The Auk 

Abstract: -Repeatability is a useful tool for the population geneticist or genetical ecologist, but several papers have carried errors in its calculation We outline the correct calculation of repeatability, point out the common mistake, show how the incorrectly calculated value relates to repeatability, and provide a method for checking published values and calculating approximate repeatability values from the F ratio (mean squares among groups/ mean squares within groups) Received 6 February 1986, accepted 25 August 1986 REPEATABILITY is a measure used in quantitative genetics to describe the proportion of variance in a character that occurs among rather than within individuals Repeatability, r, is given by: r = (VG + VEg)/ VP, (1) where VG is the genotypic variance, VEg the general environmental variance, and Vp the phenotypic variance (Falconer 1960, 1981) In addition to its use in assessing the reliability of multiple measurements on the same individual, repeatability may be used to set an upper limit to the value of heritability (Falconer 1960, 1981) and to separate, for instance, the effects of "self" and "mate" on a character such as clutch size (van Noordwijk et al 1980) Repeatability is therefore a useful statistic for population geneticists and genetical ecologists Recently, we have noticed an increasing number of published papers and unpublished manuscripts in which repeatability was miscalculated Our purpose is fivefold: (1) to outline the correct method of calculating repeatability; (2) to point out a common mistake in calculating repeatability; (3) to show how much this mistake affects values of repeatability; (4) to provide a quick way of checking published estimates, and to calculate an approximate value of repeatability from published F ratios and degrees of freedom; and (5) to make recommendations for authors, referees, editors, and readers to prevent the promulgation and propagation of incorrect repeatability values in the literature CALCULATION OF REPEATABILITY Repeatability is the intraclass correlation coefficient (Sokal and Rohlf 1981), which is based on variance components derived from a one-way analysis of variance (ANOVA) The intraclass correlation coefficient is given by some statistical packages; otherwise it can be calculated from an ANOVA ANOVA is described in most statistics textbooks (eg Sokal and Rohlf 1981; Kirk 1968 gives a detailed treatment of more complex designs of ANOVA), so we will not repeat it here, but give the general form of the results from such an analysis in Table 1 Repeatability, r, is given by r = sA / (S + SA)' (2) where S2A is the among-groups variance component and s2 is the within-group variance component These variance components are calculated from the mean squares in the analysis of variance as:

Abstract: into the body of the partly eaten bantam and replaced it in the same spot where he found it. Next morning the seemingly impossible was made a practical certainty, for he found the body of a screech owl with the claws of one foot firlnly imbedded in the body of the bantam. He very kindly presented me with the owl which, upon dissection, proved to be a female, its stomach eontalning a very considerable amount of bantam flesh and feathers, together with a great deal of wheat. (It seelns probable that the wheat was accidentally swallowed with the crop of the bantam during the feast, but there was so much that it seelns trange the owl did not discard it while eating). How a bird only 9.12 inches in length • could have dealt out such havoc in so short a time is almost in-

Abstract: Bird eggs begin to lose weight as soon as they are laid but their volume and linear dimensions do not change during incubation. The volume of an egg can be estimated within 2% from the relationship: Volume = 0.51. LB2, where L is the length and B is the breadth (maximum diameter). The fresh weight of an egg can be estimated within 2% from the relationship: Weight = Kw' LB 2, where K w is a species-specific constant that can be determined empirically or calculated from published data. Received 25 April 1978, accepted 28 October 1978. IT is frequently useful to know the fresh weight of a bird's egg. One reason is that many aspects of the biology of bird eggs can be predicted from their weight and these predicted values can be used when empirical data are lacking. Alternatively, one way to detect adaptations to unusual situations is by comparing observed values with values predicted for an "average" egg. Some of the parameters that can be predicted from weight are metabolic rate (Rahn et al. 1974), incubation period (Rahn and Ar 1974), water vapor conductance (Ar et al. 1974), the daily rate of water loss (Drent 1970), surface area, density, and shell weight (Paganelli et al. 1974), and the relation of egg weight to adult body weight (Huxley 1923-24, Rahn et al. 1975). Additionally, accurate values of fresh egg weight are required for the calculation of fractional weight loss from the daily rate of water loss (Rahn and Ar 1974) and the estimation of incubation age (Westerkov 1950). However, fresh egg weight can only be determined at the time of laying because the egg immediately begins to lose weight by diffusion of water vapor. This daily loss is proportional to the 0.74 power of egg weight (Drent 1970) and totals about 16% of the initial weight by the end of incubation (Drent 1975). As a consequence, while a great deal of information is available on egg dimensions, there are few reliable reports of fresh egg weight. Fortunately, the linear dimensions of eggs do not change during incu- bation, and in the present paper I show that they can be used to predict egg volume and fresh egg weight. Several authors have shown that the volume of a bird egg can be estimated from its linear dimensions (Bergtold 1929, Worth 1940, Westerkov 1950, Stonehouse 1963), and Preston (1974) suggested a more complex approach. In the present paper, I evaluate the accuracy with which volume (V) can be predicted from linear dimen- sions (L = length, B = breadth or maximum diameter), using the equation: V = Kv' LB 2

Abstract: -Mayfield's method for calculating the success of a group of nests is examined in detail The standard error of his estimator is developed Mayfield's assumption that destroyed nests are at risk until the midpoint of the interval between visits leads to bias if nests are visited infrequently A remedy is suggested, the Mayfield-40% method I also present a competing model, which recognizes that the actual destruction date of a failed nest is unknown Estimated daily mortality rates and standard errors are developed under this model A comparison of the original Mayfield method, the Mayfield-40% method, and the new method, which incorporates an unknown date of destruction, shows that the original or modified Mayfield method performs nearly as well as the more appropriate method and requires far easier calculations A technique for statistically comparing daily mortality rates is offered; the one proposed by Dow (1978) is claimed to be misleading Finally, I give a method for detecting heterogeneity among nests and an improved estimator, if it is found Received 5 March 1979, accepted 28 July 1979 THE well-being of an avian population lies in the delicate balance between natality and mortality Biologists attempt to infer the status of a species by estimating rates of births and deaths and, through their comparison, determining if the former are sufficient to offset the latter For most populations of wild birds, none of the crucial characteristics of population dynamics is easy to measure One component of natality that seems easy to gauge is the percentage of nests that hatch, which is often used as an indirect measure of reproduction Mayfield (1961) has demonstrated, however, serious error in the ordinary method of determining this rate: dividing the number of nests under observation into the number of those that ultimately hatch To overcome the difficulties he recognized, Mayfield (1961, 1975) developed an alternative method for calculating hatch rate In it he accounts for the fact that normally not all nests are under observation from the day of initiation but are discovered at various stages of development Nests found in a late stage are more likely to hatch than those found in an early one, because they have already survived part of the requisite time Combining all nests, regardless of stage of development, and calculating an apparent hatch rate will result in a severely biased estimator Mayfield's method places all nests on a comparable basis by using only information from the period during which a nest was under observation The length of that period he termed the exposure, although risk may be a more appropriate term Thus, a nest that was found on 10 May and was still active on 18 May had survived 8 days of exposure Had it been destroyed by 18 May, Mayfield would credit the nest with 4 days of exposure, under the assumption that it was at risk for half the period 651 The Auk 96: 651-661 October 1979 652 DOUGLAS H JOHNSON [Auk, Vol 96 From a group of nests, Mayfield calculates the total exposure in nest-days This number is divided into the number of nests that were destroyed while under observation The resultant value, expressed as losses per nest-day, is the estimated daily mortality rate of nests For example, in Mayfield's (1961: 258) analysis of Kirtland's Warbler (Dendroica kirtlandii), 154 nests seen during incubation represented a total exposure of 8825 nest-days (Mayfield's data have been reanalyzed here; some results differ slightly from his original presentation) Thirty-five nests were lost (destroyed or deserted), yielding a daily mortality rate of 35/8825 = 004 losses per nest-day To determine the probability that a nest survives the entire period of incubation, one must know the length of that period; for the Kirtland's Warbler it is 14 days The probability of survival for one day is 096 (=1 004), so the probability of surviving throughout the 14-day incubation period is 096 times itself 14 times, or 09614 = 056 Although the Mayfield method is a major advance in treating nesting data, it has been criticized (Green 1977) because of its assumption that the population is homogeneous, ie all nests are subject to the same rate of mortality In addition, Mayfield provided neither variance estimates for his mortality rate nor tests of the underlying assumptions The present paper is intended to augment Mayfield (1961 and 1975) In it I derive his estimator, which he developed heuristically, in a more formal context A standard error for his estimator can be calculated from this derivation The implications of Mayfield's assumption that nests are at risk until midway between visits are considered in detail I also propose a more realistic model, which does not require the midpoint assumption Estimators of the daily mortality rate and its standard error are obtained under this model and compared to those of Mayfield Finally, I discuss the importance of variation in daily mortality rates, from both identifiable and nonidentifiable causes Methods of detecting such variability and treating it, if it exists, are presented