## 1 Introduction

Random coefficient regression (RCR) models were originally introduced in plant and animal breeding and used for selection purposes (see e. g. Henderson (1984)). The subject of this paper is the multiple group random coefficient models, in which observational units (individuals) are allocated in several treatment groups and one control group. In each treatment group some group-special kind of treatment is available, in the control group there is no treatment. Such models are typically used for cluster randomization or cluster randomized trials (see e. g. Bland (2004) or Patton et al. (2006)).

RCR models with known population parameters were considered in detail by Gladitz and Pilz (1982). Entholzner et al. (2005) investigated optimal designs for estimation of unknown population mean parameters in RCR models, where all individuals are observed under the same regime. Analytical results for designs, which are optimal for the prediction of individual random parameters in hierarchical models with the same treatment for all individuals, have been presented in Prus and Schwabe (2016). A practical approach for computation of optimal approximate and exact designs was proposed by Harman and Prus (2018).

The fixed effects version of the multiple group models considered in this paper may be recognized as the well known one way layout model. Classical one-way layout models have been well discussed in the literature. Results for the optimal designs can be found e. g. in Bailey (2008), Rasch and Herrendörfer (1986), Wierich (1986), Schwabe (1996) or Majumdar and Notz (1983).

Optimal designs for estimation of fixed parameters in multiple group models with random coefficients were considered e. g. in Fedorov and Jones (2005), Schmelter (2007), Kunert et al. (2010), Bludowsky et al. (2015) and Lemme et al. (2015). Fedorov and Jones (2005)

worked on optimal designs, which minimize a loss function in multicentre trials models. In

Schmelter (2007) models with the same fixed number of observations in all groups were investigated. Kunert et al. (2010) proposed a design optimization method based on the generalized least squares estimation. Bludowsky et al. (2015) considered models with carryover effects. In Lemme et al. (2015) optimal designs were computed for the maximum likelihood estimation.Optimal designs for prediction of random effects in models, where the population mean parameters differ from group to group, were briefly discussed in Prus (2015).

In this paper we investigate multiple group models with the same unknown population parameters across all groups. We present analytical results for A-, D- and E-optimal designs (-optimal group sizes) based on the best linear unbiased estimation or prediction of fixed or random treatment effects, respectively.

The paper has the following structure: In Section 2 the model will be specified. In Section 3 the best linear unbiased estimation for the population parameters (mean treatment effects) and a best linear unbiased prediction for the random treatment effects of the observational units will be discussed. Section 4 provides analytical results for designs, which are optimal for estimation or prediction. The results will be illustrated by a numerical example, in which we compare the optimal group sizes in the model under investigation with optimal group sizes in the fixed effects model (one-way-layout). The paper will be concluded by a discussion of the obtained results and possible directions for the future research in Section 5.

## 2 Model Specification

We consider here a multiple group model with groups and individuals. In the first (treatment) groups individuals get group-special kinds of treatment: . Each treatment group includes individuals. The last group (group ) is a control group (no treatment) with , individuals. The -th observation at individual in the -th treatment group is given by the following formula:

(1) |

while in the control group the -th observation at the -th individual is given by

(2) |

where is the number of observations per individual, which is assumed to be the same for all individuals across all groups,

are observational errors with zero expected value and common variance

.In this work we optimize the numbers of individuals in treatment and control groups. Therefore the group allocation of individuals is not completely clear. For this reason we define the individual treatment effects for all individuals and all treatments, i. e. and . For individual in the control group the treatment effects would appear if the individual would be treated with treatment . For individual in treatment group the treatment effect would appear if the individual would get treatment instead of for , .

Individual intercepts are defined for all individual, i. e. . and have unknown expected values , and variances , for some given positive values of and . All individual parameters and and all observational errors , for , and , are uncorrelated.

For further considerations we define the regression functions

where and denote the

-th unit vector and the zero vector of length

, respectively, and the total number of individualsin the first groups, and we fix by . Then for the vector of individual random parameters the multiple group model (defined by (1) and (2)) may be rewritten in the following form:

(3) |

The parameter vectors have the expected value and the covariance matrix , where denotes the identity matrix and is the block diagonal matrix with blocks .

On the individual level the vectors for individuals in the -th group can be specified as

(4) |

where denotes the vector of length with all entries equal to and .

On the group level for the group parameters the vector of all observations at all individuals in the -th group is given by

(5) |

where

is the number of individuals in the -th group, “” denotes the Kronecker product and .

Finally, we introduce the vector of all individual random parameters (or, equivalently, ), for which the full vector of observations at all individuals in all groups is given by the following formula:

(6) |

where is the block diagonal matrix with blocks , , and .

Alternatively, for the random vector the model (6) can be represented in the form

(7) |

where for some matrices , .

Note that for , . The expected value of is zero and .

## 3 Estimation and Prediction

In this chapter we determine the best linear unbiased estimator (BLUE) for the population mean parameters

and and the best linear unbiased predictor (BLUP) for the individual random parameters and for and .We denote the mean observation in group by and the mean observation at individual in group by for all , . Then we obtain the following results for the BLUEs of the mean parameters and and the BLUPs of the treatment effects and .

###### Theorem 1.

The BLUE for the population intercept parameter is given by

(8) |

and the BLUE for the population treatment effect is given by

(9) |

###### Theorem 2.

If the -th individual is in the control group, the BLUP for the individual intercepts is given by

(10) |

otherwise, if the -th individual is in the -th treatment group, the BLUP is given by

(11) |

If the -th individual is in the -th treatment group, the BLUP for the individual treatment effect is given by

(12) |

otherwise the BLUP is given by

(13) |

For the vector of the mean treatment effects the BLUE is given by .

We denote the vector of all individual treatment effects by , where is the vector of individual treatment effects for the -th individual, . Then the BLUP of is given by , where is the BLUP for , .

The next theorems provide the covariance matrix of and the MSE matrix of .

###### Theorem 3.

The covariance matrix of the BLUE is given by

(14) |

###### Theorem 4.

The mean squared error matrix of the BLUP is given by

(15) |

where

where denotes the zero matrix and , and

## 4 Optimal Design

In this chapter we optimize the numbers and of individuals in the treatment and control groups, respectively. We define the exact experimental design as

(16) |

where the indexes , …, denote the treatment groups and the index is used for the control group.

For analytical purposes, we also define the approximate design:

(17) |

where is the weight of a treatment group and is the weight of the control group. Then only the optimal weight of a treatment group has to be determined.

### 4.1 A-criterion

For the estimation of the population treatment effects the A-criterion for an exact design is defined as the trace of the covariance matrix of the BLUE :

We determine the trace of the covariance matrix (14), replace by and by and obtain the next results for an approximate design.

###### Theorem 5.

The A-criterion for the estimation of the population treatment effects is given for an approximate design by

(18) |

###### Theorem 6.

The A-optimal weight for the estimation of the population treatment effects is given by

(19) |

Note that for large values of the intercepts variance () the optimal weight (19) tends to the value , which coincides with the optimal weight in the fixed effects model (see e.g. Bailey (2008), ch. 3 or Schwabe (1996), ch. 3) . If the treatment effects variance takes a very large value (), the limiting optimal design assigns all observations to be taken in the treatment groups: . If both variances are large and the variance ratio is fixed, the limiting optimal design is given by

The A-criterion for the prediction of the individual treatment effects is defined for an exact design as the trace of the mean squared error matrix of the BLUP :

The next theorem presents the A-criterion for an approximate design.

###### Theorem 7.

The A-criterion for the prediction of the individual treatment effects is given for an approximate design by

(20) |

Note that there is no explicit formula for the optimal weight in this case. However, it is easy to see that there is a unique solution , which may be determined numerically for given values of , , , and . To illustrate the behavior of optimal designs, we consider a numerical example.

Example 1. Let the total number of individuals be and the number of observations per individual be and the variance ratio be fixed by the values , and . The next graphics (Figure 1 and Figure 2) illustrate the behavior of the A-optimal weight in dependence on the treatment effects variance for the special cases of one (left panel) and two (right panel) treatment groups ( and , respectively). The parameter is used instead of the variance parameter to cover all values of the treatment effects variance by a finite interval. On the graphics the solid, dashed and dotted lines present the optimal weight for the values , and of the ratio . Note that the optimal weight takes all its values in the intervals and in the models with one and two treatment groups, respectively.

For the models with one and two treatment groups the optimal the optimal weights start (for ) at points and , respectively. This may be explained by the fact that the optimal designs for the fixed effects models are equal to and result in for and for . The optimal weights increase with increasing variance of the individual treatment effects with limiting values (for ) , and for and , and for for , and , respectively.

Figure 3 and Figure 4 present the efficiency of the optimal weight from the fixed effects model for the present model for one (left panel) and two (right panel) treatment groups for the values , and of the ratio .

For both particular models the efficiencies start at point and decrease with limits , and for and , and for for , and .

### 4.2 D- and E-criterion

In this section we consider D- and E-optimality criteria for the estimation and the prediction in multiple group models. We consider the general case of model (7) for the estimation of population parameters and we restrict ourselves to the special case for the prediction of individual treatment effects.

For further considerations we will use the following result.

###### Lemma 1.

The eigenvalues of the covariance matrix of the BLUE

arewith algebraic multiplicity and

with algebraic multiplicity .

###### Proof.

To determine the eigenvalues of the covariance matrix we solve the equation

(21) |

where denotes an eigenvalue of .

Then

and

are the solutions of (21). ∎

For the estimation of the population treatment effects the D-criterion is defined as the logarithm of the determinant of the covariance matrix of the BLUE :

for an exact design. We compute the determinant as the product of the eigenvalues, which are given in Lemma 1, and receive using the following result for approximate designs.

###### Theorem 8.

The D-criterion for the estimation of the population treatment effects is given for an approximate design by

(22) |

where .

###### Theorem 9.

The D-optimal weight for the estimation of the population treatment effects is given by

(23) |

where .

The E-criterion for the estimation of the population treatment effects is defined for an exact design as the largest eigenvalue of the covariance matrix of the BLUE

where denotes the largest eigenvalue of the matrix .

Using Lemma 1 we receive the following form of the E-criterion for approximate designs.

###### Theorem 10.

The E-criterion for the estimation of the population treatment effects is given for an approximate design by

(24) |

###### Theorem 11.

The E-optimal weight for the estimation of the population treatment effects is given by

(25) |

Note that also for the D- and E-criteria the optimal weights for the estimation of the population parameters ((23) and (25)) tend to the optimal weights in the fixed effects model: and , for . For large values of the treatment effects variance () all observations should to be taken in the treatment groups: . If both variances are large and , the limiting values for the optimal weights are

and

For the prediction of the individual treatment effects we consider the particular multiple group model with one treatment group and one control group (). In this case the mean squared error matrix (15) of the prediction simplifies to

(26) |

where

The approximate design (17) simplifies to

The next Lemma provides the eigenvalues of the mean squared error matrix (26).

###### Lemma 2.

The eigenvalues of the mean squared error matrix of the BLUP are

where , and

with algebraic multiplicities , , and , respectively.

For the proof see Appendix A.2.

We define the D-criterion for the prediction as the logarithm of the determinant of the mean squared error matrix of :

We compute the determinant using the results of Lemma 2 and obtain the following criterion for approximate designs.

###### Theorem 12.

The D-criterion for the prediction of the individual treatment effects is given for an approximate design by

(27) |

where .

###### Theorem 13.

The D-optimal weight for the prediction of the individual treatment effects is given by

(28) |

where .

We define the E-criterion for the prediction as the largest eigenvalue of the mean squared error matrix:

The E-criterion for approximate designs follows directly from Lemma 2.

###### Theorem 14.

The E-criterion for the prediction of the individual treatment effects is given for an approximate design by

(29) |

###### Theorem 15.

The E-optimal weight for the prediction of the individual treatment effects is given by

(30) |

Note that for both D- and E-criteria for the prediction the optimal weights tend to , which is optimal for the fixed effects model (see e.g. Wierich (1986), p. 44), for and to (all observations are being taken in the treatment group) for . For large , large and fixed ratio we observe

and

Example 2. We consider again the model with individuals and observations per individual. The variance ratio takes the values , and for both D-criterion and E-criteria. The number of treatment groups is fixed by one (). The next picture (Figure 5 and Figure 6) illustrates the behavior of the D- (left panel) and E- (right panel) optimal weights in dependence on the variance parameter .

As we can observe on the picture the optimal weights start with value , which is optimal for the fixed effects model for both criteria: , and increase with the variance parameter with limiting values , and for the D-criterion and , and for the E-criterion for , and , respectively.

On the graphics the values of the optimal designs for the D-criterion are larger than those for the E-criterion for all and . The A-optimal designs illustrated by Figure 1 turn out to take all their values between the corresponding D- and E-optimal weights.

Figure 7 and Figure 8 present the efficiency of the optimal designs from the fixed effects model for the present model for the D- and E- criteria.

The efficiencies on the picture start at point and decrease with increasing variance of the individual treatment effects with limiting values , and for the D- and , and for the E-criterion for , and , respectively.

According to these graphics and Figure 2 the efficiencies for the D-criterion are smaller than those for the E-criterion and the A-criterion is in between for all values of and .

Note also that for small values of the variance ratio the optimal designs are very close to those in the fixed effects model for all design criteria. Hence, the corresponding efficiencies are close to .

## 5 Discussion and Conclusions

In the present work multiple group RCR models with several treatment groups and a control group have been considered. We have obtained A-, D- and E-optimality criteria for the estimation of population parameters and for the prediction of individual treatment effects using the covariance matrix of the BLUE and the mean squared error matrix of the BLUP, respectively. The optimal designs (optimal group sizes) turned out to be different for the estimation and the prediction and do not coincide with those in the corresponding fixed-effects-model (one-way-layout). For large values of the treatment effects variance the optimal designs assign almost all observations to be taken in the treatment groups. If the variance of the individual intercepts is large, the optimal groups sizes tend to those in the fixed effects model.

The optimal group sizes are locally optimal, i. e. they depend on the variance parameters. To avoid this, minimax-optimal designs, which minimize the worst case for the criterion function over some reasonable region of values of the covariance matrix, or some other design criteria, which are robust with respect to the variance parameter, may be considered in the next step of this research.

For the investigated models we assumed a diagonal covariance matrix of random effects. Models with more complicated covariance structure may also be considered in the future.

## Acknowledgment

This research has been supported by grant SCHW 531/16-1 of the German Research Foundation (DFG).

## Appendix A Appendix

### a.1 Proofs of Theorems 1-4

To make use of the available results for estimation and prediction we recognize the model (7

) as a special case of the linear mixed model (see e. g.

Christensen (2002))(31) |

where and are known fixed and random effects design matrices, and are vectors of fixed and random effects, respectively. The random effects and the observational errors are assumed to be uncorrelated and to have zero means and non-singular covariance matrices

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