The Behrens–Fisher problem with covariates and baseline adjustments

Abstract

The Welch–Satterthwaite t test is one of the most prominent and often used statistical inference methods in applications. The approach is, however, not flexible with respect to adjustments for baseline values or other covariates, which may impact the response variable. Existing analysis of covariance models are typically based on the assumption of equal variances across the groups. This assumption is hard to justify in real data applications and the methods tend not to control the type-1 error rate satisfactorily under variance heteroscedasticity. In the present paper, we tackle this problem and develop unbiased variance estimators of group specific variances, and especially of the variance of the estimated adjusted treatment effect in a general analysis of covariance model. These results are used to generalize the Welch–Satterthwaite t test to covariates adjustments. Extensive simulation studies show that the method accurately controls the nominal type-1 error rate, even for very small sample sizes, moderately skewed distributions and under variance heteroscedasticity. A real data set motivates and illustrates the application of the proposed methods.

Appendix: Proofs

1.1 A.1. Proof of Theorem 1

Let $\varvec{B}_i (\varvec{B}_i'\varvec{B}_i)^{-1} \varvec{B}_i'=\varvec{P}_{n_i}$, $i=1, 2$ denote the projection matrix for each group in the linear model separately. Computing the expectation of the quadratic form yields

$$\begin{aligned}&(n_i-1-r(\varvec{M}_i))E(\widehat{\sigma }_i^2) =E(\varvec{Y}_i'\varvec{Q}_i\varvec{Y}_i) \\&\quad =E(\varvec{Y}_i'(\varvec{I}_{n_i}-\varvec{P}_{n_i})\varvec{Y}_i) \\&\quad =(\varvec{Xb}+ \varvec{Mp})'(\varvec{I}_{n_i}-\varvec{P}_{n_i})(\varvec{Xb}+ \varvec{Mp})+tr((\varvec{I}_{n_i}-\varvec{P}_{n_i})\sigma _i^2\varvec{I}) \\&\quad =(\varvec{Xb}+ \varvec{Mp})'((\varvec{Xb}+ \varvec{Mp})-(\varvec{Xb}+ \varvec{Mp}))+(n_i-1-r(\varvec{M}_i))\sigma _i^2 \\&\quad =(n_i-1-r(\varvec{M}_i))\sigma _i^2,\quad i=1, 2. \end{aligned}$$

Thus, $\widehat{\sigma }_i^2$ is an unbiased estimator of $\sigma _i^2$, $i=1, 2$. Next, the consistency of the variance estimators will be shown. We compute the variance of the quadratic form $\widehat{\sigma }_i^2$ and obtain

$$\begin{aligned} Var(\varvec{Y}_i'\varvec{Q}_i\varvec{Y}_i)=(\mu _4-3\sigma _i^4)\varvec{q}_i'\varvec{q}_i+2\sigma _i^4tr(\varvec{\varvec{Q}_i}^2)+4\sigma _i^2\varvec{\mu }_i'{\varvec{Q}_i}^2\varvec{\mu }_i+4\mu _3\varvec{\mu }_i'\varvec{Q}_i\varvec{q}_i, \end{aligned}$$

where $\varvec{\mu }_i=E(\varvec{Y}_i)$, $\varvec{q}_i=diag\{\varvec{Q}_i\}$, the vector of diagonal elements of $\varvec{Q}_i$. Here, $\mu _3$ and $\mu _4$ denote the skewness and kurtosis of the error distributions, respectively. Using the properties of projection matrix $\varvec{P}_{n_i}$, we get $0\le \varvec{q}_i'\varvec{q}_i\le n_i+tr(\varvec{P}_{n_i})=n_i+1+r(\varvec{M}_i)$ and $tr(\varvec{Q}_i^2)=tr(\varvec{Q}_i)=n_i-1-rank(\varvec{M}_i)$. Since ${\varvec{Q}_i}^2=\varvec{Q}_i, {\varvec{Q}_i}^2\varvec{\mu }_i=\varvec{Q}_i\varvec{\mu }_i=(\varvec{I}_{n_i}-\varvec{P}_{n_i})\varvec{\mu }_i=0$. Furthermore, $\varvec{\mu }_i'\varvec{Q}_i\varvec{q}_i=(\varvec{Q}_i\varvec{\mu }_i)'\varvec{q}_i=0$. In conclusion, the $L_2$-convergence follows, because

$$\begin{aligned} Var(\varvec{Y}_i'\varvec{Q}_i\varvec{Y}_i)/(n_i-1-rank(\varvec{M}_i))^2\xrightarrow {L_2}0 ,\quad n_i \rightarrow \infty . \end{aligned}$$

1.2 A.2. Derivation of (3.2)

$$\begin{aligned} {\sigma }^2_{\varvec{b}}= & {} Var(\sqrt{N}(\widehat{b}_1 - \widehat{b}_2)) \nonumber \\= & {} N\varvec{c}' \varvec{D}\varvec{\varSigma }\varvec{D}'\varvec{c}\nonumber \\= & {} N \varvec{c}' \left( \begin{array}{cc} \sum \limits _{j=1}^{n_1} d_{1j}^2\sigma _1^2 + \sum \limits _{j=n_1+1}^{N} d_{1j}^2\sigma _2^2 &{} \sum \limits _{j=1}^{n_1} A_j \sigma _1^2 + \sum \limits _{j=n_1+1}^{N} A_j\sigma _2^2\\ \sum \limits _{j=1}^{n_1} A_j \sigma _1^2 + \sum \limits _{j=n_1+1}^{N} A_j \sigma _2^2 &{} \sum \limits _{j=1}^{n_1} d_{2j}^2\sigma _1^2 + \sum \limits _{j=n_1+1}^{N} d_{2j}^2\sigma _2^2\\ \end{array} \right) \varvec{c}\nonumber \\= & {} N \left( \sigma _1^2\sum _{j=1}^{n_1} (d_{1j}-d_{2j})^2 + \sigma _2^2\sum _{j=n_1+1}^N (d_{1j}-d_{2j})^2\right) \nonumber \\\equiv & {} N\left( \sigma _1^2 n_1^*+ \sigma _2^2 n_2^*\right) . \end{aligned}$$

(7.1)