A New Linearized Formula for the Law of Total Effective Temperature and the Evaluation of Line-Fitting Methods with Both Variables Subject to Error

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1 FORUM A New Linearized Formula for the Law of Total Effective Temperature and the Evaluation of Line-Fitting Methods with Both Variables Subject to Error TAKAYA IKEMOTO 1 AND KENJI TAKAI 2 Environ. Entomol. 29(4): 671Ð682 (2000) ABSTRACT In the quantitative analysis of experimental data regarding temperature-dependent development, the so-called law of total effective temperature is sometimes expressed in the linearized equation 1: 1/D (t/k) (1/k)T. D indicates the duration of development; T, temperature; t, the estimated developmental zero temperature; and k, the effective cumulative temperature. The method of Þtting usually involves the regression of y 1/D on x T. Although the degree of Þtting of equation 1 to data within optimum temperature ranges is fairly satisfactory, we have in the current study addressed three problems regarding the use of equation 1 and methods of Þtting involving the regression of y on x. First, we found that the detection of optimum temperature ranges is frequently difþcult with equation 1. Second, in applying the method of regression of y on x with equation 1, the weights of the data points are disproportionate between those in the upper and lower parts of the line and they are not homogeneous along the temperature axis. The lower the temperature, the more disproportionate weight is burdened and the less weight is loaded. Third, in most of the data, errors in the x-variable are ignored. The second and third problems would in most cases result in a reduction in the slope of the line, a smaller t, and a larger k. Therefore, we proposed a new linearized formula: (DT) k td. We further propose the use of the reduced major axis, obtained as the solution of the functional model among bivariate errors-in-variables models, in the method of Þtting to data. We demonstrated that the majority of the problems raised above could be unraveled under this new approach based on statistical analysis. KEY WORDS temperature-dependent development, linear-þtting, errors-in-variables models, reduced major axis, geometric mean functional relationship THE LAW OF total effective temperature, applied to the temperature-dependent development of arthropods or parasites, is expressed by the equation D T t k, [1] where the product of the duration of development, D (days), and temperature T (degrees) minus t is a constant k for each strain of organism. One of the parameters, t, is designated as an estimated developmental zero temperature (Clements 1992), and the other, k, is called an effective cumulative temperature (Kiritani 1991). This law has been useful because the parameters can well characterize organisms with regard to their adaptation to temperature. Honěk and Kocourek demonstrated the existence of a negative relationship between the parameters (Honěk and Kocourek 1988, 1990; Honěk 1996a, 1996b). In addition, Kiritani (1991) uncovered two phases of relationships between the parameters: negative relationships between t and k could be observed within groups of insects, 1 Department of Microbiology, Teikyo University School of Medicine, Kaga , Itabashi-ku, Tokyo , Japan. 2 To whom correspondence should be addressed: Department of Immunology and Medical Zoology, St. Marianna University School of Medicine, Sugao , Miyamae-ku, Kawasaki , Japan. whereas a positive relationship existed among group averages of t and k. An estimation of the number of generations per year of pests is made based on equation 1 (Gomi and Takeda 1990, Morimoto and Tanahashi 1991). Parameters that are estimated using equation 1 constitute the basis for predicting the habitat expansion of pest animals under the effects of global warming (Kiritani 1991, Morimoto et al. 1998, Yamamura and Kiritani 1998). In estimating the parameters t and k, one popular method has used a linearizing transformation y 1/D for the variable D resulting in the equation 1 D t k 1 T. [2] k This is referred to as the linear degree-day model (Briére et al. 1999). A regression line of y 1/D on x T is estimated (Campbell et al. 1974; Mottram et al. 1986; Honěk and Kocourek 1988; Kiritani 1991; Ando 1993; Honěk 1996a, 1996b). We, however, address three problems regarding this estimation method with the intent of obtaining more reliable parameter values. The Þrst problem relates to the frequent difþculty in detecting the range of temperature where equation 2 is applied for the organism in question (Ito 1976). There is an optimum temper X/00/0671Ð0682$02.00/ Entomological Society of America

2 672 ENVIRONMENTAL ENTOMOLOGY Vol. 29, no. 4 ature range for each organism where equation 2 should be applied, and this optimum temperature range is deþned by higher and lower critical temperatures. Incorrect detection of the two critical temperatures would allow only unreliable estimation of the parameters. Second, the problem of Þtting a regression line of y on x with equation 2 postulates constant variances on a 1/D scale at all temperatures. This would consequently put smaller weights in the lower temperature range on a D scale. Moreover, as analyzed in this study, the weight is disproportionate between the upper and lower parts of the line on the D scale, and this distortion is more exaggerated in the lower temperature range. Third, the regression line of y on x neglects the existence of errors in the x-axis direction, that is, errors in temperature, which yield, as a well-known outcome, a smaller estimate of the slope of the line (Berkson 1950, Snedecor and Cochran 1980, Sokal and Rohlf 1995). The Þrst problem would affect t and k in an indefinite manner; the second and third problems, would result in a lower estimate of t and a larger estimate of k. Because of the problems using equation 2 with the line-þtting method of regression of y on x, we propose a new formula with a linearizing transformation. The new formula would resolve two of the above problems; it would make the detection of higher and lower critical temperatures relatively easy, and would increase weights on the points in the lower temperature range, balancing weights between the upper and lower parts of the estimated line. To resolve the third problem described above, we examined line-þtting methods with both variables subject to error in comparison with the method using the regression of y on x. The aim of this study was to demonstrate methods that can provide a more reliable estimation of parameters in the formula to represent the law of total effective temperature for arthropods and parasites. For this purpose, we have proposed a new linearized formula and a method of estimation of parameters different from the ordinary method. This new method is based on a revision of the three main drawbacks in the ordinary method of estimation of parameters in the law of total effective temperature. Materials and Methods Mathematical Formula. We are interested in the law of total effective temperature expressed as equation 1. The ordinary method of estimating the parameters of equation 1 is to use equation 2. To obtain more reliable estimates of the parameters in equation 1, however, we proposed the following new formula: DT k td, [3] derived from equation 1. This equation represents a straight line with x D and y DT. We based the evaluation of this new formula on that of the ordinary equation 2. Determination of the Optimum Temperature Range. On Þtting equation 2 or 3 to data points on the corresponding coordinate, some of the data points at higher or lower extreme temperatures need to be excluded to achieve satisfactory Þtting. This exclusion of data points determines the optimum temperature range with higher and lower critical temperatures by interpolation. Statistical Model and Analytical Method. Data for temperature-dependent development were analyzed by the ordinary regression method of y on x of equation 2, which is denoted as method 1. The data could be correctly considered under the functional model among errors-in-variables models (Cheng and Van Ness 1999). Because a reduced major axis (Kermack and Haldane 1950) provided a general solution of the model, it was applied to a line-þtting of equation 3. We designated this application as method 2. Both methods were qualitatively compared with each other with respect to detecting the efþciency of the optimum temperature range and were quantitatively compared as to the estimation results of the parameters. We further referred to another method to use equation 2 with line-þtting using the reduced major axis to be designated as method 1. Estimation of Variances of Parameters. With equation 2 (method 1), estimated parameters of the line y i x i are related to k and t by k 1/ and t /. The variances of k and t are then estimated using formulas of propagation of errors by s k k 2 2 s k 2s 2 2 s 2, 4 s t t 2 2 s 2 t 2 s s 2 2 s 2, method 2 does not require such a calculation of variances because its line parameters are the direct parameters k and t. Data Sources. To examine the general applicability of the new formula and the new method, data for the temperature-dependent development of several insects and parasites, including the mosquito, Aedes aegypti (L.) (Diptera: Culicidae) (Gilpin and McClelland 1979); the pea aphid, Acyrthosiphon pisum (Harris) (Homoptera: Aphididae) (Lamb 1992); the spruce budworm, Choristoneura fumiferana (Clemens) (Lepidoptera: Tortricidae) (Weber et al. 1999); and the human malaria parasite Plasmodium falciparum (Welch) (Coccidiina: Plasmodiidae) (Moshkovsky and Rashina 1951, Macdonald 1952, Russell et al. 1963), were selected from the literature. The graph of the developmental rate data of A. aegypti was read with a scanner and translated. The developmental duration data of one of two clones was used for A. pisum and of one of six geographical populations for C. fumiferana. Gathered data were used for the malaria parasite as above.

3 August 2000 IKEMOTO AND TAKAI: LINE-FITTING TO TEMPERATURE-DEPENDENT DEVELOPMENT 673 Fig. 1. Temperature-dependent development of the larva of the mosquito A. aegypti. Data from Gilpin and McClelland (1979). (A) A regression line of y on x is Þtted to developmental rate data (method 1). Points shown by were excluded in Þtting the line in the original analysis. For symbols a to c see text. (B) A reduced major axis of the new formula is Þtted to the same data of Fig. 1A but on different coordinate axes (method 2). Points marked by were excluded from analysis. (C) The high-temperature area squared in (B) is magniþed. Results Definite Determination of the Optimum Temperature Range with Method 2. The temperature-dependent development data of A. aegypti, A. pisum, and C. fumiferana were analyzed by methods 1 and 2. It should be emphasized that our new method (method 2) enabled determination of the optimum temperature range more precisely than the ordinary method (method 1). Fig. 1A shows the line obtained from method 1 Þtted to the data for the temperature-dependent development of the larvae of the mosquito A. aegypti (Gilpin and McClelland 1979). Four data points at extremely high temperatures signiþcantly deviate from the line; it is unclear, however, whether the two points labeled ÔaÕ and ÔbÕ should be excluded (the authors did exclude them). It appears to be permissible to include a point at the lower temperature extreme because of its apparent small deviation from the line. The authors included it. The optimum temperature range where a linear relation between T and 1/D is recognized is assumed to be from 15.5 to 30.5 C. Data points at these critical temperatures could not deþnitely be included in the linear array of points, which results in an uncertainty in the optimum temperature range as well as an unreliable estimation of parameters. In contrast, in Fig. 1B, where the same data are plotted on the D -(DT) plane (method 2), the point marked with a ÔcÕ at the lowest temperature appears to be signiþcantly deviated from the linear tendency created by the points in the optimum temperature range. In this Þgure, the points are connected by gray dotted lines in temperature order. The upper right corner corresponds to the lower temperature range, whereas the lower left corner corresponds to the higher temperature range. The area in the higher temperature range is expanded in Fig. 1C because the data points are very close to each other. The six -marked points are connected by gray dotted lines starting from the highest temperature. These points are perceived to have a tendency distinct from the points in the other temperature range. After excluding these seven -marked points at both ends of the medium temperature range, a reduced major axis is Þtted to the rest of the points. The optimum temperature range, for which the linear relation of equation 3 is applied, is between 17 and 30.5 C. Figure 2 shows a second example. The data come from Lamb (1992), in which line 2 of method 1 is Þtted to data for the temperature-dependent development of the immature of the pea aphid A. pisum (clone L). Lamb (1992) Þtted the line to data, excluding only two points at the high temperature, indicating that the optimum temperature range can be presumed to be from 5 to 25 C, although the data point at 5 C in particular appears to deviate too far from the line to be included. The same data are shown in a different manner in Fig. 2B, because they are plotted on a D -(DT) plane with a Þtted reduced major axis (method 2). Each group of -marked points from the points labeled ÔdÕ to Ôf Õ in the lower temperature range and those labeled ÔgÕ to ÔhÕ in the higher temperature range has a distinctly different tendency from the points in the medium optimum temperature range between 11.0 and 23.0 C, to which a fairly good linear Þtting is obtained. Figure 3 presents a third example. Using data for the larval temperature-dependent development of a spruce budworm, C. fumiferana (Weber et al. 1999),

4 674 ENVIRONMENTAL ENTOMOLOGY Vol. 29, no. 4 Fig. 2. Temperature-dependent development of the immature of the pea aphid A. pisum (clone L). Data from Lamb (1992). For symbols d to i denoting excluded points, see text. The explanation for (A) and (B) is the same as that in Fig. 1. method 1 is applied in Fig. 3A and method 2 is in Fig. 3B. It is difþcult in the analysis in Fig. 3A (method 1) to detect data points that signiþcantly deviate from the linear array created by the points at the intermediate optimum temperature range. However, it is clear in Fig. 3B (method 2) that three points at the lower temperature range should be excluded. The optimum temperature ranges are 11.2Ð33.0 C with method 1 and 15.0Ð33.0 C with method 2. Improvement in the Estimation of Parameters with Method 2. Data points in the three examples in the proceeding section appeared to be subject to a relatively small magnitude of error. There are other sorts of data, however, that have a much wider level of variation. Fig. 4 shows one of such data sets for the temperature-dependent development of the human malaria parasite in mosquitoes (for data source see Materials and Methods). The degree of variation within the data is demonstrated by the regression coefþcient r. All three sets of data have r 0.99 on the T - (1/D) plot, whereas the data shown in Fig. 4A have r The parasite, P. falciparum, is the pathogen of human malignant malaria, and it lives an essential portion of its life cycle in anopheline mosquitoes to achieve sporogony. This stage is temperature-dependent. Integrated data from three reports in the literature are presented in Fig. 4A to show the results of method 1. Some biased deviation can be seen in Þtting the line to the data, presumably the result of the use of 1/D as the y-variable and to neglecting the errors in the x-variable. However, Fig. 4B, which shows the results of using method 2 with the same data, clearly exhibits no apparent biased deviation of the line from the data. With our new method 2, in which the Þtting method uses the reduced major axis, what differences arose in estimating the parameters? Table 1 shows a comparison of the two methods for the four sets of data. In the Þrst data set, method 2 differed in only 1 data point from method 1 (see the sample size difference). Discrepancies in the estimates of k and t are negligible (statistically not signiþcantly different). However, in the rest of the data sets, the estimates of parameters k and t differed rather considerably: with method 2, t became 1.5 degrees higher and k became 10 degreedays (DD) smaller for A. pisum; there was a larger t by 3 degrees and a smaller k by 60 DD for C. fumiferana; and a larger t by nearly 4 degrees and a smaller k by 54 DD for P. falciparum. All differences were statistically signiþcant. In all cases but one, the standard error of the parameters became smaller in method 2 than in method 1, indicating increases in the precision of the former. We have ample circumstantial evidence that method 2 can provide more reliable estimates of parameters than method 1, and this evidence will be presented in the succeeding part of this paper.

5 August 2000 IKEMOTO AND TAKAI: LINE-FITTING TO TEMPERATURE-DEPENDENT DEVELOPMENT 675 Fig. 3. Temperature-dependent development of the spruce budworm C. fumiferana at Cypress Hills (Weber et al. 1999). The three points shown by were excluded. The explanation for (A) and (B) is the same as that in Fig. 1. Discussion Determination of the Optimum Temperature Range. Because both methods 1 and 2 use the Þtting method of straight lines, it is important to detect linearly arrayed data points to achieve a satisfactory Þtting. As described in the Results, method 2 was superior to method 1 in this respect. Briére et al. (1999) proposed a new curvilinear rate model of temperature-dependent development with parameters T 0 and T L.T 0 is the low temperature developmental threshold, and T opt. is the lethal upper threshold temperature. In the range T 0 T T L, a curve is Þtted to the data. Their model can predict an optimum temperature T L where the developmental rate becomes maximum. Our critical temperatures, to be denoted as t l and t u, limit the optimum temperature range where a linear relation between the developmental rate and temperature is perceived. Thus, neither t l nor t u is expected to correspond to T 0, T L, or T opt. In general, T 0 t l t u T opt T L, [4] or T 0 t l T opt t u T L [5] must be satisþed. We had difþculties in obtaining results from the application of equation 1 or 2 of Briére et al. (1999) to the current data (in this method it is necessary to Þt the curves by iterative nonlinear regression based on the Marquardt algorithm). We examined the data presented in Briére et al. (1999) and applied method 2 to the data of Lobesia botrana Dennis & Schiffermüller (Lepidoptera: Tortricidae) (Briére and Pracros 1998), Melanoplus sanguinipes (F.) (Orthoptera: Acrididae) (Hilbert and Logan 1983), and A. pisum (Lamb 1992). Table 2 shows a comparison between the critical temperatures t l and t u of method 2 and the parameters T 0 and T opt of equation 1 of Briére et al. (1999). Either set of inequalities four or Þve was recognized. The recognized sets of inequalities were the same when T 0 and T opt of equation 2 of Briére et al. (1999) were used. Homogeneous Variances on a 1/D Scale at All Temperatures in Method 1. Data for temperature-dependent development are collected by measuring the duration D at temperature T. D is not measured as its rate 1/D. To measure the rate, we would have to observe the degree of development within some period to be set in advance, which is not done in practice. The measurement of duration D is expected to accompany a homogeneous variance component at all temperatures in a general measurement procedure as follows. Suppose an insect larva is reared in the laboratory, and an experimenter examines it once a day when it metamorphoses to a pupa. When at day d he observes that the larva has reached the pupal stage, it would have metamorphosed just before the observation or would have metamorphosed sometime during the past day since the previous observation. Taking an average, at day (d 0.5) 0.5 the larva is thought to have

6 676 ENVIRONMENTAL ENTOMOLOGY Vol. 29, no. 4 Fig. 4. Temperature-dependent sporogony of the malaria protozoa P. falciparum. Data are from Macdonald (1952) and others (see text). An open diamond indicates a point with a number of replications. The explanation for (A) and (B) is the same as that in Fig. 1. metamorphosed. The term -0.5 is subtracted in calculating the duration D. Then D has a variance component corresponding to this measurement error, and this variance is homogeneous at all temperatures. When the variable 1/D is taken, this variance component becomes disproportionate on a 1/D scale and nonhomogeneous along the temperature axis because it involves a transformation of variables opposite to that of the procedure explained below. In Þtting the regression line of y on x with equation 2, we obtain the standard error of estimate as s 1/D s e Y i y i 2, [6] n 2 where Y i denotes the data point and y i denotes the corresponding point on the estimated line with the number of data points n. As a result, the upper and lower limiting lines are determined by 1 D t k 1 k T s e, [7] with the conþdence region of the data points covering 68% of the points. When equation 7 is expressed in the form D k T t, [8] we obtain k D T t ks e. [9] Equations 8 and 9 graphically deþne rectangular hyperbolas. We rewrite equation 8 as D D o. Equation 9 involves two curves D D l and D D u, taking the minus and plus of its double sign, respectively, D u is larger than D l at T t ks e. Each curve of D D o, D D l, and D D u has a vertical asymptotes: T t, T t ks e, and T t ks e, respectively. Then, as shown in Fig. 5, in the range t T t ks e, D o D l has a Þnite value, while D u does not exist, virtually, D u D o is inþnite. [10] Another property of the range deþned by D D l and D D u is as follows. Consider the midpoint of the lower and upper limiting lines, D D l and D D u at temperature T. The ordinate of the midpoint is expressed by (D u D l )/2. We examine f(d o ) D u D t D 2 o. [11] Eliminating t and k from D l and D u in expression 9 results in df df dd o dt dd o dt D q 2 f(d o ) D D S o e 2, 1 2 D S o e D o 2 Se2 dd o dt.

7 August 2000 IKEMOTO AND TAKAI: LINE-FITTING TO TEMPERATURE-DEPENDENT DEVELOPMENT 677 Table 1. Comparison of two methods for determination of the optimum temperature range and for estimation of parameters in temperature-dependent development In the range T t ks e,d o 1/s e (see in Fig. 5). We have dd o /dt k /(D o t) 2 0. These yield df /dt 0, which means that f (D o ) is a decreasing function with respect to T. Consequently, the magnitude of D u D o becomes larger with decreasing temperature. This tendency is exaggerated, as in expression 10. The regression line for equation 2 is Þtted by assigning evenly balanced sums of squares of upper and lower residuals from the data point to the line. However, the above results indicate that the line is Þtted to the data with smaller weights for the points below the line for decreasing temperatures. In combination with the Method 1 a (conventional method) Method 2 (current method) Difference of parameters Formula 1/D (t/k) (1/k)T (DT) k td Involved line in Þtting method regression line of y on x reduced major axis Larvae of A. aegypti: Figure Fig. 1A Fig. 1 B-C Lower critical temp ( C) 15.5? 17.0 Upper critical temp ( C) 30.5? 30.5 Sample size k SE ( C days) NS t SE ( C) NS Immatures of A. pisum: Figure Fig. 2A Fig. 2B Lower critical temp ( C) 5.0? 11.0 Upper critical temp ( C) 25.0? 23.0 Sample size k SE ( C days) *** t SE ( C) *** Larvae of C. fumiferana at Cypress Hills: Figure Fig. 3A Fig. 3B Lower critical temp ( C) 11.2? 15.0 Upper critical temp ( C) Sample size 9 6 k SE ( C days) *** t SE ( C) *** Immature protozoa to sporozoites of P. falciparum: Figure Fig. 4A Fig. 4B Lower critical temp ( C) Ñ Ñ Upper critical temp ( C) Ñ Ñ Sample size k SE ( C days) *** t SE ( C) *** SigniÞcance level: ***, P 0.001; NS, not signiþcant (WelchÐAspin test [Snedecor and Cochran 1980]). a A question mark following the critical temperature indicates uncertainty of detection (see text). Table 2. al. (1999) property of the line passing through the average point (T, Y), where Y 1/D, this would make the slope of the line smaller, resulting in a smaller t and a larger k. This constitutes one of the ßaws in equation 2 regarding the method of Þtting the regression line of y on x. There is an additional consequence of this result. If the data points in the lower temperature range are located above the estimated straight line, there will be a bias in favor of a larger slope. This is because the magnitude of D o D l is larger with decreasing temperatures as shown below; that is, a larger slope will tend to be calculated because the weights on the Comparison between the critical temperatures from method 2 and the parameters T 0 and T opt of equation 1 of Briére et Method 2 Set of Insect species Stage n T 0 T opt t inequalities a l t u n L. botrana Egg L. botrana L L. botrana L L. botrana L L. botrana L L. botrana L L. botrana Pupa M. sanguinipes Nymph A. pisum T 0, lower temperature threshold; T opt, optimum temperature threshold; t l, lower critical temperature of the optimum temperature range where formula 3 is applied; t u, upper critical temperature. a Number refers to the set of inequalities 4 or 5 in text.

8 678 ENVIRONMENTAL ENTOMOLOGY Vol. 29, no. 4 Fig. 5. ConÞdence region of data points deþned by the upper and lower lines of the standard error of estimates of the regression line of y on x in method 1. In the upper Þgure, the hatched area is expected to cover 68% of the data points around the line of equation 2. The lower Þgure is drawn to show this region on the T-D plane and demonstrates that the region is extraordinarily expanded in particular at low temperatures. Such expansion of the conþdence region yields to unreliable estimation of the line (see text). points in the lower temperature range are smaller than those on the points in the higher temperature. This yields a larger t and a smaller k, in contrast to the tendency described above. With D o and D l above, we consider g D 0 D o D l. Because it is expressed as g D o D o 2 s e, 1 D o s e we obtain dg dg dd o dt dd o dt D os e 2 D o s e dd o 1 D o s e 2 dt. From D o 0 and dd o /dt 0, we get dg /dt 0. Consequently, g (D o ) is a decreasing function with respect to T. The magnitude of g(d o ) D o D l is larger with decreasing temperature, which explains the additional consequence described above. Neglect of the x-variable Error with the Regression Line of y on x in Method 1. The regression line of y on x assumes no errors in the x-variable, that is, no errors in temperature in equation 2. This assumption may not be valid, however. It is well known that when the regression line of y on x is Þtted to experimental data, for which the x-variable is subject to error, the slope of the line is underestimated (Berkson 1950, Kendall and Stuart 1979, Snedecor and Cochran 1980). The real value 0 is estimated using variances of the ÔtrueÕ variable i and the error i for the observed x-variable. However, it is difþcult in general to obtain variances 2 and 2 or their ratio from temperature-dependent development data. Incidentally, there is also the so-called Berkson model, which furnishes unbiased in the regression line of y on x despite the x-variable being subject to error (Berkson 1950, Mandel 1964, Kendall 1976). In this model, the x variable is nonrandom and takes predetermined values. In method 1, if the temperature is predetermined and set at several values with an incubator and the development of the organisms inside is recorded, the Berkson model will be applied. The slope parameter of the regression line of y on x gives an unbiased estimate. Nevertheless, we Þnd a category of data to which this model cannot be applied with respect to the condition of setting the temperature. Such data would include situations in which the preset temperatures include more than a small degree of error because of the incubators themselves. In using an incubator, if the temperatures are remeasured, they will be close to the preset temperatures of the incubator, but the variances of the values would not be zeros. Data where temperatures are recorded in decimal values might fall into this category (Haufe and Burgess 1956, Wilkinson and Daugherty 1970, Yoshida et al. 1974, Nakamura 1983). This category of data would also include other situations in which the means of variable temperatures are used as values for the x-variable (Mogi and Okazawa 1996). Models for Linear Relations with Two Variables, Both Subject to Error, and the Reduced Major Axis. If the x-variable is subject to error in the data, some corresponding model should be applied to them. Such a model is included in the model II regression, or referred to as the measurement error model, whereas the conventional regression of y on x is called the model I regression (Sokal and Rohlf 1995). Additional assumptions regarding the model II regression distinguish structural, functional, and ultrastructural relations (Moran 1971, Dolby 1976, Mark and Church 1977, Cheng and Van Ness 1999). The Þrst relation

9 August 2000 IKEMOTO AND TAKAI: LINE-FITTING TO TEMPERATURE-DEPENDENT DEVELOPMENT 679 involves two random variables where the x-variable has only one mean. An example is found in data from the measurement of the individual lengths of two organs in an insect sample. The second relation involves a nonrandom x-variable, each measurement of which has its own mean, but there is no variance in the real variable itself. The third relation contains a nonrandom x-variable, each measurement of which has its own mean with a variance. In the temperature-dependent development in equation 2, T and 1/D are nonrandom variables. Thus, in applying the model II regression to the data, we seek to obtain a functional or ultrastructural relation. If T has errors only in measurement, the model is of a functional relation. If the Ts are the means of variable temperatures, the model is of an ultrastructural relation. This perspective of taking errors in the x-variable into account is rarely encountered in the literature related to temperature-dependent development. Model I and II regressions with roman numerals are designated in the context which Sokal and Rohlf (1995) speciþed. We denote two methods by 1 and 2 as indicated in Materials and Methods. Model II contains inclusive measurement error models, but our method 2 is restricted to the linear functional measurement model with the use of line equation 3. Method 1 is application of the model I method with the use of equation 2. Let measured values of a pair of bivariate data be X i and Y i, both subject to error, then X i i i, [12] Y i i i, [13] where i and i are the real values and i and i are errors. We assume that these values are related by the formula i i. [14] As clariþed in the preceding paragraphs, data for temperature-dependent development are analyzed under either the functional or ultrastructural model; we are restricted here, however, to only the functional model, which has the general solution for : r S yy (in case S S xy 0), [15] xx where S xx (X i X ) 2, S yy (Y i Y ) 2, and S xy (X i X )(Y i Y ) with X and Y being averages of X i and Y i, respectively (Cheng and Van Ness 1999). This solution is obtained by the maximum likelihood method, although, strictly speaking, it is not the maximum likelihood estimate but the likelihood equation estimate (Anderson and Rubin 1956, Solari 1969, Willassen 1979, Cheng and Van Ness 1999). It has been further shown that r lacks statistical consistency (Stuart et al. 1999, Cheng and Van Ness 1999); that is, when the sample size is increased, ˆ r converges to a different value from r of the model. If the ratio of the variances of i and i, 2 2 [16] is known, then is estimated as S yy S xx S yy S xx S xy. [17] 2S xy The above is the maximum likelihood solution (cf. Kendall 1976, Cheng and Van Ness 1999). In equation 17, 1 gives a major axis, while S yy [18] S xx gives the reduced major axis (Ricker 1973, 1975; Jolicoeur 1975; Sprent and Dolby 1980; Draper and Smith 1998). Some authors do not support equation 18, but instead take 1 to be a more appropriate assumption for (Jolicoeur 1975, Sprent and Dolby 1980). In addition, to the major axis having a default in which the slope is variant depending on the units of the variables, this line virtually coincides with the regression line of y on x or x on y in the case of data for temperature-dependent development (unpublished data). These regression lines are inappropriate because they ignore the x-variable or y-variable errors. If can be estimated from the experimental data, equation 17 gives a satisfactory solution. However, such data are rarely available among the data of temperature-dependent development. In considering equations 12Ð14 in the ultrastructural model, we cannot obtain any maximum likelihood solutions of even with known (Dolby 1976). Gleser (1985) has shown that the maximum likelihood solution is obtained under the boundary V( i ) 0. This is nothing but a functional model. In short, we just reduce the ultrastructural model with known to the functional model as an approximation. Interesting algebraic and geometrical characteristics of the reduced major axis include that r isthe geometrical mean of the slope parameters Y X and X Y of the regression line of y on x and x on y, respectively, and that the reduced major axis is obtained by the least sum of areas of the triangle made by the two lines parallel to the x- and y-axes and the estimated line (Teissier 1948, Hayami and Matsukuma 1971, Barker et al. 1988). Both features let us push usage of the reduced major axis in the general functional relation of the errors-in-variables models. In particular, the Þrst algebraic feature, for the sake of which the reduced major axis is called the geometric mean functional relationship estimator of the slope (Cheng and Van Ness 1999), is more or less deþnitive. In errorsin-variables models, if errors of one variable are ignored, then Y X or X Y will be obtained. Therefore, the real estimate of of the model would likely be intermediate between them, which is one of the primary reasons that we used the controversial reduced major axis. Following what Kendall (1976) has expressed for the major axis, a heuristic method for obtaining a solution of the functional model might be to take the least Ôsum of the triangle areas made of data

10 680 ENVIRONMENTAL ENTOMOLOGY Vol. 29, no. 4 points and the line,õ and it is intuitively plausible that the method yields an intermediate value of between Y X and X Y (the words interposed between the single quotation marks were in Kendall (1976) Ôsquares of the distances between data points and the line,õ which are replaced as above to Þt to referring to the reduced major axis). The property of inconsistency of r might be in part circumvented by dealing with a relatively small number of data points (Draper and Smith 1998). Adjustment of Unbalanced Weights in the Upper and Lower Parts of the Line and an Increase in Underloaded Weights at Lower Temperatures with Method 2. The degree of Þtting of the reduced major axis of equation 3 to the data are evaluated by the standard error of the residuals in the y-axis direction: s e 2 n 2 S yy S xy [19] which corresponds to equation 6 of the regression line of y on x (Tango 1988). The following formulas then determine the upper and lower lines of the standard error: DT k td s e, which limit the conþdence region of data points to cover 68% of the points, similar to the feature of equation 7. These lines are expressed in the form corresponding to line equation 9 as D k s e T t. Let the upper line of equation 8 be D D u and the lower line be D D l ; the function f(d o ) corresponding to function 11 then satisþes f D o 1 2 k s e T t k s e T t k 0. T t This indicates that equation 3 is Þtted to the data with equal weights in the upper and lower parts of the line in the D scale. Moreover, the quantities, D u D o D o D l s e T t, D u D l 2s e T t, are all small and Þnite at T t, in contrast to the inþnity of D u D l when using equation 2. This might bring about an improvement or increase in the underloaded weights of data points in the lower temperature range when applying equation 3. Method 1 or Berkson Model are Inadequate for Improvement on Method 1. If the reduced major axis is used for line-þtting of equation 2 (method 1 ), the problem of x-variable errors is expected to be resolved. Or as referred to above, if the Berkson model can be applied to equation 2, an unbiased estimation can be made with method 1 even with x-variable errors. However, the problem of homogeneous variances on a 1/D scale in equation 2 remains unresolved. This is why we have not recommended usage of the reduced major axis in equation 2. In addition, the Berkson model applied to method 1 is inadequate for the same reason. Curvilinear Fitting to Developmental Rate Data and Significance of the Study for Linear Fitting. The law of total effective temperature has long been recognized (Réaumur 1735, Bodenheimer 1926) and has attracted the attention of many researchers (Campbell et al. 1974, Gilpin and McClelland 1979, Clements 1992, Honěk 1996a, and others). Together with the simple meaning of parameters t and k, the principle of this law appears to appeal much to intuition for temperature dependent development. The ßaws of linear models include the involvement of the unrealistic developmental zero temperature as the parameter t and their inability to be used at high temperatures where developmental impairment occurs. Curvilinear Þtting to developmental rate data has been studied extensively, in part to overcome these ßaws of the linear model (Davidson 1942, 1944; Pradhan 1945, 1946; Stinner et al. 1974; Logan et al. 1976, 1991; Sharpe and DeMichele 1977; SchoolÞeld et al. 1981; Hilbert and Logan 1983; Lamb et al. 1984; Lactin et al. 1995; Briére et al. 1999). These methods involve extensive ranges of data points outside the optimum temperature. In some of the studies, the Þtted curve has had an x-intercept corresponding to a biological developmental zero temperature for the organism (Hilbert and Logan 1983, Lactin et al. 1995, Briére et al. 1999). At temperatures that are higher than optimal, some of the studies Þt the curve having an x-intercept corresponding to a biological upper limit of temperature for development (Logan et al. 1976, Hilbert and Logan 1983, Briére et al. 1999). However, it has surprised us that at optimum temperature ranges, the developmental rate has fairly satisfactory linear relations to temperature among many organisms. We are interested in a more reliable estimation of parameters to explain this phase of development. This study will contribute much to a more appropriate description as well as a precise prediction of development within this temperature range. A substantial amount of work has been carried out based on this law of total effective temperature, as referred to in the introduction. This study will also contribute to such work with respect to the speciþc or ecological differentiation of organisms. Acknowledgment We are grateful to E. Kuno (Kyoto University) for a critical review of the initial manuscript. References Cited Anderson, T. W., and H. Rubin Statistical inference in factor analysis, pp. 111Ð150. In J. Neyman [ed.], Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 5. University of California Press, Berkeley, CA.

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