Sensitivity Analysis for Nonignorable Missingness

Why Sensitivity Analysis Is Needed

A genetic association analysis may have genotype data and covariates for many participants but phenotypes for only a subset. If phenotype availability depends on the phenotype itself, even after conditioning on observed variables, the missingness is nonignorable. The phenotype values that are absent cannot be checked against the observed data, so the association estimate can depend on assumptions that the data alone do not determine.

Sensitivity analysis makes those assumptions explicit. Rather than claim that a particular unobserved-value model is known to be true, we vary a parameter over scientifically plausible values and examine how the estimated genotype association changes. The goal is to assess robustness, not to manufacture information about phenotypes that were never measured.

Definition: Let $Y$ be a phenotype, $G$ a genotype dosage, $C$ observed covariates, and $R$ an indicator equal to 1 when $Y$ is observed and 0 when it is missing. Missingness is missing at random (MAR) when, conditional on observed data such as $G$ and $C$, the probability that $Y$ is observed does not additionally depend on the unobserved value of $Y$. It is missing not at random (MNAR) when that additional dependence remains. A sensitivity analysis specifies how the distribution of missing $Y$ may differ from that of observed $Y$, or specifies a model for the dependence of $R$ on $Y$.

Here the focus is incomplete phenotypes, with genotype and the covariates used in the association model observed. The association target must be stated: for example, a covariate-adjusted mean difference per additional effect allele in the target sample. An estimate among participants with observed phenotypes need not equal that target if phenotype observation is related to $Y$.

Pattern-Mixture Shifts

A pattern-mixture approach describes the phenotype distribution separately by missingness pattern. For a simple calculation, work within a genotype and covariate stratum. Let $q(g,c)=P(R=0\mid G=g,C=c)$, and let $\mu_1(g,c)=E(Y\mid R=1,G=g,C=c)$. Define the shift $\delta(g,c)$ as the mean phenotype among participants with missing $Y$ minus the mean among those with observed $Y$, in that same stratum.

Definition: Under a pattern-mixture shift specification, $$ E(Y\mid R=0,G=g,C=c)=\mu_1(g,c)+\delta(g,c). $$ The full-data conditional mean is therefore $$ E(Y\mid G=g,C=c)=\mu_1(g,c)+q(g,c)\delta(g,c). $$ A zero shift recovers the MAR mean assumption within the stratum. A nonzero shift represents a specified departure from that assumption; it is not generally identified from the observed data.

The second equation follows by averaging the observed and missing groups: the observed group has probability $1-q(g,c)$ and mean $\mu_1(g,c)$, while the missing group has probability $q(g,c)$ and mean $\mu_1(g,c)+\delta(g,c)$. In analyses with covariates, the same idea can be applied within covariate strata or through a model for the conditional means, then standardized to the target covariate distribution. If the shift varies by genotype, it can change the genetic contrast; even a common shift can do so when missingness fractions differ across genotypes.

Worked Example: Shifts Change the Genotype Contrast

Suppose a hypothetical study has 100 participants at each dosage $G=0,1,2$. Phenotype means among those observed are 10, 12, and 14 units, respectively. The observed counts are 80, 60, and 40, so the missing fractions are $q_0=0.20$, $q_1=0.40$, and $q_2=0.60$. For illustration, define the per-allele association contrast as the difference between the dosage-2 and dosage-0 means divided by 2. Assume the pattern-mixture shifts are $\delta_0=0$, $\delta_1=-1$, and $\delta_2=-2$ phenotype units.

The reconstructed full-data means are observed mean plus missing fraction times shift:

\begin{aligned} \mu_0&=10+(0.20)(0)=10,\\ \mu_1&=12+(0.40)(-1)=11.6,\\ \mu_2&=14+(0.60)(-2)=12.8. \end{aligned}

The observed-data endpoint contrast is $(14-10)/2=2$ phenotype units per allele. Under the specified shifts, the full-data endpoint contrast is

\frac{\mu_2-\mu_0}{2} =\frac{12.8-10}{2} =1.4\text{ phenotype units per allele}.

Thus these assumptions reduce the contrast from 2 to 1.4 units per allele. This endpoint contrast is an additive dosage slope only if an additive mean relationship is appropriate; otherwise it remains the stated average difference from dosage 0 to dosage 2. The numbers illustrate a sensitivity scenario, not a result established by the observed data.

Choosing and Interpreting Sensitivity Values

A useful sensitivity analysis begins with the outcome scale and the missingness pattern, not with an arbitrary numerical grid. A shift of $-2$ might be small for a phenotype measured in hundreds of units and substantial for one measured in a narrow range. Investigators can specify shifts in natural units, as a fraction of a residual standard deviation, or as a function of genotype and covariates. They should explain why the range is plausible using measurement practice, subject-matter knowledge, external data, or a transparent range of hypothetical departures.

One convenient approach is to choose a common shift $\delta$, calculate the association estimate over a grid of values, and then expand the grid to allow genotype-specific shifts if that is scientifically credible. For each value, reconstruct or impute the missing phenotype distribution and fit the same association model, with the same genotype coding, covariate adjustment, and target population. If the result changes only under shifts that would be implausibly large, the finding is relatively robust to the examined departures. If modest shifts materially change its magnitude, uncertainty about missingness is consequential.

A tipping point is a value or combination of sensitivity parameters at which a chosen conclusion changes, such as the estimate crossing zero or a confidence interval including zero. It is a summary of dependence on the assumptions, not evidence that the tipping-point value is the true shift. Statistical uncertainty also remains: sensitivity analysis should report uncertainty conditional on each assumed model, not just a sequence of point estimates.

Worked Example: Finding a Tipping Point

Suppose a hypothetical study has observed phenotype means of 30 units at $G=0$ and 34 units at $G=2$. The missing fractions are $q_0=0.10$ and $q_2=0.50$. Use a common pattern-mixture shift $\delta$, measured in phenotype units, for both genotype groups. Define the per-allele contrast as the dosage-2 mean minus dosage-0 mean, divided by 2.

The adjusted endpoint means are $30+0.10\delta$ and $34+0.50\delta$. Therefore the sensitivity-adjusted contrast is

\hat{\beta}(\delta) =\frac{(34+0.50\delta)-(30+0.10\delta)}{2} =2+0.20\delta \quad\text{phenotype units per allele}.

At $\delta=-5$, the estimate is $2+0.20(-5)=1$ unit per allele. To find the shift at which the estimate reaches zero, solve $2+0.20\delta=0$, giving $\delta=-10$ phenotype units. Under this specification, the missing outcomes would need to average 10 units below the corresponding observed outcomes in both genotype groups for the endpoint contrast to reach zero. This conclusion depends on the common-shift assumption; allowing shifts to differ by genotype gives a different tipping-point equation.

Selection Models and Their Connection to Shifts

A selection model instead describes the probability that the phenotype is observed as a function of its value. For a continuous phenotype, one possible specification within a genotype and covariate stratum is

\operatorname{logit} P(R=1\mid Y=y,G=g,C=c) =\alpha(g,c)+\gamma y.

Here $\exp(\gamma)$ is the multiplicative change in the odds of phenotype observation for a one-unit increase in $Y$, conditional on $G$ and $C$. Thus $\gamma=0$ means response odds do not depend on $Y$ within the stratum. A nonzero $\gamma$ represents outcome-dependent selection and hence an MNAR departure in this model. The intercept may vary by stratum to allow different baseline observation rates.

The selection parameter is generally not estimable from observed phenotype values alone: the values for $R=0$ are precisely the values needed to learn how observation depends on $Y$. It is therefore varied as a sensitivity parameter, unless relevant external information or validation data are available. The following relationship to pattern-mixture shifts is useful but relies on additional distributional assumptions.

Result: Suppose, within a fixed $G,C$ stratum, the selection model has log odds $\alpha+\gamma y$, and the observed phenotype distribution is normal with mean $\mu_1$ and variance $\sigma^2$. Then the implied missing-phenotype distribution is normal with the same variance and mean $\mu_1-\gamma\sigma^2$. Consequently, the pattern-mixture mean shift is $\delta=-\gamma\sigma^2$.

Why the Selection-Model Relationship Holds

Let $f_1(y)$ and $f_0(y)$ denote the phenotype densities conditional on $R=1$ and $R=0$, respectively, within the stratum. Bayes' rule gives the ratio of these densities as the ratio of the probability of nonresponse to response, multiplied by a constant that does not depend on $y$:

\frac{f_0(y)}{f_1(y)} \ \propto\ \frac{P(R=0\mid Y=y)}{P(R=1\mid Y=y)} =\exp(-\alpha-\gamma y) \ \propto\ \exp(-\gamma y).

If $f_1(y)$ is normal with mean $\mu_1$ and variance $\sigma^2$, multiplying its density by $\exp(-\gamma y)$ and completing the square gives a normal density with mean $\mu_1-\gamma\sigma^2$ and variance $\sigma^2$. In detail, the terms involving $y$ in the exponent are

-\frac{(y-\mu_1)^2}{2\sigma^2}-\gamma y = -\frac{\{y-(\mu_1-\gamma\sigma^2)\}^2}{2\sigma^2} +\text{a term not depending on }y.

After normalization, the constant term does not affect the density's mean or variance. This derivation requires the stated normal observed distribution and log-linear response odds; it is not a general conversion from any selection model to any pattern-mixture model.

Worked Example: Translating a Selection Parameter Into an Association Shift

Consider a hypothetical quantitative phenotype measured in points. Within each genotype group, assume the observed outcomes are normally distributed with standard deviation 10 points, and suppose the log odds of observation increase by $\gamma=0.02$ for each one-point increase in the phenotype. Assume this same $\gamma$ and variance apply across genotype groups. The implied shift is

\delta=-\gamma\sigma^2 =-(0.02)(10^2) =-2\text{ points}.

For example, the response odds for a participant whose phenotype is 1 point higher are multiplied by $\exp(0.02)\approx1.020$, or about 1.020, under this model. Because higher outcomes have higher response odds, the implied mean among missing outcomes is 2 points below the observed mean within each genotype group.

Now suppose the observed means at $G=0,1,2$ are 50, 52, and 54 points, and the corresponding missing fractions are 0.20, 0.40, and 0.60. The reconstructed means are $50+(0.20)(-2)=49.6$, $52+(0.40)(-2)=51.2$, and $54+(0.60)(-2)=52.8$ points. The dosage-0-to-2 contrast is therefore

\frac{52.8-49.6}{2} =1.6\text{ points per allele},

compared with the observed endpoint contrast $(54-50)/2=2$ points per allele. The association changes because a common within-genotype shift is combined with different missing fractions across genotypes. Both the selection-model form and its parameters are assumptions for this calculation, not estimates recovered from these observed means.

Practical Analysis and Common Pitfalls

A defensible analysis makes the sensitivity specification reproducible. State the target association estimand, the observed-data model, the missingness parameters, and the range examined. For each parameter setting, account for uncertainty in estimating the observed-data distribution and in generating or integrating over missing outcomes. Multiple imputation can implement a pattern-mixture specification by drawing missing values from the shifted conditional distribution and fitting the association model in each completed data set; uncertainty is then combined across the completed analyses. The shift should be imposed on the relevant conditional distribution, not added to observed data indiscriminately.

When covariates are present, shifts may be conditional on them, and the final estimate should use a consistent covariate-adjusted model or standardization procedure across sensitivity settings. A marginal shift and a shift conditional on covariates need not yield the same result. Report estimates and intervals across the sensitivity range, and distinguish uncertainty conditional on a chosen parameter from uncertainty about which parameter is plausible.

Mistake: treating the MAR result as verified. A zero shift or $\gamma=0$ is a reference assumption, not something established merely because observed-data diagnostics look satisfactory.
Mistake: calling a sensitivity parameter estimated. Without information on unobserved outcomes or external restrictions, the shift or outcome-selection parameter is generally not identified by the observed data. Present it as an assumption and show how conclusions vary.
Mistake: assuming a common shift must cancel in a genotype comparison. The contribution to a group mean is $q_g\delta$. If $q_g$ differs across genotypes, the adjusted contrast can change even when $\delta$ is common.
Mistake: interpreting a tipping point as proof of robustness. A large numerical shift is only reassuring if it is large relative to scientifically plausible departures on the phenotype's scale. Explain the scale and rationale.

Key takeaway: Sensitivity analysis for nonignorable phenotype missingness does not identify the missing outcomes. It shows how a genetic association estimate changes under explicit, scientifically interpretable assumptions about those outcomes or about their probability of being observed.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following.

In a genotype stratum, the observed mean is 8 units, the missing fraction is 0.25, and the assumed missing-minus-observed shift is $-4$ units. What full-data mean does the pattern-mixture calculation imply?
Why can a common shift alter a genotype contrast when missing fractions differ across genotype groups?
Under the normal selection-model assumptions, what is the implied missing-phenotype mean shift if $\gamma=0.01$ per outcome unit and $\sigma=5$ outcome units?
Why is a selection parameter such as $\gamma$ ordinarily varied in a sensitivity analysis rather than estimated from observed outcomes alone?
What does a tipping point describe, and why does it not establish that the corresponding shift is true?