Is a Basketball Free-throw Sequence Nonrandom? A Group Exercise for Undergraduate Statistics Students

I describe a group exercise that I give to my undergraduate biostatistics class. The exercise involves analyzing a series of 200 consecutive basketball free-throw attempts to determine whether there is any evidence for sequential dependence in the probability of making a free-throw. The students are given the exercise before they have learned the appropriate statistical tests, so that they can come up with ideas on their own. Students spend a full class period working on the problem, with my guidance and hints. In the next class period, we discuss how each student group approached the problem. I then present several alternative ways to analyze the data, including a runs test and a contingency table analysis of transition frequencies

Estimating Maximum Performance: Effects of Intraindividual Variation

Researchers often estimate the performance capabilities of animals using a small number of trials per individual. This procedure inevitably underestimates maximum performance, but few studies have examined the magnitude of this effect. In this study we explored the effects of intraindividual variation and individual sample size on the estimation of locomotor performance parameters. We measured sprint speed of the lizard Sceloporus occidentalis at two temperatures (20 degrees C and 35 degrees C), obtaining 20 measurements per individual. Speed did not vary temporally, indicating no training or fatigue effects. About 50% of the overall variation in speed at each temperature was due to intraindividual variation. While speed was repeatable, repeatability decreased slightly with increasing separation between trials. Speeds at 20 degrees C and 35 degrees C were positively correlated, indicating repeatability across temperatures as well. We performed statistical sampling experiments in which we randomly drew a subset of each individual\u27s full set of 20 trials. As expected, the sample\u27s maximum speed increased with the number of trials per individual; for example, five trials yielded an estimate averaging 89% of the true maximum. The number of trials also influenced the sample correlation between mean speeds at 20 degrees C and 35 degrees C; for example, five trials yielded a correlation coefficient averaging 90% of the true correlation. Therefore, intraindividual variation caused underestimation of maximal speed and the correlation between speeds across temperatures. These biases declined as the number of trials per individual increased, and depended on the magnitude of intraindividual variation, as illustrated by running sampling experiments that used modified data sets

Temperature, Activity and Lizard Life Histories

Lizard life-history characteristics vary widely among species and populations. Most authors seek adaptive or phylogenetic explanations for life-history patterns, which are usually presumed to reflect genetic differences. However, lizard life histories are often phenotypically plastic, varying in response to temperature, food availability, and other environmental factors. Despite the importance of temperature to lizard ecology and physiology, its effects on life histories have received relatively little attention. We present a theoretical model predicting the proximate consequences of the thermal environment for lizard life histories. Temperature, by affecting activity times, can cause variation in annual survival rate and fecundity, leading to a negative correlation between survival rate and fecundity among populations in different thermal environments. Thus, physiological and evolutionary models predict the same qualitative pattern of life-history variation in lizards. We tested our model with published life-history data from field studies of the lizard Sceloporus undulatus, using climate and geographical data to reconstruct estimated annual activity seasons. Among populations, annual activity times were negatively correlated with annual survival rate and positively correlated with fecundity. Proximate effects of temperature may confound comparative analyses of lizard life-history variation and should be included in future evolutionary models

Why Not to Do Two-Species Comparative Studies: Limitations on Inferring Adaptation

Claims about adaptation are commonly made from comparative studies involving only two species (or only two populations of a single species). Our main purpose here is to alert practitioners to several logical and statistical problems associated with using two-species comparisons for studying adaptation and to outline some alternative approaches. Multispecies comparisons are one such alternative. However, data from multiple species may not be independent or identically distributed in the statistical sense, which violates assumptions of ordinary statistical methods (Harvey and Pagel 1991). We therefore also discuss one phylogenetically based statistical method that can be employed for valid hypothesis testing with comparative data, Felsenstein’s (1985) method of phylogenetically independent contrasts. We discuss briefly how such methods can be employed, even with incomplete phylogenetic information, and also how data for multiple populations within species can enhance comparative analyses. Phylogenetically based analyses come in a variety of flavors, and our penultimate section discusses some differences in perspectives regarding statistical hypothesis testing in a phylogenetic context. We conclude by pointing out that many of our criticisms of two-species comparisons apply also to comparisons aimed at discovering mechanisms underlying physiological differences between species

Behavioural Plasticity in an Ecological Generalist: Microhabitat Use by Western Fence Lizards

Question: What is the basis for geographic variation in microhabitat use in fence lizards? Hypothesis: Population differences in microhabitat use reflect behavioural plasticity rather than genetic or experiential differences. Organisms: Western fence lizards (Sceloporus occidentalis). Field site: Three sites (desert, valley, and mountain) in southern California, USA. Methods: We compared habitat use by free-ranging lizards in three field populations. We also collected lizards from these three populations and studied their microhabitat use in experimental enclosures at a single field site. Results: In the wild, lizards chose higher and shadier perches at the hottest (desert) site, lower and sunnier perches at the coolest (mountain) site, and intermediate perches at the thermally intermediate valley site. However, lizards collected from the three source populations did not differ in microhabitat use in experimental enclosures at a common field site, supporting our hypothesis. Microhabitat choice is an important thermoregulatory mechanism in this species

Decreased Sprint Speed as a Cost of Reproduction in the Lizard Scelopours Occidentalis: Variation Among Populations

Decreased mobility of gravid females is thought to be an important cost of reproduction in lizards. We measured sprint speeds of female western fence lizards (Sceloporus occidentalis Baird and Girard) before and after they had oviposited. Females from two California populations were about 20% slower when gravid, females from an Oregon population were about 30% slower, and females from a Washington population were about 45% slower, compared to their speeds after recovering from reproduction. The decrease in sprint speed persisted for several weeks after oviposition, suggesting that reproduction impairs sprint performance by affecting body condition in addition to the burdening effect of eggs. Oregon and Washington females carried more mass (both somatic and clutch mass) per unit body length than California females. On the shorter bodies of Oregon and Washington lizards, eggs may interfere with the mechanics of running, in addition to their effect on the total mass of the female. In addition, gravid females from Washington had significantly higher reproductive investment (mass of clutch relative to the mass of the female after oviposition) than Oregon and California populations. Greater reproductive investment by Washington females increases the burden carried per unit of body length; we suggest this further impairs sprint performance. Decrements in sprint speed were not significantly correlated with level of reproductive investment (per unit body mass) among females within any of the study populations. However, the burden carried per unit body length was correlated with the sprint speed decrement among gravid females from Oregon. Comparisons within and among populations suggest that differences in morphology among northern and southern populations interact with reproductive investment to produce interpopulation differences in sprint performance for gravid females

Gambelia Wislizenii (Long-nosed Leopard Lizard). Arboreal behavior.

In the North American southwest, leopard lizards (genus Gambelia) are characteristically ground dwelling (Stebbins 2003. A Field Guide to Western Reptiles and Amphibians, 3rd ed., Houghton-Mifflin, New York. 533 pp.; Smith 1946. Handbook of Lizards, Comstock Publishing Co., Ithaca, New York. 557 pp.). In G. wislizenii, climbing is not mentioned in several substantive ecological studies of this species (Parker and Pianka 1976. Herpetologica 32:95-114; Tanner and Krogh 1974. Herpetologica 30:63-72), and to our knowledge, only a few seconds exist of climbing in this species. Clark (1974. Trans. Kansas Acad. Sci. 77:68) observed a G. wislizenii perched 30 cm off the ground in a shrub, and Miller and Stebbins (1964. The Lives of Desert Animals in Joshua Tree National Monument,. University of California Press, Berkeley. 452 pp.) mentioned collecting an individual on top of a massive rock 30 feet from the ground. Crowley and Pietruszka (1983. Anim. Behav. 31:1055-1060) commented that this species often climbs into shrubs or onto rocks, but did not provide additional information. For this reason, we augment the limited data on climbing in G. wislizenii with an observation from the Mojave Desert of California (USA)

Xantusia Vigilis (Desert Night Lizard) and Sceloporus Magister (Desert Spiny Lizard). Predation and Diet.

Here, we report evidence of predation on Xantusia vigilis by Sceloporus magister. We collected a yearling female S. magister (71 mm SVL, mass = 13.6 g) on 24 July 1996, 5 km SE of Llano, Los Angeles County, California, USA (34°29\u27N, 117°46\u27W, elevation 1120 m). In the laboratory, on 29 July, the S. magister deposited a fecal pellet that contained part of a Xantusia vigilis body, including sections of dorsal and ventral integument and both hind limbs

Testing the Persistence of Phenotypic Plasticity After Incubation in the Western Fence Lizard, Sceloporus Occidentalis

Hypothesis: Phenotypic variation in traits induced by different incubation temperatures does not persist into the lifetime of young lizards, and therefore contributes little to variation in long-term fitness. Organism: Western fence lizard (Sceloporus occidentalis). Methods: Split-clutch laboratory incubation experiment including eggs from two different populations under two different incubation regimes, measurement of morphological traits at hatching, and tracking of morphology and temperature preference behaviour for 7 weeks after hatching. Results: Several morphological traits, including body mass, hindlimb length, inter-limb length, and tail length, initially differed between the two incubation treatments, but only the difference in tail length persisted to age 7 weeks. Thermal preference was relatively conserved, with juveniles showing no difference in mean selected body temperatures across treatments; however, warm-incubated lizards thermoregulated more precisely than their cool-incubated counterparts. Conclusion: Studies of incubation effects can reveal changes in animal phenotypes post-hatching, but if these effects do not persist, they may not be subject to natural selection and consequently be of little ecological relevance