Invariance of generalized wordlength patterns

The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang (2004) defined the $J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the $J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the $J$-characteristics are not. We briefly discuss some implications of these results.Comment: To appear in: Journal of Statistical Planning and Inferenc

Generalized wordlength patterns and strength

Xu and Wu (2001) defined the \emph{generalized wordlength pattern} $(A_1, ..., A_k)$ of an arbitrary fractional factorial design (or orthogonal array) on $k$ factors. They gave a coding-theoretic proof of the property that the design has strength $t$ if and only if $A_1 = ... = A_t = 0$. The quantities $A_i$ are defined in terms of characters of cyclic groups, and so one might seek a direct character-theoretic proof of this result. We give such a proof, in which the specific group structure (such as cyclicity) plays essentially no role. Nonabelian groups can be used if the counting function of the design satisfies one assumption, as illustrated by a couple of examples

The IDENTIFY study: the investigation and detection of urological neoplasia in patients referred with suspected urinary tract cancer - a multicentre observational study

Objective To evaluate the contemporary prevalence of urinary tract cancer (bladder cancer, upper tract urothelial cancer [UTUC] and renal cancer) in patients referred to secondary care with haematuria, adjusted for established patient risk markers and geographical variation. Patients and Methods This was an international multicentre prospective observational study. We included patients aged ≥16 years, referred to secondary care with suspected urinary tract cancer. Patients with a known or previous urological malignancy were excluded. We estimated the prevalence of bladder cancer, UTUC, renal cancer and prostate cancer; stratified by age, type of haematuria, sex, and smoking. We used a multivariable mixed-effects logistic regression to adjust cancer prevalence for age, type of haematuria, sex, smoking, hospitals, and countries. Results Of the 11 059 patients assessed for eligibility, 10 896 were included from 110 hospitals across 26 countries. The overall adjusted cancer prevalence (n = 2257) was 28.2% (95% confidence interval [CI] 22.3–34.1), bladder cancer (n = 1951) 24.7% (95% CI 19.1–30.2), UTUC (n = 128) 1.14% (95% CI 0.77–1.52), renal cancer (n = 107) 1.05% (95% CI 0.80–1.29), and prostate cancer (n = 124) 1.75% (95% CI 1.32–2.18). The odds ratios for patient risk markers in the model for all cancers were: age 1.04 (95% CI 1.03–1.05; P < 0.001), visible haematuria 3.47 (95% CI 2.90–4.15; P < 0.001), male sex 1.30 (95% CI 1.14–1.50; P < 0.001), and smoking 2.70 (95% CI 2.30–3.18; P < 0.001). Conclusions A better understanding of cancer prevalence across an international population is required to inform clinical guidelines. We are the first to report urinary tract cancer prevalence across an international population in patients referred to secondary care, adjusted for patient risk markers and geographical variation. Bladder cancer was the most prevalent disease. Visible haematuria was the strongest predictor for urinary tract cancer

Aspects of Fortet&apos;s work on reproducing kernel Hilbert spaces

We discuss two papers of Fortet relating properties of second-order stochastic processes to reproducing kernel Hilbert spaces. In the rst, Fortet explores the validity of the Gaussian Dichotomy Theorem for non-Gaussian processes. In the second, he introduces the notion of \n-dominance&quot; and uses it to study conditions under which a second-order process has its sample paths in a given reproducing kernel Hilbert space. He develops his results with no assumptions on the \time&quot; parameter set or the covariance of the process

factorial

the definition of effects in fractiona

On the Definition of Effects in Fractional Factorial Designs

This paper simplifies a previous exposition of Rao&apos;s 1947 proof of his inequalities for orthogonal arrays. A key issue in the proof is the way in which one defines effects in a fractional factorial design. Here we replace the definition used in the earlier exposition with a simpler one, based on a more obvious interpretation of what Rao wrote and more in line with common practice, and show that it still leads to the same mathematical results. As in Rao&apos;s original paper, all designs are assumed to be unblocked. Two applications are given illustrating alias patterns in certain nonregular fractional designs, the second affording an opportunity to compare this approach with an alternative one due to Box and Wilson

Optimizing selection for function-valued traits

Abstract We consider a function-valued trait z(t) whose pre-selection distribution is Gaussian, anda fitness function W that models optimizing selection, subject to certain natural assump-tions. We show that the post-selection distribution of z(t) is also Gaussian, compute theselection differential, and derive an equation that expresses the selection gradient in terms of the parameters of W and of the pre-selection distribution. We make no assumptions onthe nature of the &amp;quot;time &amp;quot; parameter t