Abstract
Linear mixed-effects models are linear models with several variance components. Models with a single random-effects factor have two variance components: the random-effects variance, i. e., the inter-subject variance, and the residual error variance, i. e., the intra-subject variance. In many applications, it is practice to report variance components as coefficients of variation. The intra- and inter-subject coefficients of variation are the square roots of the corresponding variances divided by the mean. This article proposes methods for computing confidence intervals for intra- and inter-subject coefficients of variation using generalized pivotal quantities. The methods are illustrated through two examples. In the first example, precision is assessed within and between runs in a bioanalytical method validation. In the second example, variation is estimated within and between main plots in an agricultural split-plot experiment. Coverage of generalized confidence intervals is investigated through simulation and shown to be close to the nominal value.
1 Introduction
The coefficient of variation is the sample standard deviation divided by the sample mean. This measure of dispersion is used in a wide range of applications where the standard deviation tends to increase linearly with the level of the observations. In these applications, observations cannot take negative values. Specifically, measurements are made on a ratio scale. The coefficient of variation can be used as an estimator of the population coefficient of variation, defined as the square root of the population variance, i. e., the square root of the second central moment, divided by the expected value.
Much research has been made on inference for a single coefficient of variation. When the sample consists of independent observations from a normal distribution, an exact confidence interval for the coefficient of variation can be computed using the noncentral t-distribution [1]. Approximate confidence intervals have been developed using asymptotic normality [2] and Taylor series expansion [3]. Vangel [4] and Forkman [5] advocated the use of McKay’s approximation [6, 7] for the coefficient of variation. The approximate confidence interval method proposed by Forkman [5] can be used when there are samples from populations with different expected values but with a common coefficient of variation, which is a situation that was also considered by Verrill and Johnson [8]. Hayter [9] constructed a bounded confidence interval for a coefficient of variation when observations are normally distributed. Wong and Wu [10] proposed a confidence interval using a modified signed log likelihood ratio method, which can also be used for gamma distributed observations. They showed that for small samples of normally distributed observations, this method performs as well as the Vangel [4] method.
The present article considers linear random-effects and mixed-effects models for balanced and unbalanced sets of normally distributed observations. In balanced datasets, each sub-class contains the same number of observations (the exact mathematical definition of balance is complex but has been formalized by a number of authors; for further references, see [11, p.4]). Linear random-effects and mixed-effects models include several variance components. It is often practical to express the estimates of the variances as coefficients of variation, i. e., as standard deviations divided by the mean.
The intra-subject coefficient of variation assesses the reproducibility or reliability of a measurement [12]. In bioanalytical method validation, coefficients of variation specify within- and between-run precision [13]. In crop variety testing, expressing variability as coefficients of variation facilitates comparison of variability between crops [14].
Shoukri et al. [15] derived confidence intervals, based on a variance-stabilizing transformation, for the intra-subject coefficient of variation in the balanced one-way random-effects model. The present article derives generalized confidence intervals for intra- and inter-subject coefficients of variation in balanced and unbalanced random-effects and mixed-effects models with two variance components. Generalized confidence intervals are constructed using generalized pivotal quantities [16]. This type of confidence intervals has already been suggested for some other inferential problems on the coefficient of variation: Tian [17], Behboodian and Jafari [18] and Jafari [19] proposed generalized confidence intervals for a single coefficient of variation shared by many populations, and Krishnamoorthy and Lee [20] introduced generalized confidence intervals for differences and ratios of two coefficients of variation. The generalized pivotal quantities approach may also be used for the problem of testing equality of coefficients of variation [18, 19].
In the balanced one-way random-effects model, no exact bounds for the inter-subject variance are available that are functions of complete sufficient statistics [21]. A method for construction of generalized pivotal quantities and generalized confidence intervals for variance components was developed by Chiang [22], Iyer and Mathew [23], Iyer and Patterson [24] and Hannig et al. [25]. Their method is here extended to the problem of computing generalized confidence intervals for coefficients of variation. Burdick et al. [26] presented generalized confidence intervals for functions of variance components in mixed-effects models with balanced datasets and random-effects models with balanced or unbalanced datasets. For mixed-effects models and unbalanced datasets, the approach of the present article is similar to the method that was proposed for ‘a more general model’ by Gamage et al. [27], who considered the problem of computing confidence intervals for best linear unbiased predictions. However, their method uses transformation of data, but the method proposed in the present article does not.
The objective of this article is to propose methods for computing generalized confidence intervals for intra- and inter-subject coefficients of variation in mixed-effects models with two variance components. Section 2 provides theory and proposes methods. Section 3 studies, through simulation, the performance of the proposed methods and compares with the method proposed by Shoukri et al. [15]. Section 4 gives two examples: a bioanalytical method validation exemplifies the use of a one-way random-effects model, and an agricultural field experiment exemplifies a mixed-effects split-plot model. R scripts for these examples are published online as supplementary material on the Journal’s web page.
2 Theory and methods
2.1 Generalized confidence intervals
Let
2.2 Statistical model
Consider the linear mixed-effects model
where
In a more general setting, the random variables of
2.3 Balanced datasets
Let
are central chi-square distributed with numbers of degrees of freedom depending on the design, and
where
The expressions in (4) and (5) are fiducial generalized pivotal quantities for
Generalized
2.4 Unbalanced datasets
Let
Similarly, let
The unbalanced dataset can be regarded as a subset of a corresponding balanced dataset with
where
3 Simulation studies
3.1 The one-way random-effects model
Consider the one-way random-effects model
Following Shoukri et al. [15], observations were simulated with intercept
The Newton-Raphson procedure employed in the method for unbalanced datasets used
Tables 1 and 2 present observed frequencies of confidence intervals covering the true values of
Shoukri 0.95 C.I. | Generalized 0.95 C.I. | ||||||
---|---|---|---|---|---|---|---|
Balanced | Balanced | Unbalanced | |||||
γ | γ | γ | γ | γ | γ | ||
4 | 0.01 | 0.3 | 0.906 | 0.948 | 0.961 | 0.949 | 0.947 |
0.6 | 0.907 | 0.946 | 0.954 | 0.934 | 0.959 | ||
0.8 | 0.909 | 0.954 | 0.957 | 0.940 | 0.950 | ||
0.02 | 0.3 | 0.910 | 0.949 | 0.953 | 0.946 | 0.950 | |
0.6 | 0.906 | 0.949 | 0.955 | 0.940 | 0.958 | ||
0.8 | 0.904 | 0.948 | 0.955 | 0.938 | 0.948 | ||
0.04 | 0.3 | 0.908 | 0.948 | 0.955 | 0.944 | 0.950 | |
0.6 | 0.905 | 0.950 | 0.957 | 0.940 | 0.958 | ||
0.8 | 0.906 | 0.949 | 0.952 | 0.936 | 0.954 | ||
8 | 0.01 | 0.3 | 0.933 | 0.949 | 0.953 | 0.949 | 0.945 |
0.6 | 0.933 | 0.952 | 0.952 | 0.945 | 0.953 | ||
0.8 | 0.928 | 0.953 | 0.949 | 0.956 | 0.944 | ||
0.02 | 0.3 | 0.926 | 0.946 | 0.954 | 0.952 | 0.949 | |
0.6 | 0.927 | 0.946 | 0.951 | 0.945 | 0.964 | ||
0.8 | 0.932 | 0.951 | 0.949 | 0.946 | 0.953 | ||
0.04 | 0.3 | 0.929 | 0.949 | 0.952 | 0.946 | 0.943 | |
0.6 | 0.931 | 0.950 | 0.951 | 0.959 | 0.954 | ||
0.8 | 0.929 | 0.949 | 0.951 | 0.940 | 0.957 | ||
12 | 0.01 | 0.3 | 0.937 | 0.948 | 0.949 | 0.938 | 0.936 |
0.6 | 0.938 | 0.952 | 0.948 | 0.947 | 0.943 | ||
0.8 | 0.934 | 0.949 | 0.947 | 0.928 | 0.938 | ||
0.02 | 0.3 | 0.932 | 0.947 | 0.949 | 0.939 | 0.953 | |
0.6 | 0.934 | 0.948 | 0.952 | 0.934 | 0.939 | ||
0.8 | 0.935 | 0.948 | 0.948 | 0.935 | 0.948 | ||
0.04 | 0.3 | 0.935 | 0.948 | 0.948 | 0.944 | 0.951 | |
0.6 | 0.933 | 0.948 | 0.951 | 0.946 | 0.947 | ||
0.8 | 0.937 | 0.946 | 0.952 | 0.943 | 0.944 | ||
Mean | 0.924 | 0.949 | 0.952 | 0.943 | 0.949 |
Shoukri 0.95 C.I. | Generalized 0.95 C.I. | ||||||
---|---|---|---|---|---|---|---|
Balanced | Balanced | Unbalanced | |||||
γ | γ | γ | γ | γ | γ | ||
4 | 0.01 | 0.3 | 0.940 | 0.949 | 0.947 | 0.954 | 0.952 |
0.6 | 0.942 | 0.953 | 0.948 | 0.940 | 0.945 | ||
0.8 | 0.936 | 0.945 | 0.952 | 0.952 | 0.953 | ||
0.02 | 0.3 | 0.941 | 0.949 | 0.951 | 0.956 | 0.945 | |
0.6 | 0.943 | 0.952 | 0.945 | 0.945 | 0.950 | ||
0.8 | 0.940 | 0.948 | 0.950 | 0.940 | 0.941 | ||
0.04 | 0.3 | 0.944 | 0.957 | 0.950 | 0.957 | 0.938 | |
0.6 | 0.934 | 0.947 | 0.951 | 0.941 | 0.955 | ||
0.8 | 0.940 | 0.958 | 0.953 | 0.954 | 0.946 | ||
8 | 0.01 | 0.3 | 0.944 | 0.948 | 0.951 | 0.965 | 0.951 |
0.6 | 0.947 | 0.949 | 0.952 | 0.949 | 0.943 | ||
0.8 | 0.944 | 0.950 | 0.950 | 0.938 | 0.933 | ||
0.02 | 0.3 | 0.944 | 0.945 | 0.951 | 0.938 | 0.947 | |
0.6 | 0.944 | 0.947 | 0.948 | 0.945 | 0.952 | ||
0.8 | 0.944 | 0.947 | 0.950 | 0.931 | 0.949 | ||
0.04 | 0.3 | 0.940 | 0.946 | 0.950 | 0.948 | 0.950 | |
0.6 | 0.946 | 0.953 | 0.948 | 0.944 | 0.940 | ||
0.8 | 0.947 | 0.952 | 0.947 | 0.944 | 0.950 | ||
12 | 0.01 | 0.3 | 0.946 | 0.947 | 0.949 | 0.948 | 0.937 |
0.6 | 0.945 | 0.947 | 0.950 | 0.942 | 0.939 | ||
0.8 | 0.946 | 0.948 | 0.949 | 0.949 | 0.945 | ||
0.02 | 0.3 | 0.950 | 0.953 | 0.946 | 0.941 | 0.962 | |
0.6 | 0.946 | 0.948 | 0.951 | 0.952 | 0.956 | ||
0.8 | 0.947 | 0.947 | 0.947 | 0.957 | 0.964 | ||
0.04 | 0.3 | 0.951 | 0.951 | 0.949 | 0.942 | 0.951 | |
0.6 | 0.945 | 0.948 | 0.949 | 0.946 | 0.946 | ||
0.8 | 0.945 | 0.946 | 0.947 | 0.947 | 0.955 | ||
Mean | 0.944 | 0.949 | 0.949 | 0.947 | 0.948 |
The semiparametric mixed-effects model for clustered data (Section 2.2) allows for other random-effects distributions than the normal. In a study of robustness, three other distributions were investigated: a logistic distribution, a gamma distribution with shape parameter 4, and a gamma distribution with shape parameter 16. These distributions differ with respect to skewness and kurtosis. For a random variable
Tables 3 and 4 shows estimated coverage when the random-effects distribution is non-normal, for
Shoukri 0.95 C.I. | Generalized 0.95 C.I. | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Logistic | Logistic | ||||||||||
γ | γ | γ | γ | γ | γ | γ | γ | γ | |||
4 | 0.01 | 0.3 | 0.910 | 0.909 | 0.903 | 0.949 | 0.954 | 0.950 | 0.952 | 0.944 | 0.955 |
0.6 | 0.907 | 0.908 | 0.911 | 0.949 | 0.949 | 0.949 | 0.954 | 0.951 | 0.946 | ||
0.8 | 0.907 | 0.909 | 0.907 | 0.944 | 0.941 | 0.943 | 0.952 | 0.947 | 0.934 | ||
0.02 | 0.3 | 0.906 | 0.905 | 0.903 | 0.949 | 0.950 | 0.948 | 0.948 | 0.948 | 0.949 | |
0.6 | 0.909 | 0.910 | 0.912 | 0.947 | 0.947 | 0.953 | 0.955 | 0.950 | 0.946 | ||
0.8 | 0.911 | 0.903 | 0.907 | 0.949 | 0.937 | 0.949 | 0.950 | 0.953 | 0.936 | ||
0.04 | 0.3 | 0.911 | 0.906 | 0.909 | 0.952 | 0.954 | 0.949 | 0.954 | 0.956 | 0.951 | |
0.6 | 0.906 | 0.907 | 0.908 | 0.948 | 0.944 | 0.949 | 0.957 | 0.950 | 0.950 | ||
0.8 | 0.910 | 0.908 | 0.907 | 0.956 | 0.940 | 0.950 | 0.952 | 0.951 | 0.945 | ||
8 | 0.01 | 0.3 | 0.926 | 0.925 | 0.926 | 0.949 | 0.948 | 0.946 | 0.946 | 0.947 | 0.939 |
0.6 | 0.927 | 0.930 | 0.930 | 0.947 | 0.937 | 0.948 | 0.950 | 0.948 | 0.930 | ||
0.8 | 0.930 | 0.921 | 0.929 | 0.948 | 0.923 | 0.945 | 0.939 | 0.947 | 0.921 | ||
0.02 | 0.3 | 0.930 | 0.930 | 0.925 | 0.950 | 0.948 | 0.950 | 0.952 | 0.948 | 0.945 | |
0.6 | 0.927 | 0.929 | 0.928 | 0.949 | 0.931 | 0.949 | 0.951 | 0.951 | 0.934 | ||
0.8 | 0.926 | 0.928 | 0.927 | 0.948 | 0.922 | 0.947 | 0.943 | 0.947 | 0.924 | ||
0.04 | 0.3 | 0.925 | 0.930 | 0.927 | 0.948 | 0.943 | 0.950 | 0.951 | 0.946 | 0.949 | |
0.6 | 0.928 | 0.931 | 0.926 | 0.947 | 0.933 | 0.949 | 0.947 | 0.950 | 0.933 | ||
0.8 | 0.929 | 0.931 | 0.929 | 0.948 | 0.921 | 0.949 | 0.945 | 0.949 | 0.927 | ||
12 | 0.01 | 0.3 | 0.934 | 0.936 | 0.936 | 0.948 | 0.946 | 0.947 | 0.950 | 0.949 | 0.936 |
0.6 | 0.932 | 0.934 | 0.936 | 0.946 | 0.922 | 0.945 | 0.948 | 0.952 | 0.926 | ||
0.8 | 0.934 | 0.935 | 0.940 | 0.947 | 0.913 | 0.951 | 0.940 | 0.952 | 0.907 | ||
0.02 | 0.3 | 0.934 | 0.935 | 0.940 | 0.953 | 0.946 | 0.949 | 0.944 | 0.954 | 0.939 | |
0.6 | 0.933 | 0.935 | 0.934 | 0.948 | 0.928 | 0.950 | 0.946 | 0.947 | 0.927 | ||
0.8 | 0.938 | 0.934 | 0.934 | 0.949 | 0.913 | 0.949 | 0.947 | 0.948 | 0.916 | ||
0.04 | 0.3 | 0.933 | 0.937 | 0.936 | 0.946 | 0.943 | 0.950 | 0.948 | 0.947 | 0.941 | |
0.6 | 0.935 | 0.932 | 0.936 | 0.950 | 0.931 | 0.947 | 0.945 | 0.948 | 0.926 | ||
0.8 | 0.936 | 0.937 | 0.938 | 0.949 | 0.912 | 0.951 | 0.940 | 0.951 | 0.919 | ||
Mean | 0.923 | 0.923 | 0.943 | 0.949 | 0.936 | 0.949 | 0.948 | 0.949 | 0.932 |
Shoukri 0.95 C.I. | Generalized 0.95 C.I. | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Logistic | Logistic | ||||||||||
γ | γ | γ | γ | γ | γ | γ | γ | γ | γ | ||
4 | 0.01 | 0.3 | 0.940 | 0.936 | 0.934 | 0.947 | 0.940 | 0.949 | 0.950 | 0.944 | 0.937 |
0.6 | 0.942 | 0.937 | 0.939 | 0.946 | 0.932 | 0.950 | 0.948 | 0.947 | 0.927 | ||
0.8 | 0.940 | 0.944 | 0.941 | 0.949 | 0.926 | 0.951 | 0.941 | 0.949 | 0.930 | ||
0.02 | 0.3 | 0.940 | 0.943 | 0.938 | 0.951 | 0.940 | 0.949 | 0.946 | 0.947 | 0.935 | |
0.6 | 0.946 | 0.938 | 0.941 | 0.953 | 0.931 | 0.950 | 0.941 | 0.949 | 0.935 | ||
0.8 | 0.934 | 0.941 | 0.937 | 0.950 | 0.926 | 0.951 | 0.942 | 0.949 | 0.931 | ||
0.04 | 0.3 | 0.942 | 0.938 | 0.939 | 0.953 | 0.937 | 0.949 | 0.949 | 0.949 | 0.939 | |
0.6 | 0.938 | 0.940 | 0.940 | 0.950 | 0.930 | 0.952 | 0.947 | 0.954 | 0.934 | ||
0.8 | 0.938 | 0.941 | 0.943 | 0.958 | 0.931 | 0.956 | 0.946 | 0.956 | 0.936 | ||
8 | 0.01 | 0.3 | 0.945 | 0.943 | 0.943 | 0.950 | 0.929 | 0.948 | 0.945 | 0.947 | 0.927 |
0.6 | 0.945 | 0.941 | 0.943 | 0.949 | 0.918 | 0.943 | 0.941 | 0.947 | 0.917 | ||
0.8 | 0.945 | 0.942 | 0.942 | 0.948 | 0.915 | 0.947 | 0.938 | 0.947 | 0.916 | ||
0.02 | 0.3 | 0.943 | 0.942 | 0.948 | 0.947 | 0.931 | 0.945 | 0.942 | 0.954 | 0.929 | |
0.6 | 0.945 | 0.943 | 0.945 | 0.950 | 0.921 | 0.947 | 0.937 | 0.949 | 0.917 | ||
0.8 | 0.947 | 0.946 | 0.942 | 0.950 | 0.915 | 0.950 | 0.940 | 0.949 | 0.912 | ||
0.04 | 0.3 | 0.945 | 0.943 | 0.945 | 0.950 | 0.930 | 0.946 | 0.943 | 0.950 | 0.934 | |
0.6 | 0.940 | 0.946 | 0.946 | 0.947 | 0.918 | 0.951 | 0.945 | 0.951 | 0.922 | ||
0.8 | 0.944 | 0.942 | 0.944 | 0.949 | 0.911 | 0.946 | 0.946 | 0.952 | 0.922 | ||
12 | 0.01 | 0.3 | 0.945 | 0.943 | 0.945 | 0.948 | 0.932 | 0.946 | 0.941 | 0.951 | 0.918 |
0.6 | 0.944 | 0.944 | 0.949 | 0.946 | 0.909 | 0.946 | 0.938 | 0.951 | 0.905 | ||
0.8 | 0.951 | 0.950 | 0.951 | 0.951 | 0.903 | 0.949 | 0.936 | 0.950 | 0.900 | ||
0.02 | 0.3 | 0.944 | 0.951 | 0.948 | 0.945 | 0.924 | 0.952 | 0.945 | 0.951 | 0.924 | |
0.6 | 0.945 | 0.949 | 0.949 | 0.947 | 0.913 | 0.952 | 0.939 | 0.952 | 0.908 | ||
0.8 | 0.949 | 0.950 | 0.944 | 0.949 | 0.904 | 0.950 | 0.941 | 0.946 | 0.907 | ||
0.04 | 0.3 | 0.945 | 0.943 | 0.949 | 0.945 | 0.927 | 0.947 | 0.942 | 0.953 | 0.927 | |
0.6 | 0.950 | 0.947 | 0.949 | 0.953 | 0.916 | 0.948 | 0.943 | 0.949 | 0.912 | ||
0.8 | 0.941 | 0.947 | 0.950 | 0.947 | 0.910 | 0.951 | 0.943 | 0.954 | 0.915 | ||
Mean | 0.943 | 0.943 | 0.944 | 0.949 | 0.923 | 0.949 | 0.943 | 0.950 | 0.923 |
Comparisons of Tables 3 and 4 with Tables 1 and 2, respectively, shows that generalized confidence intervals for
Coverage of 95 % generalized confidence intervals for
In the semiparametric mixed-effects model for clustered data proposed by Tao et al. [29], between-subjects effects might not be normally distributed, but error effects are. Robustness was also investigated for the reverse situation, i. e., with normally distributed between-subject effects and non-normally distributed error effects. In the supplementary material, Tables A5 and A6 show the results of this investigation, which was carried out in the same way as the other simulation studies. Coverage of 95 % generalized confidence intervals for
3.2 The mixed-effects split-plot model
A balanced split-plot experiment with main-plot factor A and subplot factor B can be analyzed [31] using the model
A simulation study was performed based on the agricultural split-plot example of Section 4.2. Based on estimates obtained using the lme function of the R package nlme, (
For the balanced datasets, estimated coverage (i. e., the observed frequency of 95 % confidence intervals covering the true parameter value) was 0.943 and 0.949 for
4 Examples
4.1 A bioanalytical method validation using a one-way random-effects model
Following the European Medicines Agency’s (EMA) guideline on bioanalytical method validation [13], ligand-binding assays or immunoassays should be studied using quality control (QC) samples assayed in at least six runs over several days. Table 5 includes measurements of concentration of a QC sample in a precision study.
Run | Concentration (mg/l) | Run | Concentration (mg/l) |
---|---|---|---|
1 | 119 | 6 | 109 |
1 | 113* | 6 | 103 |
2 | 103 | 7 | 103 |
2 | 91* | 7 | 100 |
3 | 118 | 8 | 108 |
3 | 106* | 8 | 101 |
4 | 93 | 9 | 98 |
4 | 93* | 9 | 94 |
5 | 109 | 10 | 105 |
5 | 110* | 10 | 106 |
Removed for the unbalanced example.
Generalized 95 % confidence intervals were computed based on 10,000 random samples, each comprising one observation from each of the random variables
σ | σ | μ | γ | γ | |
---|---|---|---|---|---|
One-way | |||||
Balanced | (10.6, 66.1) | (7.70, 166) | (98.8, 109) | (0.031, 0.078) | (0.027, 0.124) |
Unbalanced | (4.37, 65.9) | (3.40, 222) | (99.5, 112) | (0.020, 0.078) | (0.018, 0.142) |
Split-plot | |||||
Balanced | (11.3, 28.8) | (−3.51, 10.4) | (64.9, 67.3) | (0.051, 0.081) | (0.000, 0.049) |
Unbalanced | (8.53, 23.7) | (−1.88, 17.2) | (64.6, 67.8) | (0.044, 0.074) | (0.000, 0.063) |
Five observations were removed (Table 5). In this case, the procedure described in Section 2.4 yielded generalized 95 % confidence intervals as presented on the second row of Table 6.
4.2 An agricultural split-plot experiment analyzed using a mixed-effects model
Federer and King [32] presented an example of a split-plot agricultural field experiment with
Planting method | ||||
---|---|---|---|---|
Replicate | 1 | 2 | 3 | 4 |
Seedbed preparation 1 | ||||
1 | 81.1b | 46.2* | 78.6 | 77.7 |
2 | 72.2 | 51.6 | 70.9 | 73.6 |
3 | 72.9 | 53.6* | 69.8 | 70.3 |
4 | 74.6 | 57.0* | 69.6 | 72.3 |
Seedbed preparation 2 | ||||
1 | 74.1 | 49.1 | 72.0 | 66.1 |
2 | 76.2 | 53.8 | 71.8 | 65.5 |
3 | 71.1 | 43.7 | 67.6 | 66.2 |
4 | 67.8* | 58.8 | 60.6 | 60.6 |
Seedbed preparation 3 | ||||
1 | 68.4 | 54.5 | 72.0 | 70.6 |
2 | 68.2 | 47.6 | 76.7 | 75.4 |
3 | 67.1 | 46.4 | 70.7 | 66.2 |
4 | 65.6 | 53.3 | 65.6* | 69.2 |
Seedbed preparation 4 | ||||
1 | 71.5 | 50.9 | 76.4 | 75.1 |
2 | 70.4 | 65.0 | 75.8 | 75.8 |
3 | 72.5 | 54.9 | 67.6 | 75.2 |
4 | 67.8 | 50.2 | 65.6 | 63.3 |
Copyright 2007 by John Wiley & Sons, Inc. All rights reserved.
This number differs from the source, since it was noted that [32] actually used this number in their analysis.
Removed for the unbalanced example.
The proposed method for computing generalized 95 % confidence intervals was applied. The fiducial generalized pivotal quantities
For an example of an unbalanced dataset, five randomly selected observations were removed from the dataset (Table 7). Based on generation of 10,000 pivotal quantities
5 Discussion
This article contributed the idea of using generalized inference methodology for computation of confidence intervals for intra- and inter-subject coefficients of variation in mixed-effects models. A method was developed for unbalanced datasets in mixed-effects models. Burdick et al. [26] studied other functions of variance components than the intra- and inter-subject coefficients of variation. Their methods for unbalanced datasets apply to random-effects models only.
Simulation studies indicated good performance in terms of coverage at the 95 % confidence level, although these studies also verified that the proposed methods are approximate. Since deviations from the nominal confidence level 0.95 were generally not large, the proposed methods can be recommended for practical purposes. However, the proposed methods are sensitive to the assumptions of normal distributions. The simulation studies revealed that the 95 % confidence interval proposed by Shoukri et al. [15] for the intra-subject coefficient of variation in the balanced one-way model tends to be too narrow in small datasets. For such datasets, the generalized 95 % confidence intervals proposed in the present article should be preferred.
The postulated linear mixed-effects model (1) is a fully parametric model, which assumes normal distributions. However in practice, random effects and errors are rarely exactly normally distributed. As noted in Section 2.2, model (1) can be viewed as a special case of the more realistic semiparametric mixed-effects model that was proposed by Tao et al. [29]. For this case, when the distribution of the random effects is unknown, model (1), which assumes the normal distribution, was proposed as a working model. This approach is not unproblematic, since the performance of the method depends on how well the assumption of normality is met. In practice, the proposed generalized confidence intervals may, in the non-normal case, have coverage somewhat other than the nominal. Still, the simulation studies indicate that the proposed generalized confidence intervals for
The generalization to more than two variance components is straightforward for balanced datasets. Every variance component has a unique unbiased estimator based on sums of squares [33]. Thus, a generalized pivotal quantity can be specified for any function of the parameters of a balanced mixed-effects model [24]. The generalization to mixed-effects models with several variance components is more difficult for unbalanced datasets. Although extensions have been made to the special case of unequal cell frequencies in the last stage [27, 33, 34, 35], unbalanced cases are generally less explored in the literature on generalized inference. It is possible that the method presented here can be generalized to more complex mixed-effects models, but since the proposed method involves numerical computations, it might be a challenge to provide an efficient general algorithm.
As pointed out by Hannig et al. [25], when
Acknowledgement
This research was conducted using the resources of High Performance Computing Center North (HPC2N).
Appendix
where
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