Investigating the Causal Role of Prefrontal Control in Decision-making in Patients With Anhedonia: Complete Statistical Analysis

1. Trial at a Glance

This page covers the registry results for Investigating the Causal Role of Prefrontal Control in Decision-making in Patients With Anhedonia (NCT05084924), a randomized na study. The primary outcome is Change in the Percentage of Trials That the Participant Chooses to Perform the Hard Task.

2. Clinical Question

3. Trial Design

The registry record identifies this as a randomized, double-masked, parallel study of a device intervention. The three study arms were delta-beta tACS, theta-gamma tACS, and active-sham tACS. Enrollment was 35. The reported primary-outcome groups contained 11 participants in the delta-beta arm, 12 in the theta-gamma arm, and 12 in the active-sham arm. The study is marked completed, and its phase is recorded as NA.

The intervention was cross-frequency transcranial alternating current stimulation delivered through the NeuroConn Direct Current Stimulator Plus. The record describes theta-gamma stimulation as an active control waveform. For active sham, either delta-beta or theta-gamma stimulation was delivered for 10 seconds and then returned to baseline; the stated purpose was to mimic sensations that can occur at stimulation onset and assist blinding. The experimental-arm description is incomplete in the registry record, so its full stimulation procedure is not reported in the registry record.

The primary outcome used a task in which participants chose between an easy task with a chance at a small reward and a hard task with a chance at a greater reward. Baseline was based on the average of four blocks before stimulation, during the first hour. The intervention measure was the average of the next eight blocks during stimulation, in hours two through three. The record also lists a secondary measure of change in coupling strength between low-frequency prefrontal signals and high-frequency posterior signals.

The registry record gives the study start and primary completion dates, but does not provide the trial's original sample-size justification, detailed eligibility criteria, or full statistical analysis plan. These details are not reported in the registry record.

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

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.

6. Primary Endpoint

The registry lists the primary outcome as Change in the Percentage of Trials That the Participant Chooses to Perform the Hard Task, measured over Baseline (Hour 1), Stimulation (Hours 2 through 3).

The primary endpoint was change in the percentage of trials on which a participant chose to perform the hard task. On each trial of the Expenditure of Effort for Reward Task, the participant chose between an easy option involving low effort and a chance at a small monetary reward, or a hard option involving greater effort and a chance at a greater reward. The task therefore operationalizes a decision to expend effort in pursuit of a reward.

The registry record defines the baseline as the average across four blocks before stimulation, during the first hour. The stimulation-period measure is the average across the next eight blocks, during hours two through three. The posted arm values are mean changes, with standard deviations. The endpoint is used as an experimental measure of goal-directed behavior because it captures choices that trade effort against a possible reward. It is a task-based measure, not a direct measurement of every aspect of motivation or behavior outside the task.

The reported outcome is a change from baseline, rather than a single measurement at one time. A negative mean change indicates a lower value during the stimulation-period measurement relative to the baseline measure, according to the direction used for this outcome. The registry record does not state a minimum change considered practically important, and such a threshold is not reported in the registry record.

7. Results Posted in the Registry

• Delta-beta tACS: mean -0.30 versus -0.09 on control; difference -0.21 (95% CI -0.40 to -0.02), standardized effect d = -0.90; z-test p = 0.027.
• Theta-gamma tACS: mean -0.22 versus -0.09 on control; difference -0.13 (95% CI -0.31 to 0.05), standardized effect d = -0.59; z-test p = 0.151.

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.

Change in Coupling Strength Between Low-frequency Prefrontal Signals and High-frequency Posterior Signals

• 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:
  • Delta-beta tACS: mean 0.95 versus 0.13 on control; difference 0.82 (95% CI -0.26 to 1.90), standardized effect d = 0.60; z-test p = 0.138.
  • Theta-gamma tACS: mean -0.97 versus 0.13 on control; difference -1.10 (95% CI -2.27 to 0.07), standardized effect d = -0.75; z-test p = 0.065.
• Interpretation:
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Delta-beta tACS and the control. The interval gives the range of effects that remain compatible with the data.
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Theta-gamma tACS and the control. The interval gives the range of effects that remain compatible with the data.

The registry lists one secondary outcome: change in coupling strength between low-frequency prefrontal signals and high-frequency posterior signals. Its reported overall ANOVA across the three groups has p = 0.004. The recalculated pairwise estimates have uncertainty that includes no difference for both comparisons with active sham. Taken together, the secondary results indicate an overall group pattern in the sponsor-reported test, while the summary-based pairwise estimates do not precisely establish either specific contrast.

This outcome concerns signal coupling, not the hard-task choice measured by the primary endpoint. It is therefore related to the study's mechanistic question but is not a direct replication of the behavioral result. Differences in the patterns across outcomes are possible because the endpoints measure different things; they should not be treated as interchangeable evidence about task behavior.

Secondary outcomes generally carry less weight than a prespecified primary endpoint when interpreting a trial. In general, testing multiple outcomes or comparisons creates more opportunities for findings to arise by chance. Unadjusted secondary comparisons therefore need cautious interpretation, particularly when the registry does not report a multiplicity strategy or analysis hierarchy. The record does not state whether any multiplicity adjustment was applied.

The secondary endpoint is a change measure with group means and standard deviations, but the registry record does not report a practical-importance threshold. Its overall test and recalculated contrasts can be described statistically; they do not establish a biological mechanism or demonstrate that the observed pattern is linked causally to the primary behavioral outcome.

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

• Change in Coupling Strength Between Low-frequency Prefrontal Signals and High-frequency Posterior Signals

9. Statistical Interpretation

10. Statistical Methods Explained

What does a mean difference describe?

The primary outcome is continuous: it records the change in a percentage-of-trials measure, and the arm summaries report means and standard deviations. A mean difference compares the average change in one arm with the average change in another. Here, the recalculated pairwise estimates compare each stimulation arm with active sham. The sign indicates the direction of the estimated difference under the posted outcome coding; it does not by itself establish whether the size matters in practice.

As a general statistical point, a mean is a compact description of the center of a distribution, while a standard deviation describes variation among observations around that center. The standard deviation is not the uncertainty of the mean difference. Uncertainty also depends on how many participants contribute data and on the analysis model. The posted groups have relatively small counts, so individual observations can have meaningful influence on a group summary.

How does the sponsor's ANOVA differ from pairwise comparisons?

The registry reports an ANOVA for the three groups. In general, an ANOVA tests whether the observed group means are compatible with a common mean, considering all groups together. Its overall test does not identify which group pairs account for any evidence of differences. Pairwise comparisons instead address specific contrasts, such as delta-beta stimulation versus active sham. Those two approaches can produce different p-values because they test different hypotheses and use different arrangements of information.

The sponsor-reported analysis is the registry's reported primary analysis. The pairwise estimates and tests are unadjusted recalculations from the posted means, standard deviations, and group sizes. They are not replacements for a reported sponsor analysis. Since a discrepancy is flagged, the sponsor's own analysis takes precedence. The registry record does not report enough detail to determine the source of that discrepancy or to reconstruct the sponsor's model fully.

What do the p-values tell a reader?

In general, a p-value is calculated under a null hypothesis and a statistical model. It measures how unusual data at least as extreme as those observed would be if that null model and its assumptions held. A smaller p-value can be evidence against the null hypothesis, but it does not give the probability that the null hypothesis is true. It also does not show whether a difference is important, reproducible, or caused by the intervention.

The pairwise p-values are explicitly unadjusted. When several outcomes or comparisons are examined, repeated testing can increase the chance of at least one apparently notable result. An adjustment or a prespecified testing hierarchy can address some of this concern, but the registry record does not report whether the sponsor used a multiplicity adjustment or a hierarchy for the analyses.

Why do confidence intervals matter?

In general, a confidence interval gives a range of values compatible with the data under a specified method and its assumptions. Its width communicates precision: a wider interval represents greater uncertainty about the estimated effect. Whether an interval includes zero is often used to assess compatibility with no mean difference for a two-sided comparison, but this does not settle questions of practical importance.

The recalculated intervals are based on posted summary statistics and are unadjusted. They are not intervals from the sponsor's overall ANOVA. The sponsor analysis does not report an effect estimate or confidence interval in the registry record, so its result cannot be interpreted for magnitude and precision in the same way. The available arm summaries and recalculations are useful for understanding the posted results, while the reported sponsor analysis remains the primary analysis to prioritize given the flagged discrepancy.

What do standard deviations and over-dispersion mean?

As a general statistical point, a standard deviation describes the spread of individual observations, not the accuracy of an estimated group mean. Greater spread makes it harder to estimate a mean precisely if the group size is unchanged. Over-dispersion is a concern when data vary more than a chosen statistical model expects. A model that fails to account for excess variation can understate uncertainty and make tests appear more decisive than warranted. The registry record does not report an over-dispersion assessment for these outcomes, and the posted continuous-outcome summaries alone do not establish whether such a concern arose.

The outcome is based on repeated task choices, but the registry record does not describe the sponsor's participant-level model for handling those repeated observations. The posted arm summaries collapse the task measure into baseline and stimulation-period averages. They do not disclose whether the sponsor modeled trial-level choices, blocks, or within-participant correlation. Those details are not reported in the registry record.

How should the evidence be combined?

A careful reading distinguishes the posted group means, unadjusted pairwise recalculations, and the sponsor-reported overall test. The means describe the observed changes within arms. The recalculations provide simple pairwise contrasts and interval estimates derived from summary statistics. The ANOVA supplies the registry's sponsor-reported overall comparison. Because these analyses address different contrasts, they should be considered together without substituting one for another. None removes uncertainty arising from the small groups, unavailable analysis details, or the limited task-based outcome.

11. Confidence Intervals and Precision

Precision concerns how tightly an estimate is determined by the data. The posted primary outcome includes group means and standard deviations, while recalculated pairwise comparisons include confidence intervals. The interval for delta-beta stimulation versus active sham excludes zero; the interval for theta-gamma stimulation versus active sham includes zero. These intervals describe uncertainty in simple, unadjusted pairwise differences calculated from the posted summaries.

The interval excluding zero is not proof that the estimated difference is important or that the effect will be reproduced. An interval that includes zero does not prove that the groups are equivalent. In general, a confidence interval represents a range of values compatible with the data and the method used; it does not represent a range containing a fixed proportion of individual participants' outcomes.

The sponsor-reported ANOVA has p = 0.049, but the registry record does not report its effect estimate or confidence interval. Thus, the overall result conveys evidence of a group difference under the sponsor's analysis but does not provide a corresponding interval for the size of that overall effect. The reported pairwise recalculation intervals cannot fill that gap because they concern separate comparisons.

Small group sizes and variation in the outcome limit how precisely group differences can be estimated. The standard deviations are not themselves confidence intervals, and their magnitude should be read alongside the number of participants in each arm. The registry record does not report a clinically meaningful threshold against which to judge the interval estimates. Consequently, statistical uncertainty can be described, but the practical importance of the estimated changes cannot be settled from these results alone.

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 entries are planning illustrations for two-arm comparisons, not the number of participants observed in the trial. For these continuous outcomes, the figures are participants per arm; total-event counts apply to time-to-event planning, which is not the outcome type used. The entries show planning under the observed effect and under smaller fractions of that effect. The columns for 80% and 90% power indicate the illustrative number of participants per arm needed to have those respective chances of detecting the specified effect, if the planning assumptions hold.

The listed 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 registry record does not report the original trial-design assumptions or a sample-size justification, so these calculations should not be read as the design rationale for the completed study.

Rows planning for 100%, 75%, and 50% of the observed effect illustrate how the target effect changes the required group size. Smaller effects are harder to distinguish from random variation and therefore require more participants for the same planned power. The resulting figures depend on the observed effect used as a planning reference, which is uncertain in a small trial; they are not universal sample-size recommendations.

Power is a prospective design quantity under specified assumptions. These planning entries are not observed power, and they do not tell a reader the probability that the trial's conclusion is correct. They also do not alter the uncertainty in the posted estimates or substitute for the sponsor's reported analysis.

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 = -0.210; pooled standard deviation σ = 0.234.
2. n = 2 × (1.96 + 0.842)² × 0.234² / -0.210² = 19.5, rounded up to 20 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 <- -0.21                      # observed difference in means
sigma <- 0.234022790912908                     # 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(5084924)
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 = 20 (80% power), 27 (90% power)
 75% of observed effect: n per arm = 35 (80% power), 47 (90% power)
 50% of observed effect: n per arm = 78 (80% power), 105 (90% power)
power.t.test: 20.5 
Simulated power at n = 20 per arm: 0.787 (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. Analyses Reported by the Sponsor

The registry reports the sponsor's primary analysis as an ANOVA across delta-beta tACS, theta-gamma tACS, and active-sham tACS, with p = 0.049. It does not report an effect estimate or confidence interval for that analysis. The result is an overall comparison among the three groups, rather than a numerical estimate for one particular pair.

The pairwise recalculations use the posted arm means, standard deviations, and group sizes to estimate differences from active sham. They are simple, unadjusted analyses of summary statistics, and they do not reproduce the sponsor's reported three-group analysis. The registry record flags a discrepancy. In that situation, the sponsor's own analysis takes precedence, and the recalculations should be treated as supplementary descriptions rather than a correction of the sponsor result.

The record does not explain the source of the discrepancy or give enough detail to establish whether the analyses used different models, populations, or other procedures. Those reasons are not reported in the registry record. Readers should therefore distinguish the reported overall test from the recalculated pairwise estimates and avoid using the latter to override the former.

15. Multiplicity, Missing Data and Analysis Populations

Multiplicity arises when several hypotheses are tested, such as multiple pairwise contrasts or outcomes. In general, repeated testing increases the opportunity for at least one result to appear notable even when no underlying effect is present. A prespecified primary endpoint, testing hierarchy, or statistical adjustment can help control this issue. The registry record identifies a primary outcome and one secondary outcome, but a complete multiplicity plan is not reported in the registry record. It is also not reported whether the sponsor adjusted for multiple comparisons.

The reported primary analysis is an overall three-group ANOVA. The pairwise recalculations are unadjusted, so their p-values should not be read as if a multiplicity procedure had been applied. The same caution applies to the secondary recalculations. The registry record does not establish which specific comparisons were prespecified, whether all planned comparisons were reported, or whether any additional exploratory analyses were conducted.

Missing-data methods and analysis populations are also not described in sufficient detail. The outcome summaries give group counts, but the registry record does not report how missing or incomplete task data were handled, whether the sponsor used an intention-to-treat or other analysis population, or how participants with unavailable outcome measurements were treated. These details are not reported in the registry record and should not be inferred from the arm counts alone.

The record does not report interim analyses, stopping rules, or the sponsor's full statistical analysis plan. It is therefore not possible to assess from the registry entry whether any interim monitoring or prespecified analysis choices affected the reported results. This absence of detail limits independent assessment of analysis decisions; it does not establish that a particular procedure was or was not used.

16. 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 0% reporting threshold.

Most frequent non-serious adverse event terms

The registry lists serious adverse events for each arm. No serious adverse events are reported among the 11 participants at risk in the delta-beta arm, the 12 at risk in the theta-gamma arm, or the 12 at risk in the active-sham arm. No deaths are reported in any arm.

Other, non-serious adverse events were reported for 3 participants in the delta-beta arm, 3 in the theta-gamma arm, and 1 in the active-sham arm. These counts describe the registry's reported event summaries; they do not establish whether events were caused by stimulation. The record does not provide serious-event terms, and it does not give further detail about the nature, timing, severity, or duration of the other events.

Zero reported serious events in these small groups should not be interpreted as evidence that serious events are impossible or that the risks are fully characterized. The registry record reports counts for participants at risk, but does not provide additional context on event collection or follow-up. Safety interpretation is therefore limited to the recorded counts.

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

The study addresses whether different stimulation conditions are associated with changes in a task-based measure of effortful choice among participants with major depressive disorder and elevated anhedonia symptoms. The registry's sponsor-reported primary analysis identifies an overall difference among the three groups, with p = 0.049. The posted arm means and simple pairwise recalculations help describe the observed pattern, but the flagged discrepancy means the sponsor's own analysis takes precedence.

The finding does not establish the size or practical importance of the overall difference because the sponsor's analysis is not accompanied by an effect estimate or confidence interval in the registry record. Nor does an overall ANOVA identify which particular groups differ. Pairwise summary-based recalculations are uncertain and unadjusted, and should not be substituted for the sponsor's result.

The endpoint measures choices in a structured effort-for-reward task over a baseline and stimulation period. It does not directly establish changes in daily behavior, sustained effects, or improvement in a clinical condition. The registry record does not report follow-up evidence that would answer questions about persistence or broader functioning. The secondary signal-coupling outcome is a different measure and cannot by itself explain the behavioral result.

In general, statistical evidence is distinct from evidence of clinical importance or causation. Randomization supports a comparison between assigned conditions, but interpretation still depends on how outcomes were measured, how missing observations and analyses were handled, and how precisely effects were estimated. Several of these details are not reported in the registry record. Accordingly, the posted results describe an experimental study and its statistical findings; they do not establish a treatment recommendation or a definitive effect beyond the measured setting.

18. Why This Trial Matters Statistically

This study is statistically interesting because it connects an experimental manipulation with two distinct kinds of outcomes: a behavioral choice measure and a measure of signal coupling. The distinction matters. A statistical pattern in a mechanistic measure does not automatically establish a corresponding behavioral effect, and an overall test does not identify which pairwise contrast drives it.

The record also illustrates why the analysis label and the estimand matter. A three-group ANOVA answers an overall question, while recalculated comparisons with active sham answer pair-specific questions. A p-value from one cannot be read as if it were the result of the other. The flagged discrepancy makes it especially important to prioritize the sponsor-reported analysis without treating it as a complete account of effect magnitude, since no overall estimate or interval is reported.

The modest arm sizes and the use of a task-based endpoint make precision and design assumptions central to interpretation. The planning illustrations show how required group size depends on the effect targeted, but they do not reveal the original design rationale or guarantee that a future study would reproduce the observed effects. This is a useful reminder that sample-size planning is conditional on assumptions, while observed estimates remain uncertain.

Finally, incomplete reporting of multiplicity, missing-data handling, analysis populations, and the full statistical plan limits independent appraisal. These are not merely technical details: they shape how a reader assesses the reliability and scope of the reported findings. The registry record allows a careful account of the posted results, but not a complete reconstruction of every analytical choice.

19. 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 reported groups are small, and the arm-level estimates may be sensitive to individual observations. The registry record does not report a clinically meaningful difference for the task outcome, making it difficult to judge the practical importance of the estimated changes. The sponsor's overall primary analysis also lacks a reported effect estimate and confidence interval, which limits assessment of the magnitude and precision of that result.

A discrepancy is flagged between the sponsor analysis and the simple summary-statistic recalculations. The record does not explain the reason, preventing a full reconciliation. The sponsor's own analysis takes precedence, but the absence of further analytical detail restricts independent evaluation.

The registry record does not report the full statistical plan, the original sample-size rationale, adjustment for multiplicity, the analysis population, or methods for handling missing task data. It does not describe whether repeated task observations were modeled at the trial or block level, or how within-participant dependence was handled. The experimental-arm description is also incomplete. These reporting gaps should not be filled by assumptions.

The primary endpoint is a task measure over the first hours of the study. The record does not report whether the observed changes persisted, generalized beyond the task, or related to later clinical outcomes. The secondary measure captures signal coupling and is not equivalent to the behavioral endpoint. The adverse-event summaries provide counts but limited context about event types and collection. Together, these factors constrain conclusions to the posted outcomes and analyses.

20. Related Tutorials

21. Related Statistical Calculators

22. Sources

• ClinicalTrials.gov: Investigating the Causal Role of Prefrontal Control in Decision-making in Patients With Anhedonia — NCT05084924.
• Registry record last updated 2024-06-25; status last verified 2023-09.
• Results first posted 2024-06-25.
• 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.

23. Record Summary

The registry describes a completed, randomized, double-masked, parallel study of delta-beta tACS, theta-gamma tACS, and active-sham tACS in participants with major depressive disorder and elevated anhedonia symptoms. Enrollment was 35. The primary outcome measured change in the percentage of hard-task choices from baseline to the stimulation period. One secondary outcome measured change in coupling strength between low-frequency prefrontal and high-frequency posterior signals.

The sponsor-reported primary analysis was a three-group ANOVA with p = 0.049. Simple, unadjusted pairwise recalculations are also available from the posted summaries, but a discrepancy is flagged, so the sponsor's own analysis takes precedence. The registry record does not report an overall effect estimate or confidence interval for that ANOVA, nor does it provide the full analysis plan, original sample-size justification, multiplicity approach, or missing-data procedures.

No serious adverse events or deaths are reported in the arm summaries. These results describe the registry record and do not establish effects beyond the measured task and outcomes.