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Dimension reduction provides a useful tool for analyzing high dimensional data. The recently developed Envelope method is a parsimonious version of the classical multivariate regression model through identifying a minimal reducing subspace of the responses. However, existing envelope methods assume an independent error structure in the model. While the assumption of independence is convenient, it does not address the additional complications associated with spatial or temporal correlations in the data. In this article, we introduce a Spatial Envelope method for dimension reduction in the presence of dependencies across space. We study the asymptotic properties of the proposed estimators and show that the asymptotic variance of the estimated regression coefficients under the spatial envelope model is smaller than that from the traditional maximum likelihood estimation. Furthermore, we present a computationally efficient approach for inference. The efficacy of the new approach is investigated through simulation studies and an analysis of an Air Quality Standard (AQS) dataset from the Environmental Protection Agency (EPA).

ObjectiveMiniature Schnauzers commonly develop both inherited and diabetic cataracts. Miniature Schnauzers also often present with concurrent metabolic disease at the time of cataract surgery. The purpose of this study was to evaluate cataract surgery outcomes in Miniature Schnauzers.Materials and methodsThe Eye Care for Animals electronic medical record database (2012‐2018) was searched for “phacoemulsification” and “Miniature Schnauzer.” A minimum follow‐up time of 1 year was required for inclusion in the study. Clinical history, pre‐ and postoperative bloodwork, postoperative treatments, and visual outcomes were reviewed in all cases.ResultsCataract surgery was performed on 67 patients and 124 eyes. 93.5% (116/124) of operated eyes were visual 1 year following cataract surgery. At the time of surgery, 79.1% were diabetic, 64.2% had hypercholesterolemia, 55.2% had hypertriglyceridemia, 16.4% had hyperadrenocorticism, and 9% had a history of pancreatitis. Dorzolamide was used postoperatively in 61.2% of patients, and its use was significantly associated with postoperative cholesterol levels above reference range. Lipemic aqueous occurred in 13.4% of patients.ConclusionsMiniature Schnauzers presenting for cataract surgery are frequently affected by metabolic diseases and dyslipidemia; however, cataract surgery outcomes were not significantly affected during the study period. Dietary management, omega‐3 fatty acid supplementation, and oral fibrates may reduce the incidence of lipemic aqueous in Miniature Schnauzers.

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