Differential Missingness by Case-Control Status

Why Compare Missingness by Case-Control Status?

A variant can have a high overall call rate and still have a quality-control problem if genotypes are missing more often in cases than in controls. Such a difference matters because the groups are central to a case-control comparison: if one group has more missing genotypes, the observed genotype data may not represent the two groups in the same way. A test of differential missingness checks for this imbalance at a variant.

The test is a diagnostic, not a direct test of whether the variant affects disease. A result may reflect technical differences, such as sequencing batch or assay quality, but it may also arise from chance or other group differences. The observed pattern needs investigation before deciding whether to retain or exclude a variant.

Define the Comparison

For one variant, classify each person as having either a missing genotype call or a called genotype. Let \(p_{\mathrm{case}}\) be the probability that a case has a missing call at this variant, and let \(p_{\mathrm{control}}\) be the corresponding probability for a control. These are group-specific probabilities in the population or study-generating process represented by the analysis; they are not the observed proportions in a particular data set.

Definition: Differential missingness by case-control status means that the probability of a missing genotype call differs between cases and controls. The usual two-sided hypotheses are \(H_0:p_{\mathrm{case}}=p_{\mathrm{control}}\) and \(H_a:p_{\mathrm{case}}\ne p_{\mathrm{control}}\). For observed data, write \(a\) for missing cases, \(b\) for called cases, \(c\) for missing controls, and \(d\) for called controls. The observed missingness proportions are \(a/(a+b)\) among cases and \(c/(c+d)\) among controls.

The distinction between probabilities and sample proportions is essential. The hypotheses concern the underlying missingness probabilities; the proportions calculated from the table estimate those probabilities. The table is

\begin{array}{c|cc|c} & \text{Missing} & \text{Called} & \text{Total}\\ \hline \text{Cases} & a & b & a+b\\ \text{Controls} & c & d & c+d\\ \hline \text{Total} & a+c & b+d & N \end{array} \qquad N=a+b+c+d.

The basic sampling model treats each person’s missing/called status as an independent binary outcome within their case-control group, with a common missingness probability within each group. The two groups consist of different people, and each person contributes one status at the variant being tested. Dependence between people, such as clustering or relatedness, can undermine the usual standard errors and p-values. The test also presumes that missing and called statuses have been coded consistently and that the case-control labels are correct.

Pearson’s Chi-Square Test

Under \(H_0\), both groups share a common missingness probability. Its estimate under the null is the pooled sample proportion \(\hat{p}=(a+c)/N\). The expected count in each cell is its row total multiplied by the corresponding pooled column proportion. For example, the expected number of missing cases is \((a+b)(a+c)/N\). Pearson’s statistic sums the squared differences between observed and expected cell counts, divided by the expected count in each cell.

\chi^2 = \sum_{\text{four cells}}\frac{(O-E)^2}{E}, \qquad df=1.

There is one degree of freedom because the row and column totals determine all four expected counts under the null, leaving one independent discrepancy between the observed and expected table. Equivalently, for a two-by-two table with nonzero margins, the statistic can be calculated as

\chi^2 = \frac{N(ad-bc)^2} {(a+b)(c+d)(a+c)(b+d)}.

When the null model is true and observations are independent, the statistic is approximately distributed as a chi-square random variable with one degree of freedom if expected counts are sufficiently large. A common practical guideline is that each expected count should be at least 5; this is a guideline for the approximation, not a mathematical boundary. The p-value is the probability, under the null model and its assumptions, of a chi-square statistic at least as large as the observed value. A small p-value is evidence against equal missingness probabilities, not the probability that the null hypothesis is true.

Worked Example: A Difference With Adequate Expected Counts

Suppose a hypothetical study has 100 cases and 200 controls at one variant. Genotypes are missing for 12 cases and 10 controls. Thus, the table is:

Status	Missing	Called	Total
Cases	12	88	100
Controls	10	190	200
Total	22	278	300

The observed missingness proportions are \(12/100=0.120\), or 12.0%, for cases and \(10/200=0.0500\), or 5.00%, for controls. The observed difference is 0.0700, or 7.00 percentage points. Under the null, the pooled missingness proportion is \(22/300=0.07333\). The expected counts are 7.333 missing cases, 92.667 called cases, 14.667 missing controls, and 185.333 called controls. All exceed 5, so the usual chi-square approximation is reasonable for this table.

Using the shortcut formula,

\chi^2 = \frac{300(12\cdot190-88\cdot10)^2} {100\cdot200\cdot22\cdot278} = \frac{300(1400)^2}{122{,}320{,}000} = 4.807.

With \(df=1\), the two-sided chi-square p-value is approximately 0.0283. At the 0.05 significance level, this is evidence that missingness probabilities differ between the groups. It does not show why they differ, nor does it establish that disease status caused the difference. The 7.00-percentage-point sample difference gives the direction and size of the observed imbalance; the test quantifies how inconsistent the table is with equal probabilities under its assumptions.

When Counts Are Small

If expected cell counts are small, the chi-square reference distribution may not approximate the test statistic well. Fisher’s exact test is a standard alternative for a two-by-two table. It evaluates tables conditional on the observed row and column totals. Under the null of no association between case-control status and missingness, and conditional on those margins, the number of missing cases has a hypergeometric distribution. An exact p-value sums probabilities of outcomes at least as incompatible with the null as the observed table, according to the chosen two-sided convention.

Worked Example: Sparse Counts and Fisher’s Exact Test

In another hypothetical data set, 3 of 20 cases have a missing call, while 0 of 30 controls do. The observed proportions are \(3/20=0.150\), or 15.0%, and \(0/30=0\), respectively. The expected number of missing controls under the null is \(30(3/50)=1.8\), below 5, so the chi-square approximation is questionable.

Conditioning on the margins, there are 3 missing calls among 50 people, and 20 of the people are cases. Let \(X\) be the number of those 3 missing calls assigned to cases under the null. Then

P(X=x)=\frac{\binom{20}{x}\binom{30}{3-x}}{\binom{50}{3}}, \qquad x=0,1,2,3.

The observed table has \(X=3\). Its probability is

P(X=3) = \frac{\binom{20}{3}\binom{30}{0}}{\binom{50}{3}} = \frac{1140}{19600} = 0.05816.

For this table, the other outcome probabilities are \(P(X=0)=4060/19600\), \(P(X=1)=8700/19600\), and \(P(X=2)=5700/19600\), each greater than the observed table’s probability. Using the common two-sided probability-ordering convention, only \(X=3\) is at least as improbable as the observed outcome, so the Fisher exact p-value is approximately 0.0582. This is not below 0.05. The observed difference is notable, but this small example does not provide strong enough evidence to reject equal missingness probabilities at that level.

For sufficiently large expected counts, the Pearson test is convenient and its result for a two-by-two table is equivalent to the squared two-proportion z statistic. Fisher’s test avoids reliance on the large-sample chi-square approximation but conditions on the margins and can be conservative in some settings. If adjustment for measured covariates is needed, a regression model for missing-call status may be considered; that is a different analysis and requires its own modeling assumptions. The simple table test does not adjust for covariates.

Interpret the Diagnostic and Investigate Its Limits

A small p-value at a variant flags evidence of unequal call missingness by group. It does not identify a mechanism. For example, if cases and controls were processed in different laboratories or batches, a technical difference could be aligned with case-control status. In that situation, the pooled table may show differential missingness even when the missingness pattern is the same within each batch.

Worked Example: A Pooled Difference Caused by Batch Imbalance

Consider a hypothetical study with two processing batches. Among cases, 80 people were processed in batch A and 20 in batch B; among controls, 20 were in batch A and 80 in batch B. In batch A, 8 of 80 cases and 2 of 20 controls have missing calls, both 10.0%. In batch B, neither group has missing calls. The pooled counts are therefore 8 missing and 92 called cases, and 2 missing and 98 called controls.

The pooled sample proportions are \(8/100=0.0800\), or 8.00%, for cases and \(2/100=0.0200\), or 2.00%, for controls. The pooled chi-square statistic is

\chi^2 = \frac{200(8\cdot98-92\cdot2)^2} {100\cdot100\cdot10\cdot190} = \frac{200(600)^2}{19{,}000{,}000} = 3.789.

The pooled p-value is about 0.0516. More importantly, the batch-specific proportions show no case-control difference within either batch: both groups have 10.0% missingness in batch A and 0% in batch B. The groups differ in how many people were processed in each batch, and that imbalance produces a pooled difference. This example illustrates why a flagged variant should prompt review of processing and sample composition rather than an automatic biological interpretation. It also shows why a pooled test alone cannot rule out confounding.

In practice, review the group-specific call rates, the counts in the table, and relevant technical variables such as assay type, sequencing depth, and processing batch. The earlier material on missingness diagnostics across samples and loci helps place a variant-level group comparison in a broader quality-control review. Investigate whether the result is driven by a small number of samples or by a broader pattern. Very low missingness can make a relative difference look large while leaving few missing calls to analyze; conversely, a modest proportion difference may involve many people in a large study.

A common pitfall is to write \(H_0:a/(a+b)=c/(c+d)\) as a population-level hypothesis. Those fractions are observed sample proportions, not population probabilities. The population-level null is \(H_0:p_{\mathrm{case}}=p_{\mathrm{control}}\); the observed fractions estimate the two probabilities. Another pitfall is treating a small p-value as proof that a variant is associated with disease or that a particular assay mechanism caused missingness. The test only detects evidence of an association between case-control status and missing-call status at the variant, under the stated sampling assumptions.

Finally, a genome-wide analysis may test missingness at many variants. Even if each null hypothesis is true, some small p-values can occur by chance across a large collection of tests. Interpret individual results in light of the number of variants examined and the study’s quality-control strategy; a single unadjusted p-value is not a complete genome-wide conclusion. The test is one diagnostic signal alongside call-rate summaries and technical review, not a universal retain-or-remove rule.

Key takeaway: Compare the missing-call proportions in cases and controls using a two-by-two table. Use Pearson’s chi-square test when its approximation is appropriate and Fisher’s exact test for sparse tables. A detected difference is evidence of a QC imbalance to investigate, not an explanation of its cause.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In a study, 6 of 60 cases and 3 of 90 controls have missing calls. What are the two observed missingness proportions, and what are the corresponding population parameters in the null hypothesis?
Why is the pooled missingness proportion used to calculate expected counts under the null?
A two-by-two table has a small expected missing-control count. Which test is a standard alternative to Pearson’s chi-square test, and what does it condition on?
A variant has a small differential-missingness p-value. Does this prove that disease status causes missingness? Give one possible technical explanation.
Why can unequal batch composition create an apparent pooled case-control difference even when missingness proportions match within each batch?

Solutions. (1) The case proportion is \(6/60=0.100\), or 10.0%; the control proportion is \(3/90=0.0333\), or 3.33%. The null is \(H_0:p_{\mathrm{case}}=p_{\mathrm{control}}\), where each \(p\) is a group-specific missingness probability, not a sample fraction. (2) Under the null, both groups share a common missingness probability, estimated from all observed missing calls divided by all people; expected cell counts use that pooled estimate. (3) Fisher’s exact test is standard. It conditions on the observed row and column totals and evaluates the table under the null association model. (4) No. The result indicates evidence of unequal missingness, not causation. For example, cases and controls may have been processed in different sequencing batches with different call quality. (5) If batch-specific missingness differs and the groups have different proportions of people in each batch, the pooled rates combine those batch patterns with different weights. The pooled proportions can therefore differ even if the case and control rates agree within each batch.