Analysis of T-Cell Immune Responses as Measured by Intracellular Cytokine Staining with Application to Vaccine Clinical Trials

Abstract

Recent advances in single-cell technologies, in particular intracellular cytokine staining (ICS), have enabled multidimensional functional measurements of naturally occurring or vaccine-induced T-cell responses in clinical studies. Analysis of such increasingly multidimensional datasets presents a great challenge to statisticians. Currently, multidimensional functional cell measures are largely analyzed, either by univariate analysis of all combinations of functions individually, or by summarizing a few particular groups of functions separately. Such simple analyses do not reflect comprehensively the polyfunctional profile of the T-cell responses, nor do they allow more sophisticated statistical analysis and inference. In this paper, we introduce a new approach to statistical inference for multidimensional ICS data. We propose to reduce the dimensionality by using a weighted sum, followed by computing the minimum and maximum of the test statistic over all eligible assignments of weights which satisfy the underlying partial ordering of the data. The computation technique is presented. Statistical inference is then based on the minimum and maximum of the test statistic. We illustrate, through an example, that the technique can be useful in reducing the complexity of the multidimensional response data and providing insightful reporting of the results.

Appendix

Proof of Theorem 2.1. Let population X data be stochastically greater with respect to the partial ordering $\preceq$ than population Y data. We need to show t($\alpha$) $\ge$ 0 for all eligible weights consistent with $\preceq$.

Given any $\alpha$ consistent with the partial ordering $\preceq$, let $\alpha _{(1)} \ge \alpha _{(2)} \ge ... \ge \alpha _{(M)}$ denote them placed in descending order. The M combinations corresponding to this order can be written as $\{C_{(1)}, C_{(2)}, ..., C_{(M)}\}$ and the corresponding data as $\bar{x}^* = (x_{\cdot (1)}, ..., x_{\cdot (M)})^T$ and $\bar{y}^*= (y_{\cdot (1)}, ..., y_{\cdot (M)})^T$.

Denote a sequence of sets $\{U_{(k)}\}$ such that $U_{(k)} = \{C_{(1)}, C_{(2)}, ..., C_{(k)}\}$, $k = 1, 2, ..., M$. It is easy to see that each $U_{(k)}$ is an upper set. Then by Definition 2.2 we have $\sum _{j=1}^k \bar{x}_{\cdot (j)} \ge \sum _{j=1}^k \bar{y}_{\cdot (j)}$, $k = 1, 2, ..., M$. Let $\Delta _{(j)} = \bar{x}_{\cdot (j)} - \bar{y}_{\cdot (j)}$, we have $\sum _{j=1}^k \Delta _{(j)} \ge 0$, $k = 1, 2, ..., M$.

Given $\alpha _{(1)} \ge \alpha _{(2)} \ge ... \ge \alpha _{(M)}$, let us write $\alpha _{(1)} = \alpha _{(2)} + \delta _{(1)}$, $\alpha _{(2)} = \alpha _{(3)} + \delta _{(2)}$, ..., $\alpha _{(M-1)} = \alpha _{(M)} + \delta _{(M-1)}$, with $\delta _{(1)}, ..., \delta _{(M-1)} \ge 0$. That is, $\alpha _{(j)} = \alpha _{(M)} + \sum _{r=j}^{M-1} \delta _{(r)}$, $j = 1, 2, ..., M$. It follows that

$$\begin{aligned} \begin{aligned} \alpha ^T (\bar{x} - \bar{y})&= \sum _{j=1}^M \alpha _{(j)} (\bar{x}_{\cdot (j)} - \bar{y}_{\cdot (j)}) \\&= \sum _{j=1}^M \{ \alpha _{(M)} + \sum _{r=j}^{M-1} \delta _{(r)} \} \Delta _{(j)} \\&= \alpha _{(M)}\sum _{j=1}^M \Delta _{(j)} + \sum _{j=1}^{M-1} \delta _{(j)} \sum _{r=1}^j \Delta _{(r)} \ge 0 ,\\ \end{aligned} \end{aligned}$$

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where the inequality follows from the fact that $\alpha _{(k)} \ge 0$, $\delta _{(k)} \ge 0$, and $\sum _{j=1}^k \Delta _{(j)} \ge 0$, for $k = 1, 2, ..., M$. Hence we have proven Theorem 2.1.