# Results Reproducibility

# Case study: Evaluation of Bioequivalence for Econazole Nitrate Deposition in Porcine Skin

In the following we use the methodology developed in this paper for the evaluation of the bioequivalence of two cream products. The dataset was provided by the authors of Quartier et al. (2019), who use the cutaneous bioequivalence method detailed in their paper. It contains \(17\) pairs of comparable porcine skin samples on which measurements of econazole nitrate deposition (an antifungal medication used to treat skin infections) were collected using two bioequivalent products. These measurements were then considered on the logarithmic scale and saved as an RData file called `skin`.

The results presented in Figure 4 of Boulaguiem et al. (2023) can be reproduced as follows:

    library(cTOST)
    data(skin)

    # Empty plot
    plot(1,1,pch="",axes=FALSE,xlab="",ylab="",main="",
         ylim=c(min(unlist(skin)),log(4000)),xlim=c(0.75, 2.25))

    # Plotting and linking the paired data points of each group
    for(i in 1:nrow(skin)){
      points(c(1,2),skin[i,],col=paste0(gray(0.1),50),pch=16)
      segments(1,skin[i,1],2,skin[i,2],
               col=paste0(gray(0.1),50))
    }
    axis(1, at = c(1,2), c("Reference", "Generic"),tick=FALSE)
    axis(2, at = log(c(250,500,1000,2000,4000)), c(250,500,1000,2000,4000), las=2)
    axis(2, at = mean(c(min(unlist(skin)),log(4000))), expression(paste("ECZ deposition (ng/cm"^2*")")),padj=-4.5, tick = FALSE)

## Bioequivalence assessment

In order to assess bioequivalence, we extract each of means between the two groups, the number of degrees of freedom, and the standard error:

    # Difference in means between the two groups
    theta_hat = diff(apply(skin,2,mean))

    # Number of degrees of freedom
    nu = nrow(skin)-1

    # Standard error
    sig_hat = sd(apply(skin,1,diff))/sqrt(nu)

We consider in the paper a significance level of \(5\%\) and an equivalence limit at \(\text{delta}=log(1.25)\).

### Standard TOST

Considering these parameters, the standard TOST does not accept bioequivalence:

    res_tost = tost(theta = theta_hat, sigma = sig_hat, nu = nu, alpha = 0.05, delta = log(1.25))
    res_tost

### \(\delta\)-TOST

Using the \(\delta\)-TOST, we still cannot accept bioequivalence, even though the corrected region is larger:

    res_dtost = dtost(theta = theta_hat, sigma = sig_hat, nu = nu, alpha = 0.05, delta = log(1.25))
    res_dtost

    compare_to_tost(res_dtost)

### \(\alpha\)-TOST

However, the \(\alpha\)-TOST allows us to accept bioequivalence:

    res_atost = atost(theta = theta_hat, sigma = sig_hat, nu = nu, alpha = 0.05, delta = log(1.25))
    res_atost

    compare_to_tost(res_atost)

## Interval Inclusion Principal with the \(\alpha\)-TOST

To visually assess bioequivalence with the interval inclusion principal, we reproduce Figure 5 of Boulaguiem et al. (2023) with the following code:

    # Nominal level and zquivalence bounds
    alpha = 0.05
    delta = log(1.25)

    # Empty plot
    plot(NA, axes = F, xlim = c(-0.25, 0.25), ylim = c(0.5, 2.5), xlab = " ", ylab = " ")

    # Corrected level used by aTOST
    alpha_star = res_atost$corrected_alpha

    # TOST's CI upper and lower bound
    a1 = theta_hat + sig_hat*qt(alpha, df = nu)
    a2 = theta_hat - sig_hat*qt(alpha, df = nu)

    # aTOST's CI upper and lower bound
    b1 = theta_hat + sig_hat*qt(alpha_star, df = nu)
    b2 = theta_hat - sig_hat*qt(alpha_star, df = nu)

    # Set the colors and their transparent counterpart
    cols=c("coral","palegreen2")
    t.cols=sapply(c("coral","palegreen2"),function(x){
      rgb.val <- col2rgb(x)
      rgb(rgb.val[1], rgb.val[2], rgb.val[3],
          max = 255,
          alpha = 127.5)
    })

    # Plotting the TOST and aTOST CIs
    d = 0.1
    polygon(c(a1, a2, a2, a1), c(1-d, 1-d, 1+d, 1+d) + 1, border = NA,
            col = t.cols[1])
    polygon(c(delta, a2, a2, delta), c(1-d, 1-d, 1+d, 1+d) + 1, border = NA,
            col = cols[1], density = 20, angle = 45)
    polygon(c(delta, a2, a2, delta), c(1-d, 1-d, 1+d, 1+d) + 1, border = NA,
            col = cols[1], density = 20, angle = -45)
    polygon(c(b1, b2, b2, b1), c(1-d, 1-d, 1+d, 1+d), border = NA,
            col = t.cols[2])

    # Plotting the difference in means
    points(theta_hat, 2, col = cols[1], pch = 16, cex = 2.5)
    points(theta_hat, 1, col = cols[2], pch = 16, cex = 2.5)

    # Adding text for readability
    text(0.15, 2 + 1.75*d, "TOST CI")
    text(0.15, 1 + 1.75*d, expression(paste(alpha,"TOST CI")))

    text(0.22, 2 - 1.75*d, expression(paste("outside (-",delta,", ",delta,")")), cex = 0.8)
    abline(v = c(-0.3, -log(1.25), -0.1, 0, 0.1, log(1.25), 0.3),
           col = "grey65", lty = 2)
    axis(1, at = c(-0.3, -log(1.25), -0.1, 0, 0.1, log(1.25), 0.3),
         labels = c("-0.3", expression(paste("-",delta)), "-0.1", "0", "0.1",
                    expression(delta), "0.3"),padj=-0.15,cex.axis=1.3)
    mtext(expression(theta), side = 1, line = 2.35, cex = 1.4)

# References

Boulaguiem, Y., J. Quartier, M. Lapteva, Y. N. Kalia, M.-P. Victoria-Feser, S. Guerrier, and D.-L. Couturier. 2023. “Finite Sample Adjustments for Average Equivalence Testing.” Statistics in Medicine, 2023–03.
Quartier, J., N. Capony, M. Lapteva, and Y. N. Kalia. 2019. “Cutaneous Biodistribution: A High-Resolution Methodology to Assess Bioequivalence in Topical Skin Delivery.” Pharmaceutics 11 (9): 484.