Estimating the Null Center and Scale

Why Estimate the Null Center and Scale?

A theoretical null describes the test-statistic distribution expected when its null hypothesis is true under the specified model. For a valid, well-calibrated \(z\)-statistic, that reference is often the standard normal distribution, with center 0 and scale 1. In a large genomic analysis, however, the bulk of observed statistics may be shifted or more dispersed than this reference. An empirical null allows the null component to have an estimated center and scale, rather than fixing them at 0 and 1.

Estimating these parameters is useful only if the estimates describe the null component rather than the whole observed distribution. Many tests may be null, but the alternatives can still affect summaries of all test statistics. The method below uses robust summaries and relies on the central part of the distribution being sufficiently dominated by null statistics. That is an assumption to assess, not a guarantee supplied by the formulas.

Definition: An empirical normal null is a model in which null test statistics follow \(N(\mu_0,\sigma_0^2)\), where \(\mu_0\) is the null center and \(\sigma_0\) is the null scale. The theoretical standard-normal null is the special case \(\mu_0=0\) and \(\sigma_0=1\). Here \(\sigma_0\) is a standard deviation, not a variance.

For \(m\) genomic tests, write the observed statistics as \(z_1,\ldots,z_m\). A two-group model treats the marginal distribution as a mixture of a null component and an alternative component. The robust procedures in this tutorial estimate \(\mu_0\) and \(\sigma_0\) from all the observed statistics, on the working premise that their central region is largely null. They do not identify which individual statistics are null.

Median and Median Absolute Deviation

The sample median is a natural estimate of the center because a relatively small number of extreme values has limited effect on it. To estimate scale, first calculate each statistic’s absolute distance from the estimated center. The median of those distances is the median absolute deviation, or MAD. For a normal distribution, the MAD must be multiplied by a constant to estimate the standard deviation on its original scale.

Definition: Let \(\tilde{z}\) be the sample median of \(z_1,\ldots,z_m\). Define \(\operatorname{MAD}=\operatorname{median}_i|z_i-\tilde{z}|\). The normal-consistent robust estimates are \(\hat{\mu}_0=\tilde{z}\) and \(\hat{\sigma}_0=\operatorname{MAD}/0.67449\), where \(0.67449\) approximates the 75th percentile of a standard normal distribution.

The adjustment is not an arbitrary convention. If \(Z\sim N(\mu_0,\sigma_0^2)\), its median is \(\mu_0\). Also, for a nonnegative distance \(d\),

P(|Z-\mu_0|\le d) =\Phi\!\left(\frac{d}{\sigma_0}\right) -\Phi\!\left(-\frac{d}{\sigma_0}\right) =2\Phi\!\left(\frac{d}{\sigma_0}\right)-1.

The population median absolute deviation \(d\) satisfies \(P(|Z-\mu_0|\le d)=0.5\). Thus \(2\Phi(d/\sigma_0)-1=0.5\), so \(\Phi(d/\sigma_0)=0.75\), and \(d=\sigma_0\Phi^{-1}(0.75)\). Since \(\Phi^{-1}(0.75)\approx0.67449\), dividing the sample MAD by 0.67449 makes it consistent for \(\sigma_0\) when the null distribution is normal and the summaries represent that null.

Worked Example: Calculating the Robust Estimates

Suppose a hypothetical, small set of nine central test statistics is \(-1.2,-0.8,-0.4,0,0.1,0.2,0.4,0.8,1.2\). For illustration, assume these observations are representative of a null-dominated normal center; with only nine values, the estimates would be quite imprecise.

The ordered values have median \(\tilde{z}=0.1\), so \(\hat{\mu}_0=0.1\). The absolute deviations from 0.1 are \(1.3,0.9,0.5,0.1,0,0.1,0.3,0.7,1.1\). Ordered, they are \(0,0.1,0.1,0.3,0.5,0.7,0.9,1.1,1.3\), whose median is 0.5. Therefore,

\hat{\sigma}_0=\frac{0.5}{0.67449}=0.7413.

The fitted empirical null has estimated center 0.1 and scale 0.7413. In this small illustration the scale is below 1, but the sample is far too small to support a confident calibration conclusion. This calculation demonstrates the estimator, not a reliable genomic analysis.

Interpreting Center and Scale Estimates

A center estimate above zero indicates that the central statistics are shifted in the positive direction relative to the theoretical null; a negative estimate indicates a shift in the other direction. A scale estimate above 1 indicates greater dispersion than the theoretical standard-normal reference, while an estimate below 1 indicates less dispersion. These are descriptions relative to a reference, not diagnoses of their cause. Possible explanations include model misspecification, confounding, unaccounted heterogeneity, or a genuine structure in the tested effects. An estimated shift or scale alone cannot distinguish among these explanations.

Worked Example: Quantifying a Shift and Overdispersion

In a hypothetical genomic analysis of 20,000 tests, suppose the median test statistic is 0.12 and the MAD is 0.78. Assume the central statistics are sufficiently null-dominated for the robust estimates to be informative. The empirical-null estimates are

\hat{\mu}_0=0.12,\qquad \hat{\sigma}_0=\frac{0.78}{0.67449}=1.1564.

Relative to a theoretical scale of 1, the estimated scale is larger by

100\left(\frac{1.1564}{1}-1\right)=15.64\%.

Equivalently, using the unrounded ratio, \(100(0.78/0.67449-1)\approx15.64\%\). Thus the fitted null is shifted upward by 0.12 \(z\)-statistic units and its estimated standard deviation is about 15.64% above 1. This does not establish why the shift or dispersion occurred, nor does it show that every test statistic follows this empirical null.

For illustration, a statistic \(z=2.00\) would be centered and scaled relative to these estimates as \((2.00-0.12)/1.1564\approx1.626\). This calculation shows how the estimated parameters alter a statistic’s reference position; it is not by itself a decision rule or evidence about the cause of the observed excess dispersion.

A useful cross-check is to estimate the center and scale from symmetric quantiles, provided the quantiles being used plausibly represent the null distribution. For a normal distribution, its 25th and 75th percentiles are symmetric around its center, and their difference is \(2\Phi^{-1}(0.75)\sigma_0\approx1.34898\sigma_0\). This check can reveal disagreement with the median-and-MAD estimates. Agreement does not prove the null assumptions, and detailed procedures for matching a theoretical curve to the central distribution are a separate method.

Worked Example: Comparing Two Robust Summaries

Suppose the hypothetical central 25th and 75th percentiles of a large set of genomic statistics are \(-0.49\) and \(0.71\). The quantile midpoint and the scale estimate based on their spread are

\hat{\mu}_{Q}=\frac{-0.49+0.71}{2}=0.11,\qquad \hat{\sigma}_{Q}=\frac{0.71-(-0.49)}{1.34898} =\frac{1.20}{1.34898}=0.8896.

If the same data have median 0.12 and MAD 0.78, their median-and-MAD estimates are 0.12 and 1.1564. The centers are close, but the scale estimates differ appreciably: 0.8896 versus 1.1564. This disagreement is a diagnostic signal to examine the distribution’s shape, contamination, and heterogeneity. It is not a reason to select whichever estimate supports a preferred conclusion.

Assumptions, Diagnostics, and Failure Modes

The robust estimators have a clear normal-theory justification, but their use for an empirical null requires additional care. In particular, the center and scale computed from the full set of statistics estimate the null parameters only when the chosen summaries are not substantially distorted by alternatives or other structure. The median can be affected by sufficiently extensive one-sided alternatives. The MAD can be affected when the distribution of distances from the median is altered, even if the median itself remains stable. A high proportion of true nulls helps, but does not ensure that the central summaries are pure null summaries.

Check whether the estimated center and scale are stable under reasonable changes in the set of statistics summarized. Compare results across relevant batches or genomic strata, and inspect the distribution for asymmetry, multiple peaks, or an unusually narrow center with broad shoulders. Such patterns may indicate that a single normal null is a poor description. Because genomic tests can be dependent, the number of statistics \(m\) is not automatically the number of independent observations for uncertainty calculations. Dependence can affect the precision of estimates even when the point estimates are computed correctly.

Worked Example: Detecting Batch Heterogeneity

Suppose a hypothetical analysis reports these robust summaries for two processing batches. Assume each batch has enough tests for the estimates to be informative and that each batch’s center is sufficiently null-dominated.

Batch	Median	MAD	Estimated scale
A	-0.10	0.67	0.9933
B	0.40	0.81	1.201

For Batch A, \(0.67/0.67449=0.9933\). For Batch B, \(0.81/0.67449=1.201\). The estimated centers differ by \(0.40-(-0.10)=0.50\) \(z\)-statistic units, and the estimated scales also differ. This pattern suggests that a single pooled empirical null may conceal batch-specific behavior. It does not establish that the batch caused the differences; the analysis should investigate data processing and study structure before treating the pooled estimates as adequate.

The main alternative is to retain the theoretical null, fixing \(\mu_0=0\) and \(\sigma_0=1\). This is preferable when the test-statistic model justifies that reference and the observed data give no compelling reason to replace it. An empirical null is more flexible, but can absorb real departures or misspecification into its estimated parameters. Robust summaries are simple and resistant to some extreme values, but depend on the null-dominated-center assumption. Quantile checks provide a different summary of central spread, but also rely on the chosen quantiles representing the null. No one estimate substitutes for checking whether a single normal null is a reasonable working model.

Common Mistakes

Calling any median and MAD the null estimates. They describe the full observed distribution unless the null-dominated assumption makes them informative about the null component.
Using the MAD without its normal-consistency factor. For normal data, the raw MAD estimates about \(0.67449\sigma_0\), not \(\sigma_0\); divide by 0.67449.
Calling an estimated scale above 1 proof of biological signal. Overdispersion is a descriptive discrepancy from the theoretical reference, and it can have multiple causes.
Assuming a large number of tests guarantees precise estimates. Dependence and shared sources of heterogeneity mean that the test count alone does not determine effective information.
Interpreting the empirical null as a classification of individual tests. The fitted center and scale describe a distributional component; they do not label each statistic as null or alternative.

Key takeaway: The normal-consistent median-and-MAD estimates are \(\hat{\mu}_0=\operatorname{median}(z_i)\) and \(\hat{\sigma}_0=\operatorname{median}_i|z_i-\hat{\mu}_0|/0.67449\). Their interpretation as empirical-null parameters depends on a plausible normal null and central statistics that are not materially distorted by alternatives or heterogeneity.

Check Your Understanding

Answer each question, then compare with the solutions below.

A hypothetical test-statistic set has median \(-0.08\) and MAD \(0.60\). Calculate the normal-consistent empirical-null center and scale.
A hypothetical analysis estimates \(\hat{\mu}_0=0.05\) and \(\hat{\sigma}_0=1.10\). State what each estimate indicates relative to the theoretical standard normal, without asserting a cause.
For hypothetical 25th and 75th percentiles of \(-0.40\) and \(0.80\), calculate the quantile midpoint and scale estimate using 1.34898 as the divisor for the interquartile range.
Why can a large proportion of null tests still fail to guarantee that the sample median and MAD estimate the null parameters well?

Solutions. (1) The center estimate is \(-0.08\). The scale estimate is \(0.60/0.67449=0.8896\). (2) The center is shifted upward by 0.05 \(z\)-statistic units, and the estimated null standard deviation is 10% above the theoretical value of 1; neither estimate identifies the cause. (3) The midpoint is \((-0.40+0.80)/2=0.20\), and the scale estimate is \((0.80-(-0.40))/1.34898=1.20/1.34898=0.8896\). (4) Alternatives can be one-sided or otherwise concentrated near the center, changing the summaries; dependence or heterogeneity can also affect their stability. A high null proportion does not ensure that the particular summaries used represent the null component.