eqivalenceTest: A package for evaluating equivalence of the means of two normal distributions.

Description

We implemented two equivalence tests which evaluate equivalence in the means of two normal distributions. The first is discussed by Tsong et al. (2017) and the second by Weng et al. (2018).

Details

Let X_{I,i}\sim_{IID} N(\mu_I,\sigma_I) for I=T,R and i=1,...,n_I, where T stands for test distribution and R for reference distribution. The equivalence test here considers the following hypotheses,

H_0: |\mu_T - \mu_R| \ge \delta \;\mathrm{versus}\;H_1:|\mu_T - \mu_R| < \delta,

where \delta is the equivalence margin.

Let \hat{\mu}_I and \hat{\sigma}_I^2 be the sample mean and unbiased sample variance estimates respectively for I=T,R. Tsong et al. (2017) define the follows test statistics,

\tau_1=\frac{\hat{\mu}_T-\hat{\mu}_R+\delta}{\sqrt{\hat{\sigma}_T^2/n_T^*+\hat{\sigma}_R^2/n_R^*}},

and

\tau_2=\frac{\hat{\mu}_T-\hat{\mu}_R-\delta}{\sqrt{\hat{\sigma}_T^2/n_T^*+\hat{\sigma}_R^2/n_R^*}},

where n_T^*=min\{n_T,1.5n_R\} and n_R^*=min\{n_R,1.5n_R\} are possibly adjusted sample sizes proposed by Dong et al. (2017).

The null hypothesis H_0 is rejected at nominal size \alpha if both \tau_1 > t_{1-\alpha,df^*} and \tau_2 < -t_{1-\alpha,df^*} where t_{1-\alpha,df^*} is the (1-\alpha)-th quantile of the t-distribution with degree of freedom df^*, which is approximated by the Satterthwaite method with sample size adjusted and given as follows,

df^*=\frac{\left(\frac{\hat{\sigma}_T^2}{n_T^*}+\frac{\hat{\sigma}_R^2}{n_T^*}\right)^2}{\frac{1}{n_B-1} \left(\frac{\hat{\sigma}_T^2}{n_T^*}\right)^2+\frac{1}{n_R-1} \left(\frac{\hat{\sigma}_R^2}{n_R^*}\right)^2}.

The above assumes that \delta is a predetermined constant. However, in many studies, such constant is not available, and \delta must be determined by the study data. A popular choice is \delta=k\hat{\sigma_R}. In this case, the above test may not control type I error well.

Replacing \delta by k\sigma_R, the hypotheses becomes

H_0^\prime: |\mu_T - \mu_R| \ge k\sigma_R \;\mathrm{versus}\;H_a^\prime |\mu_T - \mu_R| < k\sigma_R.

Weng et al. (2018) proposed an improved Wald test with the following test statistics,

\tau_1^\prime=\frac{\hat{\mu}_T-\hat{\mu}_R+k\hat{\sigma}_R}{\sqrt{\frac{\tilde{\sigma}_{T,1}^2}{n_T^*}+\left(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{\sigma}_{R,1}^2}},

\tau_2^\prime=\frac{\hat{\mu}_T-\hat{\mu}_R-k\hat{\sigma}_R}{\sqrt{\frac{\tilde{\sigma}_{T,2}^2}{n_T^*}+\left(\frac{1}{n_R^*}+\frac{k^2V_{n_R}}{n_R-1}\right)\tilde{\sigma}_{R,2}^2}},

where V_{n_R} = n_R-1-2\frac{\Gamma^22(n_R/2)}{\Gamma^2((n_R-1)/2)} and \tilde{\sigma}_{T,i} and \tilde{\sigma}_{R,i} are the restricted maximum likelihood estimator of \sigma_T and \sigma_R respectively with the constraint \mu_T - \mu_R = (-1)^i \sigma_R.

The null hypothesis H_0^\prime is rejected at nominal size \alpha if both \tau_1^\prime > z_{1-\alpha} and \tau_2^\prime < -z_{1-\alpha} where z_{1-\alpha} is the (1-\alpha)-th quantile of the standard normal distribution.

For more details, see the cited reference.

References

Dong X, Weng Y, Tsong Y (2017). “Adjustment for unbalanced sample size for analytical biosimilar equivalence assessment.” Journal of biopharmaceutical statistics, 27(2), 220–232.

Tsong Y, Dong X, Shen M (2017). “Development of statistical methods for analytical similarity assessment.” Journal of biopharmaceutical statistics, 27(2), 197–205.

Weng Y, Tsong Y, Shen M, Wang C (2018). “Improved Wald Test for Equivalence Assessment of Analytical Biosimilarity.” International Journal of Clinical Biostatistics and Biometrics, 4(1), 1–10.

Perform restricted MLE (RMLE) to estimate parameters under the constraint defined by the boundary of null hypothesis

Description

Perform restricted MLE (RMLE) to estimate parameters under the constraint defined by the boundary of null hypothesis, \mu_T - \mu_R = \eta\sigma_R where \eta is the margin multiplier.

Usage

RMLE_equivTest(nT, nR, smplMuT, smplMuR, smplSigmaT, smplSigmaR, vecT,
  vecR, eta)

Arguments

Value

a list containing the RMLE for the means and standard deviations for both test and reference data

Create summary information of a dataset

Description

Create a list of summary statistics of a dataset for equivalence test.

Usage

createEquivTestSmpl(smpl)

Arguments

Value

a list of objects summarizing the dataset

Examples

vecT = rnorm(n=20)
s = createEquivTestSmpl(vecT)

Compute the Satterthwaite approximation of degree of freedom for t distribution

Description

Compute the Satterthwaite approximation of degree of freedom for t distribution.

Usage

dfSatterthwaite(s1, n1, n1s, s2, n2, n2s)

Arguments

Value

degree of freedom

Conduct the equivalence test with fixed margin

Description

Conduct the equivalence test with fixed margin.

Usage

equivTestFixedMargin(vecT, vecR, alpha = 0.05, marginX = 1.5,
  sampleSizeX = 1.5, qa = "", sigmaTOverride = NULL,
  labelT = "Proposed", labelR = "Reference", show.message = FALSE,
  method = "Fixed Margin")

Arguments

Value

a list of objects summarizing the data and test results, in particular, rslt = 1 if H_0 is rejected, and rslt = 0 if H_0 is not rejected.

References

Tsong Y, Dong X, Shen M (2017). “Development of statistical methods for analytical similarity assessment.” Journal of biopharmaceutical statistics, 27(2), 197–205.

Examples

vecT = rnorm(20,-1.5,1)
vecR = rnorm(20,0,1)
et = equivTestFixedMargin(vecT, vecR)

Provide a combined plot for equivvalence test

Description

Provide a combined plot for equivalence test, including both scatter plot of the sample data and a bar plot indicating the test result, where the null hypothesis is rejected if the red line representing the mean value of the test product lies within a grey rectangle centered at a blue line representing the mean value of the reference product.

Usage

equivTestFixedMarginCombPlot(et)

Arguments

Examples

vecR = rnorm(20,0,1)
vecT = rnorm(20,-1.5,1)
et = equivTestFixedMargin(vecT,vecR)
equivTestFixedMarginCombPlot(et)

Equivalence test by Modified Wald test with standard error estimated by RMLE (MWCMLE)

Description

Equivalence test by Modified Wald test with standard error estimated by RMLE (MWCMLE).

Usage

equivTestMWCMLE(vecT, vecR, alpha = 0.05, marginX = 1.7,
  method = "MWCMLE")

Arguments

Details

See Weng et al. (2018).

Value

a list containing the test result

References

Weng Y, Tsong Y, Shen M, Wang C (2018). “Improved Wald Test for Equivalence Assessment of Analytical Biosimilarity.” International Journal of Clinical Biostatistics and Biometrics, 4(1), 1–10.

Plot the equivalence test result

Description

Plot the equivalence test result including the margin, confidence intervals of the mean difference, and estimated mean difference.

Usage

equivTestPlot(meanDif, ci, alpha, margin, qaNameLong, testDrugName = "",
  refDrugName = "", showDrugName = FALSE, showQA = FALSE,
  showCINumbers = FALSE)

Arguments

Examples

equivTestPlot(0.623,c(-2,2),0.05,c(-9.79,9.79),
  "q a","test","reference")
equivTestPlot(0.623,c(-2,2),0.05,c(-9.79,9.79),
  "Relative Potency","test","reference",showDrugName = TRUE,showQA=TRUE,showCINumbers = TRUE)
equivTestPlot(0.5,c(-1.05,2.05),0.05,c(-9.79,9.79),
  "Relative Potency","test","reference",showQA=TRUE,showCINumbers = TRUE)

Histogram with a fitted normal density function

Description

Provide a histogram with a fitted normal density.

Usage

histWNormDensity(x, main = "")

Arguments

Examples

x = rnorm(20)
histWNormDensity(x)

Provide a side-by-side scatter plot of two or three datasets for equivalence test.

Description

Provide a side-by-side scatter plot of two samples for equivalence test.

Usage

scatterPlotEquivTestData(vecT, vecR, vecR1 = NULL, qa = "",
  labelT = "Test", labelR = "Reference", labelR1 = "Reference1")

Arguments

Examples

vecT = rnorm(20,-1.5,1)
vecR = rnorm(20,0,1)
vecR1 = rnorm(20,0,1)
scatterPlotEquivTestData(vecT,vecR,labelT="T",labelR="R",qa="potency")
scatterPlotEquivTestData(vecT,vecR,vecR1,labelT="T",labelR="R",labelR1="R1",qa="potency")

Summarize data for equivalence test

Description

Summarize data for equivalence test, can be two datasets or three datasets.

Usage

summarizeSample(vecT, labelT, vecR, labelR, vecR1 = NULL, labelR1 = "")

Arguments

Value

a data.frame consisting the sample size, min, max, mean, SD, and percentage coefficient of variation for the samples

Examples

vecT = rnorm(10,-1.5,1)
vecR = rnorm(10)
vecR1 = rnorm(15,1,2)
ss = summarizeSample(vecT,"T",vecR,"R",vecR1,"R1")