Why RMA Is a Pipeline
Affymetrix GeneChip arrays measure many short oligonucleotide probes, with multiple probes grouped into a probe set intended to measure a transcript or other target. The raw data are probe-level fluorescence intensities: a single array has many measurements, and a probe set has several measurements per array. Robust multi-array average (RMA) is a preprocessing method that converts these measurements into a normalized expression summary for each probe set on each array.
RMA is a sequence of three operations: model-based background correction, quantile normalization across arrays, and robust probe-set summarization after a log transformation. The stages address different features of the measurements. Background correction separates estimated specific signal from nonspecific intensity; quantile normalization aligns the marginal intensity distributions across arrays; median polish summarizes the multiple probes in a probe set while allowing for systematic probe effects. The introductory material on microarray background correction and probe summarization covers the broader measurement setting; here the emphasis is how the RMA stages fit together and what each stage assumes.
Stage 1: Model-Based Background Correction
The RMA background model treats an observed intensity \(Y\) as the sum of a nonnegative specific signal \(S\) and a background component \(B\): \(Y=S+B\). It assumes \(S\) has an exponential distribution with mean \(\alpha>0\), while \(B\) has a normal distribution with mean \(\mu\) and variance \(\sigma^2\), independently of \(S\). The normal background model can assign probability to negative values; this is a model for the background component, not a claim that measured fluorescence is negative.
Rather than simply subtracting a background estimate, RMA estimates the signal by its conditional expectation given the observed intensity. Let \(\phi\) and \(\Phi\) be the standard normal density and cumulative distribution function. With \(a=y-\mu-\sigma^2/\alpha\), the conditional mean is:
This estimate is positive and incorporates uncertainty about the background. The parameters are estimated from the probe-level intensity data; the calculation below uses fixed hypothetical parameters just to show how the correction works. RMA applies the correction on the intensity scale, before quantile normalization and log transformation.
Worked Example: Correcting One Hypothetical Probe Intensity
Suppose an observed intensity is \(y=18\), and the fitted background-model parameters for this illustrative calculation are \(\mu=10\), \(\sigma=2\), and \(\alpha=5\), in intensity units. First calculate
Using \(\phi(3.6)\approx0.000612\) and \(\Phi(3.6)\approx0.99984\), the corrected signal estimate is
The estimate is about 7.201 intensity units. It is not the raw intensity of 18, nor is it obtained by subtracting the background mean alone. In a full RMA analysis, corrected intensities for all probes and arrays proceed to the normalization stage.
Stage 2: Quantile Normalization Across Arrays
Quantile normalization aligns the empirical intensity distributions of the arrays. For each array, sort the corrected probe intensities from smallest to largest. At each rank, calculate the mean across arrays of the values at that rank. Replace each array's value by the mean for its rank, assigning tied observations consistently, commonly by averaging the target values over the ranks occupied by the tie. Thus, after normalization, arrays have the same empirical distribution, even though a given probe can have a different rank in different arrays.
This operation relies on the assumption that the overall distributions are comparable and that forcing them to match is appropriate for the samples being analyzed. As discussed in the earlier material on assumptions and failure modes of quantile normalization, strong global biological shifts or an unsuitable set of arrays can make that assumption implausible. Quantile normalization does not establish that arrays are biologically equivalent, and it does not correct every technical difference.
Worked Example: Normalizing Three Hypothetical Arrays
Suppose three arrays have corrected intensities for the same three probes as follows. The values are hypothetical intensity units.
| Probe | Array A | Array B | Array C |
|---|---|---|---|
| 1 | 2 | 1 | 3 |
| 2 | 4 | 5 | 6 |
| 3 | 8 | 9 | 7 |
The sorted values are A: \(2,4,8\); B: \(1,5,9\); and C: \(3,6,7\). The means at ranks 1, 2, and 3 are respectively
Each array therefore receives normalized rank values \(2,5,8\). In this example, the normalized values for probes 1–3 are \((2,5,8)\) on Array A, \((2,5,8)\) on Array B, and \((2,5,8)\) on Array C as well: each probe happened to occupy the same rank in all three arrays. In data with different rank orders, a probe's normalized value depends on its rank within its own array, not on its probe label.
Stage 3: Log Transformation and Median-Polish Summarization
After quantile normalization, RMA takes the base-2 logarithm of the corrected, normalized probe intensities. For one probe set, denote the resulting value for probe \(i\) on array \(j\) by \(Z_{ij}\). The additive model is
where \(\theta\) is an overall level, \(\rho_i\) is the effect of probe \(i\), \(\gamma_j\) is the array-specific expression effect for this probe set, and \(\varepsilon_{ij}\) is residual variation. Probe effects represent systematic differences in intensity among probes measuring the same target. The model is on the log scale: additive effects there correspond to multiplicative effects on the intensity scale.
Median polish estimates the overall, probe, and array effects by repeatedly subtracting row medians and column medians from the residual table, accumulating the subtracted medians as effects. The array-specific summary is the overall level plus the estimated effect for that array. The procedure is robust to some unusually large probe-level residuals because it uses medians rather than means, but it does not make the model immune to outliers or systematic probe failures.
Worked Example: Median-Polish Summaries for One Probe Set
Suppose the hypothetical log2 normalized intensities for two probes measured on three arrays are:
| Probe | Array 1 | Array 2 | Array 3 |
|---|---|---|---|
| 1 | 5 | 6 | 7 |
| 2 | 6 | 7 | 8 |
The grand median of the six entries \(5,6,6,7,7,8\) is \(\theta=6.5\). Subtracting it gives rows \((-1.5,-0.5,0.5)\) and \((-0.5,0.5,1.5)\). Their medians are \(-0.5\) and \(0.5\), which become the two probe effects. Subtracting those row medians leaves \((-1,0,1)\) in each row. The column medians are therefore \(-1,0,1\), the array effects.
The resulting expression summaries are the grand level plus each array effect:
These are log2-scale summaries for this probe set on the three arrays. A one-unit difference on this scale corresponds to a twofold difference in the normalized intensity scale, subject to the measurement and preprocessing assumptions.
Why the Median-Polish Calculation Works
The additive model explains why a probe-set summary can be separated from probe-specific affinity. If the model holds and the median residual in each row and column is zero after fitting, then subtracting row and column medians removes the corresponding systematic components. In the example, after removing the grand and probe effects, each row is \((-1,0,1)\); subtracting the column medians leaves zero residuals. Thus the fitted array effects recover the across-array pattern shared by the probes.
This justification depends on an approximately additive structure on the log scale and enough informative probes to estimate stable effects. The model does not say that every probe has identical intensity: probe affinity is represented by \(\rho_i\). Nor does it guarantee that every probe truly measures the intended target. Cross-hybridization, sequence variation, saturation, or poor probe-set annotation can violate the measurement interpretation even if the numerical algorithm converges.
Estimate the signal component for each observed probe intensity under the normal-plus-exponential model.
Quantile-normalize corrected intensities across the arrays being analyzed.
Take log2 normalized intensities and apply median polish to obtain one array-specific summary per probe set.
Diagnostics, Alternatives, and Limitations
RMA should be treated as a coordinated preprocessing choice, not as a guarantee that the resulting summaries are comparable or biologically valid. Before interpreting summaries, inspect whether the raw arrays have unusual intensity distributions, excessive background, saturation, or many extreme probe values. After normalization, confirm that the distributions align as intended, and investigate arrays whose rank patterns or quality measures are markedly atypical. For probe sets, check whether a few probes dominate the fit or show conspicuous residuals from the additive model. A numerical output for every probe set does not mean every summary is reliable.
The main alternatives differ in which components they estimate and how. Simple background subtraction can be easier to describe but may produce negative values and ignores uncertainty in the background estimate. Other model-based background corrections use sequence-specific information or different assumptions. For summarization, a mean on the log scale is sensitive to extreme probes, while a median can be less efficient when the additive model is appropriate; median polish offers a robust, iterative way to estimate both probe and array effects. RMA's standard use of perfect-match probes also differs from approaches that use mismatch probes in a background or signal calculation. These methods are not interchangeable merely because they all produce probe-set values.
The normalization step is especially consequential: because quantile normalization forces the distributions to match, it can obscure genuine global changes when the comparable-distribution assumption fails. RMA also does not by itself remove known batch effects, correct poor probe design, or establish that a probe set maps uniquely to a biological target. Its output is a normalized log-scale measurement summary, not an absolute transcript count or a causal estimate.
Common Mistakes
- Calling RMA a single normalization step. It combines background correction, across-array normalization, and probe-set summarization; each stage has distinct assumptions.
- Taking the logarithm before background correction. Standard RMA corrects on the intensity scale, normalizes corrected intensities, and then logs the normalized values for summarization.
- Assuming quantile normalization proves arrays are comparable. It makes their empirical distributions match by construction; it cannot validate the assumption that their underlying distributions should match.
- Interpreting a summary as absolute expression. The result is a normalized log2 probe-set measure whose meaning depends on the array platform, preprocessing, and probe-set annotation.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions below.
- In the background model, what does the corrected intensity estimate represent, and why is it not simply \(y-\mu\)?
- Three arrays have sorted intensities \((1,4)\), \((2,6)\), and \((3,8)\). What are the quantile-normalized values at ranks 1 and 2?
- Why does median polish allow probes in a probe set to have systematically different intensities?
- What does a one-unit difference between two RMA summaries mean on the log2 scale, and what does it not establish?
- Give one situation in which quantile normalization could be misleading.
Solutions. (1) It estimates \(E(S\mid Y=y)\), the expected specific signal conditional on the observed intensity under the fitted model. It accounts for uncertainty in the background and signal components, so it is not ordinary subtraction of the background mean. (2) The rank-1 mean is \((1+2+3)/3=2\), and the rank-2 mean is \((4+6+8)/3=6\); each array receives 2 and 6 at ranks 1 and 2. (3) The additive model includes a probe effect \(\rho_i\), which represents systematic probe-specific affinity on the log scale. (4) It corresponds to a twofold difference in normalized intensity, under the preprocessing scale; it does not establish an absolute transcript count or a causal effect. (5) If a biological condition causes a genuine broad shift in expression across many probe sets, forcing the array-wide distributions to match may remove part of that signal.