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The methodological discourse of mixed-methods research offers general procedures to combine quantitative and qualitative methods for investigating complex fields of research such as higher education. However, integrating different methods still poses considerable challenges. To move beyond general recommendations for mixed-methods research, this chapter proposes to discuss methodological issues with respect to a particular research domain. Taking current studies on the transition to higher education as an example, the authors first provide an overview of the potentials and limitations of quantitative and qualitative methods in the research domain. Second, they show the need for a conceptual framework grounded in the theory of the research object to guide the integration of different methods and findings. Finally, an example study that investigates transition with regard to the interplay of the individual student and the institutional context serves to illustrate the guiding role of theory. The framework integrates different theoretical perspectives on transition, informs the selection of the research methods, and defines the nexus of the two strands that constitute the mixed-methods design. As the interplay of individual and context is of concern for teaching and learning in general, the example presented may be fruitful for the wider field of higher education research.
In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.
Hotelling’s T² tests in paired and independent survey samples are compared using the traditional asymptotic efficiency concepts of Hodges–Lehmann, Bahadur and Pitman, as well as through criteria based on the volumes of corresponding confidence regions. Conditions characterizing the superiority of a procedure are given in terms of population canonical correlation type coefficients. Statistical tests for checking these conditions are developed. Test statistics based on the eigenvalues of a symmetrized sample cross-covariance matrix are suggested, as well as test statistics based on sample canonical correlation type coefficients.