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What sums are needed to calculate a correlation coefficient

what sums are needed to calculate a correlation coefficient

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© 2014 Ionan et al.; licensee BioMed Central Ltd.


The intraclass correlation coefficient (ICC) is widely used in biomedical research to assess the reproducibility of measurements between raters, labs, technicians, or devices. For example, in an inter-rater reliability study, a high ICC value means that noise variability (between-raters and within-raters) is small relative to variability from patient to patient. A confidence interval or Bayesian credible interval for the ICC is a commonly reported summary. Such intervals can be constructed employing either frequentist or Bayesian methodologies.

This study examines the performance of three different methods for constructing an interval in a two-way, crossed, random effects model without interaction: the Generalized Confidence Interval method (GCI), the Modified Large Sample method (MLS), and a Bayesian method based on a noninformative prior distribution (NIB). Guidance is provided on interval construction method selection based on study design, sample size, and normality of the data. We compare the coverage probabilities and widths of the different interval methods.

We show that, for the two-way, crossed, random effects model without interaction, care is needed in interval method selection because the interval estimates do not always have properties that the user expects. While different methods generally perform well when there are a large number of levels of each factor, large differences between the methods emerge when the number of one or more factors is limited. In addition, all methods are shown to lack robustness to certain hard-to-detect violations of normality when the sample size is limited.

Decision rules and software programs for interval construction are provided for practical implementation in the two-way, crossed, random effects model without interaction. All interval methods perform similarly when the data are normal and there are sufficient numbers of levels of each factor. The MLS and GCI methods outperform the NIB when one of the factors has a limited number of levels and the data are normally distributed or nearly normally distributed. None of the methods work well if the number of levels of a factor are limited and data are markedly non-normal. The software programs are implemented in the popular R language.


Confidence interval; Credible interval; Generalized confidence interval; Intraclass correlation coefficient; Modified large sample


Biological and physical quantities assessed for scientific studies must be measured with sufficient reproducibility for the study to produce meaningful results. For example, biological markers (“biomarkers”) are studied for many medical applications, including disease risk prediction, diagnosis, prognosis, monitoring, or optimal therapy selection. Variation in measurements occurs for numerous reasons. The measurements might have been made on different devices, may have involved subjective judgment of human raters (e.g. a pathologist assessing the number of tumor cells in a biopsy), or might have been made in different laboratories using different procedures. As another example, psychological instruments often score patients based on multi-item questionnaires completed by medical professionals. Variation in the resulting scores can be attributed to both variation among the patients and variation among the medical professionals performing the assessments. In many settings, it is not realistic to expect perfect concordance among replicate measurements, but one needs to achieve a level of reliability sufficient for the application area, such as a clinical setting. A common approach to quantify the reliability of a measurement process is to calculate the intraclass correlation coefficient (ICC) along with a confidence interval [1 -4 ].

An interval can be constructed for the ICC using frequentist or Bayesian methods. Frequentist methods assure that the probability that the interval contains the parameter if the experiment is repeated many times is the nominal confidence level (e.g. 95%). In contrast to Frequentist methods, Bayesian methods provide a probability distribution for the parameter itself, given the data and the prior uncertainty. The distribution can be summarized by a credible interval, which reflects a nominal probability (e.g. 95%) region for the distribution. When little is known about the parameter of interest a priori, then a non-informative prior, which is often provided in the statistical software, can be used to construct the interval. The relative advantages of noninformative Bayesian and frequentist approaches in general are discussed in Berger [5 ] Chapter 4, Carlin and Louis [6 ] (Section 1.4), and elsewhere. General comparisons of the different approaches are beyond the scope of this paper. This paper focuses on two issues of applied interest discussed in the next paragraph.

Two critical and inter-related characteristics of a confidence interval method are (1) the coverage probability, and (2) the interval width. The coverage probability of a method should exactly match the confidence level, such as 95%. Coverage probability is a frequentist concept since the parameter is treated as a fixed number. The interval width is important to consider when comparing intervals because one often wants the shortest possible interval that maintains the nominal coverage.

Coverage probability and interval width are important and relevant from both frequentist and objective Bayesian perspectives [7 -13 ]. Frequentist coverage probabilities are interpretable in the Bayesian framework as well [14 ].

We study two applications in detail. The first application is a study by Barzman et al. [15 ]. They evaluated the Brief Rating of Aggression by Children and Adolescents (BRACHA), a 14-item questionnaire instrument scored by emergency room staffers. BRACHA scores can be influenced by both the child being assessed and the adult performing the assessment. Interest was in whether different adult staffers scored the children in a similar way, as summarized by the intraclass correlation coefficient. These data were originally analyzed using Bayesian credible interval methods. The second application is the National Cancer Institute’s Director’s Challenge reproducibility study [16 ]. In this study, tissue samples were subdivided into separate sections, sections distributed to four laboratories, and microarray analysis performed at each laboratory. Interest was in whether different laboratories produced similar gene expression measurements for individual patients.

This paper considers the setting of a two factor, crossed, random effects model without interaction. We focus on this setting because it arises frequently in practical applications of interest [15 -17 ], and because this focus enables us to examine different aspects of study design, data distribution, and Bayesian priors, without the scope of the paper becoming unwieldy. For the purposes of this study, we assume this model is appropriate for the data; the process of selecting an appropriate statistical model and agreement measure are outside the scope of this paper and are discussed thoroughly elsewhere [18 ,19 ]. A random effects model is appropriate when each factor represents a random sample from a larger population [20 ]; for example, a factor may represent labs randomly drawn from all labs that could perform the assay. If the population of labs is small, a finite population adjustment is possible [21 ], but rarely used in practice. If for some factors random sampling is not an appropriate assumption, then fixed-effects or mixed models can be used. Reproducibility methods for fixed and mixed models are discussed elsewhere [19 ,22 ].

Confidence interval performance can be affected by both the study design used and the distribution of the data. If the study design has a limited number of levels of one or both factors, then this can impact interval performance. In practice, it is common that one factor will have a very small number of levels. The distribution of the data is assumed to be normally distributed and a violation of normality can impact coverage. Also, if one variance component is large or small relative to the others, resulting in different values of the ICC, then this can impact coverage as well. Different variance parameters and a range of model violations are studied using simulation and application. These studies lead to relatively simple and straightforward advice on which interval procedure will produce an interval with good performance characteristics. Also presented are cautionary notes about when examined methods will perform poorly.

The history of the development of the methods compared in this paper is briefly reviewed. The Modified Large Sample procedure for the two-way layout without interaction was developed in [23 ], and is based on earlier work of [24 ] using exact statistical methods. The Generalized Confidence Interval procedure for the two-way layout without interaction is presented in [25 ], and is based on a modification of a related method in [26 ], and the foundational work in [27 ]. Bayesian methods based on Markov Chain Monte Carlo are described in [28 ], were previously popularized in [29 ] and [30 ], and grow out of earlier work such as [31 ]. Bayesian intervals can be constructed with a variety of packages in R, such as MCMCglmm, or the popular software based on BUGS (Bayesian inference Using Gibbs Sampling), such as OpenBUGS [32 ], WinBUGS [33 ], or JAGS. The frequentist modified large sample (MLS) [24 ] and generalized confidence interval (GCI) [27 ] methods can be implemented using SAS version 9.3 VARCOMP procedure, or with the R programs provided with this manuscript.

This paper is organized as follows: Section 2 presents the model, briefly outlines the methods, and also presents the simulation settings. Section 3 presents the results of the Monte Carlo investigations. Section 4 presents real data applications. Section 5 presents discussion of the results. Section 6 presents conclusions. Mathematical details appear in the Additional file 1. Supplemental simulation details appear in Additional file 2.

Additional file 1. Supplement includes additional discussion, simulations, data analysis details, figures and tables.

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Additional file 2. Supplement presents the mean and standard deviation of the point estimates of the ICCb for different models and designs presented in the main paper.

Category: Forex

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