Smooth and Probabilistic PARAFAC Model with Auxiliary Covariates

Figures & data

Fig. 1 Illustration of the PARAFAC/CP decomposition method for immunoprofiles measured longitudinally. The white color represents missing data in the time direction. The three-way tensor is approximated as the sum of $K$ rank-one tensors $F^1,\dots ,F^K$. Each rank-one tensor $F^k$ $(k=1,\dots ,K)$ can be expressed using a factor along the subject direction $(U_k)$, a factor along the feature direction $(V_k)$, and a factor along the time direction $({\Phi }_k)$. The factors associated with the tensor decomposition can be used in downstream analysis.

Fig. 2 Reconstruction evaluation by the correlation between the estimates and the true signal tensor. In each subplot, the x-axis label indicates different J and observing rate, the y-axis is the achieved correlation, and the box colors represent different methods. The corresponding subplot column/row name represents the signal-to-noise ratio SNR1/SNR2.

Fig. 3 Comparison of SPACO and SPACO- for reconstructing U at J = 10, SNR2$=10$. In each subplot, the x-axis label indicates different component and observing rate, the y-axis is the achieved $(1-R^2)$, and the box colors represent different methods. The corresponding subplot column/row name represents the signal-to-noise ratio SNR1/component.

Fig. 4 Comparison of SPACO and SupCP for reconstructing $\Phi$ at J = 10. In each subplot, the x-axis label indicates different component and observing rate, the y-axis is the achieved $(1-R^2)$, and the box colors represent different methods. The corresponding subplot column/row name represents the signal-to-noise ratio SNR1/SNR2.

Fig. 5 Achieved Type I errors at observing rate r = 0.5. In each subplot, x-axis label indicates different combination of feature dimension J and targeted level $\alpha \in \{0.01,0.05\}$, while the y-axis represents the achieved Type I errors. Different bar colors represent different tests (partial or marginal). The two dashed horizontal lines correspond to levels 0.01 and 0.05.

Fig. 6 Achieved power at observing rate r = 0.5. In each subplot, the x-axis label indicates different combinations of feature dimension J and targeted level $\alpha \in \{0.01,0.05\}$, the y-axis indicates the achieved power. Different bar colors represent different tests (partial or marginal).

Fig. 7 Comparisons between subject scores estimated from SPACO and SPACO- as well as the static risk factors. Panel A displays the high concordance in the correlations between estimated subject scores from SPACO and SPACO-. Panel B shows the correlations with clinical responses (row label) using subject scores from the most distinct component, C2, estimated with SPACO and SPACO-. The associated permutation p-values, which assess the improvement in correlations with the four clinical responses using SPACO, are shown beneath the row label. Panel C shows the correlation between clinical responses and the two most significant risk factors identified through conditional independence testing, COVIDRISK_3 and BMI.

Table 1 Results from randomization test for the second component (C2).

	COVIDRISK_1	COVIDRISK_2	COVIDRISK_3	COVIDRISK_4	COVIDRISK_5	Age	sex	BMI
pval	0.958	0.809	0.001	0.268	0.73	0.518	0.762	0.07
padj	0.958	0.924	0.009	0.714	1.00	1.000	1.000	0.28
nz_count	7	24	50	23	5	NaN	47	NaN

NOTE: The column nonzero counts the number of 1 for binary covariate COVIDRISK_1 - COVIDRISK_5 and sex (1 = Male, 0 = Female). The adjusted p-values (padj) are based on BH correction.