- Research article
- Open Access

# Learning from microarray interlaboratory studies: measures of precision for gene expression

- David L Duewer
^{1}Email author, - Wendell D Jones
^{2}, - Laura H Reid
^{2}and - Marc Salit
^{3}

**10**:153

https://doi.org/10.1186/1471-2164-10-153

© Duewer et al; licensee BioMed Central Ltd. 2009

**Received:**11 July 2008**Accepted:**08 April 2009**Published:**08 April 2009

## Abstract

### Background

The ability to demonstrate the reproducibility of gene expression microarray results is a critical consideration for the use of microarray technology in clinical applications. While studies have asserted that microarray data can be "highly reproducible" under given conditions, there is little ability to quantitatively compare amongst the various metrics and terminology used to characterize and express measurement performance. Use of standardized conceptual tools can greatly facilitate communication among the user, developer, and regulator stakeholders of the microarray community. While shaped by less highly multiplexed systems, measurement science (metrology) is devoted to establishing a coherent and internationally recognized vocabulary and quantitative practice for the characterization of measurement processes.

### Results

The two independent aspects of the metrological concept of "accuracy" are "trueness" (closeness of a measurement to an accepted reference value) and "precision" (the closeness of measurement results to each other). A carefully designed collaborative study enables estimation of a variety of gene expression measurement precision metrics: repeatability, several flavors of intermediate precision, and reproducibility. The three 2004 Expression Analysis Pilot Proficiency Test collaborative studies, each with 13 to 16 participants, provide triplicate microarray measurements on each of two reference RNA pools. Using and modestly extending the consensus ISO 5725 documentary standard, we evaluate the metrological precision figures of merit for individual microarray signal measurement, building from calculations appropriate to single measurement processes, such as technical replicate expression values for individual probes on a microarray, to the estimation and display of precision functions representing all of the probes in a given platform.

### Conclusion

With only modest extensions, the established metrological framework can be fruitfully used to characterize the measurement performance of microarray and other highly multiplexed systems. Precision functions, summarizing routine precision metrics estimated from appropriately repeated measurements of one or more reference materials as functions of signal level, are demonstrated and merit further development for characterizing measurement platforms, monitoring changes in measurement system performance, and comparing performance among laboratories or analysts.

## Keywords

- Microarray Platform
- Measurement Protocol
- Intermediate Precision
- High Signal Level
- Interlaboratory Study

## Background

The ability to demonstrate the reproducibility of gene expression microarray results is a critical element in their adoption for clinical applications. Several studies have asserted that microarray data can be "highly reproducible" if probe sequences are well-mapped to the genome and standard protocols are followed [1–3]. While largely focused on comparisons among measurement platforms, these and other studies have variously characterized many aspects of microarray performance. However, the microarray community has yet to adopt a standardized terminology and practice for characterizing performance that can facilitate clear communication among the user, developer, and regulator stakeholders.

The measurement science (metrology) community is devoted to establishing a philosophically coherent terminology and practice for characterizing and communicating measurement performance [4]. As the world's largest developer and publisher of international consensus standards, the non-governmental International Organization for Standardization (ISO) plays a critical role in disseminating this guidance [5]. The documentary standard ISO 5725-1 [6] details the basic concepts and estimation techniques for assessing metrological "accuracy" which is defined as a combination of two concepts, "trueness" and "precision." These concepts are formally defined in the Vocabulary of International Metrology (VIM) [7] base document and more cogently described in ISO 3534 [8]: trueness is the closeness of a measurement to an accepted reference value and precision is the closeness of measurement results to each other. While microarrays can generate vastly more data per sample than is typical of the processes that shaped the development of these documents, we believe that this pre-existing metrological framework can be extended to microarrays and other highly multiplexed measurement processes.

Properly designed collaborative studies are one of the very best ways of obtaining the information required to characterize some aspects of measurement performance [9]. The three "rounds" of the Expression Analysis Pilot Proficiency Test evaluated replicate samples of a pair of mixed-tissue RNA pools across multiple participants from June to December of 2004; these studies provide a wealth of information relevant to the estimation of several aspects of within-platform measurement precision and among-participant measurement concordance [10]. While the known relationships between the two RNA pools used in these studies also enable evaluation of several measures of trueness in *differential* expression [11], we here evaluate only the metrological concepts of precision as applied to the underlying *direct* measurements. These concepts provide a foundation for the development of objective expectations for the consistency of microarray results. We anticipate that this foundation will facilitate improving the comparability of microarray measurements over time and place and may lead to the development of new approaches and tools for objectively demonstrating the utility of measures of differential expression and ranked lists of differentially expressed genes.

## Results

The precision of a defined measurement process can be characterized using three nested metrics: "repeatability," "intermediate precision," and "reproducibility." These measures of precision are defined in terms of the conditions that apply when the measurements are obtained, including: operator, equipment, calibrations, environmental conditions, and the period of time between measurements. Repeatability is defined as the precision of independent measurements when all conditions are assumed constant and thus do not contribute variability (eg, single participants in a given round). Reproducibility is defined as the precision observed when all conditions are permitted to vary within allowable tolerances and thus to contribute variability (eg, all participants in at least one round). Intermediate precision is a special case, where some specified factors are held constant and others are varied (eg, a single participant in two or more rounds). A variety of analysis of variance approaches are suitable for dissecting such data. ISO 5725-2 details the standard approach used in measurement science to characterize measurement precision for a specific measurement process from the results for a single material in a single study [9, 12].

### Classical precision metrics

In the context of the microarray platform measurements, the signal from each probeset of the array is a single measurement process. The ISO 5725-2 calculations for a single process in a single study are described in Methods, *Classical Precision Metrics for Single Studies*. Methods, *Classical Precision Metrics for Multiple Studies* extends the calculations to multiple studies that use nominally identical samples. The design elements of the Expression Analysis Pilot Proficiency Test studies that enable the use of these calculations are described in Methods, *Study Design*.

While tedious, none of the classical precision metrics are particularly complex or difficult to calculate. However, keeping track of the nomenclature and symbols used for the various metrics can be challenging. In quick summary: let *x*_{
ijk
}represent the *k*^{
th
}replicate measurement of a given probeset reported by the *j*^{
th
}participant in the *i*^{
th
}round. The usual mean,
, and standard deviation, *s*(*x*_{
ij
}), of the replicates estimate the value and technical variation for that probeset for the particular participant at one point in time. The value
and technical variation characteristic of the microarray platform itself are estimated by combining the
and *s*(*x*_{
ij
}) values over all participants; the metrological term for this expected technical variation is *repeatability precision* and is represented as *s*_{
ri
}. The variation among the different
estimates the *between-participant precision* and is represented as *s*_{Li}. In combination, the two sources of variation estimate the *reproducibility precision* which is represented as *s*_{
Ri
}. These single-round estimates can be regarded as characterizing the performance of the given probeset over the (typically short) duration of the study.

When nominally identical samples are analyzed in multiple studies, the resulting multiple
and *s*(*x*_{
ij
}) values can be used to estimate the participant-specific expected value,
, repeatability, *s*_{
rj
}, and *among-round precision*, *s*_{Wj}; the combination of these two sources of variation estimate the *intermediate precision over time* for the participant, *s*_{I(T)j}[13]. Combining the single-round, single-participant estimates over all participants can also estimate the long-term expected value,
, repeatability, *s*_{
r
}, between-laboratory precision, *s*_{L}, and reproducibility, *s*_{
R
}; these long-term estimates can be regarded as characterizing the intrinsic performance of the platform.

### Precision functions

The above classical precision metrics characterize performance for individual processes with multiple nominally identical samples of one material. Characterizing processes as a function of signal level can usefully identify the performance expected for "typical" samples [14]. For measurement methods involving one to a few tens of different measurement processes, this can be accomplished with interlaboratory studies involving a relatively small series of samples of similar matrix composition but with varying levels of the analytes of interest. If the many (thousands to millions) of signals typical of microarrays have very different measurement characteristics, then little simplification to this classical approach is possible. However, estimation of aggregate precision functions from analyses of a single material becomes feasible if the majority of the measurement processes share similar performance characteristics. The expected performance of a "typical" probeset with a "typical" sample can be established by compositing the multitudinous individual estimates for one or a few samples.

The Expression Analysis Pilot studies provide results for 31054 probesets in each of two mixed-tissue pools. In the following, we characterize the various precision metrics as functions of signal level from the 62108 unique combinations of probeset and material.

#### Discrete Precision Functions

*s*(

*x*

_{1,12})}. The line displays the expected value of these estimates, {< >,<

*s*(

*x*

_{1,12})>}, calculated as the medians of 100 equally sized groups of the { ,

*s*(

*x*

_{1,12})} after ordering on . While many summarization approaches could be used, this binning approach has the benefit of relative familiarity and computational simplicity.

*s*(

*x*

_{ ij })>} standard deviation functions for the individual participants, the {< >,<

*s*

_{ ri }>} repeatabilities, the {< >,<

*s*

_{Li}>} between-participant precisions, and the {< >,<

*s*

_{ Ri }>} reproducibilities. The upper panel of Figure 3 displays the {< >,<

*s*

_{ r }>} between-round repeatability, the {< >,<

*s*

_{L}>} between-participant precision, and the {< >,<

*s*

_{ R }>} reproducibility functions. The only difference in construction of these discrete functions is use of the appropriate average signal, or , rather than the participant average, , in the initial list creation step.

The pattern and magnitude of the various functions in all three rounds are very similar to each other, indicating that the precision characteristics of the method did not change much over the seven months between Round 1 and Round 3 or with the change in number and identity of the participants. This pattern of technical variability as a function of signal level (in order of lowest to highest signal level: smoothly increasing to a maximum, smoothly decreasing to a relative constant minimum until a small increase at the very highest signal levels) has been often observed with Affymetrix Microarray Suite 5 [17] (MAS5)-processed data [15, 18, 19]. This complex structure appears to be at least mostly an artifact of MAS5 processing rather than an intrinsic property of the microarray platform, since it is not observed with some other data processing approaches [15, 16, 20]. Regardless, variability estimated from short-term studies may underestimate the variability expected over longer periods: while <*s*_{
r
}> for the higher signal levels is a multiplicative factor of 2^{0.14} = 1.1 or about 10% relative, <*s*_{
R
}> is a factor of 2^{0.38} = 1.3 or about 30% relative. It is perhaps noteworthy that these lessons can only be learned through longitudinal study of measurements from a reasonably well defined population of participants.

#### Continuous precision functions

While discrete functions are efficient as graphical summaries, they are inconvenient for estimating quantitative values of an expected precision at specific signal levels. For use in further calculations, such as variance-stabilized normalization [21, 22], it is desirable to represent the various precision estimates as continuous functions,
= *f*(*x*), where
is the estimated precision value, *x* is the signal level, and *f* is some function parameterized with a relatively small set of coefficients. While a simple four-parameter sigmoidal curve captures much of the structure for most of the discrete functions for the present data, the underlying model assumption of monotonic change as the signal rises from the detection-limit to saturation-limit is not fully adequate to describe MAS5 behavior. Rather than attempting to define and fit more complete theory-based models for particular datasets, platforms, or data processing systems, interpolative empirical functions can readily capture the observed structure. While comparatively crude and unsuitable for extrapolation, even simple polynomials of modestly high-order provide a reasonable qualitative description of the dependence as well as being easily implemented and interpreted. The lower panel of Figure 3 displays 10^{th}-order polynomials parameterized to the discrete between-round precisions shown in the upper panel of Figure 3. At graphical resolution, very similar fits to the discrete functions are provided by polynomials of order 7 and above.

## Discussion

### Characterizing measurement systems

The between-round repeatability, {<
>,<*s*_{
r
}>}, between-participant precision, {<
>,<*s*_{L}>}, and reproducibility {<
>,<*s*_{
R
}>} functions displayed in Figure 3 provide one definition of the expected measurement precision of signals from one microarray platform, processed with particular software, obtained by a given group of laboratories at a particular point in time. Comparable precision characteristics for other data preprocessing approaches can be estimated by reanalysis of these data. Evaluating the expected characteristics of other platforms will require new studies, with different samples, but comparable functions can be defined given a similar experimental design and sufficient participants of comparable experience. Whether it will be possible to generalize results among related microarray platforms or across data analysis systems is yet to be assessed.

### Monitoring performance changes over time

*s*

_{ ri }>} repeatabilities, {< >,<

*s*

_{Li}>} between-participant precisions, and {< >,<

*s*

_{ Ri }>} reproducibilities that are displayed by each round in Figure 2 are redisplayed in Figure 4 by each precision component. The forms of the functions are very similar over the three studies. Curiously, while the level of within-participant repeatability shows little or no trend, the between-participant precision at signal levels above the median (2

^{8}) appears to have degraded somewhat with time. The invariant repeatability argues against any significant change in the quality of the RNA pools; this plus the distribution of microarrays from multiple lots in Round 1 but single lots in Rounds 2 and 3 argue against significant between-array variability. Thus the small increase in between-participant variability may reflect the changes in the number of participants and the processing protocols that they used; it may also indicate that somewhat less experienced analysts were involved in the later rounds.

### Comparing participant precisions

While simple contrasts of repeatability, *s*_{
r
}, and between-participant precision, *s*_{L}, (see Methods) enables identification of individual probesets that differ systematically among participants, the variation among the {<
>,<*s*(*x*_{
ij
})>} functions in Figure 2 suggests that the systematic differences among the participants are not confined to just a relatively few measurement processes. Comparison of the within-participant repeatability, {<
>,<*s*_{
rj
}>}, and among-round precision, {<
>,<*s*_{Wj}>}, functions for each participant helps identify the nature of long-term changes.

*s*

_{r 12}is quite good at the highest signal levels and

*s*

_{W12}is both quite good and roughly constant for all levels; this indicates good short-term and excellent long-term control of the measurement process. For participant 6 (middle panel),

*s*

_{r 6}is somewhat less good then

*s*

_{r 12}with large signals, and while

*s*

_{W6}is excellent for moderately high signals it is systematically less good at both low and very high levels; this suggests somewhat poorer short-term repeatability and long-term changes in the measurement process that impact high and low signals more than those at the mid-levels. We speculate that these changes may be related to scanner performance, influencing both background noise and signal saturation [23]. For participant 4 (bottom panel),

*s*

_{r 4}is generally as good to somewhat better than for

*s*

_{r 12}while

*s*

_{W4}is considerably poorer for all signal levels; this suggests excellent short-term control but significant differences in the (probably pre-scanner) measurement protocol over the course of the three rounds.

*s*

_{ rj }, and median among-round precision,

*s*

_{Wj}. To avoid estimation artifacts, the medians are evaluated over signal levels from 2

^{8}to 2

^{12}. With three exceptions, between-round precision is limited by within-round performance. The excess between-round variability for measurement systems 1B, 4, and 13 likely results from undocumented changes to the systems. The generally poorer precision for systems in protocol group "C" suggests that one or more factor in this group is not well controlled.

The structure visualized in Figure 6 is congruent with the behavior of the data for the exemplar probeset AFFX_Rat_Hexokinase_5_at discussed in Methods. The association of abstract trends with the behavior of a particular probeset in one sample may facilitate identifying root-causes. The analysis and display tools developed for traditional measurands thus can inform both the development and the interpretation of tools for interlaboratory studies of microarrays.

## Conclusion

The established metrological framework for characterizing precision can be applied to results from microarray interlaboratory studies, enabling precision characteristics of microarray results to be expressed in a way that permits comparison to those of other measurement processes. The design of the Expression Analysis Pilot Proficiency Test facilitated assessment of the nested precision metrics of repeatability, intermediate precision over time, and reproducibility – all critical figures of merit of any analytical method. Such studies and figures of merit are essential tools for objective, quantitative performance assessment of individual laboratories, the population of laboratories, and microarray platforms. We believe that continuous precision functions will prove a vital tool for characterizing and comparing measurement platforms and data processing algorithms.

The tools described here for the simplest of microarray signals, are the foundation for further work addressing precision measures for differential expression and differentially expressed gene lists. Figures of merit for these composite signals will support objective performance assessment of the measures behind the biological inference, the reason for performing the measurements in the first place.

## Methods

### Study design

Expression Analysis, Inc. coordinated three rounds of the Expression Analysis Pilot Proficiency Test in 2004. The first (Round 1) was completed in June, the second (Round 2) in September, and the third (Round 3) in December. All data from these studies are available from ArrayExpress, accession number E-MEXP-1568 http://www.ebi.ac.uk/microarray-as/ae/. The following summarizes the organizational elements pertinent to our analysis.

#### Participants

Study design, participation, and protocol groups

Protocol Groups | |||||
---|---|---|---|---|---|

Code | Round 1 6/2004 | Round 2 8/2004 | Round 3 12/2004 | # | |

1A | A | 1 | |||

1B | B | B | 2 | ||

2 | B | B | 2 | ||

3 | A | A | 2 | ||

4 | C | C | C | 3 | |

5 | A | A | A | 3 | |

6 | A | A | A | 3 | |

7 | A | A | A | 3 | |

8 | C | C | C | 3 | |

9 | C | C | C | 3 | |

10 | Y | 1 | |||

11 | A | A | 2 | ||

12 | B | B | B | 3 | |

13 | A | A | 2 | ||

14 | A | A | A | 3 | |

15 | A | A | 2 | ||

16 | A | A | A | 3 | |

17 | B | B | 2 | ||

18 | Z | 1 | |||

Total | 19 | 13 | 16 | 15 | 44 |

#### Samples

Each participant in each study analyzed three aliquots of two mixed-tissue RNA pools, Mix 1 and Mix 2. The two mixtures were prepared at Expression Analysis by combining different amounts of RNA from rat brain, kidney, liver, and testicle tissues following the design developed by Dr. Karol Thompson and her FDA colleagues [24]. The mixtures were prepared to have total RNA concentrations of 1 μg/μL. Each of the six samples was tagged with unique combinations of five polyadenylated (polyA+) bacterial gene transcripts. The three samples of each mixture were otherwise nominally identical in composition. The sample materials were stored at -80°C by Expression Analysis until shipment and by the participants after receipt. All samples were shipped on dry ice via an overnight delivery service.

#### Microarrays

Each participant received six Rat230 2.0 GeneChips (Affymetrix, Inc, Santa Clara California, USA) for use in each study, one microarray to be used for each RNA sample. Sixteen arrays were replaced because of various participant-recognized technical problems. To study the influence of lot-to-lot variation, four lots of these arrays were used in Round 1. A single array lot was used in Round 2 and a different single lot in Round 3. The GeneChips are 11-μm format. They were annotated to contain 31099 unique probesets, 27 of which were designed to respond to the polyA+ bacterial control transcripts and 18 to respond to hybridization control transcripts. The remaining 31054 probesets were designed to respond to rat RNA.

#### Measurement protocols

Each participant prepared biotin-labeled cRNA targets from each sample and hybridized the cRNA to the rat microarrays using his/her own labeling and hybridization reagents. Each participant followed his/her own standard measurement protocol, with the following restrictions: 1) it was strongly recommended that 10 μg of the RNA sample be used to prepare each target, 2) the biotinylated cRNA was to be prepared using a single round of cDNA synthesis via the Eberwine protocol [25], 3) 20 μg of the fragmented, biotinylated cRNA was to be used with 300 μL of hybridization cocktail including the oligo and eukaryotic controls, and 4) 200 μL of the hybridization cocktail was to be hybridized to each microarray. Three "protocol groups" (coded as "A", "B", and "C") were identified on the basis of the participants' choice of operating conditions and source of enzymes, purification kits, and instrumentation; only two of the 19 measurement systems used during the study were sufficiently different to be considered unique. Table 1 summarizes the use of the different protocols by participant.

#### Signal estimation and processing

Each participant measured the microarrays following their standard protocols using Affymetrix GeneChip Scanner 3000 instruments with high-resolution software. The resulting "CEL" files for each microarray were sent to Expression Analysis for data processing and evaluation where probeset signal values were evaluated using MAS5 [17]. At NIST, the 55 polyA+ and hybridization controls were deleted from all data sets and the remaining 31054 rat RNA signals were log_{2} transformed and centered to have a median log_{2}(signal) of 8 (ie, a median signal of 2^{8} = 256.) This centering value was selected as the integral power of 2 closest to the median of the raw MAS5 signals in all data sets.

#### Data sets

Forty-four sets of data for the six samples were generated over the course of the three rounds. Each data set consists of 31054 probeset signals for three replicate samples of Mix 1 and Mix 2. Table 1 lists the number of data sets returned in each study.

### Classical precision metrics for single studies

Let *P* represent some one particular measurement process, *X* the results generated by that process, and *x* a particular single result. For the data considered here, *P* is the log_{2}-transformed MAS5 evaluation of a particular probeset, *X* the set of log_{2}-transformed MAS5 results for that probeset in all arrays studied, and *x* the log_{2}-transformed MAS5 result for a particular array.

*x*can be expressed as the sum of three components [12]:

where
is either the true value or more typically a consensus estimate of the quantity being measured, *B* is the systematic difference (bias) between
and the expected result for the given participant's implementation of *P* (as estimated from replicate measurements), and *ε* is the random difference between the expectation for the implementation (ie,
+ *B*) and the given value. For a single material evaluated in a single study (that is, at some given point in time), the *ε* are assumed to follow a random distribution centered on zero with a standard deviation associated with the (metrological) repeatability precision characteristic of *P*. Likewise, the variability of the *B* among all participants is assumed to follow a random distribution centered on zero with a standard deviation associated with the (metrological) reproducibility precision of *P*. These terms are described more fully in the *Results Section*; we here detail a standard approach to their estimation.

*x*

_{ ijk }represent the

*k*

^{th}of

*N*

_{ m }replicate measurements of

*P*reported by the

*j*

^{th}participant in the

*i*

^{th}study. The standard deviation of the replicates,

*s*(

*x*

_{ ij }), estimates the random variability of

*P*as implemented by the

*j*

^{th}participant in the

*i*

^{th}study:

*P*common to all participants (

*i.e*., its repeatability precision) in the

*i*

^{th}study,

*s*

_{ ri }, can be estimated by combining the individual

*s*(

*x*

_{ ij }) over all

*N*

_{ pi }participants. While the general formulae detailed in [12] describe variable numbers of replicate measurements for different participants, for these data the same numbers of replicates were reported by every participant in every round. The appropriate formula for pooling variance in this circumstance is:

The *s*_{
ri
}can be interpreted as the "average" standard deviation expected for technical replicate measurements made by a typical laboratory, where "typical" is defined by the population of actual participants.

*i*

^{th}study,

*s*

_{Li}, is estimated from the standard deviation of the estimated participant-specific biases:

where *max*() is the "take the maximum value of the arguments" function and
is the mean of the participant mean values. The
term estimates for the repeatability contribution to the variance of the biases. With atypically noisy replicate measurements, the corrected bias variance is defined as zero – allocating the observed variance to the least-complex source. The *s*_{Li}estimates the extent of agreement among the various implementations of *P* used by the participants. Ideally, all participants will observe the same mean value for *X* and the value for *s*_{Li}will be near zero; in practice, studies involving more than one measurement protocol often (by conscious study or from hard experience) discover *s*_{Li}to be several times larger than *s*_{
ri
}.

*P*during the

*i*

^{th}study,

*s*

_{ Ri }, is estimated by combining the

*s*

_{ ri }and

*s*

_{Li}variance components:

The *s*_{
Ri
}combines all of the factors influencing *P* at the time the study was performed; the implementation of *P* in a typical laboratory is expected to yield results that agree with results of other such users within confidence limits appropriate to a normal distribution having mean
and standard deviation *s*_{
Ri
}.

*s*

_{L1}are plotted along the left-hand margin (

*s*

_{L1}= 2

^{0}= 1), a consequence of estimating variance with a relatively small number of replicate measurements.

*s*

_{L1},

*s*

_{r 1}} pairs displayed in Figure 7, where the {

*s*

_{L1},

*s*

_{r 1}} locations are marked with open circles labeled

*a*to

*d*. Exemplar 1, probeset 1379568_at of Mix 2, has very small

*s*

_{L1}and

*s*

_{r 1}; this represents results that are about the same for all participants, for all replicate samples. Exemplar 2, 1395685_at of Mix 1, has small

*s*

_{L1}but large

*s*

_{r 1}; this represents results with considerable technical variability but with averages that are about the same for all participants participants. Exemplar 3, 1371165_a_at of Mix 1, has moderate

*s*

_{L1}and

*s*

_{r 1}; this represents modest variability with some systematic differences among the participants. Exemplar 4, AFFX_Rat_Hexokinase_5_at of Mix 1, has large

*s*

_{L1}but relatively small

*s*

_{r 1}; this represents results with considerable and quite consistent systematic differences among the participants.

### Classical precision metrics for multiple studies

When results for two or more qualitatively similar interlaboratory studies are available, the individually short-term study-specific estimates can be used to define the long-term performance characteristics of the measurement process, *P*, with great confidence. It may also be possible to explore the temporal stability of participant-specific systematic bias, *B*, and random variability, *ε*. Indeed, an explicit goal of many interlaboratory studies is to help participants identify and minimize sources of systematic difference in their individual implementations of *P* and to establish tighter statistical control over its random influences [27]. Changes in individual performance will manifest may manifest as changes in the study-specific repeatability and reproducibility estimates [28].

*N*

_{ s }studies that evaluate identical samples, laboratory-specific repeatabilities can be estimated for all participants that use nominally identical implementations of

*P*in at least two of the studies. For the

*j*

^{th}such laboratory,

*s*

_{ rj }is estimated by combining the simple standard deviations,

*s*(

*x*

_{ ij }), over the

*N*

_{ sj }studies in which they participated. Since here the same number of replicates were evaluated in each round, the general pooling formula again simplifies to:

*j*

^{th}participant,

*s*

_{Wj}, is calculated:

*s*

_{I(T)j}[13], is the combination of

*s*

_{ rj }and

*s*

_{Wj}:

Additional file 4 summarizes these long-term within-participant precision calculations for the four exemplar probesets.

*X*, , and the total number of sets of

*X*values reported by all participants in all of the studies,

*N*

_{ t }, can be calculated across the

*N*

_{ p }participants or across the

*N*

_{ s }studies:

*s*

_{ r }, is directly calculated from the participation-weighted average of the laboratory-specific repeatability variances; however, the same value is obtained by pooling the study-specific repeatability estimates:

*s*

_{L}, is calculated from the study size-weighted average of the between-laboratory precision variances:

*s*

_{ R }, can be calculated from the study size-weighted average of the study-specific reproducibility variances or by combining

*s*

_{ r }and

*s*

_{L}:

*s*

_{L}= 2

^{0}= 1. Note that the locations of the exemplar probesets relative to {

*s*

_{L},

*s*

_{ r }} are unchanged after three rounds. Figure 10 displays all of the above precision estimates for the four exemplar probesets.

### "Outliers"

It is often necessary to identify and remove grossly aberrant values and/or use robust statistical estimation techniques to enable sensible summarization of a given data set. Other than the arrays that were replaced because of participant-recognized technical problems, no "outlier arrays" (in the sense of the majority of MAS5 values for one of the three replicate arrays being quite different from those of its siblings) were identified in the final data. Robust methods, including those advocated in ISO 5725-5 [29], were evaluated and found to yield estimates very similar to those described above.

## Declarations

### Acknowledgements

We thank the participants of the interlaboratory studies for their cooperation and support.

Certain commercial equipment, instruments, and materials are identified in order to specify experimental procedures as completely as possible. In no case does such identification imply a recommendation or endorsement by the National Institute of Standards and Technology nor does it imply that any of the materials, instruments, or equipment identified are necessarily the best available for the purpose.

## Authors’ Affiliations

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