The Efficient Covariate-Adaptive Design for high-order balancing of quantitative and qualitative covariates

Abstract

In the context of sequential treatment comparisons, the acquisition of covariate information about the statistical units is crucial for the validity of the trial. Furthermore, balancing the assignments among covariates is of primary importance, since the potential imbalance of the covariate distributions across the groups can severely undermine the statistical analysis. For this reason, several covariate-adaptive randomization procedures have been suggested in the literature, but most of them only apply to categorical factors. In this paper we propose a new class of rules, called the Efficient Covariate-Adaptive Design, which is high-order balanced regardless of the number of factors and their nature (qualitative and/or quantitative), also accounting for every order covariate effects and interactions. The suggested procedure performs very well, is flexible and simple to implement. The advantages of our proposal are also analyzed via simulations and its finite sample properties are compared with those of other well-known rules, by also including the redesign of a real clinical trial.

Appendix A Proofs

1.1 A.1 Optimality of balance for estimating/testing the treatment difference $\mu _A-\mu _B$

For simplicity of notation in this Appendix we omit the subscript n. After n allocations, let $\hat{\varvec{\gamma }}$ be the LSE of $\varvec{\gamma }=(\mu _A,\mu _B,\varvec{\beta })^{t}$ and $a^t=(1;-1;{\varvec{0}}_q^t)$, then $\text {{var}}(a^t\hat{\varvec{\gamma }})=\sigma ^2a^t(n{\mathbb {M}})^{-1}a$, where

$$\begin{aligned} {\mathbb {M}}=\frac{1}{n}\left( \begin{array}{ccc} n_A &{} 0 &{} \varvec{\delta }^{t}{\mathbb {F}} \\ 0 &{} n_B &{} ({\varvec{1}}-\varvec{\delta })^{t}{\mathbb {F}} \\ {\mathbb {F}}^{t}\varvec{\delta } &{} {\mathbb {F}}^{t}({\varvec{1}}- \varvec{\delta }) &{} {\mathbb {F}}^{t}{\mathbb {F}} \end{array} \right) . \end{aligned}$$

Let ${\textbf{x}}^t={\textbf{1}} ^t{\mathbb {F}}$ and ${\textbf{y}}^t=(2\varvec{\delta }-{\varvec{1}})^{t}{\mathbb {F}}$, then $n{\mathbb {M}}= \left( \begin{array} {c | c} \text {{diag}}\left( n_A,n_B\right) &{} {\mathbb {D}}^t\\ \hline {\mathbb {D}} &{} n{\mathbb {A}} \end{array}\right)$, with $\mathbb {D}=2^{-1}\left( {\textbf{x}}+{\textbf{y}}\mid {\textbf{x}}-{\textbf{y}}\right)$. By letting $\mathbb {T}=\text {{diag}}\left( n_A,n_B\right) -{\mathbb {D}}^t (n{\mathbb {A}})^{-1} {\mathbb {D}}$, then $a^t(n{\mathbb {M}})^{-1}a=(1,-1)\mathbb {T}^{-1}\left( \begin{array} {c} 1\\ -1 \end{array}\right)$, where

$$\begin{aligned} \mathbb {T}=\left( \begin{array} {c | c} n_A-(4n)^{-1}({\textbf{x}}+{\textbf{y}})^t{\mathbb {A}}^{-1}({\textbf{x}}+{\textbf{y}}) &{} -(4n)^{-1}({\textbf{x}}+{\textbf{y}})^t{\mathbb {A}}^{-1}({\textbf{x}}-{\textbf{y}})\\ \hline -(4n)^{-1}({\textbf{x}}-{\textbf{y}})^t{\mathbb {A}}^{-1}({\textbf{x}}+{\textbf{y}}) &{} n_B-(4n)^{-1}({\textbf{x}}-{\textbf{y}})^t{\mathbb {A}}^{-1}({\textbf{x}}-{\textbf{y}}) \end{array}\right) . \end{aligned}$$

After some algebra, $\det \mathbb {T}=(n-{\textbf{x}}^t(n{\mathbb {A}})^{-1}{\textbf{x}})(n-\ell )/4$, so that

$$\begin{aligned} a^t(n{\mathbb {M}})^{-1}a=\frac{n-{\textbf{x}}^t(n{\mathbb {A}})^{-1}{\textbf{x}}}{(n-{\textbf{x}}^t(n{\mathbb {A}})^{-1}{\textbf{x}})(n-\ell )/4}= \frac{4}{n-\ell }= \frac{4}{n} \left( 1 - \frac{\ell }{n}\right) ^{-1}. \end{aligned}$$

(A.1)

Taking into account hypothesis testing, under well-known regularity conditions $\sqrt{n}(\hat{\varvec{\gamma }}-\varvec{\gamma })\overset{d}{\longrightarrow } \text {N}({\varvec{0}}_{q+2}, \sigma ^{2} {\mathbb {M}}^{-1})$, so that $\sqrt{n}a^t(\hat{\varvec{\gamma }}-\varvec{\gamma })\overset{d}{\longrightarrow } \text {N}(0, \sigma ^{2}\Vert a\Vert _{{\mathbb {M}}^{-1}}^2)$. Assuming $\sigma ^2$ known, the classical Wald statistic is $W=n\hat{\varvec{\gamma }}^t a[\sigma ^2\Vert a\Vert _{{\mathbb {M}}^{-1}}^2]^{-1}a^t\hat{\varvec{\gamma }}=n({\hat{\mu }} _{A}-{\hat{\mu }} _{B})^2[\sigma \Vert a\Vert _{{\mathbb {M}}^{-1}}]^{-2}$. Under $H_0$, $W \overset{d}{\longrightarrow }\ \chi _{1}^2$, namely it converges to a (central) $\chi ^2$ with 1 degree of freedom (dof); whereas, under the alternative W converges to a non-central $\chi ^2_1$ with non-centrality parameter $n(\mu _{A}-\mu _{B})^2[\sigma \Vert a\Vert _{{\mathbb {M}}^{-1}}]^{-2}$. From (A.1), $a^t{\mathbb {M}}^{-1}a=4 \left( 1- \ell /n\right) ^{-1}$, so the non-centrality parameter is equal to $(2\sigma )^{-2} n(\mu _{A}-\mu _{B})^2(1 - \ell /n)$. For fixed dof the non-central $\chi ^2$ is stochastically increasing in the non-centrality parameter. Thus, for every sample size the power is an increasing function of it and is maximized when $\ell =0$.

1.2 A.2 Proof of Theorem 1

Following Theorem 4.5 of Meyn and Tweedie (1992), a Markov chain $\{{\textbf{X}}_n\}_{n\in \mathbb {N}}$ on a general state-space $\mathbb {X}$ is bounded in probability if i) ${\textbf{X}}_n$ is a T-chain and ii) ${\textbf{X}}_n$ satisfies a positive drift condition, namely there exists a norm-like function $V:\mathbb {X} \longrightarrow {\mathbb {R}}^+$ such that, for some $\varepsilon >0$ and a compact set ${\mathcal {C}}\in {\mathcal {B}}(\mathbb {X})$ (where ${\mathcal {B}}(\mathbb {X})$ is the Borel sigma algebra), we have

$$\begin{aligned} \begin{aligned} (D)&\quad \Delta V({\textbf{X}}_n):={\mathbb {E}} [V({\textbf{X}}_{n+1}) \mid {\textbf{X}}_n]-V({\textbf{X}}_n)\le -\varepsilon , \qquad {\textbf{X}}_n \in {\mathcal {C}}^c.\\ \end{aligned} \end{aligned}$$

Since a Markov chain on a countable state space is always a T-chain (Meyn and Tweedie 1992, p. 548) in what follows we just need to show that condition (D) is satisfied by ${\textbf{D}}_n$ and ${\textbf{b}}_n$. Let ${\mathbb {C}}={\mathbb {H}}^t{\mathbb {W}}{\mathbb {H}}$, then ${\mathbb {C}}=(c_{ij})_{i,j=1,\ldots ,s}$ is symmetric and positive-definite, since ${\mathbb {H}}$ is a full (row) rank matrix. Moreover, by letting $\tilde{{\textbf{D}}}_n={\mathbb {H}}^t{\mathbb {W}}{\mathbb {H}} {\textbf{D}}_n$, the linear transformation $\tilde{{\textbf{D}}}_{n}={\mathbb {C}}{\textbf{D}}_n$ is an isomorphism and therefore the behavior of the Markov chain $\{{\textbf{D}}_n\}_{n \in \mathbb {N}}$ is equivalent to the one of $\{\tilde{{\textbf{D}}}_n\}_{n \in \mathbb {N}}$ (although defined in a proper transformed space) and their roles in the proof could be naturally exchanged. We show that condition (D) holds by setting $V({\textbf{D}}_n)={\textbf{D}}_n^t{\mathbb {C}}{\textbf{D}}_n$ and ${\mathcal {C}}=\{{\textbf{D}}_n: \Vert {\mathbb {C}} {\textbf{D}}_n\Vert _{\infty } \le \kappa \}$, where $\kappa >0$. Note that $\Vert {\mathbb {C}} {\textbf{D}}_n\Vert _{\infty }$ is still a norm of the vector ${\textbf{D}}_n$, since ${\mathbb {C}}$ is invertible (namely the corresponding linear transformation is injective). Due to the isomorphism, the compact set could be analogously expressed by ${\mathcal {C}}=\{\tilde{{\textbf{D}}}_n: \Vert \tilde{{\textbf{D}}}_n\Vert _{\infty }\le \kappa \}$, so ${\mathcal {C}}^c=\{\tilde{{\textbf{D}}}_n: \max _{j=1,\ldots ,s} \vert \tilde{{D}}_{nj}\vert > \kappa \}$. From (6),

$$\begin{aligned} \Delta V({\textbf{D}}_n)=\sum _{j=1}^s {\mathbb {E}} [V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n) \mid {\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j] \Pr ({\textbf{x}}_{n+1}={\textbf{e}}_j), \end{aligned}$$

(A.2)

where

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} [V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n)\mid {\textbf{D}}_n , {\textbf{x}}_{n+1}={\textbf{e}}_j]=\\&\Pr (\delta _{n+1}=1\mid {\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j)\, {\mathbb {E}}[V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n)\mid {\textbf{D}}_n,{\textbf{x}}_{n+1}={\textbf{e}}_j, \delta _{n+1}=1]+\\&\Pr (\delta _{n+1}=0\mid {\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j) \,{\mathbb {E}}[V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n)\mid {\textbf{D}}_n,{\textbf{x}}_{n+1}={\textbf{e}}_j, \delta _{n+1}=0].\\ \end{aligned}\nonumber \\ \end{aligned}$$

(A.3)

Moreover,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n)\mid&{\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j, \delta _{n+1}=1]=\\ =&({\textbf{D}}_n+{\textbf{e}}_j)^t {\mathbb {C}}({\textbf{D}}_n+{\textbf{e}}_j)-{\textbf{D}}_n^t {\mathbb {C}} {\textbf{D}}_n=c_{jj}+2{\textbf{e}}_j^t{\mathbb {C}}{\textbf{D}}_n \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[V({\textbf{D}}_{n+1}) -V({\textbf{D}}_n)\mid&{\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j, \delta _{n+1}=0]=\\ =&({\textbf{D}}_n-{\textbf{e}}_j)^t {\mathbb {C}}({\textbf{D}}_n-{\textbf{e}}_j)-{\textbf{D}}_n^t {\mathbb {C}} {\textbf{D}}_n=c_{jj}-2{\textbf{e}}_j^t{\mathbb {C}}{\textbf{D}}_n, \end{aligned} \end{aligned}$$

where $c_{jj}={\textbf{e}}_j^t{\mathbb {C}}{\textbf{e}}_j>0$. Thus, recalling that $\tilde{{\textbf{D}}}_n={\mathbb {C}}{\textbf{D}}_n$, Eq. (A.3) becomes

$$\begin{aligned} \begin{aligned} {\mathbb {E}} [V({\textbf{D}}_{n+1})&-V({\textbf{D}}_n)\mid {\textbf{D}}_n, {\textbf{x}}_{n+1}={\textbf{e}}_j]= h({\textbf{e}}_j^t\tilde{{\textbf{D}}}_n)[c_{jj}+2{\textbf{e}}_j^t\tilde{{\textbf{D}}}_n]+\\ +&[1-h({\textbf{e}}_j^t\tilde{{\textbf{D}}}_n)] [c_{jj}-2{\textbf{e}}_j^t\tilde{{\textbf{D}}}_n]=4\tilde{{D}}_{nj}\left[ h(\tilde{{D}}_{nj})-\frac{1}{2}\right] +c_{jj}. \end{aligned} \end{aligned}$$

(A.4)

Therefore, from (A.2) and (A.4), $\Delta V({\textbf{D}}_n)=4\sum _{j=1}^s\tilde{{D}}_{nj}[h(\tilde{{D}}_{nj})-1/2]p_j+\sum _{j=1}^s c_ {jj} p_j$, where the first term is always non-positive since, if $\tilde{{D}}_{nj}\ge 0$ then $h(\tilde{{D}}_{nj})\le 1/2$ and when $\tilde{{D}}_{nj}<0$ then $h(\tilde{{D}}_{nj})>1/2$. Condition (D) is equivalent to

$$\begin{aligned} \sum _{j=1}^s \vert \tilde{{D}}_{nj} \vert [ h(-\vert \tilde{{D}}_{nj} \vert )- 1/2] p_j\ge (\varepsilon + \sum _{j=1}^s c_ {jj} p_j)/4. \end{aligned}$$

(A.5)

By the definition of ${\mathcal {C}}^c$, there exists at least one stratum ${\tilde{j}}$ such that $\vert \tilde{{D}}_{n{\tilde{j}}}\vert > \kappa$ and therefore $\sum _{j=1}^s \vert \tilde{{D}}_{nj} \vert [ h(-\vert \tilde{{D}}_{nj} \vert )- 1/2] p_j\ge \kappa [h(-\vert \kappa \vert )-1/2]p_{{\tilde{j}}}$, which is an increasing function of $\kappa$ since $h(-\vert \kappa \vert )>1/2$ and $p_{{\tilde{j}}}>0$. Thus, for every ${\textbf{D}}_n \in {\mathcal {C}}^c$ condition (D) is verified, since the RHS of (A.5) is bounded.

In addition, since ${\textbf{b}}_n={\mathbb {H}}{\textbf{D}}_n$, then ${\textbf{b}}_n=O_p(1)$ as a bounded linear combination of ${\textbf{D}}_n$; thus, from (2) and (4), $\ell _n=o_p(1)$, since $\Vert {\textbf{b}}_n \Vert _{{\mathbb {P}}_n^{-1}}^2$ is asymptotic equivalent to $\Vert {\textbf{b}}_n \Vert _{{\mathbb {P}}^{-1}}^2=O_p(1)$ (recalling that ${\mathbb {P}}_n-{\mathbb {P}}=o_{a.s.}(1)$). At the same time ${\textbf{b}}_n=O_p(1)$, then $n_A\bar{{\textbf{f}}}^A_{n} -n_B\bar{{\textbf{f}}}^B_{n}=O_p(1)$ and $n^{-1}D_n=o_p(1)$; by letting $\pi _n=n_A/n$, then $\pi _n-1/2=o_p(1)$ and $\Vert n_A\bar{{\textbf{f}}}^A_{n} -n_B\bar{{\textbf{f}}}^B_{n}\Vert ^2_{{\mathbb {A}}_n^{-1}}=O_p(1)$, so that $n^{-1}\Vert n_A\bar{{\textbf{f}}}^A_{n} -n_B\bar{{\textbf{f}}}^B_{n}\Vert ^2_{{\mathbb {A}}_n^{-1}}=n \Vert \pi _n\bar{{\textbf{f}}}^A_{n} -(1-\pi _n)\bar{{\textbf{f}}}^B_{n}\Vert ^2_{{\mathbb {A}}_n^{-1}}=o_p(1)$. Since $\pi _n - 1/2=o_p(1)$ and $\varvec{{\mathbb {A}}}_n-\varvec{{\mathbb {A}}}=o_{a.s.}(1)$, then $n \Vert \pi _n\bar{{\textbf{f}}}^A_{n} -(1-\pi _n)\bar{{\textbf{f}}}^B_{n}\Vert ^2_{{\mathbb {A}}_n^{-1}}$ is asymptotic equivalent to $4^{-1}n \Vert \bar{{\textbf{f}}}^A_{n} -\bar{{\textbf{f}}}^B_{n}\Vert ^2_{{\mathbb {A}}^{-1}}=o_p(1)$, namely ${\mathcal {M}}_n=o_p(1)$. Thus, under the ECADE, $\ell _n$ is asymptotic equivalent to ${\mathcal {M}}_n$. By the same arguments $\ell _n$ and ${\mathcal {M}}_n$ are asymptotic equivalent for every CA rule under which $\pi _n-1/2=o_p(1)$.

1.3 A.3 Proof of Theorem 2

Let ${\mathbb {C}}_n={\mathbb {H}}^t{\mathbb {W}}_n {\mathbb {H}}$, since ${\mathbb {W}}_n\longrightarrow {\mathbb {W}}$ a.s., then ${\mathbb {C}}_n \longrightarrow {\mathbb {C}}={\mathbb {H}}^t{\mathbb {W}}{\mathbb {H}}$ a.s. from Slutshy’s theorem. Consider the function $W({\textbf{D}}_n)={\textbf{D}}_n^t {\mathbb {C}}_{n} {\textbf{D}}_n=U({\textbf{D}}_n)+V({\textbf{D}}_n)$, where $U({\textbf{D}}_n)={\textbf{D}}_n^t ({\mathbb {C}}_{n}-{\mathbb {C}}) {\textbf{D}}_n$, while $V(\cdot )$ and the compact set ${\mathcal {C}}$ are the same as in the proof of Theorem 1 in Sect. 1. Thus we show that condition (D) is satisfied by $\Delta W({\textbf{D}}_n)= {\mathbb {E}} [W({\textbf{D}}_{n+1}) -W({\textbf{D}}_{n})\mid {\textbf{D}}_n]=\Delta U({\textbf{D}}_n) +\Delta V({\textbf{D}}_n)$. Since ${\mathbb {C}}_{n}-{\mathbb {C}}=o_{a.s.}(1)$, then $\Delta U({\textbf{D}}_n)={\mathbb {E}}[{\textbf{D}}_{n+1}^t ({\mathbb {C}}_{n+1}-{\mathbb {C}}) {\textbf{D}}_{n+1}-{\textbf{D}}_{n}^t ({\mathbb {C}}_{n}-{\mathbb {C}}) {\textbf{D}}_{n}\mid {\textbf{D}}_{n}]$ tends to be negligible for a sufficiently large n. Moreover, from Sect. 1, $\Delta V({\textbf{D}}_n)$ satisfies condition (D); thus the Markov chain induced by ${\mathbb {W}}_n$ is asymptotically equivalent to the one corresponding to ${\mathbb {W}}$, and therefore ${\textbf{D}}_{n}=O_p(1)$.

1.4 A.4 Proof of Theorem 3

Under the ECADE with quantitative factors ${\textbf{b}}_{n+1}={\textbf{b}}_n+(2\delta _{n+1}-1) (1;{\textbf{f}}({\varvec{Z}}_{n+1})^t)^t$ a.s. for every n (with ${\textbf{b}}_0={\textbf{0}}_{q+1}$) and therefore $\{{\textbf{b}}_n\}_{n \in {\mathbb {N}}}$ is Markov chain on $\mathbb {X}=\mathbb {Z}\times \mathbb {R}^q$ with one step transition kernel

$$\begin{aligned} \begin{aligned} P({\textbf{x}}, A)&=\Pr ({\textbf{b}}_{n+1}\in A \mid {\textbf{b}}_n={\textbf{x}})\\&=\int \Pr ({\textbf{b}}_{n+1}\in A \mid {\textbf{b}}_n={\textbf{x}},{\textbf{Z}}_{n+1}={\varvec{z}} ) {\mathcal {L}}({\varvec{z}}) d{\varvec{z}}\\&= \int \Big \{ h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{x}}\right) I\{{\textbf{x}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in A \} + \\&\quad \quad + \left[ 1- h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{x}}\right) \right] I\{{\textbf{x}}-(1;{\textbf{f}}({\varvec{z}})^t)^t \in A\}\Big \} {\mathcal {L}}({\varvec{z}}) d{\varvec{z}}. \end{aligned} \end{aligned}$$

(A.6)

Following Theorem 4.5 of Meyn and Tweedie (1992), we firstly show that $\{{\textbf{b}}_{n}\}_{n \in \mathbb {N}}$ is a T-chain and then we prove that condition (D) is satisfied. As regards the T-chain property, we need to show that there exists a sampling distribution a and a substochastic transition kernel $T({\textbf{x}},\cdot )$ such that for any $A \in {\mathcal {B}}(\mathbb {X})$, $K_\alpha ({\textbf{x}},A)= \sum _{i=1}^\infty P^i({\textbf{x}},A)\alpha (i) \ge T({\textbf{x}},A)$, where $T(\cdot , A)$ is a lower semicontinuous (LSC) function with $T({\textbf{x}}, \mathbb {X})>0$ for all ${\textbf{x}} \in \mathbb {X}$. By taking $\alpha (1)=1$ and 0 otherwise, then $K_\alpha ({\textbf{x}},A)=P({\textbf{x}}, A)$; if we set $T({\textbf{x}},A)=e\int I\{{\textbf{x}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in A \} {\mathcal {L}}({\varvec{z}}) d{\varvec{z}}$, then $P({\textbf{x}}, A) \ge T({\textbf{x}},A)$, recalling that $h(x)\ge e$ for any $x \in \mathbb {R}$. Since the indicator function of any open set is LSC, $T({\textbf{x}}, A)$ is always LSC. Indeed, if A is an open subset then

$$\begin{aligned} \begin{aligned} \lim _{{\textbf{y}} \rightarrow {\textbf{x}}} \inf T({\textbf{y}}, A)&=e \lim _{{\textbf{y}} \rightarrow {\textbf{x}}} \inf \int I\{{\textbf{y}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in A \}{\mathcal {L}}({\varvec{z}}) d{\varvec{z}} \\&\ge e \int \lim _{{\textbf{y}} \rightarrow {\textbf{x}}} \inf I\{{\textbf{y}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in A \} {\mathcal {L}}({\varvec{z}}) d{\varvec{z}} \\&\ge e \int I\{{\textbf{x}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in A \} {\mathcal {L}}({\varvec{z}}) d{\varvec{z}}=T({\textbf{x}}, A). \end{aligned} \end{aligned}$$

(A.7)

Moreover, (A.7) holds even if A is not open; indeed, $T({\textbf{x}},A)$ does not change since the closure of A has zero Lebesgue measure. Finally notice that $T({\textbf{x}},\mathbb {X})>0$ since ${\textbf{x}}+(1;{\textbf{f}}({\varvec{z}})^t)^t \in \mathbb {Z}\times \mathbb {R}^q$ a.s. Thus, $\{{\textbf{b}}_{n}\}_{n \in \mathbb {N}}$ is a T-chain.

We now show that condition (D) is satisfied by choosing $V({\textbf{b}}_n)={\textbf{b}}_n^t{\mathbb {W}}{\textbf{b}}_n$. The one-step drift is

$$\begin{aligned} \begin{aligned} \Delta V({\textbf{b}}_n)&={\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n]\\&={\mathbb {E}} \left[ {\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}})\mid {\textbf{b}}_n={\textbf{b}}, {\textbf{Z}}_{n+1}={\varvec{z}} ]\right] \\&=\int {\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}}]{\mathcal {L}}({\varvec{z}}) d{\varvec{z}}, \end{aligned} \end{aligned}$$

where the inner expectation is

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}}]\\&\quad ={\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}},\delta _{n+1}=1] h((1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n)\\&\qquad +{\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}},\delta _{n+1}=0] [1-h((1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n)]. \end{aligned} \end{aligned}$$

Since ${\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n, {\textbf{Z}}_{n+1}= {\varvec{z}},\delta _{n+1}=1]=2 (1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}{\textbf{b}}_n+(1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}(1;{\textbf{f}}({\varvec{z}} )^t)^t$ and ${\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}},\delta _{n+1}=0]=-2 (1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}{\textbf{b}}_n+(1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}(1;{\textbf{f}}({\varvec{z}})^t)^t$, then ${\mathbb {E}} [V({\textbf{b}}_{n+1}) -V({\textbf{b}}_n)\mid {\textbf{b}}_n,{\textbf{Z}}_{n+1}={\varvec{z}}]= 4(1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\left[ h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\right) -\frac{1}{2}\right] +(1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}(1;{\textbf{f}}({\varvec{z}} )^t)^t$, so that

$$\begin{aligned} \begin{aligned} \Delta V({\textbf{b}}_n)&=4\int (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n \left[ h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\right) -\frac{1}{2}\right] {\mathcal {L}}({\varvec{z}})d{\varvec{z}}+\\&+\int (1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}(1;{\textbf{f}}({\varvec{z}})^t)^t {\mathcal {L}}({\varvec{z}})d{\varvec{z}}. \end{aligned} \end{aligned}$$

Notice that $(1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n \left[ h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\right) -1/2\right] \le 0$, since $(1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n \left[ h\left( (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\right) -1/2\right] =0$ if and only if $(1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n=0$ and it is negative otherwise. To verify condition (D), we need to show that, for a compact set ${\mathcal {C}}$, $\Delta V({\textbf{b}}_n)\le -\varepsilon$, namely

$$\begin{aligned} \begin{aligned} \int \vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert&\left[ h\left( -\vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert \right) -\frac{1}{2}\right] {\mathcal {L}}({\varvec{z}})d{\varvec{z}} \ge \\&\frac{\varepsilon + \int (1;{\textbf{f}}({\varvec{z}} )^t){\mathbb {W}}(1;{\textbf{f}}({\varvec{z}})^t)^t {\mathcal {L}}({\varvec{z}})d{\varvec{z}}}{4}, \quad \text {on} \,\,\, {\mathcal {C}}^c. \end{aligned} \end{aligned}$$

(A.8)

Let ${\mathcal {Z}}^*=\{ {\varvec{z}} : \vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert >0\}\subset \mathbb {R}^p$, then $\Pr {\left( {\varvec{z}}\in {\mathcal {Z}}^*\right) >0}$ and the LHS of (A.8) becomes

$$\begin{aligned} \int _{{\mathcal {Z}}^*} \vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert \left[ h\left( -\vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert \right) -\frac{1}{2}\right] {\mathcal {L}}({\varvec{z}})d{\varvec{z}}. \end{aligned}$$

Let ${\mathcal {C}}=\{{\textbf{b}}_n : \max \vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n \vert \le \kappa , {\varvec{z}}\in {\mathcal {Z}}^*\}$ be the compact set (in ${\mathcal {Z}}^*$ the linear transformation $(1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n$ is injective and corresponds to an induced norm of ${\textbf{b}}_n$), for every ${\textbf{b}}_n\in {\mathcal {C}}^c$,

$$\begin{aligned}{} & {} \int _{{\mathcal {Z}}^*} \vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert \left[ h\left( -\vert (1;{\textbf{f}}({\varvec{z}})^t){\mathbb {W}}{\textbf{b}}_n\vert \right) -\frac{1}{2}\right] {\mathcal {L}}({\varvec{z}})d{\varvec{z}}\\{} & {} \quad >\kappa \left[ h(-\vert \kappa \vert )-\frac{1}{2} \right] \Pr {\left( {\varvec{z}}\in {\mathcal {Z}}^*\right) }; \end{aligned}$$

so condition (D) is verified since $\kappa \left[ h(-\vert \kappa \vert )-\frac{1}{2} \right] \Pr {\left( {\varvec{z}}\in {\mathcal {Z}}^*\right) }$ increases in $\kappa$, while the RHS of (A.8) is bounded. Finally, the last statement follows from Theorem 2.

Under the mixed scenario, through the usual factorization of the joint distribution of mixed random variables, ${\mathcal {L}}({\varvec{z}})$ in (A.6) will be substituted by the product of the joint probability distribution of $p_1$ qualitative covariates and the conditional density function of the remaining $p_2$ factors. Thus, the T-chain property is preserved and, under the same choice of the function V and the compact set ${\mathcal {C}}$, the positive drift condition is also satisfied, so $\{{\textbf{b}}_n\}_{n \in \mathbb {N}}$ is bounded in probability.