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Statistical inference on the location of the optima (global maxima or minima) is one of the main goals in the area of Response Surface Methodology, with many applications in engineering and science. While there exist previous methods for computing confidence regions on the location of optima, these are for linear models based on a Normal distribution assumption, and do not address specifically the difficulties associated with guaranteeing global optimality. This paper describes distribution-free methods for the computation of confidence regions on the location of the global optima of response surface models. The methods are based on bootstrapping and Tukey's data depth, and therefore their performance does not rely on distributional assumptions about the errors affecting the response. An R language implementation, the package \code{OptimaRegion}, is described. Both parametric (quadratic and cubic polynomials in up to 5 covariates) and nonparametric models (thin plate splines in 2 covariates) are supported. A coverage analysis is presented demonstrating the quality of the regions found. The package also contains an R implementation of the Gloptipoly algorithm for the global optimization of polynomial responses subject to bounds. 

Life history theory is based on the assumption that resources are finite so that traits competing for this common pool of resources will experience a trade-off. The shared resource most commonly studied is food and studies typically manipulate resource acquisition by varying diet quantity or quality without considering the specific nutrients involved. Recent studies using the Geometric Framework (GF), however, suggest that life-history trade-offs are often regulated by the intake of specific nutrients. Despite this, a robust framework documenting the existence and quantifying the strength of nutritionally based trade-offs currently does not exist for studies using the GF. Here, we provide a conceptual framework showing that such trade-offs occur when life-history traits are maximised in different regions of nutrient space and that this divergence can be quantified by the overlap in the 95% confidence region (CR) of the global maxima, the angle (θ) between the linear nutritional vectors and the Euclidean distance (d) between the global maxima for each trait. We then empirically tested this framework by examining the effects of protein (P) and carbohydrate (C) intake on the trade-off between reproduction and immune function in male and female decorated crickets (Gryllodes sigillatus). Encapsulation ability and egg production in females increased with the intake of both nutrients, being maximised at a P:C ratio of 1.04:1 and 1:1.17, respectively. In contrast, encapsulation ability in males only increased with the intake of P being maximised at a P:C ratio of 5.14:1, whereas calling effort increased with the intake of C but decreased with the intake of P and was maximized at a P:C ratio of 1:7.08. Consequently, the trade-off between reproduction and encapsulation ability is much larger in males than females, a view supported by the non-overlapping 95% CRs on the global maxima for these traits in males and the larger estimates of θ and d. The sexes regulated their intake of nutrients in a similar way under dietary choice, at a P:C ratio of 1:2 and 1:1.84 in males and females, respectively. Although this ratio was more closely aligned with the optima for immune function and reproduction in females than males, neither sex optimally regulated their nutrient intake. Collectively, our study highlights that greater consideration should be given to the intake of specific nutrients when examining nutritionally based life-history trade-offs and how this varies across the sexes. 

Life-history theory assumes that resources are finite and that there may be trade-offs between traits competing for this common resource pool. The limiting resource most commonly studied is food and studies typically manipulate resource acquisition by varying diet quantity or quality without considering the specific nutrients involved. Recent studies using the Geometric Framework (GF) however, suggest that life-history trade-offs are often regulated by the intake of specific nutrients. Despite this, we lack a robust framework identifying and quantifying the strength of nutritionally based trade-offs using the GF. Here, we provide a conceptual framework showing that such trade-offs occur when life-history traits are maximised in different regions of nutrient space and that this divergence can be quantified by the overlap in the 95% confidence region (CR) of the global maxima, the angle (θ) between the linear nutritional vectors, and the Euclidean distance (d) between the global maximum for each trait. We then empirically test this framework by examining the effects of protein (P) and carbohydrate (C) intake on the trade-off between reproduction and immune function in male and female decorated crickets (Gryllodes sigillatus). Encapsulation ability and egg production in females increased with the intake of both nutrients, being maximised at a P:C ratio of 1.04:1 and 1:1.17, respectively. In contrast, encapsulation ability in males only increased with the intake of P being maximised at a P:C ratio of 5.14:1, whereas calling effort increased with C intake but decreased with P intake and was maximized at a P:C ratio of 1:7.08. Consequently, the trade-off between reproduction and encapsulation ability is much larger in males than females, a view supported by the non-overlapping 95% CRs on the global maxima for these traits in males but not females and the larger estimates of θ and d. When given dietary choice, the sexes regulated their nutrient intake in a similar way, at a P:C ratio of 1:2 and 1:1.84 in males and females, respectively. Although this ratio was more closely aligned with the optima for immune function and reproduction in females than males, neither sex optimally regulated their nutrient intake to maximise the expression of reproductive effort or immune function or, in the case of males, allow for moderate expression of both traits. Collectively, our study highlights that greater consideration should be given to the intake of specific nutrients when examining nutritionally based life-history trade-offs and how this varies between the sexes.