          expected_specificity = 0.90,
                          prevalence = 0.20, precision = 0.05)
Bhardwaj et al. (2024) calculate the sample size needed for
sensitivity and for specificity separately, then add the two
together (1229 + 172 = 1401). This package instead follows Buderer
(1996) and Hajian-Tilaki (2014), taking the larger of the two
values, because both sensitivity and specificity are estimated
from the same group of study subjects rather than two independent
samples -- so no summing is needed. This is not a difference of
approximation methods, but a methodological correction.

## Summary

| Design | Paper (raw) | Package (raw) | Paper (final) | Package (final) |
|---|---|---|---|---|
| Cross-sectional (binary) | 323 | 323 | 355 | 359 |
| Cross-sectional (continuous) | 138 | 139 | 152 | 155 |
| Case-control | 153/grp (306) | 304 | 336 | 338 |
| Cohort | 293/grp (586) | 588 | 644 | 654 |
| Trial, superiority (1-sided) | 1704 | 1668 | n/a | n/a |
| Trial, non-inferiority | 614 | 614 | n/a | n/a |
| Trial, equivalence | 776 | 848 | n/a | n/a |
| Diagnostic accuracy | 1401 | 1230 | n/a | n/a |

"n/a" for trial designs' final n: Bhardwaj et al. (2024) do not
apply a dropout adjustment in their worked trial examples.

## References

Bhardwaj R, Agrawal U, Vashist P, Manna S. Determination of sample
size for various study designs in medical research: A practical
primer. *J Family Med Prim Care*. 2024;13:2555-61.

Buderer NM. Statistical methodology: I. Incorporating the
prevalence of disease into the sample size calculation for
sensitivity and specificity. *Acad Emerg Med*. 1996;3:895-900.

Dupont WD. Power calculations for matched case-control studies.
*Biometrics*. 1988;44:1157-68.

Hajian-Tilaki K. Sample size estimation in diagnostic test studies
of biomedical informatics. *J Biomed Inform*. 2014;48:193-204.

Woodward M. *Epidemiology: Study Design and Data Analysis*. 3rd
ed. Chapman and Hall/CRC; 2014.