Confidence intervals for a product of proportions: application to importance values

Transcription

1 Confidence intervals for a product of proportions: application to importance values KEN A. AHO AND R. TERRY BOWYER Department of Biological Sciences, Idaho State University, 921 South 8th Avenue, Pocatello, Idaho USA Citation: Aho, K. A., and R. T. Bowyer Confidence intervals for a product of proportions: application to importance values. Ecosphere 6(11): Abstract. Importance values (proportional use 3 proportional availability of a resource) add a useful component to studies of resource selection. The merit of the metric is that it identifies commonly available resources that are nonetheless critical components in the ecology of an organism. Such resources would not be identified as being selected in comparisons of use relative to availability (e.g., use/availability). Importance values have received limited use in the ecological literature because they have been descriptive in nature and lack a framework for hypothesis testing. In this paper we present a simple asymptotic oneand two-tailed method for calculating confidence intervals for the product of proportions. The approach is easily extended to importance values. Results from simulations indicate that our method is effective across a wide range of sample sizes, but performs poorly when designs are unbalanced. As a result, we recommend rarifying availability data in matched-case designs for determining resource selection, before calculating confidence intervals for importance values. Applications of our method extend beyond the calculation of confidence intervals for importance values. Indeed, the approach is applicable for any situation in which confidence intervals are needed for the product of proportions, including product of more than two proportions. Key words: ratio. confidence interval; importance value; product of proportions; relative risk; resource selection; selection Received 6 July 2015; accepted 13 July 2015; published 16 November Corresponding Editor: D. P. C. Peters. Copyright: Ó 2015 Aho and Bowyer. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. ahoken@isu.edu INTRODUCTION The calculation of importance values to accompany metrics of resource selection (use relative to availability) was proposed initially by Bowyer and Bleich (1984), and subsequently noted by Manly et al. (2010) as a commonly used index of selectivity. Unlike selection ratios (relative risk in immunology), in which proportional use is divided by proportional availability (use/ availability), importance values are the product of proportional use and proportional availability (use 3 availability). Importance values help to identify habitats (or other resources) that are critical to organisms, but appear unselected (use. availability) because they are highly available on the landscape (Anderson et al. 2012). Importance values, however, have received only limited use by ecologists studying resource selection (Bowyer and Bleich 1984, Stewart et al. 2002, 2006, 2010, Anderson et al. 2012). A potential reason for their sparse application is that importance values are currently descriptive in nature and lack a formal framework for hypothesis testing. Thus, we believe that methods for hypothesis testing and confidence interval estimation would make importance values far more useful for ecologists studying resource selection. Previously we developed and evaluated confiv 1 November 2015 v Volume 6(11) v Article 230

2 dence interval methods for the ratio of proportions (Aho and Bowyer 2015), while ignoring products of proportions. We emphasize that these metrics are statistically distinct. Selection ratios have the range [0, ] with avoided resources in the narrow interval [0, 1), and selected resources in the infinite span (1, ]. Conversely, importance values are strictly bounded in [0, 1]. Sampling distributions for estimators of these metrics will reflect their distinct properties. As a result, importance values are likely to augment (not repeat) ecological information from selection ratios. Little effort has been directed toward the development of inferential methods (e.g., confidence intervals) for the product of two proportions. Indeed, we know of only one paper (Buehler 1957) that specifically addresses this problem. Moreover, that work considered only one-sided confidence intervals, and required that proportions were both much less than one. Herein, we present a simple asymptotic one- or two-tailed method for calculating confidence interval for the product of proportions. The approach is easily extended to importance values. As noted, an importance value, in its simplest form, is the product of the proportional use and proportional availability of a resource. Values close to 1, or 100 after multiplication by 100, (we use the percentage format here), indicate that the resource is both dominant on the landscape and used regularly by an organism. Values close to zero indicate that the resource is rare, avoided by an organism, or both. Let Y 1 and Y 2 be multinomial random variables with parameters n 1, p 1i and n 2, p 2i, respectively; where i ¼ 1, 2,..., r. We define the true importance value for the ith resource of r total resources to be h i ¼ p 1i p 2i ð1þ where p 1i and p 2i represent the proportional use and availability of the ith resource, respectively, and define h to be a set of all importance values fh i, h 2,..., h r g. The maximum likelihood estimators for p 1i and p 2i are the sample proportions where y 1i and y 2i are the observed counts for use and availability for the ith resource, respectively. The estimator for h i is h^i ¼ p^1i p^2i : ð3þ Two-tailed confidence interval, p 2i unknown Under a delta derivation, the natural log of the products of p 1i and p 2i (or the natural log of a product of p 1i and p 2i and a constant) is asymptotically normal with mean ln(h i ) and variance (1 p 1i )/p 1i n 1 þ (1 p 2i )/p 2i n 2 (see Appendix). Thus, an approximate (1 a)100% confidence interval for h i is given by where h^i 3 expð6 z 1 ða=2þ r^lnðh^ iþ Þ r^lnðh^ iþ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p^1i 1 p^2i þ p^1i n 1 p^2i n 2 ð4þ and z 1 (a/2) is the standard normal inverse cumulative distribution function at probability 1 (a/2). We note that our variance estimator is identical to the well-known delta-derived variance estimator for the logarithm of relative risk (i.e., the ratio of two proportions; Aho and Bowyer 2015). This is because of the mathematical underpinnings of those estimators. Specifically, let A and B be random variables, then Var(log(A/B)) ¼ Var(log(A)) þ Var(log(B)), but also Var(log(A 3 B)) ¼ Var(log(A)) þ Var(log(B)). One-tailed confidence interval Our approach is easily extended to one-tailed intervals. Specifically, the lower one-sided confidence interval (1 a)100% confidence bound for h i is given by h^i 3 expð z 1 a r^lnðh^ iþ Þ ð5þ and the upper one-sided (1 a)100% confidence bound for h i is given by h^i 3 expð z 1 a r^lnðh^ iþ Þ; ð6þ where z 1 a is the standard normal inverse cumulative distribution function at probability 1 a. Both one- and two-sided methods assume the independence of the variables Y 1 and Y 2. ˆp 1i ¼ y 1i n 1 ; ˆp 2i ¼ y 2i n 2 ; ð2þ v 2 November 2015 v Volume 6(11) v Article 230

3 Two-tailed confidence interval, p 2i known In many instances the proportional availability of a resource on a landscape may be truly or essentially known. For instance, in remotely sensed imagery, the areal coverage of resources often can be accurately quantified, at even large spatial scales (Aho and Bowyer 2015). If p 2i is known then, because the variance of lnðp^1þ is unaffected by multiplication by a constant (see Appendix), an approximate (1 a)100% confidence interval for h i is again given by Eq. 4. Now, however, h^i ¼ p^1i p 2i (or h^i ¼ p^1i 3 p 2i if a proportional expression of importance is to be used), and METHODS r^ 2 1 p^1 ¼ : lnðh^þ p^1n 1 We evaluated our new method by assessing its coverage properties. Coverage is the proportion of times that a confidence-interval method will, with repeated sampling, include the true parameter value, given a specified (nominal) rate. Thus, coverage defines the central purpose of a confidence interval. A method with perfect coverage, and with the nominal confidence 0.95, will include the true parameter value exactly 95% of the time. Conservative coverages will include the parameter at a rate greater than the nominal level, and inflate type II error. Anticonservative coverages will include the parameter at a rate less than the nominal level, and inflate type I error (Aho and Bowyer 2015). Simulations were conducted by sampling from binomial random variables with defined levels of p 1i, p 2i, n 1 and n 2. We considered five sampling configurations: small-balanced n 1 ¼ n 2 ¼ 10; moderate-unbalanced n 1 ¼ 40, n 2 ¼ 50; moderately large-unbalanced n 1 ¼ 95, n 2 ¼ 100; moderate-balanced: n 1 ¼ n 2 ¼ 50; and largebalanced n 1 ¼ n 2 ¼ 200. Resources availability levels (p 2i s) were fixed at 10 levels from 0.1 to 1.0, representing a continuum from very rare to monodominant. For each of those levels, 100 different levels of proportional use (p 1i s) from 0.01 to 1.0 were assessed. For each combination of sample size, resource level, and use level, 10,000 simulations were conducted. The nominal confidence level 1 a ¼ 0.95 was used for all assessments. As an ecological example we considered a dataset from Marcum and Loftsgaarden (1980) describing elk (Cervus elaphus) habitat selection among tree-canopy cover classes in Western Montana. These data were used previously by Aho and Bowyer (2015) and Manly et al. (2010) to describe properties of resource-selection indices. We conducted all analyses using the R computational environment (R Core Team 2014) with heavy reliance on the package asbio (Aho 2014, 2015). The asbio package (vers ) includes the function impt.ci which calculates confidence limits for importance values with the method described herein. Code for the function is included in the Supplement. Asbio is freely available at packages/asbio/index.html. RESULTS In simulations our new method performed well for small-, medium-, and large-balanced sampling configurations (Fig. 1). Coverage properties improved with sample size because of the convergence to normality of ln( p 1i 3 p 2i ). Intervals, however, were increasingly variable as the product of proportions approached one. The method had relatively large errors given unbalanced sample sizes for use and availability. This outcome was particularly evident when resources were dominant and use was high (Fig. 2). Specifically, if a smaller proportion has the larger sample size, the interval will range from conservative (for small values of the second proportion) to anticonservative (for large values of the second proportion). Conversely, if the larger proportion has the larger sample size, this will shrink r^h^ i and result in intervals that may be extremely anticonservative. These issues become more problematic as sample sizes decrease or the degree of imbalance increases (Fig. 2). DISCUSSION The use of confidence intervals for importance values is helpful in understanding habitat selection, especially for type II (home-range level) designs (Johnson 1980). Animals may make decisions about the size and location of home v 3 November 2015 v Volume 6(11) v Article 230

4 Fig. 1. Coverages of 95% confidence intervals for importance values (h i ) for 10 values of p 2i, and balanced sampling configurations: n 1 ¼ n 2 ¼ 10; n 1 ¼ n 2 ¼ 50; n 1 ¼ n 2 ¼ 200. The nominal confidence level (0.95) is indicated. ranges based on resources located outside of the home range (Kie et al. 2002). Under such circumstances, resources within the home range already have been selected at a scale greater than the home range, yet use may approach availability within the home range, indicating that those resources are not selected. Importance values reflect the essential nature of common but critical resources within the home range, and also have merit at larger scales (e.g., type III selection; Johnson 1980). As expected, confidence intervals for importance values provided supplementary information to that given by selection ratios in the elk dataset (Table 1). Notably, importance values for all cover classes were statistically distinct from each other (i.e., no importance value confidence intervals overlapped), although this was not true for corresponding selection ratios (Table 1). This outcome indicates that elk resource selection for cover types in the study was described better by importance values than selection ratios. This result would be expected when resource use largely parallels resource availability, as in our example (Table 1). Our confidence interval approach is effective for making inferences concerning the true importance value, although users should be aware of its limitations when (1) the value of h i is close to its absolute bounds (0 or 1), or (2) sample sizes for use and availability are unbalanced. Limitations of the first type exist for the confidence intervals of all strictly bounded e.g., [0, 1] parameters, including correlation coefficients (see Aho 2014, Ch. 8). The problem will be of decreasing importance as sample sizes grow large because the sampling distribution of h^i will become increasingly normal. Lack of balance in sample sizes for use and availability may be inherently problematic in studies of resource selection when employing logistic regression with a matched-case design, wherein multiple random sites are paired with a single used site (Boyce 2006, Lendrum et al. 2012, Villepique et al. 2015). When a match-case design is used for determining resource selection, we recommend rarifying data on availability to obtain a balanced design when calculating confidence intervals for importance values. v 4 November 2015 v Volume 6(11) v Article 230

5 Fig. 2. Coverages of 95% confidence intervals for importance values (h i ) for 10 values of p 2i, and unbalanced sampling configurations: n 1 ¼ 40, n 2 ¼ 50; n 1 ¼ 50, n 2 ¼ 40; n 1 ¼ 95, n 2 ¼ 100; n 1 ¼ 100, n 2 ¼ 95. The nominal confidence level (0.95) is indicated. As an alternative to the confidence formula given in Eq. 4, one could adapt the Wald confidence interval for the binomial probability of success, p. The sampling distribution of the estimator p^ is asymptotically normal with variance p(1 p)/n. Thus, if c is a constant, the variance of cp^ is c 2 p(1 p)/n. Under this logic, an adjusted (1 a)100% Wald confidence interval for h i when p 2i is known, is simply Table 1. Selectivity data for 325 elk along with importance values (proportional use 3 proportional availability 3 100) and selection ratios (proportional use/proportional availability) and corresponding 95% confidence intervals (lower bound ¼ 2.5%, upper bound ¼ 97.5%). Cover class y 1i y 2i Estimate 2.5% 97.5% Estimate 2.5% 97.5% Importance values Selection ratios 0% cover % cover % cover % cover Note: Selection ratio confidence intervals were derived using the iterative method of Koopman (1984). Intervals are adjusted for simultaneous inference using the Bonferroni-corrected confidence 1 (0.05/4) ¼ Use and availability total counts were n 1 ¼ X y 1i ¼ 325 and n 2 ¼ 200. v 5 November 2015 v Volume 6(11) v Article 230

6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h^i 6 z 1 ða=2þ p 2 2i p^1ið1 p^1i Þ : ð7þ When using raw proportions instead of percentages, the term should, of course, be dropped. In simulations when p 2i is known, however, Eq. 7 had inferior coverage to our proposed method (Eq. 4), and performed very poorly whenever p 1i was near 0 or 1, across a wide range of sample sizes. This distinction undoubtedly occurs because the log of the product of proportions (utilized by our method) converges more rapidly to normality than the product itself (Agresti 2002:73). Applications of our method extend far beyond importance values. For instance, consider a biological process that relies on the simultaneous functioning of independent components (e.g., DNA replication), each with a non-zero probability of failure (Finkelstein 2008). The probability for system failure would be the product of the marginal probabilities, and a confidence interval for the true probability of failure would be possible with our method. In general, the product of probabilities represents the probability for the intersection of their associated events under independence. Our method allows inferential statements about the true probability of this intersection based on samples. Our basic approach can also be applied to products of more than two proportions. Specifically, the confidence interval for the product of c independent proportions j ¼ 1, 2,..., c can be calculated by applying Eq. 4 after obtaining qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ĥ ¼ P c j¼1 ˆp ji and ˆr lnð ĥ i Þ ¼ R c j¼1 ½ð1 ˆp jiþ=ðˆp ji n j ÞŠ: This property clearly distinguishes our method from existing inferential approaches for relative risk. ACKNOWLEDGMENTS We are indebted to Drs. DeWayne Derryberry and Ann Gironella for useful comments on our manuscript. Funding was provided by the Department of Biological Sciences at Idaho State University. LITERATURE CITED Agresti, A Categorical data analysis. Wiley, Hoboken, New Jersey, USA. n 1 Aho, K Foundational and applied statistics for biologists using R. CRC Press, Boca Raton, Florida, USA. Aho, K asbio: a collection of statistical tools for biologists. R package version r-project.org/web/packages/asbio/index.html Aho, K., and R. T. Bowyer Confidence intervals for ratios of proportions: implications for selection ratios. Methods in Ecology and Evolution 6: Anderson, E. D., R. A. Long, M. P. Atwood, J. G. Kie, T. R. Thomas, P. Zager, and R. T. Bowyer Winter resource selection by female mule deer Odocoileus hemionus: functional response to spatiotemporal changes in habitat. Wildlife Biology 18: Bowyer, R. T., and V. C. Bleich Effects of cattle grazing on selected habitats of southern mule deer. California Fish and Game 70: Boyce, M. S Scale for resource selection functions. Diversity and Distributions 12: Buehler, R. J Confidence intervals for the product of two binomial parameters. Journal of the American Statistical Association 52: Finkelstein, M Failure rate modelling for reliability and risk. Springer, London, UK. Johnson, D. H The comparison of usage and availability measurements for evaluating resource preference. Ecology 61: Kie, G. J., R. T. Bowyer, M. C. Nicholson, B. B. Boroski, and E. R. Loft Landscape heterogeneity at differing scales: effects on spatial distribution of mule deer. Ecology 83: Koopman, P. A. R Confidence intervals for the ratio of two binomial proportions. Biometrics 40: Lendrum, P. E., C. R. Anderson, Jr., R. A. Long, J. G. Kie, and R. T. Bowyer Habitat selection by mule deer during migration: effects of landscape structure and natural-gas development. Ecosphere 3:art82. Manly, B. F. J., L. L. McDonald, D. L. Thomas, T. L. McDonald, and W. P. Erickson Resource selection by animals: statistical design and anaysis for field studies. Second edition. Kluwer, Boston, Massachusetts, USA. Marcum, C. L., and D. O. Loftsgaarden A nonmapping technique for studying habitat preferences. Journal of Wildlife Management 44:963. R Core Team R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Stewart, K. M., R. T. Bowyer, J. G. Kie, N. J. Cimon, and B. K. Johnson Temporospatial distriubtions of elk, mule deer, and cattle: resource partitioning and competitive displacement. Journal of Mammalogy 83: v 6 November 2015 v Volume 6(11) v Article 230

7 Stewart, K. M., R. T. Bowyer, J. G. Kie, and M. A. Hurley Spatial distriubtions of mule deer, and North American elk: resource partitioning in a sage-steppe environment. American Midland Naturalist 163: Stewart, K. M., R. T. Bowyer, R. W. Ruess, B. L. Dick, and J. G. Kie Herbivore optimization by North American elk: consequences for theory and management. Wildlife Monographs 167:1 24. Villepique, J. T., B. M. Pierce, V. C. Bleich, A. Andic, and R. T. Bowyer Resource selection by an endangered ungulate: a test of predator-induced range abandonment. Advances in Ecology 2015: SUPPLEMENTAL MATERIAL ECOLOGICAL ARCHIVES The Appendix and the Supplement are available online: v 7 November 2015 v Volume 6(11) v Article 230