Random sample sizes in orthogonal mixed models with stability

Abstract

In this work, we present a new approach that considers orthogonal mixed models, under situations of stability, when the sample dimensions are not known in advance. In this case, sample sizes are considered realizations of independent random variables. We apply this methodology to the case where there is an upper bound for the sample dimensions, which may not be attained since failures may occur. Based on this, we assume that sample sizes are binomially distributed. We consider an application on the incidence of unemployed persons in the European Union to illustrate the proposed methodology. A simulation study is also conducted. The obtained results show the relevance of the proposed approach in avoiding false rejections.

1 INTRODUCTION

Situations where the sample dimensions are not known in advance are common when planning a study. An illustrative example of this was presented by Moreira et al1 and Nunes et al,2, 3 where the collection of observations took place during a fixed time period in a study from patients with several pathologies arriving at a hospital. Another example is the approach proposed by Nunes et al,4 where it was considered for the comparison of pathologies in the case where one of the pathologies was rare. In these situations, samples with dimensions n₁,…,n_m are assumed as realizations of independent random variables N₁,…,N_m in models with random sample sizes. In the aforementioned papers, it was assumed that the collection of observations was given by a Poisson counting process, so the sample sizes were considered as realizations of independent Poisson variables.

Nonetheless, the earliest studies in which the number of observations was considered as random go back to the 1980s, with the work of Nguyen5 and Singh and Gupta.6 In the work of Nguyen,5 the random number of observations was modeled by a Poisson process, while in the work of Singh and Gupta,6 a random number of observations was used to estimate the mean.

More recently, in the work of Esquível et al,7 some classical statistical inference was extended to the case of a random number of observations. The authors also described some models for the number of observations, obtained by truncation or translation of usual models (Poisson, binomial, geometric, and negative binomial).

The aim of the present paper is to extend the theory of mixed models to the case of random sample sizes, considering stable statistics. In these models, the test statistics have the same distribution whether referring to the fixed or the random part of the model when the tested hypothesis holds (see the work of Ferreira8).

We will use the flexibility of the model to consider experiments performed under conditions where there are many observation failures. Therefore, the dimensions of the samples cannot be fixed but only ensure minimal and maximum dimensions for the samples. So, there exist upper bounds for the sample sizes r₁,…,r_m, which may not always be attained, due to the occurrence of observations failures (see Nunes et al9, 10). In these situations, we consider that the binomial distribution is the correct choice for the sample dimensions, which lead us to assume that N₁,…,N_m are binomially distributed with parameters r₁,…,r_m and 1−p (where p denotes the probability of an observation failure, obtained from previous results). We put

Moreover, is a realization of the following random variable:

and since N_i, i=1,…,m are independent,

with . Furthermore, the vector n=(n₁,…,n_m)^t will be a realization of N=(N₁,…,N_m)^t.

We consider a mixed model with orthogonal structure. So, with , which is the mean vector of the vector of sample means, we have

where ⊗ denotes the orthogonal direct sum of subspaces and ▽_j, j=1,…,w′, represents a subspace into space Ω. We also have

where j=1,…,w′ corresponds to the fixed effect part and j=w′+1,…,w to the random effect part of the model.

The remainder of this paper is organized as follows. In Section 2, it is shown how we may test hypotheses in effects and interaction on mixed models, assuming normality. It also shows how the conditional distribution of the test statistics is obtained. Considering that we have random sample sizes, the unconditional distributions of the statistics are presented in Section 3. In Section 4, an application considering the incidence of unemployed persons in the European Union is developed to illustrated this approach. In Section 5, a simulation study is provided to analyze the proportion of false rejections, which may be avoid considering the proposed methodology. Finally, in Section 6, we make some concluding comments.

2 MODEL, HYPOTHESES, AND TEST STATISTICS

In this section, we assume that we have fixed sample sizes n₁,…,n_m. The total number of observations will be .

Let us assume that the row vectors of matrix A_j constitute an orthonormal basis for ▽_j. The orthogonal projection of a vector of ▽_j, , is null if and only if

which is equivalent to belonging to the orthogonal complement of ▽_j, .

Let us now consider the observations vector given by

where

with being a blockwise diagonal matrix with principal elements , and

with , normal independent with null mean vectors and covariance matrices , j=w′+1,…,w, and g_j=rank(A_j). The , j=w′+1,…,w, are independent from

So, the vector of the sample means is

where the components of are the means of the components of , corresponding to the different samples. Therefore,

Thus, will be independent of

with , where I_r denotes the identity matrix of order r.

Therefore,

and

So, for the fixed effect part of the model, we consider the hypotheses

which may be rewritten as

while for the random effect part, we have

which are equivalent to

We can write all the hypotheses (whether for fixed and random effects part of the model) as

Thus, we have stability in these situations (see the work of Ferreira8).

So, when H_0, j holds,

will be the product by σ² of a central chi‐square with g_j degrees of freedom, (see Mexia11).

Since is independent of , S_j,j=1,…,w, will be independent of

with

For example, see the work of Khuri et al12 and Searle et al.13

Thus, when H_0,j, j=1,…,w, holds, the statistics

have central F distributions with g_j, j=1,…,w, and n−m degrees of freedom F(.|g_j,n−m).

In the fixed effect case, when H_0,j, j=1,…,w′, does not hold, S_j will be the product by σ² of a noncentral chi‐square with g_j degrees of freedom and noncentrality parameter

, and consequently ℑ_j will have a noncentral F distribution with g_j and n−m degrees of freedom and noncentrality parameter δ_j, F(.|g_j,n−m,δ_j). The power of the F test increases with δ_j, j=1,…,w′ (see the work of Mexia11).

On the other hand, in the random effect case, when H_0,j,j=w′+1,…,w, does not hold, we have

and ℑ_j will be the product by of a random variable with central F distribution with g_j and n−m degrees of freedom. In this case, the test power increases with γ_j, j=w′+1,…,w (see the work of Lehmann and Romano14).

3 UNCONDITIONAL DISTRIBUTIONS

When we consider random sample sizes, we concentrate our study on the distribution when H_0,j,j=1,…,w, holds. So, the conditional distribution of the statistics ℑ_j, j=1,…,w, will be

In this section, we will uncondition the results obtained in the previous section to the random variables N₁,…,N_m. To perform the analyses, it is necessary to have at least one observation per treatment, which means that N_i ≥ 1, i=1,…,m. To accomplish inference, we also need to admit that N>m. On the other hand, to avoid highly unbalanced cases, we will consider a global minimum dimension for the samples n^•, with n^•>m (see the work of Mexia et al15 and Nunes et al2, 3).

So, we have N_i∼B(r_i,1−p), where r₁,…,r_m denote the upper bounds for the sample sizes and p the probability of an observation failure. We assume that this probability is known from previous results. Through the independence of N_i, i=1,…,m, we also have N∼B(r,1−p), with . To estimate r (and consequently r_i, i=1,…,m), we can assume that we have a fixed probability q of taking an observation, with r being the least value of the parameter in the binomial distribution that ensures

Considering several global minimum dimensions for the samples (n^•=9, n^•=13, n^•=20, n^•=32) and a probability q=0.95, we obtained the minimum value of r presented in Table 1. The execution time (in seconds) is also provided. The results show that, for different values of probability p, we have reasonable upper bounds for the sample sizes.

Table 1. Minimum value of r for a probability q=0.95

n^•	p = 0.1	p = 0.2	p = 0.3	p = 0.4	p = 0.5	time (s)
9	12	14	17	21	26	0.103
13	17	20	24	28	35	0.104
20	25	29	35	42	51	0.107
32	39	46	53	64	78	0.108

Therefore, for a known value of p and setting the global minimum dimension n^•, we will take

(1)

and the unconditional distribution of ℑ_j will be given by (see the work of Nunes et al3)

4 AN APPLICATION

In this section, we will apply the presented methodology to the incidence of unemployment in the European Union. The data comes from PORDATA16 and gathers information regarding the age of the unemployed persons.

Given the sizes of the samples, we worked with asymptotic normal distributions to ensure the use of F tests (see the work of Ito17). We will consider a two‐way model with one fixed effect and one random effect factors. The fixed effect factor will be the time periods, with two levels: 2006‐2007 (before financial crisis) and 2012‐2013 (during financial crisis). The random effect factor will be the Countries. Due to the large number of countries, we resorted to the simple random sampling method to select four different countries from the available list. This leads to m=2×4=8 different treatments. Table 2 illustrates the sample mean age for each selected country considering the two time periods.

Table 2. Sample mean ages

		Time periods (fixed effect factor)
		2006‐2007	2012‐2013
Countries (random effect factor)	Slovakia	35.47186	33.96895
	Spain	35.33543	35.88430
	Greece	30.58468	34.92408
	Czech Republic	34.85264	35.03557

We will test the hypotheses

which correspond to the hypotheses of absences of fixed effects, random effects, and interaction.

We will consider the balanced case, which is robust in the presence of non‐normality and heteroscedasticity (see the work of Scheffé18). In this case, n_i=6, i=1,…,8, which correspond to the number of different classes for the age of the unemployed persons. So, .

Therefore, when H_0,j, j=1,2,3, holds, assuming N=n, we have

The unconditional distribution of ℑ_j is now given by

where is defined by 1.

Moreover, with n<n′, we have (see the work of Mexia et al15)

and consequently,

Thus, F(z|g_j,n^•−8) gives us a lower bound for , j=1,2,3, from which we can obtain upper bounds for the quantiles of , j=1,2,3.

When we consider these upper bounds as critical values, the derived test's sizes do not exceed the theoretical values (see the work of Nunes et al3). We can use these upper bounds for a preliminary test. If the statistic's value exceeds the upper bounds, it also exceeds the real critical value (obtaining considering random sample sizes), and in this case we reject the null hypothesis. When the statistic's value is lower than the upper bound, we must compute the real critical values or calculate the minimum value of n^•, which leads to reject the null hypothesis (eg, see the work of Nunes et al3, 19).

The interaction belongs to the random effect part of the model. Although the analysis usually starts with an interaction test and follows with the tests to the main effects, whenever it is not significant, we do not follow this approach since we are interested in showing how these tests could be carried out through unconditioning (see the work of Nunes et al3).

All computational procedures, namely, the quantiles of the conditional distributions and the upper bounds for the quantiles of the unconditional distributions, as well as the computations in Section 3 and 5, were conducted through R software.

4.1 Fixed effect factor

For the fixed effect factor, we have

and g₁=rank(A₁)=1. We obtain

for the numerator of the statistics ℑ₁, where has as components the values presented in Table 2.

When N=n, we have for the denominator of the statistics. In this example, we obtain

So, ℑ₁∼F(z|1,40), and its observed value is given by

The quantiles of the conditional distribution of ℑ₁, z_1−α, provided in Table 3, lead to the nonrejection of H_0,1 considering α=0.10, 0.05, and 0.01.

Table 3. Conditional quantiles and upper bounds for unconditional quantiles of ℑ₁

α	0.10	0.05	0.01
z1−α	2.8354	4.0847	7.3141
	4.0604	6.6079	16.2582

Let us now assume that N ≥ 13, which means that n^•=13. Table 3 also presents the upper bounds for the quantiles for probability 1−α of the unconditional distribution of ℑ₁, . So, assuming n^•=13, we decide do not reject H_0,1 for the usual levels of significance.

The quantiles for random sample sizes (obtained when using the unconditional distribution) will exceed the classical ones (obtained when using common conditional distribution) since a new source of variation is considered. So, since in this case we do not reject the null hypothesis using the classical quantiles, we take the same decision using the quantiles for random sample sizes and consequently considering the upper bound approach. Therefore, we conclude that the fixed effect factor is not significant.

4.2 Random effect factor

For the random effect factor, we have the matrix

where g₂=rank(A₂)=3, and for the numerator of the statistics, we have

In this case, ℑ₂∼F(z|3,40), and its observed value is

Considering the quantiles of the conditional distribution of ℑ₂, z_1−α, given in Table 4, we decide to reject H_0,2 for the usual values of α.

Table 4. Conditional quantiles and upper bounds for unconditional quantiles of ℑ₂ and ℑ₃

α	0.1	0.05	0.01
z1−α	2.2261	2.8387	4.3126
	3.6195	5.4095	12.0600

Assuming that the global minimum dimension is n^•=13, the upper bounds for the quantiles for the unconditional distribution of ℑ₂, , are presented in Table 4. These results lead us to reject the hypothesis for α=0.1 and do not reject for 0.05 and 0.01, which means that the decision made for both approaches are not concordant for α=0.05 and 0.01.

Table 5 presents the total sample sizes needed to reject the hypothesis as well as the execution time (in seconds). So, we have for all n^• ≥ 27 (see Nunes et al3). In this case, we reject H_0,2 considering the usual values of α, and the random effect factor is significant, which means that the unemployed's age is significantly different for different countries.

Table 5. Minimum value of n^• that leads to reject the hypotheses H_0,2 and H_0,3

α	0.1	0.05	0.01	time (s)
n^•	12	14	27	0.036

4.3 Interaction between factors

For the interaction, we have the matrix

with g₃=rank(A₃)=3. For the numerator of the statistics, we have

(2)

and consequently, the observed value of the statistic ℑ_3,Obs is given by

Considering the quantiles for the conditional distribution presented in Table 4, we decide to reject H_0,3 for α=0.10, 0.05, and 0.01.

Through the upper bounds for the quantiles, , of the unconditional distribution presented in Table 4, obtained considering n^•=13, we decide to reject H_0,3 for α=0.1 and do not reject for 0.05 and 0.01. So, these results show that the decision depends on the approach when we consider α=0.05 and 0.01.

We should have the total sample sizes presented in Table 5 for ensuring rejection. Therefore, we have for all n^• ≥ 27, which means that we reject H_0,3 considering the usual values of α. In this case, the interaction between factors is significant.

Our discussion shows the following:

• • The fixed effect factor is not significant, which means that the unemployed's age is not significantly different for these two periods.
• • Considering a sufficiently large dimension for the samples, the random effect factor and interaction are significant.
• • For the random effect factor and interaction, we conclude that the common approach may drives in a false rejection for α = 0.05 and 0.01. A rejection is considered a false rejection whenever the null hypothesis is rejected by the conditional but not rejected by the unconditional approach.

5 A SIMULATION STUDY

In this section we carry out to a simulation study to verify the proposed method and analyze the proportion of false rejections which may be avoided considering our approach.

In line with Section 4, we considered a mixed model with a fixed effect factor (with two levels) and a random effect factor (with four levels). The samples dimensions, for the eight different treatments, were randomly generated, considering integers from a uniform distribution in the interval . Next, the observed value of the statistics was computed for the fixed effect factor, random effect factor, and interaction, as well as the respective P values considering the conditional and unconditional approaches (assuming n^•=10, n^•=30, n^•=50, and n^•=100). The program ran 1000 times.

Table 6 show the obtained results. The values in bold style correspond to the cases were the null hypothesis is rejected at the 5% significant level. The execution time (in seconds) of this process is also provided. We conclude that, for n^•=10, the decisions taken considering both approaches are not concordant for fixed effect factor and interaction.

Table 6. The values P values for conditional and unconditional approaches

		Conditional approach	Unconditional approach				time (s)
		(p‐value)	(p‐value)
	ℑj,Obs	n=8133	n^•=10	n^•=30	n^•=50	n^•=100
Fixed effects	17.2170	<0.001	0.0535	<0.001	<0.001	<0.001
Random effects	48.1637	<0.001	0.0204	<0.001	<0.001	<0.001	6.767
Interaction	9.9682	<0.001	0.0925	<0.001	<0.001	<0.001

The number of rejections was also computed, considering both approaches, as well as the proportion of false rejections, which can be avoided when the proposed methodology is considered. Through the results presented in Table 7, we conclude that the proportion of false rejections is very high for n^•=10, being lower for high values of n^•.

Table 7. Number of rejections and proportion of false rejections

	Conditional approach	Unconditional approach				Number of false rejections/
	(number of rejections)	(number of rejections)				proportion of false rejections
		n^•=10	n^•=30	n^•=50	n^•=100	n^•=10	n^•=30	n^•=50	n^•=100
Fixed effects	642	295	609	624	635	347 / 0.540	33 / 0.051	18 / 0.028	7 / 0.011
Random effects	927	364	910	920	923	563 / 0.607	17 / 0.018	7 / 0.008	4 / 0.004
Interaction	942	384	922	936	940	558 / 0.592	20 / 0.022	6 / 0.006	2 / 0.002

The results obtained agree with our previous comments on the relevance of our approach in avoid false rejections. So, we suggest the use of this approach whenever the sample dimensions are not known in advance.

6 FINAL REMARKS

In this work, we try to open a new field in the application of orthogonal mixed models to situations where sample dimensions are unknown. We point out the interesting fact that the test behaviors under null hypothesis are the same for the fixed effect and the random effect parts of the models. This situation of stability was first considered by Ferreira.8 We resorted to the case where the samples dimensions are binomial distributed, which is considered the appropriate choice when observation failures may occur. This approach leads to interesting applications in several research areas, namely in economic research, as can be seen through the application. A simulation study was conducted to verify and relate the proposed methodology with the classical one. The obtained results confirm the relevance of the proposed approach in avoiding false rejections.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous referees for useful comments and suggestions. This work was supported in part by the FCT‐ Fundação para a Ciência e Tecnologia, through projects UID/MAT/00212/2019 and UID/MAT/00297/2019.

CONFLICT OF INTEREST STATEMENT

The authors declare that there is no conflict of interests regarding the publication of this article.

Biographies

• 

  Célia Nunes is an Assistant Professor at the Department of Mathematics, University of Beira Interior (UBI, Covilhã, Portugal), and a member of the Center of Mathematics and Applications, UBI. She received the BSc degree in mathematics and the MSc degree in applied mathematics from the University of Évora (Évora, Portugal), and the PhD degree in mathematics from UBI. She has published research in the fields of probability and statistics and applied statistics. Her research interests include statistical inference in linear models and distribution theory. She has been involved in several research projects and is a guest reviewer of several international journals, being also a scientific committee member of international conferences. For comprehensive access to her work, please refer to: http://orcid.org/0000-0003-0167-4851.

• 

  Anacleto Mário is a student in the Mathematics and Applications PhD program of the University of Beira Interior (UBI, Covilhã, Portugal), and is a collaborator in the Center of Mathematics and Applications, UBI. He received the BSc degree in science of education from Agostinho Neto University (Luanda, Angola), and the MSc degree in financial auditing from the Polytechnic University of Madrid (Madrid, Spain). His doctoral research focuses on the extension of the theory of mixed effects models to the case of random sample sizes.

• 

  Dário Ferreira is an Assistant Professor at the Department of Mathematics, University of Beira Interior (UBI, Covilhã, Portugal), a member of the Center of Mathematics, UBI, and an active member of the Portuguese Society of Statistics (SPE). He has graduated in mathematics at the University of Évora (Évora, Portugal) and received the PhD degree in mathematics and statistics from UBI. His general research area is in linear models. More specifically, he works in the estimation of variance components using algebraic and stochastic methods. He has been publishing research in this field, invited to participate in some research projects, and cochaired several international conferences and workshops, being also a program committee member of several international conferences. For more details, please visit http://dar364.wix.com/dario.

• 

  Sandra S. Ferreira is a Professor at the University of Beira Interior, (UBI, Covilhã, Portugal). She received the PhD degree from UBI in 2006, where she also teaches courses on basic statistics, quantitative methods, hierarchical linear models and multivariate analysis. She is member of the working group (WG) CMStatistics (this WG focuses on all computational and methodological aspects of statistics) and a member of the IEOM Society and serves as an editorial board member of several journals. Her publications and current research interests focus on statistical inference for estimable functions and variance components, in linear mixed models with commutative orthogonal block structure. ORCID: 0000‐0002‐9209‐7772.

• 

  João T. Mexia is an Emeritus Full Professor at the Faculty of Science and Technology, New University of Lisbon, (FCT/UNL, Lisbon, Portugal). He received the BSc degree in forestry from the Technical University of Lisbon (Lisbon, Portugal) and the Habilitation in Mathematics and PhD degree in statistics from FCT/UNL. He has published many research papers in reputed international journals in the fields of probability and statistics, and his actual research interest is in linear statistical inference. He has supervised more than 40 PhD students. He has also supervised research projects and participated in the scientific committees of international conferences.