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Healthy Subjects Phase 3 Continuous Endpoint NCT00708123

In Situ Caries of Fluoride Toothpastes: Complete Statistical Analysis

This record examines a randomized, double-masked crossover study of fluoride toothpastes using an in situ enamel caries model. The primary outcome was percentage surface microhardness recovery over baseline to 14 days.

Trial start: 2007-11  ·  Enrollment: 60  ·  Sponsor: GlaxoSmithKline
Scope of this record

This page provides an independent statistical analysis and educational interpretation of publicly reported results. ClinicalTrials.gov provides the official trial registry record. Numerical trial results on this page are restricted to the results posted in the ClinicalTrials.gov record; calculations added by Clinical Biostats are labelled as such.

1. Trial at a Glance

This page covers the registry results for In Situ Caries of Fluoride Toothpastes (NCT00708123), a randomized phase 3 study. The primary outcome is Mean Percentage Surface Microhardness Recovery (%SMHR) of Enamel Specimens Exposed to NaF/Carbopol Toothpaste (1400 ppmF), NaF Toothpaste (1350ppmF) Compared to NaMFP/NaF Toothpaste (1450ppmF).

60
Enrollment
Phase 3
3
Arms
In the primary outcome
1.69
Mean difference
NaF Toothpaste (1350 Ppm F)
-3.11–6.49
95% CI
Two-sided
FeatureRegistry value
ClinicalTrials.gov identifierNCT00708123
PhasePhase 3
ConditionHealthy Subjects, Partial Denture Wearers, Caries
AllocationRandomized
Design modelCrossover
MaskingDouble
Enrollment60
Arms in the primary outcome3
Lead sponsorGlaxoSmithKline
Results postedYes

2. Clinical Question

Population

Healthy subjects who wore partial dentures and had natural teeth were studied. The primary efficacy population was defined as randomized participants with at least one post-baseline efficacy assessment; missing data were not imputed. The registry reports an enrollment of 60, while efficacy group counts vary by treatment because of participant dropouts.

Intervention

The study compared fluoride toothpastes with different formulations and concentrations, including sodium fluoride (NaF), NaF with carbopol, and a combination of sodium monofluorophosphate and NaF. A fluoride-free placebo toothpaste was also included. Participants were instructed to brush twice daily for one timed minute after removing their partial denture.

Comparator

The primary outcome description focuses on NaF/Carbopol toothpaste (1400 ppm F) and NaF toothpaste (1350 ppm F) compared with NaMFP/NaF toothpaste (1450 ppm F). The broader set of treatments also included NaF toothpaste (250 ppm F) and placebo toothpaste (0 ppm F).

Primary question

Over baseline to 14 days, how did mean percentage surface microhardness recovery (%SMHR) of enamel specimens differ across the specified fluoride toothpaste formulations? The measure was obtained from enamel specimens exposed in an in situ caries model, rather than from a clinical caries outcome in participants.

3. Trial Design

The registry describes a completed Phase 3 randomized, double-masked crossover study sponsored by GlaxoSmithKline. The study began in 2007-11 and reached primary completion in 2008-04. It enrolled 60 participants. Its stated purpose was to evaluate the effect of fluoride dentifrices on enamel with artificial caries lesions using an in situ model.

In a crossover design, participants receive more than one treatment over the course of a study, so comparisons can use information from the same participant under different treatment conditions. This can reduce the influence of stable differences between participants. That general feature does not establish how this particular study handled treatment periods, sequence, or possible carryover. Those protocol details are not reported in the registry record.

The treatment descriptions specify twice-daily brushing with a full ribbon of toothpaste for one timed minute, after removing the partial denture. The formulations included NaF/Carbopol toothpaste (1400 ppm F), NaF toothpaste (1350 ppm F), NaMFP/NaF toothpaste (1450 ppm F), NaF toothpaste (250 ppm F), and fluoride-free placebo toothpaste (0 ppm F). The record describes the primary endpoint as comparing three formulations, while secondary efficacy information also covers the broader set of treatments.

The primary outcome analysis population was defined as all randomized participants who had at least one post-baseline efficacy assessment. Missing data were not imputed, and the record notes differences in treatment-group counts because of dropouts. The original treatment sequence, period structure, washout arrangements, and trial-specific sample-size justification 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.

Before assignment: Two days, prior to each treatment visit ,participants used a fluoride- free dentifrice twice daily to avoid any carry over effect.

Period 1

StageSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
STARTED1212121212
COMPLETED1211111112
NOT COMPLETED01110

Reasons for not completing this period:

ReasonSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
Protocol Violation01010
Adverse Event00100

Period 2

StageSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
STARTED1112121111
COMPLETED1012121011
NOT COMPLETED10010

Reasons for not completing this period:

ReasonSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
Protocol Violation00010
Other Reason10000

Period 3

StageSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
STARTED1210111210
COMPLETED1210101110
NOT COMPLETED00110

Reasons for not completing this period:

ReasonSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
Lost to Follow-up00110

Period 4

StageSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
STARTED1012101011
COMPLETED1011101011
NOT COMPLETED01000

Reasons for not completing this period:

ReasonSodium Fluoride (NaF) Toothpaste[1350 Parts Per Million(Ppm)F]NaF/Carbopol Toothpaste (1400 Ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
Protocol Violation01000

Analysis population for the primary outcome, as defined in the registry: Intent to treat population: All randomized participants who had at least one post baseline efficacy assessment. Missing data was not imputed. Due to subject drop outs, there are differences in the "n" per treatment group.

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.

CharacteristicUnitAll Study Participants
Participants analysed60
Age Continuousyears67.31 (SD 9.166)
Sex: Female, Male: FemaleParticipants33
Sex: Female, Male: MaleParticipants27
Region of Enrollment: United Statesparticipants60

6. Primary Endpoint

The registry lists the primary outcome as Mean Percentage Surface Microhardness Recovery (%SMHR) of Enamel Specimens Exposed to NaF/Carbopol Toothpaste (1400 ppmF), NaF Toothpaste (1350ppmF) Compared to NaMFP/NaF Toothpaste (1450ppmF), measured over Baseline to 14 days.

ElementRegistry entry
Result typeLeast Squares Mean
Measure of spreadStandard Error
Unit%SMHR
Endpoint classificationContinuous Endpoint

The primary endpoint was mean percentage surface microhardness recovery (%SMHR) of enamel specimens from baseline to 14 days. Surface microhardness is a laboratory measure of how resistant enamel is to indentation. In the study description, the length of indentations was measured with a Wilson 2100 Hardness tester. Longer indentations relative to baseline indicate softening, while shorter indentations indicate rehardening of the enamel surface.

The registry states that %SMHR was calculated from specimen indentation values at baseline, after intra-oral exposure, and after in-vitro demineralization. The endpoint is therefore intended to reflect mineralization status after exposure to the toothpaste regimen in an in situ model. It provides a continuous measure, allowing group means and differences in means to describe the observed enamel response.

In general, a continuous surrogate measure can be useful for detecting and quantifying differences in a biological process, but it is not the same as observing caries development or other clinical outcomes in participants. The registry record reports specimen-based %SMHR, not a clinical caries event endpoint. The numerical result should therefore be interpreted as a comparison of this enamel measure over the stated timeframe, not as a direct estimate of a clinical benefit.

7. Results Posted in the Registry

Mean difference: NaF Toothpaste (1350 Ppm F)

1.69

95% CI: -3.11 to 6.49

Recalculated from the posted summary statistics

Primary outcome by arm, with 95% confidence intervals
Primary outcome by arm with 95% confidence intervals25.027.930.933.936.939.9NaF/Carbopol Toothpaste (1400… (n=54)NaF Toothpaste (1350 Ppm F) (n=53)NaMFP/NaF Toothpaste (1450 Pp… (n=54)%SMHR (95% CI)
Treatment effect versus the control arm
Forest plot of mean difference vs control (95% CI)NaF Toothpaste (1350 Ppm F)NaMFP/NaF Toothpaste (1450 Pp…-9.96-6.42-2.890.644.177.70mean difference vs control (95% CI)

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.

Secondary outcomeTypeEffect measureEstimate (95% CI)Planning size, 80% power
Mean %SMHR of Enamel Specimens Exposed to NaF/Carbopol Toothpaste (1400 ppmF), NaF Toothpaste (1350ppmF), NaMFP/NaF Toothpaste (1450ppmF), NaF Toothpaste (250ppmF) and Placebo Toothpaste (0ppmF)Continuous EndpointMean difference11.49 (6.71 to 16.27)20 per arm
Adjusted Mean Change From Baseline in Enamel Fluoride UptakeContinuous EndpointMean difference847.55 (649.27 to 1,046)7 per arm

Mean %SMHR of Enamel Specimens Exposed to NaF/Carbopol Toothpaste (1400 ppmF), NaF Toothpaste (1350ppmF), NaMFP/NaF Toothpaste (1450ppmF), NaF Toothpaste (250ppmF) and Placebo Toothpaste (0ppmF)

Treatment effect versus the control arm
Forest plot of mean difference vs control (95% CI)NaF/Carbopol Toothpaste (1400…NaF Toothpaste (1350ppm F)-1.442.736.91111519mean difference vs control (95% CI)

Adjusted Mean Change From Baseline in Enamel Fluoride Uptake

Treatment effect versus the control arm
Forest plot of mean difference vs control (95% CI)NaF/ Carbopol Toothpaste (140…NaF Toothpaste (1350ppm F)-881674216759301,184mean difference vs control (95% CI)

The secondary outcomes extend the enamel assessment beyond the primary comparison. One reports %SMHR across a broader set of toothpaste formulations, including lower-fluoride and placebo products; another measures adjusted mean change from baseline in enamel fluoride uptake. Across these outcomes, the broad pattern is that the fluoride formulations are separated from the fluoride-free placebo on the posted enamel measures, while the magnitude and ordering depend on the specific formulation and endpoint.

This pattern is directionally compatible with the study's focus on fluoride delivery and enamel response, but it does not independently resolve the primary-result discrepancy. The primary endpoint's reported ANOVA comparisons concern selected formulations, while secondary analyses use different treatment contrasts and a separate outcome. They should be considered as related but distinct evidence, not as interchangeable confirmations of the same estimate.

Some secondary comparisons are reported by the sponsor with adjusted ANOVA results, while the recalculations are unadjusted. Those results should be interpreted in light of the same distinction between model-based analyses and summary-statistic calculations. The registry record does not report a multiplicity adjustment or a prespecified hierarchy for secondary testing. When multiple outcomes or contrasts are examined, the chance of at least one apparently notable result can increase. General statistical practice therefore gives unadjusted secondary findings less weight than a prespecified primary analysis, particularly when multiplicity handling is not reported.

The secondary outcomes help describe the broader pattern of enamel response and fluoride uptake in this model. They do not establish clinical caries outcomes, and the registry record does not provide a stated threshold for clinical importance for either measure.

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

9. Statistical Interpretation

Clinical Biostats interpretation

The posted primary group means were 33.54 for NaF/Carbopol toothpaste (1400 ppm F), 35.23 for NaF toothpaste (1350 ppm F), and 29.57 for NaMFP/NaF toothpaste (1450 ppm F). The reported ANOVA analyses give adjusted mean differences of 3.97 for NaF/Carbopol compared with NaMFP/NaF, with a 95% confidence interval from 0.47 to 7.48 and p = 0.026. For NaF toothpaste (1350 ppm F) compared with NaMFP/NaF, the adjusted mean difference is 5.66, with a 95% confidence interval from 2.15 to 9.18 and p = 0.002.

Those sponsor-reported analyses indicate statistical evidence of differences in adjusted mean %SMHR for the two reported comparisons. The confidence intervals give a range of values compatible with the fitted analyses under their assumptions. Both reported intervals exclude zero, which is consistent with the reported p-values. The registry does not report a threshold for a clinically important difference in %SMHR, so statistical evidence alone does not establish whether the differences matter clinically.

Separate unadjusted recalculations from posted summary statistics do not reproduce the sponsor's reported primary comparisons in all respects. For example, the recalculated comparison of NaF/Carbopol and NaMFP/NaF has an estimate of -3.97, a 95% interval from -8.738 to 0.797, and p = 0.103, whereas the sponsor reports an adjusted difference of 3.97 with a 95% interval from 0.47 to 7.48 and p = 0.026. The discrepancy means the sponsor's own analysis takes precedence. The recalculations are approximate, unadjusted comparisons based on posted summary statistics and should not replace the trial's reported analysis.

The crossover structure also matters to interpretation. In general, analyses of crossover studies may account for within-participant comparisons, treatment period, and treatment sequence. The registry lists ANOVA for the sponsor's analyses but does not report the model details needed to determine how those design features were handled. It also does not report enough information to assess the clinical relevance of the specimen-based differences or the possibility of period or carryover effects.

Overall, the registry reports differences between selected toothpaste formulations on the primary enamel measure, with positive sponsor-reported adjusted mean differences for the two stated contrasts. This is evidence about %SMHR in an in situ model over baseline to 14 days. It is not direct evidence about clinical caries outcomes or a basis for treatment advice.

10. Statistical Methods Explained

The primary endpoint is a mean of a continuous measure: %SMHR. A mean describes the central value of observations in a group, while a standard error describes the statistical uncertainty in an estimated mean. The registry reports least-squares means and standard errors for the treatment groups. Least-squares means are model-based averages; they can differ from simple arithmetic averages when the analysis adjusts for other terms or uses a particular model structure. The registry does not give full model specifications.

What does ANOVA contribute?

ANOVA, or analysis of variance, is a general method for comparing means across groups. The reported analyses give adjusted mean differences, confidence intervals, and p-values for specified group contrasts. An adjusted contrast is based on the model used for the analysis, rather than necessarily being a direct subtraction of the posted group means. The registry record does not state all adjustment variables, the covariance structure, or the exact ANOVA model. Those details should not be inferred from the method label alone.

In general, the p-value describes how incompatible the observed data are with a specified null hypothesis under the statistical model. It is not the probability that the null hypothesis is true, nor does it quantify the size or clinical importance of a difference. A confidence interval conveys information about the range of estimates reasonably compatible with the data and model. Both measures depend on the analysis assumptions and the way comparisons were selected or adjusted.

How should recalculated estimates be understood?

The recalculated comparisons use posted summary statistics to make simple unadjusted comparisons. Their pooled standard deviations are approximate, as indicated in the calculations, and the calculations use the displayed group means and standard errors. They do not recreate the sponsor's ANOVA. In particular, summary statistics do not disclose the participant-level outcome pattern or the within-participant covariance that may be important in a crossover study. General statistical practice would account for correlated observations when the same participant contributes outcomes under multiple treatments; the registry record does not provide enough detail to determine whether or how this was done in the sponsor analysis.

Accordingly, the recalculated estimates are useful as an independent descriptive check, not as a replacement for the sponsor's analysis. The registry flags a discrepancy between the reported analyses and recalculations. In that circumstance, the sponsor's own analysis takes precedence for describing the trial's reported inferential result, while the recalculation highlights that the published summary information does not fully reproduce it.

Why might variability matter?

In general, variability describes how much measured outcomes differ among observations. Standard errors reflect uncertainty in estimated means, whereas a standard deviation describes variation among individual observations. These quantities are not interchangeable. A standard error can be used to form an interval for a mean when the analysis assumptions are appropriate; it should not be read as the range of individual responses. For a crossover study, the variability of within-participant differences can differ from the variability between people, so independent-group approximations may not represent the study design well.

Over-dispersion is a general concern when observed variation exceeds what a model assumes. If the model understates variation, standard errors can be too small and intervals too narrow; if it overstates variation, intervals can be wider than necessary. The registry record does not report evidence of over-dispersion or a diagnostic assessment. It is therefore appropriate to explain the concept as a general modeling consideration, not to claim it occurred in this trial.

What is not specified?

The record identifies randomization, double masking, and a crossover model, but the original statistical analysis plan is not reported in the registry record. Details about the randomization sequence, allocation concealment, period effects, carryover assessment, model covariates, and missing-data handling beyond the statement that missing data were not imputed are also not reported. The record does not report an interim-analysis plan or a sample-size justification for the original trial. Without those details, the posted estimates can be described, but important design and analysis assumptions cannot be independently evaluated.

The registry defines the primary efficacy population as randomized participants with at least one post-baseline efficacy assessment. This definition excludes randomized participants who had no such assessment from the reported efficacy population, and the record says missing data were not imputed. In general, excluding participants with missing outcomes can affect interpretation if the reasons for missingness are related to treatment or outcomes. The registry does not give the information needed to assess that possibility. The treatment-specific efficacy counts vary, which the record attributes to dropouts.

11. Confidence Intervals and Precision

Precision concerns how narrowly an estimate is determined by the data and analysis. A confidence interval expresses uncertainty around an estimated contrast: a narrower interval indicates greater precision under the model, while a wider interval allows a broader range of compatible values. It does not guarantee that the true value lies within the interval, and it does not account for sources of bias or model misspecification.

For the sponsor-reported primary comparison of NaF/Carbopol toothpaste with NaMFP/NaF toothpaste, the adjusted mean difference is 3.97 and the 95% confidence interval is 0.47 to 7.48. For NaF toothpaste (1350 ppm F) compared with NaMFP/NaF toothpaste, the adjusted mean difference is 5.66 and the 95% confidence interval is 2.15 to 9.18. Both reported intervals are above zero. Their widths show that the registry does not identify a single exact contrast, even though the reported analyses provide evidence against a zero difference under the model.

Precision should be considered alongside the discrepancy between the sponsor analyses and the unadjusted recalculations. For the recalculated NaF/Carbopol versus NaMFP/NaF comparison, the 95% interval is -8.738 to 0.797, which includes zero. This differs from the sponsor's reported interval and reinforces that the recalculation is not a substitute for the model-based analysis. In general, a confidence interval describes sampling uncertainty conditional on the analysis assumptions; it does not resolve uncertainty about the model, missing data, or clinical relevance.

The standard errors accompanying the reported group means also describe uncertainty in group mean estimates, not the spread of individual participant responses. The registry does not give a clinically meaningful threshold for %SMHR, so the intervals cannot by themselves establish whether a plausible contrast is important in clinical terms.

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.

Effect to detectParticipants per arm, 80% powerParticipants per arm, 90% power
100% of the observed effect8801,179
75% of the observed effect1,5652,095
50% of the observed effect3,5204,713
Sample size needed versus the size of the effect to detect
Sample size needed versus size of effect to detect5001,0002,0005,00010,00040%60%80%100%80% power90% powersize of effect to detect, as % of the effect observed (log vertical scale: participants per arm)
Treat these figures cautiously. The observed difference is imprecise: its confidence interval includes no difference at all. A sample size built on a noisy estimate can be far too small or too large, so the smaller-effect rows are the more cautious guide.
Method and assumptions. Two-sided α = 0.05, 1:1 allocation, two-sample comparison of means with the pooled standard deviation. Standard deviation was approximated from model-based standard errors, so treat it as rough. Calculated for the comparison: NaF Toothpaste (1350 Ppm F).

The planning figures are participants per arm, not total events. They illustrate the sample size that would be needed under stated assumptions to detect a chosen fraction of an observed effect. The 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 specified as a fraction of the observed effect. These are planning illustrations based on the observed effect, not the original trial's design assumptions; the trial's own sample-size justification is not reported in the registry record.

The 80% power column means the illustrated design would have an 80% chance of detecting the selected effect at the stated significance level if the assumptions held and the effect were real. The 90% power column uses a higher target chance of detection under those same assumptions. Power is a prospective design quantity, not a measure of what the completed study achieved.

The rows for 100% of the observed effect, 75% of the observed effect, and 50% of the observed effect show how the planning estimate changes when the target effect is reduced. Smaller effects need more participants to detect with the same power and significance level. Because the observed effect can be unstable, especially when estimated from a small study, these figures should be read as illustrations rather than firm recommendations. They do not account for the crossover structure unless explicitly included in the stated assumptions, and the listed assumptions specify equal allocation between two arms rather than a trial-specific crossover calculation.

This is not observed power. The figures are not a retrospective score of whether the trial was adequate; they are conditional planning illustrations, not evidence about the trial's achieved power.

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 z1−α/2 = 1.960) and the desired power (z1−β = 0.842 for 80% and 1.282 for 90%).

Formula

n = 2 (z1−α/2 + z1−β)2 σ2 / Δ2

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 = 1.690; pooled standard deviation σ = 12.653.
  2. n = 2 × (1.96 + 0.842)2 × 12.6532 / 1.6902 = 880.0, rounded up to 880 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 <- 1.69                      # observed difference in means
sigma <- 12.6532534494944                     # 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(708123)
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 = 880 (80% power), 1,179 (90% power)
 75% of observed effect: n per arm = 1,565 (80% power), 2,095 (90% power)
 50% of observed effect: n per arm = 3,520 (80% power), 4,713 (90% power)
power.t.test: 880.9 
Simulated power at n = 880 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

MethodEstimate95% CIP-value
ANOVAAdjusted mean difference 3.970.47 to 7.480.026
ANOVAAdjusted mean difference 5.662.15 to 9.180.002
Where the two analyses differ, rely on the sponsor's. The simple re-calculation above reaches a different conclusion from the sponsor's analysis, usually because the sponsor adjusts for covariates, stratification or repeated measures, or (for event rates) uses a negative binomial model that allows for over-dispersion and so gives wider intervals.

The sponsor's reported primary analyses use ANOVA and give adjusted mean differences, 95% confidence intervals, and p-values for two comparisons against NaMFP/NaF toothpaste. Those analyses are the trial's own inferential results. The registry's separate recalculations use unadjusted comparisons derived from posted group summaries and approximate pooled variability; they do not reproduce every aspect of the sponsor's model.

A visible discrepancy occurs for the comparison of NaF/Carbopol toothpaste and NaMFP/NaF toothpaste. The recalculated estimate is -3.97, with a 95% interval from -8.738 to 0.797 and p = 0.103. The sponsor reports an adjusted mean difference of 3.97, with a 95% interval from 0.47 to 7.48 and p = 0.026. The estimate direction and inferential results differ. Differences in adjustment, model structure, contrast coding, or use of the crossover data could matter in general, but the registry does not provide enough detail to identify the explanation.

Because the record flags a discrepancy, the sponsor's own analysis takes precedence. The recalculation should be treated as a check on reproducibility from summary information, not as a corrected analysis or a competing primary result. The information needed to reconstruct the sponsor's model is not reported in the registry record.

15. Multiplicity, Missing Data and Analysis Populations

Multiplicity arises when multiple outcomes, treatment contrasts, or statistical tests are considered. In general, performing more tests can increase the chance of at least one nominally unusual result even when no true differences exist. Methods such as a prespecified testing hierarchy or adjustment of significance levels can address this issue, but the registry record does not report whether either approach was used.

The record identifies a primary endpoint and lists secondary outcomes, and it reports more than one sponsor analysis. However, the original statistical analysis plan, the full planned comparison set, and any multiplicity strategy are not reported in the registry record. It is therefore not possible to determine whether the reported p-values were adjusted across comparisons or how confirmatory the secondary analyses were intended to be. The stated p-values should be read as reported, without assuming either that multiplicity was controlled or that it was ignored in the trial's original plan.

The primary efficacy population included randomized participants with at least one post-baseline efficacy assessment. Missing data were not imputed. The registry notes that treatment-group counts differ because of participant dropouts. This is important because an analysis limited to participants with an observed post-baseline outcome may be affected if missingness is related to treatment response or another factor connected to the outcome. The record does not report the reasons for missing observations or an analysis of sensitivity to missing data.

In general, an intent-to-treat label can refer to different practical analysis rules, so the population definition matters. Here the registry specifies the included participants rather than reporting an imputation method for missing outcomes. The treatment-group counts and stated population definition should guide interpretation; a broader claim that every randomized participant contributed to the endpoint analysis is not supported by the record.

The record also does not report an interim-analysis plan, original sample-size justification, or detailed statistical analysis plan. These omissions limit assessment of whether the reported analyses followed a prespecified strategy. They do not invalidate the posted results, but they constrain how confidently the analysis choices and multiplicity can be evaluated.

16. Safety Results

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

ArmParticipants with serious adverse eventsParticipants at riskPercentage
NaF/ Carbopol Toothpaste (1400 Ppm F)0560.0%
NaF Toothpaste (1350ppm F)0540.0%
NaMFP/NaF Toothpaste (1450 Ppm F)0550.0%
NaF Toothpaste (250 Ppm F)0560.0%
Placebo Toothpaste (0 Ppm F)0540.0%

Other (non-serious) adverse events

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

ArmParticipants affected of at risk
NaF/ Carbopol Toothpaste (1400 Ppm F)4 of 56
NaF Toothpaste (1350ppm F)4 of 54
NaMFP/NaF Toothpaste (1450 Ppm F)1 of 55
NaF Toothpaste (250 Ppm F)1 of 56
Placebo Toothpaste (0 Ppm F)2 of 54

Most frequent non-serious adverse event terms

Event termOrgan systemNaF/ Carbopol Toothpaste (1400 Ppm F)NaF Toothpaste (1350ppm F)NaMFP/NaF Toothpaste (1450 Ppm F)NaF Toothpaste (250 Ppm F)Placebo Toothpaste (0 Ppm F)
NasopharyngitisInfections and infestations4 of 564 of 541 of 551 of 562 of 54
Do not overextend the safety comparison. Serious-adverse-event counts are safety observations, not efficacy outcomes, and should not be combined with the primary result into a single benefit-risk number.

The registry lists serious adverse events by treatment group and reports 0 participants affected by serious adverse events in each listed group. The figures at risk are 56 for NaF/Carbopol toothpaste (1400 ppm F), 54 for NaF toothpaste (1350 ppm F), 55 for NaMFP/NaF toothpaste (1450 ppm F), 56 for NaF toothpaste (250 ppm F), and 54 for placebo toothpaste (0 ppm F). No serious adverse event terms are listed.

These figures describe the serious adverse events recorded in the registry for the stated groups and denominators. They do not establish that the treatments have no serious risks; an absence of recorded serious events in this study is not the same as proof of absence. The registry does not report death counts, so deaths are not reported in the registry record. Non-serious adverse events are listed separately, and their presence does not change the interpretation of the serious-event figures.

The safety counts should be read as recorded observations within this study, not as a comparative estimate of long-term safety. The registry does not report follow-up beyond the stated study timeframe or enough detail to make broader safety conclusions.

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

The result concerns the recovery of surface microhardness in enamel specimens exposed in an in situ caries model over baseline to 14 days. The sponsor's ANOVA reports adjusted mean differences between selected fluoride toothpaste formulations, with confidence intervals and p-values indicating statistical evidence of differences for those contrasts under the fitted analysis.

It does not establish that one toothpaste prevents clinical caries better than another. The endpoint is an enamel-specimen measure, not a count of caries in participants or a measure of symptoms, function, or quality of life. Nor does the record establish that the measured differences are clinically important: no minimally important difference for %SMHR is reported.

The posted values should also be interpreted with the discrepancy between sponsor-reported analyses and summary-statistic recalculations in mind. The sponsor's own analysis takes precedence, but the discrepancy makes it harder to independently reconstruct the reported results from the registry summaries. The registry does not provide the full model specification or crossover-period details needed to resolve that difference.

Finally, the study population and model limit generalization. Participants were healthy subjects who wore partial dentures, and the outcome was assessed using an in situ enamel model. The registry record does not establish that the results apply to other populations, use patterns, or clinical settings. These are statistical and evidential boundaries, not recommendations about which product to use.

18. Why This Trial Matters Statistically

This study illustrates why the estimand and measurement scale matter. A statistically significant difference in a laboratory-based enamel outcome is not automatically a difference in a clinical event. Readers need to distinguish the biological measure being analyzed from the clinical question that might motivate it.

It also demonstrates the importance of analysis structure. A crossover study can generate correlated observations because participants may receive multiple treatments. A simple unadjusted comparison based on group summaries may not capture that structure. When the sponsor's adjusted ANOVA and an approximate recalculation differ, the full analysis specification becomes essential for understanding how the reported inference was produced.

Confidence intervals add useful context beyond p-values by showing the range of estimates compatible with the fitted analysis. At the same time, interval interpretation depends on the model and cannot settle questions about clinical importance, missingness, or multiplicity. The registry does not report a clinical threshold for the primary endpoint or enough detail to assess all of those issues.

Finally, the record highlights the value of transparent reporting. Information on treatment periods, missing-data handling, prespecification, sample-size rationale, and multiplicity would help readers evaluate the design and analysis more fully. The posted results still support a clear statistical description of the reported comparisons, while leaving important questions about reproducibility and broader meaning unresolved.

19. Limitations

Several trial-specific limitations affect interpretation. The study uses an in situ caries model and an enamel specimen endpoint, so the primary result is not a direct clinical caries outcome. The registry describes a crossover design but does not report the treatment sequence, period structure, washout arrangements, or how the sponsor's model handled within-participant correlation, period effects, or carryover.

The sponsor's adjusted primary results differ from recalculations based on posted summary statistics. Because the registry does not give a full analysis model or participant-level data, the reason for the discrepancy cannot be established from the record. The recalculations are approximate and unadjusted, and the record flags the discrepancy; the sponsor's own analysis takes precedence.

The analysis population excludes randomized participants without a post-baseline efficacy assessment, and missing data were not imputed. The record attributes differences in treatment-group counts to dropouts but does not report reasons for missingness or sensitivity analyses. It also does not report the original sample-size justification, interim analyses, or the full statistical analysis plan.

Multiplicity handling is not reported despite multiple endpoints and comparisons. A clinical-importance threshold for %SMHR is also not reported, limiting interpretation of the size of the differences. The registry's listed serious-event observations are limited to the reported groups and study period; death counts are not reported in the registry record. These limitations narrow what can be concluded from the record without negating the reported outcome measurements.

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23. Record Summary

The registry describes a completed randomized, double-masked Phase 3 crossover study of fluoride toothpaste formulations in an in situ enamel caries model. The primary outcome was mean %SMHR from baseline to 14 days. The sponsor reported adjusted ANOVA differences for two comparisons against NaMFP/NaF toothpaste. Recalculations from posted summaries are approximate, unadjusted, and discrepant; the sponsor's own analysis takes precedence. The record does not report the full model, multiplicity strategy, original sample-size rationale, or several crossover details. Results describe enamel measures and do not directly establish clinical caries outcomes.

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