A likelihood-based approach for multivariate one-sided tests with missing data

<p>Inequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.</p

Overview of the CSD-based algorithm.

<p>Initially the PPG signal is segmented into windows (60 s or 120 s) with 50% of overlap. In the subsequent step the CSD is applied to calculate the spectrum of the windowed signals. The HR is estimated by detecting the maximum frequency peak within the cardiac frequency band. The signal is then low pass filtered and the RR is estimated by detecting the maximum frequency peak within the respiratory frequency band.</p

CSD applied to an in-vivo signal.

<p>CSD and PSD performance applied to an in-vivo signal (1 min) of one infant subject: (A) reference HR (dotted red line with markers) and mean HR represented by a dotted grey line, (B) reference RR (dotted red line with markers) and mean RR represented by a dotted grey line, (C) ECG signal, (D) capnometry, (E) PPG signal, (F) CSD and (G) PSD applied to the PPG signal. In addition, the average CSD and PSD spectrum of the database’s population is illustrated in the background on (F) and (G) respectively, where the cardiac component is represented in dark grey and the filtered signal that corresponds to respiration in light grey.</p

RMS error estimating RR and HR with different methods.

<p>RMS error median (quartiles) estimating RR and HR using different methods. The statistical significant difference (0.05) of the RMS error obtained with the CSD-based algorithm in comparison to other methods is indicated (asterisk *).</p

Time-varying CSD of 8-min PPG signal.

<p>Both respiratory and cardiac frequency peaks reflect RR and HR, respectively. Respiratory frequency peak is around 0.3 Hz (18 breaths/min) and cardiac frequency peak around 1.25 Hz (75 beats/min).</p

CSD applied to a simulated signal.

<p>(A) Simulated signal with 0.2 Hz modulation respiratory frequency (12 breaths/min), 1 Hz cardiac frequency (60 beats/min), and , (B) same simulated signal with some outliers randomly added, (C) and (D) the CSD of the simulated signal with and without outliers, and (E) and (F) the PSD of the simulated signal with and without outliers, respectively. CSD analysis provides a clearer and more robust against outliers respiratory frequency peak than conventional PSD.</p

CSD sensitivity to the kernel parameter.

<p>CSD-based algorithm’s perfomance estimating RR is illustrated for the kernel values: , , and . The is calculated by Silverman’s rule.</p

Scatter plot, error per subject.

<p>Scatter plot showing the median value of estimated and reference values of (A) RR and (B) HR for each subject using 60-s time window. The respiratory and cardiac frequency peaks are detected around the extended RR and HR range. Observations with artifacts are included. The dotted line represents the optimal performance.</p

Time-varying estimated and manually labeled reference RR and HR.

<p>Estimated (solid blue with * markers) and manually labeled (dotted red with+markers) reference RR in (A) and HR in (B). For this subject the RMS errors estimating RR and HR are 0.25 breaths/min and 0.35 beats/min, respectively.</p

Scatter plot, error per time window.

<p>Scatter plot showing the estimated and reference values of (A) RR and (B) HR for each 60-s time window (represented by blue +) and for each 120-s time window (represented by black *). The respiratory and cardiac frequency peaks are detected around the extended RR and HR range. Observations with artifacts are included. The dotted line represents the optimal performance.</p