# Abstract

This article describes an extension of multiple contrast tests to the case of multiple, correlated endpoints. These endpoints are assumed to be normally distributed with different scales and variances. Unlike in previous articles, the covariance matrices are also assumed to be unequal for the treatment groups. Approximate multivariate *t*-distributions are used to obtain multiplicity-adjusted *p*-values and quantiles for test decisions or simultaneous confidence intervals. Simulation results show that this approach controls the family-wise error type I over both the comparisons and the endpoints in an admissible range. The approach will be applied to a semi-synthetic example data set of a randomized, placebo-controlled phase IIb dose-finding study of a novel anti-muscarinic agent for five continuous endpoints. A related R package is available.

## 1 Introduction

Randomized clinical trials and pre-clinical studies often cover many correlated endpoints. The scales of these endpoints are often different. The experimental goal is not only to clarify which treatment groups differ but also for which endpoints. Hence, it is not clear a priori, for which endpoints differences between the treatment groups can be expected. These endpoints must be detected by the analysis a posteriori, so that they must be evaluated jointly – not separately. Multiplicity adjustment must then take both the number of treatment comparisons and the number of endpoints into account. The family-wise error type I (FWE) must be maintained with regard to all endpoints simultaneously. In addition, their correlations are important to be considered. First, the degree of conservatism of the elementary test decisions is reduced by taking them into account. Second, effects may be erroneously ignored or masked when analysing the endpoints separately. And third, the degree of correlation is essential. For example, highly correlated endpoints do not give the same amount of information about the data as uncorrelated ones.

Hasler and Hothorn [1] have described an extension of the Dunnett procedure [2] to the case of multiple, correlated endpoints. The focus is on simultaneous confidence intervals (SCIs) for differences of means. In a further paper [3], the authors have presented an extension of the trend test by Williams [4] and Bretz [5], respectively. Here the focus is on SCIs for ratio of means (based on Dilba et al. [6, 7]). Both procedures take the correlations of the endpoints into account, and the multiplicity adjustment includes both the number of endpoints and the common treatment comparisons. A related approximate multivariate *t*-distribution is used to obtain quantiles for SCIs and test decisions or to obtain multiplicity-adjusted *p*-values. The FWE is maintained in the strong sense in an admissible range. The procedures assume multivariate normally distributed endpoints with possibly different scales, allowing endpoint-specific variances.

However, one further assumption is that the covariance matrices – containing the covariances of the endpoints – are equal for all treatments. This is not fulfilled in situations when variances or correlations differ due to the treatment groups. This problem is briefly addressed by Hasler and Hothorn [1, 3], but a solution is only suggested. In addition, the suggestions made in these two papers are slightly different. Moreover, many-to-one comparisons based on Dunnett [2], or trend test based on Williams [4] and Bretz [5] are special cases of multiple contrast tests (MCTs), which allow the evaluation of a broad class of linear testing problems, such as the all-pair comparison of Tukey [8], or any user-defined contrast tests. This article presents an extension of the methods of Hasler and Hothorn [1, 3] to the general case of MCTs for multivariate normally distributed endpoints with unequal covariance matrices for the treatment groups. Former suggestions are considered in detail. In Section 2, the testing problem is formulated, approximate distributions of the test statistics are derived for several approaches. Section 3 shows results of simulations concerning the FWE. SCIs are presented in Section 4. An example is given in Section 5, conclusions and a discussion in Section 6.

## 2 Testing problem and test procedures

### 2.1 Testing problem

For *j*th observation on the *i*th endpoint under the *h*th treatment in a one-way layout. Each endpoint is measured for all *k*-variate normal distributions with mean vectors

Let

Of interest are the contrasts

representing linear combinations of the means

where *i*. The method described here is sufficiently general to allow for both comparison-specific and also for endpoint-specific contrast coefficients and thresholds. If the test direction is to be reversed for some endpoints, the corresponding test statistics have to be multiplied by minus one. Testing problem (1) is a union–intersection test because the overall null hypothesis of interest can be expressed as an intersection of the local null hypotheses, i.e.

This means that the overall null hypothesis

### 2.2 Test procedures

For the case of equal covariance matrices,

where *i*. Hasler and Hothorn [1] have shown that the joint distribution of *qk*-variate *t*-distribution. The authors considered contrasts that were related specifically to many-to-one comparisons based on Dunnett [2], but their conclusions are also valid here for general contrasts *qk*-variate *t*-distribution with degree of freedom

and correlation matrix

The submatrices *l*th and the

where

For the case of unequal covariance matrices, i.e. there exist at least two groups

Under *t*-distributed with degree of freedom

according to Satterthwaite [9]. The normal strategy would be now to derive the (approximate) joint distribution of all test statistics *qk*-variate normal distribution. However, in contrast to the homoscedastic case, the covariance matrices here additionally are unequal for the treatment groups. For that reason, the plug-in procedure (PI) for heterogeneous variances of Hasler and Hothorn [10] will be adopted and extended to the case of multiple endpoints, as was suggested by Hasler and Hothorn [1]. Therefore, *qk* different approximate *qk*-variate *t*-distributions must be applied, each with a contrast- and endpoint-specific degree of freedom according to eq. (6). Hence, each test statistic *p*-value is calculated based on its own *qk*-variate *t*-distribution. The correlation matrix *qk*-variate *t*-distributions has the same structure as

This procedure is referred to here as the CE procedure, indicating that contrast- and endpoint-specific degrees of freedom are used.

As an alternative to the above-mentioned approach, Hasler and Hothorn [3] suggested another version, namely the same test statistics and correlation matrix as for the CE procedure but only contrast-specific degrees of freedom. Therefore define

For each contrast, the minimum of the degrees of freedom (6) is taken over the endpoints. Hence, *q* different *qk*-variate *t*-distributions will be applied, and

Considering the elements of the correlation matrix belonging to CE and MIN, one can see that the correlations of MCTs in the presence of heteroscedasticity [10] are recovered for

The decision rule for testing problem (1) is to reject *qk*-variate *t*-distribution. For the computation of the quantiles or adjusted *p*-values, one may resort to the numerical integration routines of Genz and Bretz [11] and Bretz et al. [12], which are available in the package

mvtnorm[13, 14] of the statistical software R [15].

## 3 Simulations concerning the FWE

As already in the case of equal covariance matrices [1], the derivation of the exact joint distribution of the test statistics would be a challenging problem. The endpoints have different scales, their covariances must be estimated, and the covariance matrices are unequal. In this article, approximations are used based on multivariate *t*-distributions. Therefore, a validation was done by simulations. Three and five treatment groups, respectively, have been compared in a simulation study. The first group was regarded as the control. The study had different numbers of endpoints with related expected values:

mvtnorm[13, 14] and

SimComp[16].

Tables 1 and 2 show the simulated

According to Xu et al. [17] and Liu et al. [18], applying multivariate *t*-distributions in the context of multiple endpoints and using the method of Genz and Bretz [11] may lead to slightly liberal test decisions. Also for that reason, the degrees of freedom for the MIN procedure according to eq. (8) are defined in a conservative manner. For each contrast, the minimum of the degrees of freedom (6) is taken over the endpoints.

Groups | Endpoints | Correlations | Procedures | |||

MIN | CE | BON | HOM | |||

0.049 | 0.049 | 0.048 | 0.141 | |||

0.050 | 0.052 | 0.049 | 0.135 | |||

0.052 | 0.053 | 0.042 | 0.121 | |||

0.050 | 0.052 | 0.045 | 0.131 | |||

0.053 | 0.058 | 0.056 | 0.200 | |||

0.051 | 0.056 | 0.052 | 0.190 | |||

0.050 | 0.052 | 0.031 | 0.131 | |||

0.050 | 0.054 | 0.046 | 0.173 | |||

0.048 | 0.056 | 0.053 | 0.267 | |||

0.047 | 0.055 | 0.050 | 0.250 | |||

0.057 | 0.061 | 0.030 | 0.158 | |||

0.047 | 0.054 | 0.044 | 0.230 | |||

0.053 | 0.053 | 0.047 | 0.160 | |||

0.050 | 0.051 | 0.044 | 0.157 | |||

0.051 | 0.052 | 0.038 | 0.126 | |||

0.054 | 0.055 | 0.046 | 0.148 | |||

0.052 | 0.058 | 0.051 | 0.225 | |||

0.056 | 0.060 | 0.050 | 0.208 | |||

0.051 | 0.053 | 0.030 | 0.145 | |||

0.047 | 0.051 | 0.041 | 0.192 | |||

0.051 | 0.059 | 0.052 | 0.318 | |||

0.048 | 0.055 | 0.048 | 0.300 | |||

0.058 | 0.062 | 0.028 | 0.166 | |||

0.055 | 0.061 | 0.049 | 0.280 |

Groups | Endpoints | Correlations | Procedures | |||

MIN | CE | BON | HOM | |||

0.049 | 0.049 | 0.038 | 0.151 | |||

0.047 | 0.050 | 0.040 | 0.146 | |||

0.050 | 0.051 | 0.033 | 0.128 | |||

0.048 | 0.050 | 0.037 | 0.137 | |||

0.044 | 0.050 | 0.040 | 0.219 | |||

0.050 | 0.056 | 0.043 | 0.205 | |||

0.052 | 0.055 | 0.028 | 0.146 | |||

0.049 | 0.053 | 0.037 | 0.189 | |||

0.049 | 0.059 | 0.045 | 0.300 | |||

0.050 | 0.060 | 0.043 | 0.275 | |||

0.054 | 0.059 | 0.023 | 0.164 | |||

0.049 | 0.060 | 0.040 | 0.258 | |||

0.050 | 0.050 | 0.034 | 0.183 | |||

0.050 | 0.052 | 0.036 | 0.170 | |||

0.052 | 0.053 | 0.033 | 0.140 | |||

0.052 | 0.053 | 0.033 | 0.167 | |||

0.050 | 0.056 | 0.039 | 0.268 | |||

0.048 | 0.054 | 0.038 | 0.251 | |||

0.054 | 0.057 | 0.025 | 0.166 | |||

0.052 | 0.056 | 0.035 | 0.216 | |||

0.050 | 0.059 | 0.040 | 0.371 | |||

0.051 | 0.059 | 0.040 | 0.355 | |||

0.062 | 0.066 | 0.026 | 0.188 | |||

0.051 | 0.061 | 0.038 | 0.312 |

Although the procedures presented allow unequal covariance matrices for the treatment groups, the multivariate normal distribution is still a strong assumption. Therefore, a further simulation study was done to check how sensitive the proposed methods are to violations of the multivariate normal distribution. Similarly to the above study, three and five treatment groups, respectively, have been compared. The first group was regarded as the control. The study had different numbers of correlated log-normally distributed endpoints based on multivariate normally distributed data with related parameters:

Tables 3 and 4 show the simulated

Groups | Endpoints | Procedures | |||

MIN | CE | BON | HOM | ||

0.028 | 0.028 | 0.024 | 0.077 | ||

0.025 | 0.026 | 0.023 | 0.093 | ||

0.027 | 0.029 | 0.024 | 0.113 | ||

0.034 | 0.035 | 0.028 | 0.099 | ||

0.034 | 0.035 | 0.028 | 0.106 | ||

0.032 | 0.035 | 0.026 | 0.150 |

Groups | Endpoints | Procedures | |||

MIN | CE | BON | HOM | ||

0.025 | 0.025 | 0.018 | 0.074 | ||

0.021 | 0.022 | 0.015 | 0.089 | ||

0.021 | 0.024 | 0.017 | 0.117 | ||

0.028 | 0.029 | 0.018 | 0.093 | ||

0.031 | 0.034 | 0.021 | 0.123 | ||

0.030 | 0.035 | 0.022 | 0.162 |

## 4 Simultaneous confidence intervals

For test decisions, as well as for the estimation of the contrasts

where *qk*-variate *t*-distribution, and

## 5 Example

Homma et al. [19] published summary data for five multiple, continuous endpoints of the randomized, placebo-controlled phase IIb dose-finding study of a novel anti-muscarinic agent. For a one-way layout with a zero-dose placebo group (*incontinence episodes per week (Iepw)*, (2) *urgency incontinence episodes per week (Uiepw)*, (3) *micturitions per day (Mpd)*, (4) *urgency episodes per day (Uepd)*, and (5) *urine volume voids per micturition (Uvvpm)*. Although these endpoints are baseline-adjusted ratios, they are treated as approximately normally distributed endpoints. The authors published means, standard deviations, sample sizes, and adjusted *p*-values of separate Dunnett [2] procedures. Because the raw data were not available, multivariate normally distributed data were generated covering Table 2 of Homma et al. [19] by using the package

SimComp[16], command

ermvnorm(), of the statistical software R [15]. For these data, the summary statistics are given in Table 5. The correlation matrix was not given by Homma et al. [19]. Therefore, the semi-synthetic example data are based on the same theoretical correlation matrix for all treatment groups, namely

representing plausible highly positive (e.g. 0.8 for Uiepw and Uepd) and lightly negative (e.g. –0.3 for Mpd and Uvvpm) correlations for these multiple urinary endpoints. Note that means and standard deviations of the generated and of the original data set are exactly the same. Actually, some changes to the baseline were negative. The related data were multiplied by minus one, so that all endpoints have the same positive direction and higher values indicate a better treatment effect. The standard deviations per endpoint clearly differ depending on the treatment group. For example, endpoint *Uiepw* has standard deviations 272.76 (Placebo), 72.88 (Imid0.1), 41.11 (Imid0.2), and 93.45 (Imid0.5).

Endpoint | Placebo | Imid0.1 | Imid0.2 | Imid0.5 | ||||

Iepw | 42.86 | (70.17) | 59.81 | (61.48) | 71.61 | (43.95) | 82.19 | (28.68) |

Uiepw | 18.94 | (272.76) | 57.07 | (72.88) | 75.67 | (41.11) | 74.20 | (93.45) |

Mpd | 1.07 | (1.93) | 1.72 | (2.11) | 1.59 | (1.89) | 2.33 | (2.20) |

Uepd | 38.12 | (62.58) | 60.29 | (43.51) | 57.37 | (53.28) | 62.31 | (32.64) |

Uvvpm | 2.29 | (42.70) | 14.06 | (37.50) | 9.89 | (37.64) | 26.11 | (43.79) |

Sample size | 95 | 91 | 93 | 76 |

These data are the same as already used in Hasler and Hothorn [3]. The authors considered SCIs for ratios of means, and contrasts related to the trend test of Williams [4]. Here, SCIs for differences of means, and contrasts related to Dunnett [2] are applied. The Dunnett-type contrast matrix is given by

where the rows represent the comparisons versus the placebo, and the columns represent the treatment groups. The differences of interest are

and the hypotheses to be tested are given by

Table 6 shows the estimated differences to the placebo (Estimate), *p*-values of Dunnett tests, separately for the endpoints and assuming homogeneous variances for the dose groups according to Homma et al. [19] (*p*-val. (prev.), adjusted *p*-values according to the multivariate MIN procedure described (*p*-val. (adj.), and related lower simultaneous confidence limits (Lower limit) for each dose and each endpoint. By definition, the multivariate procedure is more conservative than elementary Dunnett tests separately for the endpoints, each at level *p*-val. (prev.)). Consequently, column *p*-val. (adj.) in Table 6 has substantially higher values than column *p*-val. (prev.), i.e. there is a price to pay for selecting at least one of *k* endpoints. This fulfils the conservativeness principle of claims in randomized clinical trials. Furthermore, Homma et al. [19] have assumed homogeneous variances for the dose groups, which does not seem to be fulfilled and causes biased test decisions. This example shows how much the assumptions about the data and the method for incorporating the endpoints can influence test decisions. All dose effects for endpoint *Uiepw* are insignificant, whereas according to Homma et al. [19] the Imid0.2 effect is significant. Except for *Uiepw*, Imid0.5 shows a significant effect for all endpoints. For example, Imid0.5 causes at least 18.55 percent more change from baseline compared to placebo for Iepw. The two lower doses, Imid0.2 and Imid0.1, show significances for *Iepw* and *Uepd*, respectively.

Dose | Endpoint | Estimate | p-val. (prev.) | p-val. (adj.) | Lower limit |

Imid0.1 | Iepw | 16.95 | 0.0906 | 0.2978 | −8.46 |

Imid0.1 | Uiepw | 38.13 | 0.2287 | 0.5300 | −38.12 |

Imid0.1 | Mpd | 0.65 | 0.0791 | 0.1356 | −0.13 |

Imid0.1 | Uepd | 22.17 | 0.0079 | 0.0312 | 1.46 |

Imid0.1 | Uvvpm | 11.77 | 0.1367 | 0.1965 | −3.71 |

Imid0.2 | Iepw | 28.75 | 0.0010 | 0.0063 | 6.31 |

Imid0.2 | Uiepw | 56.73 | 0.0335 | 0.1950 | −17.81 |

Imid0.2 | Mpd | 0.52 | 0.1920 | 0.2485 | −0.21 |

Imid0.2 | Uepd | 19.25 | 0.0246 | 0.1147 | −3.05 |

Imid0.2 | Uvvpm | 7.60 | 0.4653 | 0.5412 | −7.85 |

Imid0.5 | Iepw | 39.33 | <0.0001 | <0.0001 | 18.55 |

Imid0.5 | Uiepw | 55.26 | 0.0541 | 0.2571 | −23.42 |

Imid0.5 | Mpd | 1.26 | 0.0002 | 0.0009 | 0.42 |

Imid0.5 | Uepd | 24.19 | 0.0053 | 0.0086 | 4.68 |

Imid0.5 | Uvvpm | 23.82 | 0.0006 | 0.0031 | 6.33 |

The package

SimComp[16] of the statistical software R [15] provides calculations concerning simultaneous tests and confidence intervals for both difference- and ratio-based contrasts of normal means for data with possibly more than one primary endpoint. The covariance matrices – containing the covariances between the endpoints – may be assumed to be equal or possibly unequal for the different groups. (The MIN procedure is realized for the latter case.) This package was used to re-generate and to analyse the example data. It is available at http://www.r-project.org. The input for the results of Table 6 is approximately:

SimCiDiff(data=data object, grp="name of the grouping variable", resp=c("name of endpoint 1"," name of endpoint 2","..."), type="Dunnett", base=alphanumerical number of the control group, alternative="greater", covar.equal=FALSE).

For the related *p*-values use the command

SimTestDiff(), and for ratio-based testing and intervals

SimTestRat()and

SimCiRat(), respectively.

## 6 Conclusions and discussion

For the special cases of contrasts related to Dunnett [2] and Williams [4], Hasler and Hothorn [1, 3] had proposed extensions to the case of multiple endpoints. Their approaches have been generalized in this article to any MCTs and to the situation of heteroscedasticity, i.e. unequal covariance matrices *t*-distributions will be applied, based on the approach of Hasler and Hothorn [10]. Correlations among both the contrasts and the endpoints will be taken into account. Test decisions – adjusted *p*-values as well as SCIs – are available for all contrasts and all endpoints. The intervals and tests may be one- or two-sided. An adaption to a formulation based on ratios of means is possible.

In the presence of heteroscedasticity, the CE procedure with contrast- and endpoint-specific degrees of freedom tends to liberal test decisions, whereas the MIN procedure with (minimized) contrast-specific degrees of freedom maintains the FWE in the strong sense in a passable range. This was shown by simulations. Possible slight variations around the nominal FWE

If the assumption of a multivariate normal distribution for the data is not fulfilled, the procedures considered cannot be recommended without caution. The simulation results show that they yield conservative test decisions if the data follow distributions with a positive skew. In this case or for other non-normal distributions, non-parametric procedures – like those of Munzel and Brunner [22]; Bathke and Harrar [23]; Harrar and Bathke [24, 25] – should be used instead. Note that these procedures do not provide multiplicity-adjusted *p*-values or SCIs for each contrast-endpoint combination as they use ANOVA-type *F* statistics, respectively.

The main advantage of usual MCTs over other testing procedures is that the correlations of the contrasts are taken into account. These correlations mainly depend on whether and which treatment means are involved in the contrasts. All treatment means, however, which are involved in the same contrast are independent. Consequently, the MIN procedure can also be used for the analysis of repeated measures by simply regarding the time points as endpoints, as long as the condition

A solution to the following problem might be a task for the future: the MIN procedure presented generally assumes unequal covariance matrices which occur if variances or correlations of some endpoints differ depending on the treatment groups. The procedure does not distinguish whether unequal correlations or unequal variances lead to unequal covariances. In practice, however, it can occur that just the variances differ, and the correlation structure stays the same for the treatment groups (or the other way round).

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**Published Online:**2014-2-11

**Published in Print:**2014-5-1

© 2014 by Walter de Gruyter Berlin / Boston