Jerrold H. Zar

Significance Testing of the Spearman Rank Correlation Coefficient

Abstract A table of critical values of the Spearman rank correlation coefficient, r8, is given for n = 4(1)50(2) 100, for nine levels of significance: α = 0.50, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001.

Calculation and Miscalculation of the Allometric Equation as a Model in Biological Data

The simple linear regression model, (1) Y= 0o+ 31X, where Y is the dependent variable, X is the independent variable, and P0 and P1 are population parameters to be estimated empirically, often does not adequately describe or represent the relationship between the two variables. A commonly used procedure to arrive at a more descriptive model is to fit a higher degree polynomial, (2) Y=f3o+1 1X1+32X22+ 03X33 +... +RX,. Since the polynomial regression is simply a multiple linear regression, it can be fitted easily by ordinary least squares techniques, especially with the aid of a digital computer. At times, however, the researcher may believe that his data are better fit by a model where the parameters do not enter linearly. There are several nonlinear regression models in use in biology; what follows is a discussion of one such model, emphasizing the caution which must be exercised in fitting the model by the common approximate method.

Environmental temperature and the fatty acid compositions of house sparrow (Passer domesticus) muscle and brain

1. 1. Summer- and winter-caught House Sparrows (Passer domesticus) were acclimated to either 35, 20, 5, or -20°C, and brain and muscle fatty acid compositions were determined. 2. 2. Muscle lipids were composed mainly of fatty acids 16:0, 18:1, 18:0, and 18:2. In the brain were mainly 16:0, 18:0, 18:1, 22:6, and 20:4. There was no significant difference in fatty acid composition between sexes or between seasons of capture. 3. 3. Mean upper and lower lethal temperatures were found to be 45 and -27°C, respectively. 4. 4. Cold-stressed and starved birds had less 16:0, 16:1, and 18:1 in their muscles, and more 18:0 and 22:6, likely because of depletion of neutral lipids in that tissue. At high temperature there were somewhat greater amounts of 16:1 and 18:1, and less 18:0. 5. 5. Brain fatty acid composition did not change with sub-lethal temperature stress, but contained significantly high 18:0 at the upper lethal temperature. Brains of starved birds contained significantly high 20:4.

The effect of malathion on the temperature selection response of the common shiner,Notropis cornutus (Mitchill)

Warm-(17°C) and cold-acclimated (8°C) young-of-the-year, 1-year-old, and 2-year-old common shiners,Notropis cornutus (Mitchill) were exposed for 24 hr to sublethal doses (1.0, 0.25, and 0.05μg/L) of malathion to determine the effect of environmental concentrations of this compound on temperature selection. All three factors studied (acclimation temperature, age, and concentration) were found to affect the temperature selection response of malathion-treated fish. Malathion produced dose-dependent decreases in selected temperature (1.9° to 4.3°C below controls) in 17°C-acclimated fish but not in 8°C-acclimated fish. Two-year-olds treated with 1.0μg/L displayed the greatest lowering of selected temperature, followed by two-year-olds treated with 0.25μg/L. One-year-olds treated with 1.0μg/L showed a slight lowering, while no downward shifts in selected temperature were found for young-of-the-year. Exposure to 0.05μg/L had no effect on any age group.Two-year-olds acclimated to 17°C and exposed to 1.0μg/L malathion for 24 hr were placed in clean water and allowed to recover from the pesticide. It was found that the lowered selected temperature induced by malathion returned to that of the controls in 24 hr.

A Basic Program to Compute the Sensitivity, Specificity, and Predictive Values of Screening Tests

A BASIC microcomputer program is presented, which calculates several indices of reliability of a screening test: sensitivity, specificity, positive predictive value, and negative predictive value.

Probabilities of rayleigh’s test statistics for circular data

~he Kolmogorov-Srnirnov one-sample test is appropriate for testing goodness of fit of a hypothesized population distribution to a sampled population distribution (Bradley, 1968; Siegel, 1956; lar, 1974). If F(X) is the hypothesized cumulative frequency of the variable X [i.e., F(X) is the number of cases expected to be less than or equal to XJ, and S(X) is the observed cumulative frequency of X [i.e., S(X) is the number of cases in the sample which are less than or equal to X], then

Fatty acid composition of emperor penguin (aptenodytes forsteri) lipids.

Abstract 1. 1. The fatty acid compositions of the adipose tissue, brain, and foot skin were determined for an Emperor Penguin. 2. 2. In the adipose tissue, fatty acids 20:1, 20:4, 14:0, 16:0, and 18:1 were the major components, primarily 20:1 and 20:4. Part of this is correlated with the animals food, and comparisons with the Adelie Penguin are made. 3. 3. The foot skin lipids were comprised primarily of 16:1, 18:1, 20:1, and 20:4. This is discussed as an adaptation to cold environmental temperature. 4. 4. The predominant brain fatty acids were 18:0, 18:1, 16:0, and 22:6 acids, mainly the C-18 acids. This is compared with mammalian brain composition.

A fast and efficient algorithm for the Fisher exact test

which has the product of two binomial coefficients in the numerator and a binomial coefficient in the denominator. It is efficient, therefore, to have a computer subroutine for calculating a binomial coefficient. The computation of a factorial (call it y !) is very timeconsuming, and for y > 33 the result is > 10 8 , a number larger than can typically be accommodated by microcomputers. We can avoid the necessity of computing three factorials for each binomial coefficient (i.e., nine for the Fisher exact test probability) by calculating the binomial coefficient N! X!(N-X)! n as

Fatty acid composition of brown fat and brain fat of the little brown bat, Myotis lucifugus, during hibernation

Abstract 1. 1. Male and female little brown bats were collected on 7 dates during hibernation and on one date prior to hibernation. 2. 2. The predominant fatty acids in brown fat were 16:0 and 18:1, with lesser proportions of 16:1, 18:0 and 18:2, and small amounts of others; 16:0 decreased, while 18:1 increased, during hibernation. 3. 3. The major fatty acids in the brain were 16:0, 18:0 and 18:1, with lesser amounts of 16:1, 18:2 and 22:6, and small proportions of others; there were no significant changes in major fatty acids during hibernation. 4. 4. No significant fatty acid differences were found between sexes.

Watson’s nonparametric two-sample test

Watson (1962) introduced a distribution-free test of the null hypothesis that two samples come from the same population. The computed statistic, U , depends on the relative ranks of the observations in the two samples, and not on the starting point of the measurement scale. Thus, while it may also be used for data measured on a linear scale, the test is especially useful for data on a circle, for which conventional hypothesis testing is inappropriate. Common data from a circular scale of measurement are compass directions or clock times; parametric testing for such data is possible (see Batschelet, 1965, 1972; Mardia, 1972; Zar, 1974, 1976). Watsons nonparametric test is described in Batschelet (1965, pp. 35·36), Mardia (1972, pp. 201-203), and Zar (1974, pp. 324.326). A computer program has been developed that performs Watsons (1962) calculation of U , appro, priate for either nontied data or data tied within and/or between samples. The most extensive tables of critical values of U are in Zar (1974, pp.575-576). Stephens (1965) gives Pearson curve approximations for some large sample sizes. For large samples, a normal deviate may be calculated as: