PreRule staircase power analysis:

0. Project context and intended analysis

PreRule evaluates implementation of a prehospital accelerated diagnostic pathway for patients with possible acute coronary syndrome, including point-of-care troponin testing in ambulances and Emergency Primary Care Centres (EPCCs). The clinical aim is to reduce unnecessary referral or presentation to hospital emergency departments while preserving patient safety.

This note focuses on the power calculation for the primary effectiveness endpoint: whether the intervention reduces the proportion of included patients who present to, or are referred to, the hospital ED. The planned primary analysis is a patient-level GLMM with a logit link, fixed categorical period effects, an intervention indicator, and a random intercept for each site/EPCC. The random intercept allows each site to have its own underlying ED-referral tendency.

The analytic power calculations are performed using the SteppedPower R package. The calculation is a planning approximation to the intended GLMM analysis, using the explicit staircase design matrix, site-specific expected patient numbers, and assumed correlation parameters.

1. Design definition

The planned design is a balanced incomplete stepped-wedge, also called a staircase design. We denote it as:

SC(4, 2, 3, 3)

This means four rollout sequences, two EPCC/ambulance-sector clusters per sequence, three observed two-month control periods per site, and three observed two-month intervention periods per site. In other words, each site contributes 12 months of data. The design spans 18 months of calendar time.

The current fixed allocation is deliberately size-balanced: larger sites are paired with smaller sites so that expected recruitment is not too concentrated in a single rollout sequence.

Rollout sequence Sites Expected N over
12 observed months		 Expected N per
2-month observed period
1 Bergen + Kvam 2,600 433.3
2 Sandnes + Hå 1,500 250.0
3 Øygarden + Klepp 1,300 216.7
4 Nordhordaland + Bjørnafjorden 1,250 208.3

The expected patient numbers by centre are:

EPCC / ambulance sector Rollout sequence Expected N over
12 observed months		 Expected N over
6 intervention months		 Expected N per
2-month observed period
Bergen 1 2,500 1,250 416.7
Kvam 1 100 50 16.7
Sandnes 2 1,000 500 166.7
Hå 2 500 250 83.3
Øygarden 3 800 400 133.3
Klepp 3 500 250 83.3
Nordhordaland 4 750 375 125.0
Bjørnafjorden 4 500 250 83.3

The expected total sample size is 6,650 patients, split equally into 3,325 expected control observations and 3,325 expected intervention observations.

Figure 0. PreRule staircase design grid
Figure 0. PreRule staircase design grid

In the design grid, 0 denotes an observed control cluster-period, 1 denotes an observed intervention cluster-period, and blank cells denote periods where the site is not observed.

2. Model parameters and clinical interpretation

The power analysis uses three main clinical/statistical inputs.

First, the baseline ED referral rate is the expected proportion of included patients who would present to hospital ED under usual care. The main grid evaluates 10%, 20%, 30%, 40%, 50%, and 60%, with the 40-60% range treated as the most clinically relevant range for ED referral.

Second, the odds ratio is the intervention effect. OR 0.70 means a 30% reduction in the odds of ED presentation. In absolute terms, OR 0.70 corresponds to approximately an 8-9 percentage-point reduction when the baseline ED referral rate is 40-60%. For example, if 50% would normally be referred to ED, OR 0.70 corresponds to an intervention ED referral rate of about 41%, i.e. about 9 fewer ED referrals per 100 included patients. OR 0.80 is a smaller effect, corresponding to roughly a 5-6 percentage-point reduction in this same baseline range.

Third, the analysis uses ICC and Autocorrelation to describe clustering. ICC is the degree to which outcomes from patients within the same site are more similar than outcomes from patients at different sites. Higher ICC means the effective information is smaller than the raw patient count would suggest. Autocorrelation describes how stable each site’s ED-referral tendency is over time. High Autocorrelation means that a site with relatively high ED referral in one period tends to remain relatively high in later periods; low Autocorrelation means that site-level referral patterns fluctuate more over time. In a staircase design, Autocorrelation is important because the design relies partly on comparing each site before and after implementation.

In the scripts, the repeated cross-sectional correlation structure is passed to SteppedPower as:

alpha_0_1_2 = c(ICC, ICC * autocorrelation, ICC * autocorrelation)

The main analysis assumes high Autocorrelation, set to 1.00. This is favourable for the staircase design, so it is explicitly stress-tested in the sensitivity analysis.

3. Main power analysis

The main power analysis uses the current fixed, size-balanced allocation shown above. It is the favourable planned allocation, where larger and smaller sites are paired, for example Bergen with Kvam. The parameter grid is:

primary_baseline_risks <- c(0.10, 0.20, 0.30, 0.40, 0.50, 0.60)
primary_odds_ratios <- c(0.40, 0.50, 0.60, 0.70, 0.80, 0.90)
primary_iccs <- c(0.01, 0.05, 0.10, 0.20)
primary_autocorrelation <- 1.00
Figure 1. Main power analysis
Figure 1. Main power analysis

The main interpretation is that the design is adequately powered for an odds ratio around 0.70 when the ED referral rate is in the clinically plausible 40-60% range and Autocorrelation is high. Clinically, this corresponds to detecting about an 8-9 percentage-point absolute reduction in ED referral. The design is not adequately powered for OR 0.80, which would represent a smaller reduction of about 5-6 percentage points.

4. Focused sensitivity analysis: where the conclusion breaks

The sensitivity analysis tests the two assumptions most relevant to the conclusion.

The first sensitivity varies Autocorrelation while holding the effect at OR 0.70. This asks whether the conclusion still holds if site-level ED-referral patterns are less stable over time. The second sensitivity varies recruitment by 80%, 100%, and 120% of the expected sample size while keeping Autocorrelation at 1.00.

baseline_risk <- c(0.40, 0.50, 0.60)
odds_ratio <- 0.70
ICC <- c(0.01, 0.05, 0.10, 0.20)
autocorrelation <- c(0.25, 0.50, 0.75, 1.00)
recruitment_multiplier <- c(0.80, 1.00, 1.20)
Figure 2. Autocorrelation and recruitment sensitivity
Figure 2. Autocorrelation and recruitment sensitivity

The conclusion is mainly sensitive to Autocorrelation. If Autocorrelation is low, the OR 0.70 conclusion can break, especially when ICC is high. Recruitment uncertainty is more manageable: under high Autocorrelation, reducing expected recruitment to 80% weakens power but is less damaging than low Autocorrelation.

5. Allocation sensitivity

The allocation sensitivity asks whether the rollout order matters. The main power analysis uses the current fixed, size-balanced allocation. The sensitivity analysis compares this with two possible randomisation strategies.

Size-blocked randomisation means first dividing the sites into four larger sites and four smaller sites based on expected recruitment, then randomly assigning one larger and one smaller site to each rollout sequence. In this project, the larger sites are Bergen, Sandnes, Øygarden, and Nordhordaland; the smaller sites are Bjørnafjorden, Hå, Klepp, and Kvam. This preserves randomisation while preventing two very large sites from being placed in the same sequence.

Unrestricted randomisation means complete randomisation of the eight sites into four pairs, with no constraint on site size. This can produce unfavourable allocations, for example placing too much expected recruitment in one rollout sequence and too little in another.

The allocation sensitivity uses a deliberately difficult stress-test scenario:

baseline_risk <- 0.40
odds_ratio <- 0.70
ICC <- 0.20
autocorrelation <- 1.00
Figure 3. Allocation sensitivity
Figure 3. Allocation sensitivity

Each point is one allocation. The x-axis is sequence-size imbalance, defined as the largest expected sequence N minus the smallest expected sequence N. Higher values mean more uneven expected recruitment across rollout sequences. The current fixed allocation is highlighted. Vertical bands occur because many allocations have the same imbalance value; residual vertical spread reflects the order and exact composition of the rollout sequences.

The key interpretation is that unrestricted randomisation can generate poorly balanced allocations that fall below 80% power. The current fixed allocation and size-blocked randomisation are safer because they avoid concentrating too much recruitment in one sequence.

6. Compact conclusion table

topic finding clinical_interpretation
Project and estimand Primary endpoint is ED presentation/referral among included patients. The power calculation targets the conditional odds ratio for ED referral from a binomial logit model.
Design timing SC(4,2,3,3): 8 EPCC clusters, 9 two-month calendar periods, and 48 observed cluster-periods. The calendar spans 18 months, but each site contributes only six observed two-month periods: three control periods followed by three intervention periods.
Expected recruitment Expected N = 6650 total: 3325 control and 3325 intervention observations. Expected patient counts are site-specific and are used as a cluster-period N matrix in SteppedPower.
Analysis model Analytic power is calculated using SteppedPower for a conditional binomial GLMM approximation with fixed period effects and a random intercept for site/EPCC. The random intercept allows each site to have its own underlying ED-referral tendency.
Main target effect For ED referral rates 40-60%, OR 0.70 gives power 85.8-93.1% across ICC 1-20% when Autocorrelation = 1.00. This corresponds to approximately a 8.2-8.8 percentage-point absolute reduction in ED referral, i.e. about 8-9 percentage points.
Smaller effect For ED referral rates 40-60%, OR 0.80 gives power 49.1-57.7% across ICC 1-20% when Autocorrelation = 1.00. OR 0.80 corresponds to only about a 5.2-5.6 percentage-point absolute reduction and is not adequately powered.
Autocorrelation sensitivity Power for OR 0.70 falls below 80% in several scenarios when Autocorrelation is materially below 1.00, especially at higher ICC. This is the critical modelling assumption: the conclusion relies on site-level ED-referral patterns being reasonably stable over time.
Recruitment sensitivity Recruitment multipliers of 80%, 100% and 120% have much smaller impact than Autocorrelation under the high-Autocorrelation assumption. Recruitment uncertainty is manageable for OR 0.70, but extra recruitment does not make OR 0.80 adequately powered.
Allocation sensitivity Stress-test power: current fixed allocation 85.8%; minimum size-blocked 81.6%; minimum unrestricted 74.6%. Use the current fixed allocation or constrained/size-blocked randomisation; avoid unrestricted randomisation because it can place too much recruitment in one rollout sequence.

7. Draft conclusion

The planned PreRule staircase design is adequately powered to detect an odds ratio of approximately 0.70 for ED presentation if the baseline ED referral rate is around 40-60%, expected recruitment is achieved, and Autocorrelation is high. In clinical terms, this corresponds to detecting about a 9 percentage-point reduction in ED referral.

The design is not powered for OR 0.80, corresponding to a smaller absolute reduction of approximately 5-6 percentage points. Recruitment uncertainty is manageable, but the Autocorrelation assumption is critical. The allocation should either remain fixed as currently proposed or be randomised only under size-balance constraints. Unrestricted randomisation can produce sequence-size imbalance and reduce power below 80% in plausible stress-test scenarios.