From Scanner Intensities to Expression Values
A microarray scanner records fluorescence at many probe locations. The recorded intensity is not a direct measurement of transcript abundance: it can include fluorescence from the optical system, local surface signal, nonspecific binding, and measurement noise. Before comparing samples, analysts therefore make preprocessing choices that turn probe-level observations into one value per sample and gene-associated feature.
This tutorial focuses on two linked operations: background correction, which attempts to remove an unwanted intensity component, and probe summarization, which combines measurements from several probes intended to represent the same target. These choices affect the values in the final sample-by-feature expression matrix. The introductory material on normalization targets and expression scale established why the scale and target of normalization matter; here the concern is the construction of the expression values that may later be normalized.
The distinction between optical background and nonspecific hybridization is important. Optical background arises from scanner, slide, or local surface fluorescence. Nonspecific hybridization is biological material binding to a probe despite not being the intended target. A correction intended for optical background does not automatically remove nonspecific binding, nor does it guarantee that the remaining intensity is proportional to transcript abundance.
Optical-Background Correction
A simple additive measurement model is
where \(S_{si}\) is the target-related signal, \(B_{si}\) is background fluorescence, and \(\varepsilon_{si}\) is measurement error. This model motivates subtraction, but it is an approximation: the background may vary over the array, and the error may depend on intensity. A local background estimate can be obtained from nearby pixels or dedicated background measurements, depending on the platform and image-processing procedure.
Direct subtraction relies on the estimated background being relevant to the probe location and on the additive approximation being adequate. If the background estimate is too high, corrected values can be negative or unstable near zero. Taking a logarithm of a negative value is undefined; replacing every such value by zero and then logging it is also undefined. A floor or alternative correction must therefore be explicit, rather than introduced silently after inspecting results.
Worked Example: Subtracting Local Background
Suppose three probes in one hypothetical sample have observed intensities of 120, 95, and 65 scanner units. Their local background estimates are 20, 25, and 30 units. Under the additive model, the corrected intensities are calculated probe by probe:
All three corrected values are positive, so no floor is needed for this example. If a log base 2 expression scale is desired, their transformed values are \(\log_2(100)\approx6.644\), \(\log_2(70)\approx6.129\), and \(\log_2(35)\approx5.129\). The arithmetic subtraction occurs before taking logarithms. These numbers are hypothetical scanner-unit measurements, not a claim that corrected intensity is a perfectly calibrated measure of transcript abundance.
A key diagnostic is the relationship between observed intensity and estimated background. Inspect the distribution of corrected values, the fraction close to zero or below zero before any floor, and whether background estimates vary systematically across array regions or samples. A large fraction of near-background measurements means that small changes in the background estimate can materially change the corrected values and their logs.
Summarizing Probes into a Feature Value
A probeset often contains multiple probes intended to measure the same target. Even after correction, those probes can differ because of probe affinity, sequence composition, cross-hybridization, and technical noise. A simple summary takes the mean or median of corrected probe intensities within each sample. If expression values are analyzed on a log scale, a common simple summary is the median of the probe-level log intensities. The mean of log intensities corresponds to the log of a geometric mean on the original intensity scale, not the log of the arithmetic mean.
Worked Example: Median Summary on the Log Scale
Consider a hypothetical probeset with three probes measured in two samples. After background correction, the intensities are as follows:
| Probe | Sample A (intensity) | Sample B (intensity) |
|---|---|---|
| 1 | 64 | 128 |
| 2 | 128 | 256 |
| 3 | 256 | 1,024 |
Taking \(\log_2\) gives Sample A values \(6,7,8\) and Sample B values \(7,8,10\). The median summaries are therefore
The resulting feature value is one log-scale number per sample, and the difference between the summaries is \(8-7=1\) log base 2 unit. The third probe in Sample B is higher than the other two; the median is less affected by that value than the mean, which would be \(25/3\approx8.333\) for Sample B. This robustness is useful only if the probes mostly measure the same target; the median cannot repair a probeset whose probes consistently target different sequences.
A more structured summary models probe affinity separately from sample-level expression. For feature \(k\), a common additive representation on the log-intensity scale is
where \(Z_{si}\) is the corrected log intensity for probe \(i\) in sample \(s\), \(\alpha_i\) is a probe effect, \(\gamma_s\) is the sample-level expression effect, and \(\varepsilon_{si}\) is residual variation. The probe effect captures persistent differences in intensity among probes; the sample effect is the quantity used as the feature summary. For a well-behaved probeset, probe effects are reasonably stable across samples and the residuals do not show a dominant probe-by-sample pattern.
The model needs an identifiability convention because adding a constant to every \(\alpha_i\) and subtracting it from \(\theta\) leaves the fitted values unchanged. One standard convention is to center probe effects so that \(\sum_i\alpha_i=0\), and to center sample effects so that \(\sum_s\gamma_s=0\); the grand level \(\theta\) then accounts for the overall location. Robust fitting, such as median-based iterative fitting, reduces the influence of a small number of extreme probe measurements. It does not make the additive model true by construction.
Worked Example: Separating Probe and Sample Effects
Suppose a hypothetical probeset has the following log base 2 intensities. The last measurement is unusually high.
| Probe | Sample A | Sample B | Sample C |
|---|---|---|---|
| 1 | 6 | 7 | 8 |
| 2 | 7 | 8 | 9 |
| 3 | 8 | 9 | 15 |
A plausible additive fit has grand level \(\theta=8\), probe effects \(\alpha_1=-1,\alpha_2=0,\alpha_3=1\), and sample effects \(\gamma_A=-1,\gamma_B=0,\gamma_C=1\). Both sets of effects sum to zero, as required by the centering convention. For example, the fitted value for probe 2 in Sample C is
The observed value is 9, so its residual is 0. For probe 3 in Sample C, the fitted value is \(8+1+1=10\), while the observation is 15, giving residual \(15-10=5\). The sample summaries from this fit are \(\theta+\gamma_s\): \(7\) for Sample A, \(8\) for Sample B, and \(9\) for Sample C. The large residual flags a probe-sample discrepancy instead of forcing the entire Sample C summary upward to 15. These fitted values illustrate the decomposition; in practice, robust fitting estimates the effects from the full probe-by-sample table.
From Probe Values to the Expression Matrix
After correction and summarization, the data can be represented by a matrix \(X\) with samples in rows and features in columns. The entry \(X_{sk}\) is the summary for feature \(k\) in sample \(s\), on the stated scale. This is the matrix used as input to later analyses; the exact values depend on the choices made upstream.
Worked Example: How Preprocessing Choices Change a Matrix Entry
For one hypothetical feature, suppose two probes in Sample A have observed intensities 40 and 80, and both have background estimates of 10. Direct subtraction gives corrected intensities 30 and 70. The log base 2 values are approximately 4.907 and 6.129, so their median (the midpoint of the two values) is approximately 5.518. In Sample B, suppose the corresponding observed intensities are 80 and 160 with the same background estimates. The corrected values are 70 and 150, with logs approximately 6.129 and 7.229; their median is approximately 6.679.
Thus, this chosen procedure places the feature’s two entries at about 5.518 and 6.679 log base 2 units. If the analyst instead skipped background subtraction, the log-scale probe values would differ, especially for the weaker measurements: in Sample A they would be \(\log_2(40)\approx5.322\) and \(\log_2(80)\approx6.322\), with median 5.822. Both the background correction and the summary rule can therefore change a matrix entry. The difference is a preprocessing consequence, not by itself evidence of a biological change.
The comparison also illustrates why preprocessing must be applied consistently. Different background rules or probe summaries across samples can create technical differences that look like sample-level expression differences. Record the intensity scale, background method, handling of nonpositive values, probe-to-feature mapping, and summary method alongside the resulting matrix.
Assumptions, Diagnostics, and Alternatives
Background correction and summarization are not assumption-free. The additive subtraction model assumes that the estimated background is relevant to the probe and represents an additive contribution. A probe-summary model assumes that the probes assigned to a feature provide sufficiently comparable information, and an additive probe/sample model further assumes that persistent probe effects can be separated from sample effects. Violations are suggested by spatial patterns in background estimates, large numbers of corrected values near zero, probes that disagree sharply across samples, or residuals dominated by particular probes.
| Choice | Strength | Risk or limitation |
|---|---|---|
| Direct background subtraction | Simple and transparent when the additive background estimate is credible | Can produce negative or unstable low signals; sensitive to background estimation error |
| Mean of probe values | Uses all probes and is easy to calculate | Can be strongly influenced by an outlying or cross-hybridizing probe |
| Median of probe values | Resists a small number of extreme probe values | Discards some information and may hide systematic disagreement among probes |
| Additive probe-effect summary | Separates stable probe affinity from sample-level expression | Depends on an adequate additive model and stable probe effects across samples |
Model-based background correction is an alternative when direct subtraction is unstable, but it replaces a transparent subtraction with assumptions about signal, background, and measurement error. It should be assessed by its behavior near the detection floor and by whether it introduces undesirable changes in low-intensity variation. No correction method can rescue poor image quality or an unsuitable probe-to-feature mapping.
A frequent mistake is to subtract background after logging, or to interpret every corrected value as pure biological signal. The first is mathematically inconsistent with an additive intensity model: generally, \(\log_2(Y-\widehat{B})\ne\log_2(Y)-\log_2(\widehat{B})\). The second ignores nonspecific binding, cross-hybridization, and uncertainty in the background estimate. Correct on the scale implied by the measurement model, inspect the result, and describe the final expression matrix as a processed measurement rather than a direct count of transcripts.
Check Your Understanding
Answer the questions, then compare your reasoning with the solutions.
- An observed probe intensity is 54 scanner units and its estimated background is 18 units. What is the corrected intensity under direct subtraction?
- Why is it generally invalid to subtract a background estimate from a logged intensity when the assumed measurement model is additive on the original intensity scale?
- Three log-scale probe values for one sample are 5, 6, and 12. What are the mean and median summaries, and which is less affected by the extreme value?
- In the additive probe-effect model, what does the sample effect represent, and why is a centering convention needed?
- Name two diagnostics that could reveal that background correction or probe summarization is unreliable.
Solutions. (1) \(54-18=36\) scanner units. (2) Logging is nonlinear, so \(\log_2(Y-\widehat{B})\) is not the difference of the logged quantities; the additive model applies before logging. (3) The mean is \((5+6+12)/3=23/3\approx7.667\), and the median is 6; the median is less affected by the value 12. (4) The sample effect represents the sample-level feature expression after accounting for probe effects. A centering convention is needed because the grand level and probe effects can otherwise trade off by an arbitrary constant without changing fitted values. (5) Examples include many corrected values at or below zero before any floor, spatial patterns in background estimates, or a probe with consistently large residuals across samples.