Technical Summary: Testing the Missing Completely at Random Assumption for Functional Data

Problem Statement

Functional Data Analysis (FDA) traditionally assumes observations are available over a common domain. However, in many practical applications, data are "partially observed," meaning each function is only observable on a subset $O$ of the domain $I$, with values missing on $I \setminus O$. A critical prerequisite for most existing reconstruction and inference methods for such data is the assumption that the missingness mechanism is Missing Completely at Random (MCAR). This assumption posits that the observed function $X$ and the observation set $O$ are independent.

While numerous methods exist to recover missing functional data under the MCAR assumption (e.g., Delaigle and Hall, 2016; Kraus, 2015), there is a lack of general statistical procedures to validate this assumption. Existing tests for MCAR in multivariate settings (e.g., Little, 1988) rely on grouping data by distinct missingness patterns. This approach is infeasible for functional data because the number of distinct observation sets is uncountably infinite, and in practice, every observation pattern may be unique. The paper addresses the gap of developing a general testing framework to determine whether $X$ and $O$ are independent.

Methodology

1. Partitioning Observation Sets

The core of the proposed framework is to partition the space of observation sets $\mathcal{K}$ into two disjoint subsets, $\mathcal{A}$ and $\mathcal{B}$, such that $\mathcal{K} = \mathcal{A} \dot{\cup} \mathcal{B}$.

• Ad Hoc Partitioning: A simple approach divides data into "complete" and "incomplete" functions.
• Systematic Clustering: The authors propose a centroid-based clustering approach. They define a distance between sets $S_1, S_2$ using the symmetric difference measure $d(S_1, S_2) = \lambda(S_1 \triangle S_2)$, which corresponds to the $L^1$ distance between their indicator functions.
• Optimal Centers: The partition is determined by finding two optimal centers (medians) $M_A$ and $M_B$ that minimize the expected distance to the nearest center. The sets $\mathcal{A}$ and $\mathcal{B}$ are defined as the Voronoi cells associated with these centers.
• Domain Restriction: To ensure valid comparison, the analysis is restricted to a subdomain $I \subset [0, 1]$ where both groups have sufficient coverage (specifically, $P(t \in O | \mathcal{A}) > 0$ and $P(t \in O | \mathcal{B}) > 0$).

2. Hypothesis Testing

The null hypothesis $H_0$ (MCAR) implies that $X$ and $O$ are independent. The testing framework evaluates a stronger hypothesis $H_0^F$:
$$H_0^F: P(X(t) \le z | O \in \mathcal{A}) = P(X(t) \le z | O \in \mathcal{B}), \quad \forall z \in \mathbb{R}, t \in [0, 1].$$
Under $H_0^F$, the conditional distribution of the process given the observation set group should be identical. The authors propose two test statistics:

• Mean Comparison ($T_\mu$):
  This statistic compares the empirical mean functions of the two groups, $\hat{\mu}_{\mathcal{A}}(t)$ and $\hat{\mu}_{\mathcal{B}}(t)$.
  $$T_\mu = \sqrt{n} \| \hat{\mu}_{\mathcal{A}} - \hat{\mu}_{\mathcal{B}} \|_\infty$$
  The authors establish that under MCAR, $T_\mu$ converges to the supremum of a Gaussian process. A key theoretical contribution is handling the fact that empirical mean functions of partially observed data are generally discontinuous (càdlàg), requiring the use of the Skorokhod space $D[0, 1]$ rather than $L^2[0, 1]$ or $C[0, 1]$.

• Distribution Comparison ($T_F$):
  This statistic compares the empirical conditional distribution functions $\hat{F}_{\mathcal{A}}(t, z)$ and $\hat{F}_{\mathcal{B}}(t, z)$.
  $$T_F = n \int_{[0,1] \times \Xi} (\hat{F}_{\mathcal{A}}(t, z) - \hat{F}_{\mathcal{B}}(t, z))^2 d(\lambda \otimes \nu)(t, z)$$
  This is a Cramér–von Mises type statistic. It is shown to be consistent against a broader class of alternatives than the mean comparison, as it detects differences in the entire distribution, not just the mean.

3. Asymptotic Theory and Implementation

The paper derives the asymptotic distributions for both test statistics under the null hypothesis. The limiting distributions involve Gaussian processes with specific covariance structures that depend on the observation patterns.

• Covariance Estimation: The authors provide estimators for the covariance functions required to approximate the null distribution.
• Simultaneous Confidence Bands: A method is introduced to construct simultaneous confidence bands for the difference in mean functions, providing a graphical tool to visualize potential violations of MCAR.
• Consistency: Theoretical results (Theorem 3) establish that the tests are consistent against alternatives where the mean functions or distribution functions differ between the groups.

Key Results

Simulation Study

The methodology was evaluated using Brownian motion simulations:

• Type I Error: Under MCAR, both tests ($T_\mu$ and $T_F$) maintained error rates close to the nominal level (0.05) across various sample sizes ($n=100, 250, 500$).
• Power:
  • In scenarios where MCAR was violated due to censoring (Missing Not At Random - MNAR), the distribution test $T_F$ demonstrated higher power than the mean test $T_\mu$.
  • Specifically, in symmetric censoring scenarios where the conditional means of the groups were identical ($\|\mu_A - \mu_B\|_\infty = 0$), $T_\mu$ failed to reject the null, while $T_F$ successfully detected the violation.
  • The systematic clustering approach performed comparably to or better than deterministic partitions (e.g., complete vs. incomplete) in many scenarios, particularly for smaller sample sizes.

Real Data Applications

The framework was applied to three datasets:

1. Heart Rate Data (Swiss Kidney Project): Analysis of 878 heart rate profiles. The tests did not reject the MCAR assumption ($p$-values: 0.765 for $T_\mu$, 0.911 for $T_F$). This supports the epidemiological view that missingness (due to device removal or failure) is random.
2. Electricity Price Data: Analysis of 583 weekly price curves. The tests strongly rejected MCAR ($p < 0.001$). The mean difference plot showed significant deviations, aligning with the market structure where price levels correlate with the duration of observation (larger observation sets correspond to higher prices).
3. Temperature Data: Analysis of 76 daily temperature curves. The tests rejected MCAR ($p$-values: 0.036 for $T_\mu$, 0.054 for $T_F$). The results suggested a potential technical defect in the sensor (e.g., sensitivity to high temperatures or power issues) rather than random missingness.

Significance and Contributions

The paper claims the following contributions:

1. Novel Framework: It introduces the first general testing framework for the MCAR assumption in functional data, overcoming the infeasibility of pattern-based grouping used in multivariate settings.
2. Systematic Partitioning: It proposes a data-adaptive clustering method based on the Fréchet-Nikodym-Aronszajn distance to systematically divide observation sets, rather than relying solely on ad hoc definitions.
3. Theoretical Advancement: It establishes asymptotic results for functional data in the Skorokhod space $D[0, 1]$, accommodating the discontinuities inherent in partially observed mean functions.
4. Superiority of Distribution Tests: The authors demonstrate that comparing full distributions ($T_F$) is more robust and powerful than comparing means ($T_\mu$), particularly in cases where missingness affects higher-order moments or distribution shapes but not the mean.
5. Practical Utility: The methodology provides a diagnostic tool for researchers to validate the assumptions underlying their functional data reconstruction methods, illustrated by its ability to detect sensor defects and market-driven missingness in real-world examples.

The authors note that while their approach is general, the power of the test depends on the choice of partition. They suggest that for small sample sizes, expert-guided deterministic partitions might be preferable, and future work could explore supervised classification methods to optimize group separation.