3 Analysis plan
This chapter is written before any outcome is examined. Changes after that point are logged in the decision record with a reason. It was revised after plan review, before any data were seen (D-013 to D-017).
3.1 Population and unit of analysis
All randomized patients, analysed in the group they were randomized to (intention to treat). Patients with a sore throat before surgery (preOp_pain) are kept. The unit of analysis is the patient (D-005).
3.2 Outcomes
All outcomes are any sore throat at rest, defined as a score above zero on the zero-to-ten scale.
- Primary: thirty minutes after PACU arrival (
pacu30min_throatPain; D-001). - Secondary: ninety minutes after PACU arrival (
pacu90min_throatPain), four hours after surgery (postOp4hour_throatPain) and the first postoperative morning (pod1am_throatPain; D-002).
Cough is not analysed (D-004). Swallowing pain is an open question for the client (Chapter 1).
3.3 Estimands
Every comparison is licorice against sugar water in all randomized patients; because the analyses are complete case, this holds only if missing outcomes are missing completely at random (see Missing data). The odds ratios that answer the client’s questions are population-averaged (marginal): they compare the odds of sore throat had everyone received licorice with the odds had everyone received sugar water (D-002). Each odds ratio is reported with the proportion with any sore throat in each arm and the risk difference, because the odds ratio overstates the relative risk when the outcome is common (D-006).
| Question | Population | Outcome | Summary measure |
|---|---|---|---|
| Primary: did it reduce sore throat? | All randomized | Any sore throat at rest, thirty minutes after PACU arrival | Marginal odds ratio; risk difference |
| Secondary: does it last? | All randomized | The same outcome at each later assessment | Marginal odds ratio and risk difference at each assessment |
| Summary across time (does not answer “does it last”) | All randomized | The same outcome at all four assessments | Common marginal odds ratio, assuming one common effect |
| Supporting: patient-level effect | All randomized | The same outcome at all four assessments | Common conditional odds ratio (mixed model) |
| Sensitivity: covariate adjustment | All randomized | Primary outcome | Covariate-conditional odds ratio |
Two of these odds ratios are conditional, and both are reported only beside the marginal estimate with a label saying so. The mixed-model odds ratio compares the odds for the same patient under licorice and under sugar water; it is further from one than the marginal odds ratio whenever patients differ in their underlying risk, even with no change in the treatment effect (D-008). The adjusted odds ratio compares patients with the same covariate values; it differs from the unadjusted one even when the arms are perfectly balanced, for the same reason (odds ratios are non-collapsible), so a difference is not by itself evidence of confounding (D-009).
Does it last? Durability is judged from the per-time odds ratios, risk differences and their 95% CIs, read together. There is no formal test across time points. A non-significant result at a later assessment is not evidence that the effect wore off: it means the data cannot rule out no effect at that time, and the CI shows which effects remain plausible (D-003, D-013).
The common odds ratio assumes one treatment effect shared by all four assessments, including the primary one at thirty minutes. If the effect changes over time, it is a weighted average that leans toward the early, better-observed assessments. It is a summary only and does not answer “does it last”; the per-time estimates do. The same holds for the mixed-model common odds ratio (D-014).
3.4 Models
- Primary and each assessment. Unadjusted logistic regression of any sore throat on treatment, fitted separately at each assessment with all patients observed at that assessment. With treatment as the only term, its odds ratio is the marginal odds ratio. The risk difference is the difference in observed proportions, with a Wald 95% CI (D-006, D-007).
- Common marginal odds ratio. A generalized estimating equation (GEE) with a logit link, treatment and assessment (as a factor) as terms, an exchangeable working correlation within patient and robust (sandwich) standard errors, fitted with
geepack(D-007). There is no treatment-by-assessment term, so the model assumes one common effect (D-014). - Common conditional odds ratio. A logistic mixed model with the same terms and a random intercept for patient, fitted with
lme4. Labeled as patient-level in every table and sentence where it appears (D-008). - Adjusted sensitivity analysis. Logistic regression of the primary outcome on treatment, sex, age, BMI, ASA status, smoking and surgery size, plus preoperative sore throat unless fewer than ten patients have it (counted from baseline data only). Age and BMI enter as linear terms; ASA status, smoking and surgery size as factors. Any covariate category with fewer than ten patients is merged into its neighbour, decided from baseline counts only. Mallampati class is not included: it acts through intubation difficulty, which is not measured, and leaving it out keeps the model small. The adjusted odds ratio is conditional on the covariates (D-009, D-011, D-016).
All CIs are Wald 95% CIs (D-007). Warnings from any fit are reported as findings, not suppressed.
3.5 Missing data
There is no variable for “not assessable”; any missing score is treated as missing. Each analysis uses the patients observed at that assessment (complete case). The GEE and the mixed model use every observed assessment for each patient (D-010). The complete-case analyses assume outcomes are missing completely at random (MCAR) (D-015).
For the primary outcome, best- and worst-case bounds: refit the primary model with every missing outcome in the licorice arm set to no sore throat and every missing outcome in the sugar-water arm set to sore throat (best case for licorice), then the reverse (worst case). If the primary conclusion holds at both bounds, the missing patients cannot change it.
Bounds are computed for the primary outcome only (D-010, D-015).
Added after results were seen (post hoc; D-021): these bounds cover only the patients in the delivered file. A further worst case adds every enrolled patient absent from the file to the licorice arm with a sore throat.
The number of patients missing each outcome, by arm, is reported in the data chapter (D-012). If more than five percent of patients are missing the outcome at any assessment, the missing-data approach is revisited before any model is fitted, with a decision entry (D-015).
3.6 Verification before trusting a model
Each model function must recover a known effect from simulated data with the same structure before it is run on the study data (tests/testthat/test-models.R). For the GEE and the mixed model, the simulation draws correlated repeated outcomes per patient, and each model must recover its own estimand: the marginal and the conditional odds ratio differ for the same simulated data. When outcomes are simulated from a random-intercept model, the true marginal odds ratio is not a model parameter; it is computed by numerical integration over the random-intercept distribution (D-017).
The risk-difference Wald CI gets a known-answer test (D-017).
Added after the plan was locked at plan-v1, before any data were seen (D-018): the recovery test for the common odds ratio, from both the GEE and the mixed model, simulates a treatment effect that is constant over time, since only then does the common odds ratio have a single true value. The best- and worst-case bounds procedure gets a known-answer test.
3.7 Reporting
Effect sizes with 95% CI; p-values to three decimals. There is no multiplicity adjustment: there is one primary comparison, and the secondary analyses are descriptive (D-003). The marginal odds ratios and risk differences are the reported effects; conditional odds ratios always appear labeled, next to the marginal estimate.