Update vignette

Author: NightlordTW

R/helper.r

@@ -21,11 +21,12 @@ print.simss <- function(x, ...) {
   output <- data.frame(Parameter = c("Total Sample Size", "Achieved Power", "Power Confidence Interval"),
                        Value = c(sst, 100*power, paste0(100*lpower, " - ", 100*upower)))
 
-  message(cat("Sample Size Calculation Results"))
-  message(cat("-------------------------------------------------------------"))
-  message(cat(paste0("Study Design: ", x$param.d$dtype, " trial targeting ",100*tpower,"% power with a ",100*alpha, "% type-I error.")))
-
+  message("Sample Size Calculation Results")
+  cat("-------------------------------------------------------------\n")
+  cat(paste0("Study Design: ", x$param.d$dtype, " trial targeting ",100*tpower,"% power with a ",100*alpha, "% type-I error.\n"))
+  cat("-------------------------------------------------------------\n")
   print(output, row.names = FALSE)  # Suppress row numbers
+  #cat("-------------------------------------------------------------\n")
   #print(x$response[,-1], row.names = FALSE)
 }
 

vignettes/sampleSize_parallel_2A3E.Rmd

@@ -64,12 +64,12 @@ For both endpoints, the analysis assumes that:
 
 To evaluate bioequivalence, we apply the $80\%/125\%$ rule, which defines equivalence bounds relative to the reference mean. The evaluation is conducted using a one-sided significance level of 5\%, with a target statistical power of 90\%.
 
-# Testing the Difference of Means (DOM) for multiple Co-primary Endpoints
-In biosimilar development, it is important to demonstrate equivalence across all relevant doses, routes of administration, patient populations, and endpoints. To establish equivalence between two treatments, the difference in means for each endpoint, $\mu_{T}^{(j)} - \mu_{R}^{(j)}$, must lie within a predefined equivalence margin around zero for all primary endpoints.
 
 ## Hypotheses
 The null and alternative hypotheses for the equivalence test are as follows:
 
+### Difference of Means (DOM)
+
 Null Hypothesis ($H_0$): At least one endpoint does not meet the equivalence criteria:
 
 $$H_0: \mu_T^{(j)} - \mu_R^{(j)} \le E_L ~~ \text{or}~~ \mu_T^{(j)} - \mu_R^{(j)} \ge E_U \quad \text{for at least one}\;j$$
@@ -78,14 +78,39 @@ Alternative Hypothesis ($H_1$): All endpoints meet the equivalence criteria:
 
 $$H_1: E_L<\mu_{T}^{(j)}-\mu_{R}^{(j)} < E_U \quad\text{for all}\;j$$
 
-The null hypothesis ($H_0$) is rejected if, and only if, all null hypotheses associated with the $K$ primary endpoints are rejected at a significance level of $\alpha$. This ensures that equivalence is established across all endpoints simultaneously.
+The null hypothesis ($H_0$) is rejected if, and only if, all null hypotheses associated with the $K$ primary endpoints are rejected at a significance level of $\alpha$. This ensures that equivalence is established across all endpoints simultaneously. 
+
+### Ratio of Means (ROM)
+The equivalence hypotheses can also be expressed as a Ratio of Means (ROM), which is often used in bioequivalence studies:
+
+Null Hypothesis ($H_0$): At least one endpoint does not meet the equivalence criteria:
+ 
+$$H_0: \frac{\mu_T^{(j)}}{\mu_R^{(j)}} \le \log(E_L) ~~ \text{or}~~ \frac{\mu_T^{(j)}}{\mu_R^{(j)}} \ge \log(E_U) \quad \text{for at least one}\;j$$
+
+Alternative Hypothesis ($H_1$): All endpoints meet the equivalence criteria:
+
+$$H_1: \log(E_L)< \frac{\mu_{T}^{(j)}}{\mu_{R}^{(j)}} < \log(E_U) \quad\text{for all}\;j$$
+
 
 ## Statistical Considerations
 
-* **Type I Error Control**: Since rejection of $H_0$ requires all individual null hypotheses to be rejected, there is no need for multiplicity adjustments. The Type I error rate is controlled by the design.
-* **Impact on Power**: Requiring equivalence across multiple endpoints decreases the overall power of the test. The Type II error increases as the number of primary endpoints ($K$) grows, which can make equivalence testing more challenging [@mielke_sample_2018].
+### Consistency Across Endpoints
+For equivalence to be established, all primary endpoints must simultaneously satisfy the equivalence criteria. This applies whether the criteria are expressed as:
 
-## Independent Testing of Pharmacokinetic (PK) Measures
+  * The Difference of Means (DOM) approach measures absolute differences between treatment means.
+  * The Ratio of Means (ROM) approach captures relative differences and is commonly used when analyzing log-transformed data, such as in pharmacokinetic studies.
+
+### Type I Error Control
+Rejection of the null hypothesis ($H_0$) requires that all individual null hypotheses across endpoints be rejected. Since the test is designed to achieve equivalence simultaneously for all endpoints, there is no need for multiplicity adjustments, and the Type I error rate is controlled by the study design.
+
+### Impact on Power
+Requiring equivalence across multiple endpoints reduces the overall power of the test. Specifically:
+
+* The Type II error increases as the number of primary endpoints ($K$) grows.
+* This makes equivalence testing more challenging for studies with multiple endpoints, as additional endpoints require larger sample sizes or stronger effect sizes to achieve sufficient power [@mielke_sample_2018].
+
+
+## Independent Testing of PK Measures
 If each pharmacokinetic (PK) measure is tested independently, the following sample sizes would be required for each endpoint to achieve a 5\% significance level:
 
 ```{r}
@@ -135,8 +160,14 @@ library(SimTOST)
 ))
 ```
 
+If we were to test each PK measure independently, we would find a total sample size of `r sim_AUCinf$response$n_total` for AUCinf, `r sim_AUClast$response$n_total` for AUClast, and `r sim_Cmax$response$n_total` for Cmax. This means that we would have to enroll `r sim_AUCinf$response$n_total` + `r sim_AUClast$response$n_total` + `r sim_Cmax$response$n_total` = `r sim_AUCinf$response$n_total + sim_AUClast$response$n_total + sim_Cmax$response$n_total` patients in order to reject $H_0$ at a significance level of 5\%. For context, the original trial was a randomized, single-blind, three-arm, parallel-group study conducted in 159 healthy subjects, slightly more than the `r sim_AUCinf$response$n_total + sim_AUClast$response$n_total + sim_Cmax$response$n_total` patients estimated as necessary. This suggests that the original trial had a small buffer above the calculated sample size requirements.
+
+## Simultaneous Testing of PK Measures with Independent Endpoints
+This approach focuses on simultaneous testing of pharmacokinetic (PK) measures while assuming independence between endpoints. Unlike the previous approach, which evaluated each PK measure independently, this method integrates comparisons across multiple endpoints, accounting for correlations (or lack thereof) between them. By doing so, it enables simultaneous testing for equivalence without inflating the overall Type I error rate.
+
+In this setting, equivalence is required for at least one endpoint rather than all endpoints, reducing the overall sample size compared to independent testing. Furthermore, this approach allows for greater flexibility by enabling users to specify correlation structures or work with uncorrelated endpoints as a default assumption.
+
 
-If we were to test each PK measure independently, we would find a total sample size of `r sim_AUCinf$response$n_total` for AUCinf, `r sim_AUClast$response$n_total` for AUClast, and `r sim_Cmax$response$n_total` for Cmax. This means that we would have to enroll `r sim_AUCinf$response$n_total` + `r sim_AUClast$response$n_total` + `r sim_Cmax$response$n_total` = `r sim_AUCinf$response$n_total + sim_AUClast$response$n_total + sim_Cmax$response$n_total`$ patients in order to reject $H_0$ at a significance level of 5\%.