A Non-Invasive Neuromodulation Device for Treatment of Migraine Headache: Complete Statistical Analysis

1. Trial at a Glance

This page covers the registry results for A Non-Invasive Neuromodulation Device for Treatment of Migraine Headache (NCT01899040), a randomized na study. The primary outcome is Change in Number of Monthly Migraine Headache Days.

2. Clinical Question

3. Trial Design

The registry describes a randomized, triple-masked, parallel-group study of a non-invasive neuromodulation device for episodic migraine headache. The study was sponsored by Scion NeuroStim, had a listed enrollment of 100, and is marked completed. The registry lists its start as 2013-08 and primary completion as 2016-05. Its phase is listed as NA.

The active-device group received a standardized active neuromodulation waveform, and the fully inactive placebo group received a standardized placebo waveform. Both devices were to be used twice daily. The registry describes a randomized, blinded study but does not report further details about how the randomization sequence was generated, how allocation was concealed, or which parties were masked. The masking category is recorded as triple; the registry record does not specify the three masked roles.

The primary endpoint results include 38 participants in the active-device group and 19 in the fully inactive placebo group. These are the analyzed counts for this outcome, not the overall enrollment. The outcome description restricts the analysis population to participants who met eligibility criteria, received a device, and completed the outcome effectiveness measures. The record also notes additional missing data, but its description is truncated and does not explain the amount or handling of that missingness.

Safety counts use different group denominators and include a partially active placebo-device group. The record does not explain how those safety groups relate to the randomized assignment or why their denominators differ from the primary endpoint analysis counts. It also does not report the trial’s original sample-size assumptions, a detailed statistical plan, or reported analysis results distinct from the posted arm-level outcome data.

4. Participant Flow and Accounting

The registry reports how participants moved through each stage of the study. All counts below are exactly as posted; missing counts are not inferred from other stages.

Overall Study

Reasons for not completing this period:

Analysis population for the primary outcome, as defined in the registry: Only participants who were confirmed to meet eligibility criteria, received either placebo (completely inactive) or fully active devices, and completed this outcome effectiveness measures were included in the overall number of participants analyzed for this outcome. Additional missing data comes from participants not being fully compliant with headache diary.

5. Baseline Characteristics

The table reproduces the baseline characteristics posted in the registry, which describe the participants analysed at baseline and help show whether the randomized groups were comparable.

Population: Baseline analysis populations include the fifty (50) participants who were randomized BID Active, the thirty (30) participants who were randomized to BID Placebo and received BID Placebo (fully inactive), and the twenty (20) who were randomized to BID Placebo BUT received Partially-Active devices.

6. Primary Endpoint

The registry lists the primary outcome as Change in Number of Monthly Migraine Headache Days, measured over after 84 days of Device use.

The primary endpoint is change in the number of monthly migraine headache days after 84 days of device use. A migraine headache day is described as a day in the month when a participant experienced a migraine. The change was assessed against a pre-device baseline period of 1 month and reported using headache-diary information. The registry specifies the mean as the summary measure and the standard deviation as the measure of dispersion.

Change scores make the direction of the outcome important. A negative value means fewer migraine headache days than during the baseline period; a more negative group mean therefore represents a larger average reduction. Comparing group means describes the average difference between the active-device and fully inactive placebo groups, while the standard deviations describe how participant-level changes vary within each group.

This endpoint is relevant to the study question because it directly counts the monthly days on which participants experienced migraine, rather than relying only on a broader symptom or quality-of-life measure. The diary-based measure also gives the outcome a defined reference period. However, the record does not report diary completion rates, the detailed rules for classifying a migraine day, or how incomplete diary entries were handled. Those details matter when judging how consistently the endpoint was measured across participants.

7. Results Posted in the Registry

• BID Active Device: mean -3.50 versus -1.10 on control; difference -2.40 (95% CI -2.73 to -2.07), standardized effect d = -4.00; z-test p = <0.001.

8. Secondary Efficacy Endpoints

The registry also posts results for secondary outcomes. For each outcome below, the endpoint type, the test applied to the posted summary statistics, the result and its interpretation are given in turn. Secondary endpoints are usually not adjusted for multiple testing, so they support rather than confirm the primary finding.

Number of Monthly Migraine Headache Days (Reduction by by 50% or More)

• Endpoint type: Binary: the number of participants with the event, compared as proportions.
• Test: Unadjusted two-proportion z-test, with a Wald 95% confidence interval for the risk difference.
• Result:
  • Active Device: 52.6% (20/38) versus 31.6% (6/19) on control; risk difference 21.1 percentage points (95% CI -5.2 to 47.3), risk ratio 1.67 (0.81–3.45); unadjusted two-proportion z-test p = 0.132.
• Interpretation:
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Active Device and the control. The interval gives the range of effects that remain compatible with the data.

Total Monthly Headache Pain Score

• Endpoint type: Continuous: a mean value per participant, compared as a difference in means.
• Test: Unadjusted two-sample z-test for the difference in means, using the posted standard deviations.
• Result:
  • Active Device: mean -19.30 versus -10.30 on control; difference -9.00 (95% CI -10.91 to -7.09), standardized effect d = -2.55; z-test p = <0.001.
• Interpretation:
  • The 95% confidence interval excludes no difference, so the mean is statistically lower for Active Device than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.

Average of Monthly Treated Headaches (Acute Anti-migraine Prescription Drug Intake)

• Endpoint type: Continuous: a mean value per participant, compared as a difference in means.
• Test: Unadjusted two-sample z-test for the difference in means, using the posted standard deviations.
• Result:
  • BID Active Device: mean -3.50 versus -1.80 on control; difference -1.70 (95% CI -2.11 to -1.29), standardized effect d = -2.53; z-test p = <0.001.
• Interpretation:
  • The 95% confidence interval excludes no difference, so the mean is statistically lower for BID Active Device than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.

The posted secondary results include a binary measure of participants whose monthly migraine headache days were reduced by 50% or more, a continuous total monthly headache pain score, and average monthly treated headaches. The continuous outcomes have mean changes in the same general direction as the primary result, while the observed proportion meeting the binary threshold is higher in the active-device group. The binary comparison is less precise: its confidence interval includes no difference, and its unadjusted p-value is 0.132.

These secondary measures address related but distinct aspects of the study. The binary threshold recasts change in migraine days as whether an individual crosses a specified cutoff. The pain score measures a different dimension, and treated-headache intake concerns acute prescription medication use. Agreement in direction can provide context for the primary result, but these outcomes are not interchangeable and do not constitute independent replications.

The registry lists additional secondary outcomes concerning mood, cognition, dizziness, memory, and medication intake. Numeric results for every listed outcome are not reported in the registry record. The posted secondary results therefore do not provide a complete account of all listed measures.

Unadjusted secondary results carry less weight because testing multiple outcomes increases the chance that at least one comparison will appear notable by chance. The registry record does not report a multiplicity adjustment, a prespecified hierarchy among secondary outcomes, or a complete analysis plan. The secondary comparisons should consequently be treated as exploratory context rather than as separate confirmatory evidence. Their interpretation also depends on the populations analyzed and missing-data handling, which are not fully described.

Secondary outcomes registered for this trial, for reference (only those above were recalculated):

• Number of Monthly Migraine Headache Days (Reduction by by 50% or More)
• Total Monthly Headache Pain Score
• (Safety Measure) Mood and Cognition - Change in Beck Depression Index Score
• (Safety Measure) Dizziness: Verify the Absence of Change in Material Dizziness (Berg Balance)
• (Safety Measure) Mood and Cognition - Change in Digit Symbol Coding Cognitive Speed
• (Safety Measure) Mood and Cognition Measure - Change in Short Term Memory Test Scores
• Average of Monthly Treated Headaches (Acute Anti-migraine Prescription Drug Intake)

9. Statistical Interpretation

10. Statistical Methods Explained

The primary measure is a continuous change score summarized by its mean and standard deviation. A mean describes the average change across participants in a group. The standard deviation describes the spread of participant-level changes around that group mean; it is not the uncertainty around the mean itself. Here, the change is defined relative to the baseline period, and negative values indicate fewer monthly migraine headache days after device use.

What does a mean difference describe?

The mean difference compares the active-device group’s average change with the fully inactive placebo group’s average change. The posted recalculated contrast is -2.4 migraine days. Its negative direction follows from the active group having the more negative mean change. This difference is expressed in the endpoint’s own units, which makes it directly interpretable as a contrast in average change. It does not describe the change experienced by every participant, and it does not show the distribution of individual responses.

What do the confidence interval and p-value add?

A confidence interval gives a range of estimates compatible with the observed data under the method used. For the primary recalculated contrast, the interval runs from -2.73 to -2.07. It conveys precision as well as direction: a narrower interval indicates less statistical uncertainty about the contrast under that calculation. The unadjusted p-value, p < 0.001, measures how unusual a result at least as incompatible with a no-difference reference would be under the assumptions of the test. It is not the probability that the null hypothesis is true, and it does not measure the clinical importance of a difference.

The posted contrast statistics are unadjusted recalculations from arm-level summaries. The registry record does not report the specific inferential model or test used in the trial’s original analysis. It also does not report whether an adjusted analysis was planned, how baseline values were incorporated beyond defining the change score, or whether any covariates were included. Those details cannot be inferred from the arm means and standard deviations.

Why do denominators matter?

The primary endpoint analysis includes 38 active-device participants and 19 fully inactive placebo participants. The endpoint description says participants had to meet eligibility criteria, receive a device, and complete the outcome effectiveness measures to be included. That restriction identifies the analyzed population but does not reveal how many randomized participants were excluded, the reasons for exclusion, or whether the exclusions differed by group. The record’s statement about additional missing data is incomplete; the amount and handling of missing values are not reported in the registry record.

Safety counts use separate denominators: the registry lists active-device, fully inactive placebo-device, and partially active placebo-device groups. These are not interchangeable with the primary endpoint analysis counts. Analyses based on different populations can answer different questions, so a reader should not assume that the safety and effectiveness summaries use the same participant set.

How should binary secondary outcomes be read?

A binary outcome classifies participants according to whether they meet a stated criterion. For the reduction of 50% or more in monthly migraine headache days, the registry reports counts in each group, and the recalculation expresses the contrast as a risk difference. A risk difference is the difference between the proportions meeting the criterion in the two groups. Its confidence interval includes values on both sides of no difference, and its unadjusted p-value is 0.132. This means the posted data do not estimate that secondary contrast with the same precision as a result whose interval excludes no difference. It does not establish that the groups are equivalent.

General statistical principle: with binary outcomes, the number of participants meeting the criterion and the number assessed both affect precision. A small analysis population can leave considerable uncertainty even when the observed proportions differ. Relative measures such as risk ratios or odds ratios express the comparison differently from a risk difference; no single scale captures every aspect of the data.

Why does over-dispersion matter?

General statistical principle: over-dispersion occurs when observed variability is greater than a model assumes. If a model understates variability, standard errors may be too small and confidence intervals too narrow. The registry lists standard deviations for the continuous outcomes, but it does not report whether the original analysis assessed over-dispersion or used a model requiring that assessment. Because the primary outcome is summarized as a continuous change score, the registry’s summaries alone do not establish whether over-dispersion is relevant to the original analysis.

For planning illustrations, the registry record states a two-sided significance level of 0.05, equal allocation between two arms, and a normal approximation, with target effects set as stated fractions of the observed effect. Those are assumptions for the planning illustrations, not documented assumptions from the trial’s original design. The original sample-size justification and analysis plan are not reported in the registry record.

11. Confidence Intervals and Precision

The primary recalculated mean difference is -2.4 migraine days, with a 95% confidence interval from -2.73 to -2.07. The interval is entirely below zero, so the estimated contrast is on the side corresponding to a larger average reduction in the active-device group. The interval also communicates uncertainty: the data are compatible with a range of mean differences, rather than identifying one exact value.

Confidence intervals are not guarantees that the true value lies within the displayed limits, nor do they describe the range of individual participant outcomes. They quantify uncertainty about a group-level estimate under the calculation’s assumptions. The registry record does not report the original inferential method, so the interval should be understood as part of the unadjusted recalculation from posted summaries.

Precision depends in part on the amount of information available and the variability of the outcome. The primary endpoint analysis includes 38 participants in the active-device group and 19 in the fully inactive placebo group. The record’s inclusion description and incomplete note about additional missing data mean that the analyzed counts do not explain all participant flow or missingness. This limits what can be concluded about the stability of the estimate under other handling of unavailable outcomes.

For the secondary binary endpoint, the confidence interval for the risk difference extends from -0.052 to 0.473. Because it includes zero, the data remain compatible with no group difference as well as with differences in either direction. This interval illustrates why the point estimate alone is insufficient: uncertainty around that estimate is substantial. The intervals for the other posted continuous outcomes are available in their respective results, but secondary-outcome uncertainty must also be considered in light of multiple outcomes and the absence of a reported multiplicity plan.

12. Planning a Similar Trial: Sample Size

This is a planning illustration, not a re-evaluation of the original trial. It asks: if a new study wanted to detect a difference like the one observed, or a smaller one, how large would it need to be? It is deliberately not “observed power” of the finished trial, which is not a meaningful quantity.

The sample-size illustrations translate observed contrasts into hypothetical planning targets. Each row considers a stated fraction of the observed effect, and the columns describe participants per arm for 80% power and participants per arm for 90% power. A higher power target means planning for a greater chance of detecting the specified effect if that effect is present, under the stated assumptions. It does not describe the probability that the study result is correct.

The planning assumptions are a two-sided significance level of 0.05, equal allocation between the two arms, and a normal approximation. The effect to detect is the stated fraction of the observed effect. The figures are participants per arm for the outcomes shown; the registry’s note says time-to-event illustrations would instead give total events, but no time-to-event outcome is included among these planning results.

Smaller assumed effects require more participants because they are harder to distinguish from random variation. The rows based on reduced fractions of the observed effect therefore call for larger planning totals than rows based on the full observed effect. These calculations use effects estimated from this study’s posted data, so they are illustrative rather than independent evidence about what a future study will observe. This is not observed power.

The registry record does not report the assumptions used to plan the original trial or its original sample-size justification. These illustrations should not be read as evidence that the trial was planned with the listed targets, or that its enrollment was based on the observed effects.

13. How the Sample Size Is Derived

Power is the probability that a study of a given size rejects the null hypothesis when a specific true effect exists. It depends on four things: the effect to detect, the variability of the outcome, the significance level (here a two-sided α of 0.05, so z_1−α/2 = 1.960) and the desired power (z_1−β = 0.842 for 80% and 1.282 for 90%).

Formula

n is the number of participants in each arm, Δ is the difference in means to detect and σ is the common standard deviation.

Worked calculation for this trial

1. Observed difference Δ = -2.400; pooled standard deviation σ = 0.600.
2. n = 2 × (1.96 + 0.842)² × 0.600² / -2.400² = 1.0, rounded up to 1 participants per arm for 80% power.

The smaller-effect rows in the table above repeat the same calculation with the effect multiplied by 0.75 and 0.5 (for ratios, applied to the logarithm of the ratio).

R code

alpha <- 0.05
z_a   <- qnorm(1 - alpha / 2)          # two-sided
delta <- -2.4                      # observed difference in means
sigma <- 0.6                     # pooled standard deviation
n_per_arm <- function(power, frac = 1) {
  ceiling(round(2 * (z_a + qnorm(power))^2 * sigma^2 / (delta * frac)^2, 6))
}
for (f in c(1, 0.75, 0.5))
  cat(sprintf("%3.0f%% of observed effect: n per arm = %s (80%% power), %s (90%% power)\n", 100 * f,
              format(n_per_arm(0.80, f), big.mark = ","), format(n_per_arm(0.90, f), big.mark = ",")))

# Check against R's built-in function (t-based, so very slightly larger)
cat("power.t.test:", round(power.t.test(delta = delta, sd = sigma, power = 0.80)$n, 1), "\n")

# Simulation: exact two-sample t-test using simulated means and variances
set.seed(1899040)
n    <- n_per_arm(0.80)
reps <- 20000
m_diff <- rnorm(reps, delta, sigma * sqrt(2 / n))             # difference in sample means
s2     <- sigma^2 * rchisq(reps, 2 * (n - 1)) / (2 * (n - 1))   # pooled sample variance
tstat  <- m_diff / sqrt(s2 * 2 / n)
cat(sprintf("Simulated power at n = %s per arm: %.3f (%d replicates)\n", format(n, big.mark = ","), mean(abs(tstat) > qt(0.975, 2 * (n - 1))), reps))

Output from running the code

100% of observed effect: n per arm = 1 (80% power), 2 (90% power)
 75% of observed effect: n per arm = 2 (80% power), 3 (90% power)
 50% of observed effect: n per arm = 4 (80% power), 6 (90% power)
power.t.test: 2.4 
Simulated power at n = 1 per arm: NA (20000 replicates)

The simulation draws many sets of group means and variances at the planned sample size and counts how often the two-sample t-test rejects the null. It should land close to the target power; at very small required sample sizes it can fall below it, because the normal approximation is rough there.

14. Multiplicity, Missing Data and Analysis Populations

The registry lists several secondary outcomes in addition to the primary endpoint. Testing multiple outcomes creates a multiplicity concern: the more comparisons considered, the greater the opportunity for chance findings if each is interpreted separately. The registry record does not report whether the trial adjusted for multiple comparisons, designated a hierarchy among endpoints, or treated particular secondary outcomes as confirmatory. The posted secondary p-values are unadjusted recalculations, so they should not be read as controlling the overall false-positive risk across all outcomes.

The endpoint description identifies an effectiveness population consisting of participants who met eligibility criteria, received a device, and completed the outcome measures. It also mentions additional missing data, but the text is incomplete. The number of participants randomized to each group, the reasons for missing outcomes, and the method used to address missing values are not reported in the registry record. It is therefore not possible to assess from the registry alone whether the analysis used all randomized participants or whether missingness could have affected the results.

The record does not report sensitivity analyses, imputation methods, or an intention-to-treat analysis definition. These omissions matter because an analysis restricted to participants with completed measures can differ from one that includes all randomized participants under a stated missing-data approach. No particular method should be assumed.

There is also a distinction between outcome and safety populations. The primary endpoint uses 38 active-device and 19 fully inactive placebo participants, while safety summaries list 50, 30, and 20 participants across three groups. The registry record does not explain the relationship between those populations or the reason for the differing counts. The original statistical-analysis plan and detailed population definitions are not reported in the registry record.

15. Safety Results

The registry reports serious adverse events as affected participants divided by participants at risk.

Deaths

Other (non-serious) adverse events

Participants with at least one non-serious event at or above the registry's 5% reporting threshold.

Most frequent non-serious adverse event terms

The registry reports zero serious adverse events among 50 participants in the Active Device group, zero among 30 participants in the Placebo Device (Fully-inactive Device) group, and zero among 20 participants in the Placebo Device (Partially Active Device) group. It also reports zero deaths in each of these groups. No top serious adverse-event terms are listed.

These are counts within the safety groups and denominators recorded for those groups. They should not be confused with the smaller analyzed populations for the primary endpoint. The registry also records other, non-serious adverse events: 14 among 50 in the Active Device group, 7 among 30 in the fully inactive placebo group, and 5 among 20 in the partially active placebo group. These figures are counts, not a complete description of event type, timing, severity, or causal attribution.

Zero recorded serious events does not establish that no serious events could occur, particularly when the available group sizes are limited. The registry record does not report detailed adverse-event definitions, follow-up completeness, or adjudication procedures. Safety counts should be interpreted as registry-recorded observations, not as a stand-alone assessment of overall safety.

16. What the Result Does, and Does Not, Mean

The primary result describes an average difference in change in monthly migraine headache days between participants analyzed in the active-device and fully inactive placebo groups. The active-device group’s mean change was more negative, and the unadjusted recalculated confidence interval for the mean difference lies below zero. These statistics describe the direction and precision of the posted group-level contrast.

They do not show that every participant had a reduction, or that the average difference applies to people beyond the analyzed study population. A group mean can conceal variation in individual outcomes. The record does not report the distribution of participant-level changes, the full participant flow, or how missing outcome data were handled. Those gaps limit assessment of the result’s robustness and generalizability.

The finding should also be distinguished from the secondary results. The continuous secondary outcomes point in a similar direction, but the binary 50% reduction comparison is uncertain, and numeric results for all listed secondary outcomes are not reported in the registry record. Because the multiplicity plan is not reported, the secondary results cannot be treated as independent confirmatory tests.

The registry’s completed status and posted results do not replace details of a full analysis report. The original statistical plan, original sample-size justification, randomization procedures, masking roles, and missing-data methods are not reported in the registry record. The posted figures therefore support a description of the observed data and recalculated comparisons, but they do not resolve every question about bias or analysis choices.

No clinical recommendation follows from these statistical summaries alone. The data concern a specific randomized study, its recorded outcome definitions, analyzed participants, and available follow-up. Broader judgments require consideration of the full study record and other relevant evidence, neither of which is established by these estimates alone.

17. Why This Trial Matters Statistically

This trial illustrates why a clinical result needs to be read through both its estimate and its uncertainty. The primary outcome is expressed in the endpoint’s own units, and the group means make the direction of change clear because negative values represent fewer monthly migraine headache days. The recalculated difference and confidence interval then quantify the group contrast and its statistical precision rather than reducing the result to a p-value alone.

It also illustrates the importance of distinguishing randomized enrollment, endpoint analysis counts, and safety denominators. The registry lists enrollment of 100, while the primary outcome and safety summaries use different group counts. Without participant-flow details, those figures cannot be treated as the same analysis population. This is a central issue when evaluating whether posted results represent all randomized participants.

The record also shows why secondary outcomes need a different interpretive status from a primary endpoint. Several related measures are listed, but the multiplicity strategy is not reported, and the binary secondary comparison is imprecise. Similar directions across measures may be informative, but they do not remove the need for a prespecified analysis hierarchy or careful accounting for multiple testing.

Finally, planning illustrations derived from observed effects are not the same as an original sample-size justification. They can show how planning targets change under different assumed effect sizes, but they do not document how this completed trial was designed. Taken together, these distinctions help readers separate what the posted statistics directly show from questions that remain unresolved in the registry record.

18. Limitations

• The posted summary statistics are the only data these results rest on; patient-level data, full protocols and statistical analysis plans are not available.
• Recalculated results are simple, unadjusted calculations from posted summaries. They do not replace the sponsor's prespecified analysis.

The registry record leaves several trial-specific uncertainties. The primary endpoint analysis is limited to participants who met eligibility criteria, received a device, and completed the outcome measures. The complete number randomized to each group, reasons for exclusions, and details of missing outcomes are not reported in the registry record. The record’s statement about additional missing data is incomplete, so its impact cannot be assessed.

The primary endpoint comparison uses 38 active-device participants and 19 fully inactive placebo participants, while safety counts use different denominators and include a partially active placebo-device group. The registry record does not explain how these groups correspond to randomization or why the analyzed counts differ. This limits direct comparison across outcome types.

The record identifies randomization and triple masking, but does not describe sequence generation, allocation concealment, the masked roles, or whether masking was maintained. It does not report detailed device-use adherence, diary completion, or the rules used to classify migraine days. Those omissions make it harder to judge implementation and measurement quality.

The original statistical plan, sample-size justification, handling of multiple outcomes, analysis population definitions, and sensitivity analyses are not reported in the registry record. The posted comparisons are unadjusted recalculations from summary data and cannot answer whether alternative prespecified analyses would yield similar estimates. Numeric results for all listed secondary outcomes are also not reported in the registry record.

19. Related Tutorials

20. Related Statistical Calculators

21. Sources

• ClinicalTrials.gov: A Non-Invasive Neuromodulation Device for Treatment of Migraine Headache — NCT01899040.
• Registry record last updated 2025-05-31; status last verified 2025-05.
• Results first posted 2025-05-31.
• Data retrieved from ClinicalTrials.gov on 2026-11-09.
• Group sizes, means, event counts and sponsor-reported p-values are reproduced as posted. Effect estimates, confidence intervals and p-values not attributed to the sponsor are independently calculated from the posted summary statistics.

22. Record Summary

NCT01899040 is a completed randomized, triple-masked, parallel-group study of a non-invasive neuromodulation device for episodic migraine headache, sponsored by Scion NeuroStim. The primary endpoint was mean change in monthly migraine headache days after 84 days of device use, relative to a baseline period of 1 month. Negative change indicates fewer migraine days.

The posted primary group means were -3.5 for the active device and -1.1 for the fully inactive placebo device. The unadjusted recalculated mean difference was -2.4, with a 95% confidence interval from -2.73 to -2.07 and p < 0.001. The primary analysis included 38 and 19 participants, respectively. The record also lists secondary outcomes and safety counts, but does not report the original analysis plan, sample-size justification, or complete missing-data methods.