Smell in Covid-19 and Efficacy of Nasal Theophylline: Complete Statistical Analysis

1. Trial at a Glance

This page covers the registry results for Smell in Covid-19 and Efficacy of Nasal Theophylline (NCT04789499), a randomized phase 2 study. The primary outcome is UPSIT.

2. Clinical Question

3. Trial Design

This completed phase II trial compared theophylline with placebo in a randomized, double-blind, parallel-group design. The registry describes it as a single-site study. Enrollment was 51. The study start was 2021-03-15, and primary completion was 2021-12-30. The sponsor was Washington University School of Medicine.

Participants assigned to theophylline used a 400mg capsule diluted in 240 mL isotonic nasal saline lavage twice daily for six weeks. Participants assigned to placebo used a 500mg lactose capsule in the same volume and schedule. The registry record identifies both interventions as drugs. Randomization and double blinding are recorded, but the method used to generate or conceal the allocation sequence and the procedures used to maintain masking are not reported in the registry record.

The primary comparison concerned response at 6 weeks post-intervention from baseline. Response was defined using the Clinical Global Improvement Scale: participants choosing Very Much Improved, Much Improved, or Minimally Improved were classified as responders. The primary endpoint record says it could be assessed only for participants who completed that scale at 6 weeks. The result therefore describes this assessed group rather than automatically representing every enrolled participant.

The registry lists four secondary outcomes: the Questionnaire for Olfactory Dysfunction, the SF-36, the Olfactory Dysfunction Outcomes Rating, and UPSIT. The record lists five outcome measures in total, but no secondary analysis results are posted in the information available for this trial. The original sample-size rationale, allocation ratio planned by the investigators, analysis populations, interim analyses, and detailed statistical analysis plan 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

Analysis population for the primary outcome, as defined in the registry: This outcome measure could be assessed only from the participants that completed CGI-I at 6 weeks post intervention.

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 UPSIT, measured over Comparison of response rate at 6 weeks post-intervention from baseline between the 2 study groups.

The primary endpoint is titled UPSIT and is expressed as a percentage of participants. The registry describes UPSIT as a 0-40 measure on which higher scores indicate better results. However, the reported response definition is not a particular change in UPSIT score: it is the proportion in each group who self-reported at least slightly better smell on the Clinical Global Improvement Scale at 6 weeks post-intervention compared with baseline, before nasal irrigations.

The scale has seven response options ranging from Very Much Improved to Very Much Worsened. A participant was classified as a responder if the rating was Very Much Improved, Much Improved, or Minimally Improved. Participants with other ratings were not counted as responders under this definition. Thus, the endpoint converts a participant's global rating into a binary outcome: responder or not responder.

A responder proportion is useful because it gives a direct account of how many assessed participants crossed the stated improvement threshold. It is also straightforward to compare between randomized groups. It does not describe the size of improvement among responders, distinguish degrees of improvement within the responder category, or substitute for a comparison of measured UPSIT score changes. The registry record does not report a separate numerical UPSIT score result for this primary endpoint. Assessment was limited to participants who completed the Clinical Global Improvement Scale at 6 weeks post-intervention, which is important when interpreting the denominators.

7. Results Posted in the Registry

• Theophylline: 59.1% (13/22) versus 43.5% (10/23) on control; risk difference 15.6 percentage points (95% CI -13.2 to 44.5), risk ratio 1.36 (0.76–2.43); unadjusted two-proportion z-test p = 0.295.

8. Statistical Interpretation

9. Statistical Methods Explained

The primary result is a binary-outcome comparison. Each assessed participant is classified as either a responder or not a responder according to the prespecified description in the registry: a response of 1, 2, or 3 on the Clinical Global Improvement Scale counts as improvement. The group response percentages express the number classified as responders relative to the number assessed in that group. This kind of outcome differs from a continuous score comparison, which would describe the magnitude of scores or score changes.

What does the risk difference express?

The risk difference subtracts the response proportion in the placebo group from the response proportion in the theophylline group. A value above zero corresponds to a higher observed response proportion in the theophylline group, while a value below zero corresponds to a lower observed response proportion. The estimated risk difference is 0.156. It is an absolute contrast in proportions, rather than a relative measure. The confidence interval from -0.132 to 0.445 communicates uncertainty around that estimate and includes zero.

How do relative measures differ?

The relative risk compares the response proportions as a ratio; the odds ratio compares the odds of response. The posted relative risk is 1.359, with a confidence interval from 0.760 to 2.431. The posted odds ratio is 1.878, with a confidence interval from 0.575 to 6.136. A ratio of 1 represents no relative difference. Both intervals include 1, so these results also leave substantial uncertainty about the relative contrast. Odds ratios and relative risks answer related but distinct questions and should not be treated as interchangeable, especially when the event is not rare.

What does the p-value add?

The unadjusted p-value is 0.295. In general, a p-value describes how compatible the observed data are with a specified null model, under its assumptions. It is not the probability that the null hypothesis is true, the probability that the result occurred by chance, or a measure of clinical importance. The value should be considered alongside the point estimate, confidence interval, endpoint definition, and available sample. The registry record does not report a multiplicity adjustment or a detailed analysis plan for this test.

Why does the assessed population matter?

The primary endpoint was available only for participants who completed the Clinical Global Improvement Scale at 6 weeks post-intervention. That makes the outcome denominators essential context for the percentages. The registry record lists 22 assessed participants in the theophylline group and 23 in the placebo group, against enrollment of 51. It does not report the reasons for the difference or how missing endpoint assessments were handled. In general, if outcome availability is related to treatment assignment or participant outcomes, an analysis limited to observed assessments may not fully represent the randomized groups. Whether that concern applies cannot be determined from this record.

What does randomized, double-blind design contribute?

In general, random assignment is intended to make groups comparable in expectation, while masking is intended to limit effects of knowing the assigned intervention on participant experience, care, or outcome assessment. These design features can reduce some sources of bias, but they do not eliminate uncertainty from a small assessed sample, missing outcomes, measurement choices, or other limitations. The registry records randomization and double blinding; the detailed implementation and any checks on masking are not reported in the registry record.

The relative risk, odds ratio, risk difference, and p-value are all ways of analyzing the same underlying responder counts. They do not provide separate replications of the result. The primary endpoint's reliance on a subjective global rating also differs from an objective comparison of UPSIT score changes, even though the outcome title is UPSIT. No conclusion about a score-based endpoint can be drawn from these responder proportions alone.

10. Confidence Intervals and Precision

The confidence interval around the risk difference extends from -0.132 to 0.445, while the point estimate is 0.156. In general, a confidence interval gives a range of effect values reasonably compatible with the observed data under the statistical method and its assumptions. It is not a guarantee that the true effect lies in the interval, and it does not assign probabilities to individual values inside it.

For this outcome, the interval includes zero, corresponding to no absolute difference in response proportions, and also extends in both directions around that value. Thus, the estimated contrast is imprecise: the posted data do not pin down the magnitude or direction of the population difference. The interval is consistent with a potentially lower response proportion for theophylline as well as a potentially higher one. The point estimate should not be read in isolation from this uncertainty.

The relative measures tell a similar precision story. The relative risk interval, 0.760 to 2.431, includes 1, as does the odds ratio interval, 0.575 to 6.136. A ratio of 1 is the no-relative-difference value. The width of these intervals reflects the limited information in the assessed responder counts; it is not evidence that every value in the range is equally plausible.

In general, precision depends on the amount of information in the data, the outcome frequency, and the analysis method. Here, the registry reports 22 assessed participants in one group and 23 in the other for the primary comparison. The confidence intervals do not resolve whether the observed response contrast reflects a meaningful underlying difference. The result is best characterized as uncertain rather than as proof of equivalence or proof of a treatment effect.

11. 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 planning values are participants per arm, not total events. They are illustrations based on the observed effect and are not the trial's original sample-size design. The listed assumptions are a two-sided significance level of 0.05, equal allocation between the two arms, and a normal approximation; each planning scenario targets the stated fraction of the observed effect.

For planning at the full observed effect, the display gives 160 participants per arm for 80% power and 214 participants per arm for 90% power. For planning at 75% of the observed effect, it gives 285 participants per arm for 80% power and 382 participants per arm for 90% power. For planning at 50% of the observed effect, it gives 642 participants per arm for 80% power and 858 participants per arm for 90% power.

In general, power is the chance that a study will detect an effect of the specified size under the planning assumptions, if that effect is present. The 80% and 90% columns therefore describe different planning targets: the 90% target seeks a higher chance of detection and requires more participants per arm for each listed effect. Smaller effects are harder to distinguish from random variation, so planning for a smaller effect requires more participants. The graphical display represents the planning relationship; its values should be interpreted under the same assumptions rather than as outcomes observed in this trial.

These figures are not observed power. The registry record does not report the investigators' original sample-size assumptions or justification, so the planning illustrations cannot establish what the trial was designed to detect.

12. 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, p₀ and p₁ are the control and treatment proportions, and p̄ is their average.

Worked calculation for this trial

1. Control proportion p₀ = 0.435; treatment proportion p₁ = 0.591; average p̄ = 0.513.
2. First term: z_1−α/2 × √[2p̄(1−p̄)] = 1.96 × 0.7069 = 1.3854.
3. Second term: z_1−β × √[p₀(1−p₀) + p₁(1−p₁)] = 0.842 × 0.6982 = 0.5876.
4. n = (1.3854 + 0.5876)² / (0.1561)² = 159.7, rounded up to 160 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
p0 <- 0.434782608695652                          # control proportion
p1 <- 0.590909090909091                          # treatment proportion
n_per_arm <- function(power, frac = 1) {
  q1 <- p0 + (p1 - p0) * frac               # smaller effect = fraction of the observed difference
  pbar <- (p0 + q1) / 2
  z_b <- qnorm(power)
  ceiling(round((z_a * sqrt(2 * pbar * (1 - pbar)) + z_b * sqrt(p0 * (1 - p0) + q1 * (1 - q1)))^2 / (q1 - p0)^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 (unrounded n per arm)
cat("power.prop.test:", round(power.prop.test(p1 = p0, p2 = p1, power = 0.80)$n, 1), "\n")

# Simulation: how often does the z-test reject at the 80%-power sample size?
set.seed(4789499)
n    <- n_per_arm(0.80)
reps <- 20000
x0 <- rbinom(reps, n, p0); x1 <- rbinom(reps, n, p1)
ph <- (x0 + x1) / (2 * n)
z  <- ((x1 - x0) / n) / sqrt(ph * (1 - ph) * 2 / n)
cat(sprintf("Simulated power at n = %s per arm: %.3f (%d replicates)\n", format(n, big.mark = ","), mean(abs(z) > z_a, na.rm = TRUE), reps))

Output from running the code

100% of observed effect: n per arm = 160 (80% power), 214 (90% power)
 75% of observed effect: n per arm = 285 (80% power), 382 (90% power)
 50% of observed effect: n per arm = 642 (80% power), 858 (90% power)
power.prop.test: 159.7 
Simulated power at n = 160 per arm: 0.802 (20000 replicates)

The simulation draws many pairs of event counts at the planned sample size and counts how often the two-proportion z-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.

13. Multiplicity, Missing Data and Analysis Populations

The registry lists a primary endpoint and four secondary outcomes: UPSIT, change in the Questionnaire for Olfactory Dysfunction from baseline to 6 weeks post-intervention, SF-36, and ODOR. It reports five outcome measures in total. The record does not include posted analysis entries, and it does not provide results for the listed secondary outcomes in the information available for this trial.

When several outcomes or comparisons are tested, the chance of obtaining at least one apparently unusual result can increase. In general, multiplicity methods can address that issue by adjusting inference or by defining a hierarchy of outcomes. The registry record does not report whether any such method was planned or used, so no adjustment should be presumed. The primary comparison's p-value is listed as unadjusted.

The endpoint population is also restricted: the registry states that the primary outcome could be assessed only among participants who completed the Clinical Global Improvement Scale at 6 weeks post-intervention. The comparison uses denominators of 22 and 23. The registry record does not state how missing assessments were handled, why participants lacked assessments, or whether any imputation or sensitivity analysis was conducted. It also does not describe an intention-to-treat, per-protocol, or other named analysis population.

In general, missing outcome data can affect interpretation when the likelihood of having an observed result is related to assignment or to the outcome itself. The available record does not provide enough information to judge whether missingness altered the comparison. The original statistical analysis plan, interim analyses, adjustment strategy, and detailed analysis-population definitions are not reported in the registry record. These absences limit the ability to assess how fully the posted comparison reflects the trial's intended analysis.

14. 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 safety figures distinguish serious adverse events, other adverse events, and deaths. The registry lists zero serious adverse events in the theophylline group among 26 at risk and zero in the placebo group among 25 at risk. It lists 7 other adverse events among 26 at risk in the theophylline group and 11 among 25 at risk in the placebo group. Deaths are listed as zero in both groups.

These counts describe events recorded in the registry for the stated at-risk groups. They are not the primary efficacy outcome, and the group sizes used for these safety summaries differ from the primary outcome denominators. The record does not provide event descriptions for the other adverse events, severity detail, exposure-adjusted rates, or information about the attribution of those events to treatment. No serious-event terms are listed.

In general, a zero count in a trial does not establish that an event cannot occur; the ability to characterize uncommon harms depends on the amount and duration of observation. The posted figures should therefore be read as descriptive safety counts for this trial, not as a definitive comparison of safety between treatments.

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

The result answers a narrow question: among participants who completed a global improvement rating at 6 weeks post-intervention, what proportion met the registry's responder definition in each randomized group? The posted percentages are 59.0 for theophylline and 43.0 for placebo. The unadjusted risk difference is 0.156, but its confidence interval, -0.132 to 0.445, includes zero and spans differences in either direction. The unadjusted p-value is 0.295.

These results do not establish that the groups differ, and they do not establish that the groups are equivalent. The confidence intervals leave uncertainty about both the direction and size of a possible contrast. A non-significant p-value is not proof of no effect; conversely, the point estimate favoring theophylline is not by itself proof of benefit. The registry record supports a description of the observed responder proportions and their uncertainty, not a definitive claim about treatment effect.

The outcome definition matters. Although the primary endpoint is titled UPSIT and the registry notes that higher UPSIT scores indicate better results, response was defined by a self-reported Clinical Global Improvement Scale rating. The comparison therefore does not report a difference in mean UPSIT scores or in UPSIT score change. It also does not describe how much improvement occurred among responders.

The assessment was available only for those who completed the scale at 6 weeks, and the registry does not explain the gap between enrollment and the outcome denominators. The posted result may not represent every enrolled participant. The four listed secondary outcomes do not have analysis results in the record. These limitations mean the primary result should be interpreted as one endpoint-specific, follow-up-specific comparison with incomplete information about missing assessments and other analyses.

No medical decision should be based on these statistics alone. The registry results characterize the trial data; they do not settle how an individual should be treated or establish broader effects beyond the population and outcome assessed.

16. Why This Trial Matters Statistically

This trial illustrates several statistical issues that arise when a clinical outcome is converted into a responder proportion. A responder definition can make a result easy to communicate, but it depends on a threshold: here, three of the seven global improvement responses count as improvement. Participants just inside and outside that threshold are classified differently even if their experiences are similar. The proportion also does not retain information about the degree of improvement within the responder group.

The endpoint label and operational definition should be read together. UPSIT is described as a 0-40 measure with higher scores indicating better results, while the reported primary response uses a participant's global rating. This distinction matters because a reader might otherwise interpret the result as a change in measured UPSIT score. Clear alignment between an endpoint name, its measurement instrument, and its analysis is central to interpreting clinical evidence.

The confidence intervals are as important as the point estimate. The risk difference estimate is 0.156, yet the interval ranges from -0.132 to 0.445. That breadth shows why the observed direction alone is not enough to characterize the evidence. The relative risk and odds ratio intervals likewise include their null values. Different effect measures describe the same underlying data from different perspectives; none removes the uncertainty.

Finally, the denominator restriction and missing information about assessment completion matter for interpretation. The registry states that the primary outcome was assessed only among participants completing the scale, but does not report why the endpoint denominators are smaller than enrollment or how missing outcomes were handled. Statistical conclusions depend not only on the test and effect measure, but also on who contributes outcome data and how the outcome is defined.

17. 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.
• Participant counts were reconstructed from posted percentages and arm sizes, so small rounding differences are possible.

Several trial-specific limitations are evident from the registry record. The primary responder outcome uses a self-reported global improvement rating rather than reporting a numerical UPSIT score change, despite the endpoint title being UPSIT. The result therefore cannot be read as a direct comparison of UPSIT score levels or changes.

The endpoint was assessed only for participants who completed the Clinical Global Improvement Scale at 6 weeks post-intervention. The primary comparison includes 22 participants in the theophylline group and 23 in the placebo group, while enrollment was 51. The registry record does not explain the difference, give reasons for missing assessments, or describe any missing-data approach. This limits assessment of how representative the endpoint results are of the enrolled randomized groups.

The primary comparison has wide confidence intervals, including values corresponding to no difference and effects in either direction. The registry also lists four secondary outcomes but does not provide their analysis results in the available record. No analysis entries are posted. The record does not report the original sample-size justification, detailed statistical analysis plan, interim analyses, multiplicity approach, or definitions of analysis populations.

Although randomization and double blinding are recorded, details of sequence generation, allocation concealment, and masking procedures are not reported in the registry record. The record identifies the study as single-site and phase II, but does not provide additional information that would support judging how broadly the findings apply. These limitations do not erase the posted results; they define the uncertainty around their interpretation.

18. Related Tutorials

19. Related Statistical Calculators

20. Sources

• ClinicalTrials.gov: Smell in Covid-19 and Efficacy of Nasal Theophylline — NCT04789499.
• Registry record last updated 2023-05-24; status last verified 2023-04.
• Results first posted 2023-05-24.
• 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.

21. Record Summary

NCT04789499 is a completed phase II, randomized, double-blind, parallel-group, single-site trial sponsored by Washington University School of Medicine. Enrollment was 51. The study compared twice-daily intranasal theophylline with a lactose placebo, each diluted in 240 mL isotonic nasal saline lavage for six weeks.

The primary endpoint was a responder proportion at 6 weeks post-intervention based on a self-reported Clinical Global Improvement Scale rating. The posted response percentages were 59.0 for theophylline and 43.0 for placebo, based on 13 of 22 and 10 of 23 assessed participants, respectively. The unadjusted risk difference was 0.156, with a 95% confidence interval from -0.132 to 0.445 and an unadjusted p-value of 0.295.

The result is uncertain and applies to the defined responder outcome among participants with a completed assessment. The registry record does not report the original sample-size rationale, detailed missing-data methods, or results for the listed secondary outcomes.