Selenium and Immune Function: Complete Statistical Analysis

1. Trial at a Glance

This page covers the registry results for Selenium and Immune Function (NCT00279812), a randomized na study. The primary outcome is Cellular and Humoral Immune Response.

2. Clinical Question

3. Trial Design

The registry describes Selenium and Immune Function as a randomized, double-masked, parallel-group study. It lists enrollment of 133 and a completed status. The study start is recorded as 2005-04, with primary completion in 2008-08. The registry lists the phase as NA.

Six conditions are described: placebo; three doses of selenium-enriched yeast; control onion; and enriched onion. The yeast interventions are described as containing 60% selenomethionine, at 50ug/d, 100ug/d, and 200ug/d. Control onion meals were equivalent to 4ug/d Se, while enriched onion meals were equivalent to 50ug/d Se; both were served 3 meals/wk. The registry classifies these as dietary supplements, except placebo, which is listed as another intervention type.

The posted primary outcome is total glutathione peroxidase 1 activity in platelets after 12 weeks. It is a continuous measure summarized as a mean and standard deviation. The record says the outcome data came from 117 subjects because of an incomplete time-course experiment for one subject and baseline plasma selenium concentration for one subject. It does not report the full analysis population definition or a detailed account of how missing measurements were handled.

The reported sponsor analysis uses ANOVA across the six groups. The record does not report the trial's original sample-size assumptions or a detailed statistical analysis plan. The posted pairwise recalculations compare each non-placebo group with placebo and should not be mistaken for the sponsor's overall analysis.

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: Data from 117 subjects because of incomplete time-course experiment for one subject and baseline plasma selenium concentration for one subject

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.

Only the first 6 of 7 groups are shown.

6. Primary Endpoint

The registry lists the primary outcome as Cellular and Humoral Immune Response, measured over 12 weeks.

The primary endpoint is total glutathione peroxidase 1 activity in platelets, measured after 12 weeks of supplementation and expressed in umol/min. The registry identifies it as part of the cellular and humoral immune response. The posted group summaries report means and standard deviations, so the outcome is treated as a continuous measurement.

In general, a continuous biomarker can describe the level of a biological process on a scale that supports comparison of group averages. Glutathione peroxidase 1 activity is used as a functional marker related to selenium status and immune function, consistent with the study aim to examine dose and form of selenium and identify functional markers. This does not make the endpoint a direct measure of overall health or clinical benefit; it is a specific laboratory outcome.

The registry records the primary measurement at 12 weeks. It does not report a clinically important threshold for this measure, nor does it explain how a change in platelet activity should be translated into a patient-important outcome. The results therefore describe the measured biomarker, not a demonstrated change in symptoms, disease risk, or other clinical outcome.

7. Results Posted in the Registry

• 50ug Selenium Enriched Yeast: mean 0.34 versus 0.29 on control; difference 0.05 (95% CI -0.03 to 0.13), standardized effect d = 0.41; z-test p = 0.221.
• 100ug Selenium Enriched Yeast: mean 0.34 versus 0.29 on control; difference 0.05 (95% CI -0.02 to 0.12), standardized effect d = 0.42; z-test p = 0.168.
• 200ug Selenium Enriched Yeast: mean 0.29 versus 0.29 on control; difference 0.00 (95% CI -0.05 to 0.05), standardized effect d = 0.00; z-test p = 1.000.
• Control Onion: mean 0.35 versus 0.29 on control; difference 0.06 (95% CI -0.02 to 0.14), standardized effect d = 0.53; z-test p = 0.130.

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.

Selenium Status

• 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:
  • 50ug Selenium Enriched Yeast: mean 118.30 versus 93.70 on control; difference 24.60 (95% CI 15.17 to 34.03), standardized effect d = 1.64; z-test p = <0.001.
  • 100ug Selenium Enriched Yeast: mean 152.00 versus 93.70 on control; difference 58.30 (95% CI 45.64 to 70.96), standardized effect d = 2.79; z-test p = <0.001.
• Interpretation:
  • The 95% confidence interval excludes no difference, so the mean is statistically higher for 50ug Selenium Enriched Yeast than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.
  • The 95% confidence interval excludes no difference, so the mean is statistically higher for 100ug Selenium Enriched Yeast than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.

Selenoproteins and Se-biomarkers

• 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:
  • 50ug Selenium Enriched Yeast: mean 6.17 versus 4.96 on control; difference 1.21 (95% CI 0.67 to 1.75), standardized effect d = 1.41; z-test p = <0.001.
  • 100ug Selenium Enriched Yeast: mean 6.73 versus 4.96 on control; difference 1.77 (95% CI 1.20 to 2.34), standardized effect d = 1.88; z-test p = <0.001.
• Interpretation:
  • The 95% confidence interval excludes no difference, so the mean is statistically higher for 50ug Selenium Enriched Yeast than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.
  • The 95% confidence interval excludes no difference, so the mean is statistically higher for 100ug Selenium Enriched Yeast than for the control. This is an unadjusted, secondary comparison, so it is suggestive rather than confirmatory.

The secondary outcomes are Selenium Status and Selenoproteins and Se-biomarkers, both measured at 10 weeks. Across the posted group summaries, selenium-enriched yeast groups have higher mean values than placebo on both measures, and the enriched onion group is also higher than the control onion group in the summaries. The control onion and placebo means are similar on these secondary measures. The pattern is therefore more consistent across these biomarkers than the primary platelet activity pattern, which does not rise consistently across the yeast doses.

The pairwise recalculations for some yeast comparisons against placebo show intervals above zero and unadjusted p-values reported as <0.001. These are secondary biomarker findings, not additional primary-outcome results. The sponsor's reported secondary analyses are mixed models across the six groups, with p reported as 0.05 for each outcome. The record does not provide the model details or a clear account of how those overall results relate to the recalculated pairwise contrasts.

These secondary outcomes help describe biological measures related to selenium status, but they do not establish that the primary immune-function endpoint changed in the same way. The difference in pattern across endpoints emphasizes that a marker of selenium exposure or status is not the same as a measure of immune function. Nor do these biomarkers establish a clinical outcome.

Unadjusted secondary results carry less weight because multiple outcomes and comparisons create more opportunities for apparently notable findings. The registry record does not report a multiplicity plan. The secondary findings should therefore be interpreted as supportive, exploratory evidence about the measured biomarkers rather than as independent confirmation of a clinical effect.

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

• Selenium Status
• Selenoproteins and Se-biomarkers

9. Statistical Interpretation

10. Statistical Methods Explained

What do the group summaries describe?

A mean is the arithmetic average of the observed measurements in a group. The standard deviation describes how spread out individual measurements are around that group's mean. In general, a mean gives a compact description of the group's central level, while the standard deviation indicates variability among the observations. Neither quantity alone shows the uncertainty around a comparison between groups.

The registry presents group sizes, means, and standard deviations for the primary endpoint. The recalculated comparisons use the placebo group as the reference and express each estimate as a mean difference. A positive difference means the comparison group's observed mean is higher than the placebo mean on this scale; a difference of zero means the group means coincide in the reported summary. These are unadjusted recalculations from the posted summaries, not the sponsor's own pairwise analysis.

What is the role of the ANOVA?

The sponsor reports an ANOVA across the six primary-outcome groups, with p <0.05. In general, an analysis of variance tests whether the data are compatible with equal group means across the set of groups. An overall result does not identify which particular groups differ, or the size of any pairwise difference. It also does not answer whether one dose is better than another unless those specific contrasts are analyzed and reported.

The pairwise recalculations ask a narrower question: how does each non-placebo group's mean compare with the placebo mean? Because five such comparisons are presented, they should not be treated as if they were the sponsor's single overall ANOVA. The registry does not report a detailed account of the planned contrasts, adjustment for multiple comparisons, or the analysis choices behind the sponsor's result. The discrepancy between the overall reported result and recalculated pairwise results means the sponsor's analysis takes precedence.

How should a p-value be read?

In general, a p-value describes how unusual the observed data, or data more extreme, would be under a specified null model. It is not the probability that the null hypothesis is true, and it does not measure the size or clinical importance of a difference. The pairwise p-values in the posted recalculations are explicitly unadjusted. The reported p <0.05 for the ANOVA is an overall result, not evidence that every intervention differs from placebo.

Why use a confidence interval?

In general, a confidence interval communicates statistical uncertainty around an estimate. It gives a range of values compatible with the data and the method used to construct the interval. For the mean differences, intervals that include zero indicate that no difference is among the values compatible with the interval, but they do not prove equality. The intervals also show which magnitudes of difference remain compatible with the data. Their width depends on the variability and the amount of information in the compared groups.

What does a recalculation represent?

The posted pairwise estimates are simple unadjusted recalculations from the group summaries. They do not incorporate any adjustment that might have been used in the sponsor's analysis, and the record does not report whether the sponsor's analysis adjusted for baseline values or other covariates. The recalculations are useful for describing the raw group-mean contrasts and their approximate uncertainty, but they cannot replace the trial's own analysis. As the record flags a discrepancy, the sponsor's analysis takes precedence when describing the trial's reported inferential result.

What assumptions underlie the continuous-outcome comparisons?

In general, a mean-based comparison is most straightforward to interpret when the measurement is meaningful on a continuous scale and the statistical method adequately represents the outcome's variability. These recalculations use the posted means, standard deviations, and group sizes. The registry does not report diagnostic checks for distributional assumptions, handling of outliers, a transformation, or an adjusted model for the primary outcome. Those details should not be inferred from the summary statistics.

The endpoint data were reported for 117 subjects, and the registry gives two reasons related to missing information. It does not report whether the analysis used complete cases, imputation, or another missing-data approach. In general, missing-data methods matter because results can depend on whether missingness is related to observed or unobserved outcomes. The record does not give enough detail to assess that issue beyond the stated reasons.

What does the overall result not establish?

An overall group test and a set of pairwise comparisons answer different statistical questions. Neither by itself establishes clinical importance, a dose-response pattern, or generalizability to populations beyond those studied. The registry record does not report the trial's original statistical plan, sample-size justification, or multiplicity strategy. Those absences limit how fully the reported p-values can be contextualized.

11. Confidence Intervals and Precision

The pairwise confidence intervals quantify uncertainty around the recalculated mean differences between each intervention group and placebo. For the primary outcome, every interval includes zero. This means that the posted summaries are compatible with no mean difference, as well as with differences in either direction across the ranges shown. It does not demonstrate that the groups are identical.

The interval widths also matter. Wider intervals indicate less precision: the data are compatible with a broader range of mean differences. The primary outcome groups are modest in size, and the standard deviations vary across them. In general, smaller groups and more variable measurements tend to produce less precise estimates. The confidence intervals should therefore be read alongside the group means, spreads, and numbers contributing to each comparison, rather than as a binary pass-or-fail rule.

The posted recalculations are unadjusted. Their intervals do not account for any multiplicity adjustment, and the registry does not report whether the sponsor's ANOVA used additional modeling choices. The overall ANOVA result, p <0.05, is not itself a confidence interval for a particular group difference. It supports a different inferential question from the pairwise intervals.

Precision is not the same as clinical relevance. The registry does not report a clinically important difference for platelet glutathione peroxidase 1 activity, so the intervals cannot be judged against a stated clinical threshold. They describe uncertainty in the biomarker contrast, not uncertainty about a patient-important benefit.

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 figures are planning illustrations based on the observed effect, not the trial's original design assumptions. For these continuous outcomes, the figures are participants per arm, not total events. The stated planning assumptions are a two-sided significance level of 0.05, equal allocation between the two arms, and a normal approximation. The target effect is the stated fraction of the observed effect.

The 80% power column gives the planned number of participants per arm under those assumptions for an 80% chance of detecting the target effect if it is present. The 90% power column gives the corresponding number per arm for a 90% chance. In general, higher target power requires more participants when other assumptions are held constant.

Rows for 100% of the observed effect use the recalculated effect as the planning target; rows for 75% and 50% use smaller targets. Smaller effects need more participants to detect, which is why the planning figures increase as the target fraction decreases. The illustrations are specific to the observed effect and variability in the comparison. They should not be read as the sample sizes that the completed trial was designed to recruit, and they are not observed power.

These figures rely on simplified assumptions and do not account for features that may affect a real trial's design, such as multiple comparisons or missing outcomes. The registry record does not report the original sample-size justification or the assumptions used in the trial's design.

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 = 0.050; pooled standard deviation σ = 0.121.
2. n = 2 × (1.96 + 0.842)² × 0.121² / 0.050² = 92.1, rounded up to 93 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 <- 0.05                      # observed difference in means
sigma <- 0.121140414395857                     # 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(279812)
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 = 93 (80% power), 124 (90% power)
 75% of observed effect: n per arm = 164 (80% power), 220 (90% power)
 50% of observed effect: n per arm = 369 (80% power), 494 (90% power)
power.t.test: 93.1 
Simulated power at n = 93 per arm: 0.799 (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

For the primary outcome, the sponsor reports an ANOVA across all six groups with p <0.05. The posted recalculations instead compare each of the five non-placebo groups with placebo, using group means and standard deviations. Those unadjusted pairwise comparisons have confidence intervals that include zero and p-values ranging from 0.081 to 1.0.

These results are not directly interchangeable. The sponsor's ANOVA tests an overall question about the set of group means; it does not specify which pairwise comparison accounts for the reported result. The recalculations describe individual mean differences against placebo and use no reported multiplicity adjustment. The registry does not provide sufficient details to reconstruct the sponsor's analysis or explain the discrepancy.

Because the discrepancy is flagged in the record, the sponsor's own analysis takes precedence when reporting the trial's inferential result. The recalculations remain useful as descriptive, unadjusted comparisons from posted summary statistics, but they should not be presented as a replacement for the sponsor's analysis.

15. Multiplicity, Missing Data and Analysis Populations

The trial includes multiple intervention groups, a primary endpoint, and secondary outcomes. The sponsor's primary analysis is an overall ANOVA, while the posted recalculations include individual comparisons with placebo. Considering multiple comparisons matters because, in general, testing more hypotheses increases the chance of obtaining an apparently unusual result by chance when all null hypotheses hold.

The registry record does not report whether the original analysis plan specified a hierarchy of outcomes, planned contrasts, or a method for adjusting for multiplicity. The recalculated pairwise p-values are unadjusted. The reported overall ANOVA is a separate analysis and should not be treated as a correction for the multiple pairwise comparisons. For the secondary endpoints, the record reports mixed models across the six groups but does not describe their specification or how the multiple outcomes were handled.

The primary outcome record states that data from 117 subjects were used, citing an incomplete time-course experiment for one subject and a missing baseline plasma selenium concentration for one subject. It does not explain which observations were omitted for each analysis, whether those participants were excluded from all outcomes, or how incomplete measurements were handled. The analysis population definition and missing-data method are not reported in the registry record.

In general, missing-data approaches rely on assumptions about why outcomes are missing. Without a stated method or further information, it is not possible to determine whether the reported analysis could be affected by those assumptions. The record also does not report an interim-analysis plan or the trial's original statistical plan. These missing details limit assessment of selective analysis choices and the context for the reported results.

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 no serious adverse events in any of the six groups, and deaths are recorded as 0 in every group. The listed numbers at risk differ by group: the record gives 21 for placebo, 26 for 50ug selenium-enriched yeast, 24 for 100ug selenium-enriched yeast, 23 for 200ug selenium-enriched yeast, 20 for control onion, and 19 for enriched onion.

Other adverse events are recorded as 0 for placebo, 1 for each of the 50ug and 100ug yeast groups, 0 for the 200ug yeast group, 2 for control onion, and 1 for enriched onion. These are the registry's recorded counts; the record does not provide serious-event terms or further details about event type, timing, or attribution.

Zero recorded serious events or deaths in these groups does not establish that no harm is possible. The safety counts are descriptive, and the registry record does not report enough detail to characterize uncommon events or compare safety with precision.

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

The posted data describe measured biomarker outcomes in a randomized study of healthy participants assigned to selenium-related conditions or placebo. For the primary platelet activity endpoint, the recalculated group means and pairwise comparisons do not show clear evidence of a difference from placebo. The sponsor's overall ANOVA reports p <0.05, and the registry flags a discrepancy; the sponsor's analysis takes precedence over the recalculations.

This finding does not establish that all groups are the same, nor does an overall difference establish that every intervention differs from placebo. The pairwise confidence intervals remain compatible with zero, and the record gives no clinically important threshold for platelet glutathione peroxidase 1 activity. The primary endpoint is a laboratory measure, so the results do not demonstrate a change in symptoms, illness, or another patient-important outcome.

The secondary biomarkers show a more consistent pattern in the selenium-enriched yeast groups than the primary endpoint does. Those findings can inform interpretation of selenium status and related biomarkers, but they are not interchangeable with immune function or clinical outcomes. Their unadjusted nature and the unreported multiplicity strategy also call for caution.

The registry record does not establish the original sample-size justification, detailed analysis plan, or handling of missing data. The primary outcome data came from 117 subjects, with stated issues involving a time-course experiment and baseline plasma selenium. These limitations affect how confidently the estimates can be generalized or interpreted beyond the recorded summaries.

The results should be understood as evidence about the outcomes measured in this particular trial, not as a basis for medical advice. They do not establish effects in populations other than the healthy participants described by the registry, or effects on outcomes that were not reported.

18. Why This Trial Matters Statistically

This trial is statistically informative because it combines several intervention forms and doses with both an overall group analysis and posted comparisons against placebo. Those analyses illustrate why an overall test and individual contrasts can lead to different readings: an overall result can indicate variation somewhere across the groups without identifying which comparison drives it.

It also illustrates the importance of separating biomarker domains. The secondary selenium-status measures show a pattern across the selenium-enriched conditions, while the primary platelet activity measure has a less consistent pattern across yeast doses. A change in a marker associated with exposure or status should not automatically be interpreted as a change in a functional endpoint, much less as a clinical benefit.

The record also highlights the role of precision and multiplicity. The primary pairwise intervals include zero, and the recalculated p-values are unadjusted. The sponsor's reported overall analysis has a different scope, and the discrepancy means it should take precedence for the sponsor's inferential result. Without the original analysis plan and multiplicity details, readers cannot fully reconcile these analyses.

Finally, the sample-size illustrations show how planning depends on the effect targeted and the assumptions used, but they do not recover the trial's original design rationale. These distinctions matter whenever trial summaries are interpreted from registry results: estimates, uncertainty, analysis scope, and clinical meaning are related but separate questions.

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 registry describes a healthy population but does not report further participant characteristics in this record. This limits assessment of how well the findings might apply to other populations. The study compared several selenium sources and doses, so a result for one condition should not automatically be generalized to another formulation or dose.

The primary outcome data are reported for 117 subjects, with the record citing an incomplete time-course experiment for one subject and missing baseline plasma selenium concentration for one subject. The registry does not describe the full analysis population, the disposition of missing measurements across outcomes, or the method used to handle them. It also does not report detailed outcome measurement procedures beyond the endpoint description and timeframe.

The sponsor reports an overall ANOVA result that differs in appearance from the recalculated unadjusted pairwise results. The record does not include sufficient model details or a full statistical analysis plan to explain the discrepancy. Multiplicity handling, baseline adjustment, interim analyses, and the original sample-size justification are not reported in the registry record.

The primary endpoint is a laboratory biomarker, and the registry does not report a clinically important threshold. The study therefore does not establish the clinical importance of the observed mean differences. The safety record includes no serious events or deaths, but the event details are limited and do not allow firm conclusions about rare harms.

20. Related Tutorials

21. Related Statistical Calculators

22. Sources

• ClinicalTrials.gov: Selenium and Immune Function — NCT00279812.
• Registry record last updated 2018-07-16; status last verified 2017-09.
• Results first posted 2018-07-16.
• 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

Selenium and Immune Function is recorded as a completed, randomized, double-masked, parallel-group trial with enrollment of 133. It compared placebo with three selenium-enriched yeast doses, control onion, and enriched onion in healthy participants. The primary endpoint was total glutathione peroxidase 1 activity in platelets after 12 weeks.

The sponsor reported an overall ANOVA result of p <0.05. Posted unadjusted pairwise recalculations versus placebo have confidence intervals that include zero, and the record flags a discrepancy; the sponsor's analysis takes precedence. Secondary outcomes were selenium status and selenoproteins and Se-biomarkers at 10 weeks. The registry does not report the original sample-size justification, detailed analysis plan, or multiplicity and missing-data methods.