Modeling and Analysis of Method Comparison Data with Skewness and Heavy Tails

Abstract

The analysis of method comparison data is mainly concerned with evaluating agreement between methods of measuring a continuous variable. The methodology commonly assumes normally distributed data, which are usually modeled using a standard linear mixed model that assumes normality for both random effects and errors. In practice, however, the data often exhibit skewness and have tails heavier than those of a normal distribution, possibly due to outlying observations. When such data are analyzed using the standard mixed model, the non-normality may become apparent from model diagnostics. This article develops a methodology for agreement evaluation by modeling data using a recent robust mixed model that assumes a skew-t distribution for random effects and an independent t-distribution for errors. As the standard model is a special case of the robust model, the new methodology offers a unified framework for analyzing data with skewness and heavy tails as well as normally distributed data. The methodology is presented for both unreplicated and replicated data. A real example is used for illustration.

Appendix

1.1 A.1 Definitions

Let $\mathbf {Y}$ be a $J \times 1$ random vector with $\varvec{\mu }$ as a $J \times 1$ location parameter vector and $\varvec{\varSigma }$ as a $J \times J$ scale matrix. Define

$$\begin{aligned} \mathbf {Y}^* = \varvec{\varSigma }^{-1/2}(\mathbf {Y}-\varvec{\mu }). \end{aligned}$$

Also let $\phi _J( \cdot | \varvec{\mu }, \varvec{\varSigma })$ be the density function of a $\mathcal {N}_{J}(\varvec{\mu }, \varvec{\varSigma })$ distribution, and $\tau (\cdot ,\nu )$ be the distribution function of a univariate t-distribution with $\nu$ degrees of freedom. The J-dimensional skew-normal, t and skew-t distributions are defined as follows.

Definition 1

$\mathbf {Y} \sim \mathcal {SN}_J (\varvec{\mu }, \varvec{\varSigma }, \varvec{\lambda })$ if its density function is

$$\begin{aligned} f(\mathbf {y} | \varvec{\mu }, \varvec{\varSigma }, \varvec{\lambda })= 2 \phi _J (\mathbf {y}|\varvec{\mu },\varvec{\varSigma }) \Phi (\varvec{\lambda }^{\prime } \mathbf {y}^*), ~~ \mathbf {y} \in \mathbb {R}^J. \end{aligned}$$

Definition 2

$\mathbf {Y} \sim t_J (\varvec{\mu }, \varvec{\varSigma },\nu )$ if its density function is

$$\begin{aligned} f(\mathbf {y} | \varvec{\mu }, \varvec{\varSigma }, \nu )= (\nu \pi )^{-J/2} \, \frac{\text {gam}( (\nu +J)/2)}{\text {gam}(\nu /2)} | \varvec{\varSigma } | ^{-1/2} \left( 1 + \mathbf {y}^{*\prime } \mathbf {y}^*/\nu \right) ^{-(\nu +J)/2}, ~~ \mathbf {y} \in \mathbb {R}^J, \end{aligned}$$

with $\text {gam} (\cdot )$ as the gamma function.

Definition 3

$\mathbf {Y} \sim \mathcal {ST}_J (\varvec{\mu }, \varvec{\varSigma }, \varvec{\lambda }, \nu )$ if its density function is

$$\begin{aligned} f(\mathbf {y} | \varvec{\mu }, \varvec{\varSigma }, \varvec{\lambda }, \nu )=2\, f_t (\mathbf {y} | \varvec{\mu }, \varvec{\varSigma }, \nu ) \, \tau \! \left( \varvec{\lambda }^{\prime } \mathbf {y}^* \{ (\nu +J)/(\nu + \mathbf {y}^{*\prime } \mathbf {y}^*) \}^{1/2} \mid \nu +J \right) \!\!, ~~ \mathbf {y} \in \mathbb {R}^J, \end{aligned}$$

where $f_t (\mathbf {y} | \varvec{\mu }, \varvec{\varSigma }, \nu )$ is the density function of a $t_J(\varvec{\mu },\varvec{\varSigma },\nu )$ distribution.

1.2 A.2 Hierarchical Representation for a GST Mixed Model

Let $\mathbf {Y}$ be an M-vector obtained by dropping the subscript i in $\mathbf {Y}_i$ defined by (1). From (3), the GST mixed model for $\mathbf {Y}$ can be written as

$$\begin{aligned} \mathbf {Y} = \mathbf {X} \varvec{\beta } + \mathbf {Z} \mathbf {b} + \mathbf {e}, ~~ \mathbf {b} \sim \mathcal {ST}_q (\mathbf {0}, \varvec{\varPsi }, \varvec{\lambda }, \nu _b), ~~ \mathbf {e} \sim t_n (\mathbf {0}, \varvec{\varSigma }, \nu _e), \end{aligned}$$

(A.1)

where $\mathbf {b}$ and $\mathbf {e}$ are mutually independent. For a hierarchical representation of this model, define for $v > 0$,

$$\begin{aligned}&\varvec{\Pi }_v = ( \mathbf {Z} \varvec{\varPsi } \mathbf {Z}^\prime + v \varvec{\varSigma }), \nonumber \end{aligned}$$

$$\begin{aligned}&\varvec{\lambda }_v = \frac{ \varvec{\Pi }_v ^{-1/2} \mathbf {Z} \varvec{\varPsi }^{1/2} \varvec{\lambda }}{ \big ( 1 + \varvec{\lambda }^\prime \varvec{\varPsi }^{-1/2} (\varvec{\varPsi }^{-1} + \mathbf {Z}^\prime \varvec{\varSigma }^{-1} \mathbf {Z} /v )^{-1} \varvec{\varPsi }^{-1/2} \varvec{\lambda } \big )^{1/2}}, \end{aligned}$$

(A.2)

and let $\mathcal {G} (\alpha , \beta )$ denote a gamma distribution with parameters $\alpha , \beta > 0$, and density

$$\begin{aligned} f (y | \alpha , \beta ) = \frac{\beta ^\alpha }{\text {gam}(\alpha )} y^{\alpha - 1} \exp (- \beta y), ~~ y >0. \end{aligned}$$

(A.3)

Now from [18], the model (A.1) can be represented as

$$\begin{aligned} \mathbf {Y} | U, V \sim \mathcal {SN}_n (\mathbf {X} \varvec{\beta }, \varvec{\Pi }_V/U, \varvec{\lambda }_V), \, U \sim \mathcal {G} (\nu _b/2, \nu _b/2), \, U/V \sim \mathcal {G} (\nu _e/2, \nu _e/2). \end{aligned}$$

(A.4)

1.3 A.3 Linear Combination of Skew-Normals

Proposition 1

Let $\mathbf {Y} \sim \mathcal {SN}_q (\varvec{\beta }, \varvec{\varPsi }, \varvec{\lambda })$ and consider the quantities defined in (4). Let $\mathbf {a} \in \mathbb {R}^q$ with at least one non-zero element. Then

$$ \mathbf {a}^\prime \mathbf {Y} \sim \mathcal {SN}_1 \big (\mathbf {a}^\prime \varvec{\beta }, \mathbf {a}^\prime \varvec{\varPsi } \mathbf {a}, \mathbf {a}^\prime \varvec{\varPsi }^{1/2} \varvec{\delta }/( \mathbf {a}^\prime \varvec{\varGamma } \mathbf {a})^{1/2} \big ). $$

Proof

The proof relies on a stochastic representation of a skew-normal variate. Let $\mathbf {Y}^* \sim \mathcal {SN}_q (\mathbf {0}, \varvec{\varPsi }, \varvec{\lambda })$. Then, from [1],

$$\begin{aligned} \mathbf {Y}^* \overset{d}{=} \varvec{\varPsi }^{1/2} \varvec{\delta } |G_1^*| + \varvec{\varPsi }^{1/2} (\mathbf {I}_q - \varvec{\delta } \varvec{\delta }^\prime ) \mathbf {G}_2^*, \end{aligned}$$

(A.5)

where $G_1^* \sim \mathcal {N}_1 (0, 1)$, $\mathbf {G}_2^* \sim \mathcal {N}_q (\mathbf {0}, \mathbf {I}_q)$ independently of $G_1^*$, and the notation “$\overset{d}{=}$” means “equal in distribution.” Using (A.5), we can write

$$\begin{aligned} \mathbf {a}^\prime \mathbf {Y} \overset{d}{=} \mathbf {a}^\prime \varvec{\beta } + \mathbf {a}^\prime \varvec{\varPsi }^{1/2} \varvec{\delta } |G_1^*| + (\mathbf {a}^\prime \varvec{\varGamma } \mathbf {a})^{1/2} G^*, \end{aligned}$$

where $G^* \sim \mathcal {N}_1 (0, 1)$ independently of $G_1^*$. Define

$$ \lambda ^* = \mathbf {a}^\prime \varvec{\varPsi }^{1/2} \varvec{\delta }/( \mathbf {a}^\prime \varvec{\varGamma } \mathbf {a})^{1/2}, ~~ \delta ^* = \lambda ^*/(1 + \lambda ^{*2})^{1/2}. $$

From an application of (4), we have $(\mathbf {a}^\prime \varvec{\varPsi }^{1/2} \varvec{\delta })^2 + \mathbf {a}^\prime \varvec{\varGamma } \mathbf {a} = \mathbf {a}^\prime \varvec{\varPsi } \mathbf {a}$, implying

$$\begin{aligned} \mathbf {a}^\prime \varvec{\varPsi }^{1/2} \varvec{\delta } = (\mathbf {a}^\prime \varvec{\varPsi } \mathbf {a})^{1/2} \delta ^*, ~~ (\mathbf {a}^\prime \varvec{\varGamma } \mathbf {a})^{1/2} = (\mathbf {a}^\prime \varvec{\varPsi } \mathbf {a})^{1/2} (1-\delta ^{*2})^{1/2}. \end{aligned}$$

This allows us to write

$$\begin{aligned} \mathbf {a}^\prime \mathbf {Y} \overset{d}{=} \mathbf {a}^\prime \varvec{\beta } + (\mathbf {a}^\prime \varvec{\varPsi } \mathbf {a})^{1/2} \delta ^* |G_1^*| + (\mathbf {a}^\prime \varvec{\varPsi } \mathbf {a})^{1/2} (1-\delta ^{*2})^{1/2} G^*. \end{aligned}$$

Now the result follows from the representation (A.5) for the univariate case.