Choice of Degree of Smoothing in Fitting Nonparametric Regression ...

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Journal of Health Science, 54(2) 143–153 (2008)

Choice of Degree of Smoothing in Fitting Nonparametric Regression Models for Temperature-mortality Relation in Japan Based on a Priori Knowledge Victoria Nikolaeuna Likhvar a and Yasushi Honda∗, b a

Doctoral Program in Human-Care Sciences, Graduate School of Comprehensive Human Sciences, University of Tsukuba, 1–1–1 Tennoudai, Tsukuba 305–8577, Japan and b Department of Health Care Policy and Management, Graduate School of Comprehensive Human Sciences, University of Tsukuba, 1–1–1 Tennoudai, Tsukuba 305–8577, Japan (Received June 30, 2007; Accepted January 31, 2008; Published online February 5, 2008)

The objectives of this study were to determine the extent of smoothness for some selected nonparametric models, and to examine the suitability of the default method for evaluating temperature-mortality relation in Japan. Our analysis was conducted for Japanese aged 65 and older, from 1972 to 1994. The models we selected were smoothing spline and locally-weighted scatterplot smoothing (LOWESS/LOESS). Firstly, we determined the degrees of smoothness by an “a priori” approach. After exhaustively drawing curves of the relation between daily maximum temperature and sex-specific mortality rate for each prefecture using a wide range of smoothing parameters, we selected the degrees of smoothing for each prefecture, based on a priori knowledge. This assumes that we a priori know that the relation between temperature and mortality is V-shaped, i.e., between two temperature extremes the curve should have a minimum mortality rate at a certain temperature (optimum temperature = OT), which is an absolute minimum with no local extremes. For the cases in which no OT was observed for any of the degree of smoothing, we did not assign an OT. Among selected degrees of smoothing, we further selected “best” degrees of smoothing for the models such that the degrees of smoothing yielded OTs for all the prefectures (except for those with non-OT). Next, based on the “best” degrees of smoothing, we examined the generalized cross-validation (GCV) method, which is one of the most successful “default” methods for selecting a smoothing parameter, and which is a default method in R statistical language. For most of the prefectures, the relation between daily maximum temperatures and mortality rates were V-shaped. The OTs varied among prefectures and tended to be higher for southern prefectures. Some of the estimates based on GCV method, in particular for the LOESS models, yielded non-OT type relations even when the “a priori” approach yielded OTs. LOESS model showed more sensitivity to the value of span (the parameter of smoothness); an average difference in OT levels within the “best” selected range of spans was 0.5◦ C, while that for the smoothing spline model was 0.3◦ C. This study suggests that, for evaluating the relation between daily mortality and temperature, the smoothing spline model with degree of freedom being 6– 7.5 was the most appropriate model for the Japanese data, and that blind use of the default method was problematic in this case. Key words —— temperature-mortality relation, model choice, nonparametric regression, smoothing

INTRODUCTION It is well known that the relation between ambient temperature and mortality appears graphically as a “U” or “V” shape, with the average temperature for the lowest mortality ranging between 15◦ C to 25◦ C.1, 2) Investigations carried out in a large num∗

To whom correspondence should be addressed: Department of Health Care Policy and Management, Graduate School of Comprehensive Human Sciences, University of Tsukuba, 1–1– 1 Tennoudai, Tsukuba 305–8577, Japan. Tel. & Fax: +81-29853-2627; E-mail: [email protected]

ber of cities have shown that the optimum temperature (OT) level corresponding to the minimum mortality level varies from place to place and country to country according to the climate of each zone, and probably reflects adaptations of the population to the usual range of temperatures.3) Our ultimate goal is to evaluate the effect of global warming using this V-shaped relation and its relation with climate. For this goal, we need to explore the temperaturemortality relation model that can be applicable to wide range of the world. In this paper, we were trying to explore the applicability of some nonpara-

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metric models using Japanese data, which may be the basis for the world model. There have been studies conducted on modeling mortality across temperatures using spline functions to describe the relations between temperature and mortality, allowing different slopes for different parts of the temperature distribution.2) Nonetheless, the nonparametric smoothing techniques were highly recommended as having enough flexibility to describe the nonlinear relation4, 5) and are well known as producing an optimal, smooth set of mortality predictions as a function of temperature or air pollution level. The most commonly used nonparametric models are smoothing spline6, 7) and LOESS (former LOWESS lowess, which is short for locally-weighted scatterplot smoothing).8–11, 23) Smoothing splines balance between the two contradicting tasks, that is, matching the data and producing a smooth function. The idea of smoothing spline is to set some penalty, so that a higher penalty is assigned to rougher curves [Appendix A]. The smoothing parameter, which allows adjustment of the smoothness of the result, in this case corresponds to the penalty. Using a penalty generates similar looking sequences of smoothed locations.12) The LOESS function is obtained by an iterative procedure, combining the ideas of local polynomial smoothing with robustness to outliers. In other words this method is locally (in neighborhoods) adjusting polynomials of low degree, whose smoother estimates are obtained by the regression between two variables by means of weighted least square in each neighborhood [Appendix B]. To carry out a nonparametric regression, it is necessary first to determine the smoothness of the fit. With statistical software (such as SAS, S-plus, or R) it is possible to specify a particular value for a smoothing parameter, or to specify the approximate number of parameters (degree of freedom or span) for a smoother (smoothing spline or LOESS). The data then smoothed to estimate the regression curve. A large degree of freedom results in a rough curve. In case of LOESS, the large values of span produce smoother curves. By default, the optimal smoothness for the models (smoothing spline or LOESS) is estimated based on mean square error using generalized cross-validation (GCV). GCV, which is a weighted version of cross validation, developed in Wahba’s article,13) has been identified as one of the most successful computational methods for estimating the smoothing parameter.14, 15) This estimation is carried out for a number of different values of the

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smoothing parameter, and the value that minimizes the estimated mean square error is selected (Hastie and Tibshirani, 1990). However, blind use of default methods can be problematic, as exemplified in Dominici’s article, 2002.16) The main goal of this study was to select the most appropriate method to model the relation between temperature and mortality in Japan and to obtain the OT level corresponding to the minimum mortality. Hence, the primary objective was to determine the optimal degree of smoothing for selected nonparametric models, i.e., smoothing spline and LOESS. Rather than using the GCV method, we used an “a priori” approach based on an a priori knowledge, which assumes that we a priori know that the relation between temperature and mortality is V-shaped, i.e., between two extreme temperatures a function should take an absolute minimum of a mortality rate at a certain temperature (optimum temperature = OT) with no local extremes. In other words, the slopes of the curve below and above the OT should be monotonous. The secondary objective was to examine the performance of the GCV method for determining smoothness and to compare it to the “a priori” method. For our further analysis we will use the results of this study to obtain OT change over time, in order to investigate the relation between OT and climate change and estimate the effect of adaptation to a certain climate in Japan.

MATERIALS AND METHODS The daily mortality data for 1972–1994 period for 47 prefectures of Japan (1973–1994 for Okinawa), which include gender, date of birth and area codes, were provided by the Ministry of Health and Welfare of Japan. We restricted these mortality data by age of persons 65 years and older, because this group of people is known to be the most vulnerable to heat.17, 18) The corresponding population data were provided by the Prime Minister’s Office of Japan. These data were stratified by gender, age, year and area code. The meteorological data for the capital cities were provided by Japan Meteorological Agency. These data included daily maximum, mean and minimum temperatures reported in degrees Centigrade and relative humidity in each capital city. When the capital cities did not have an

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observatory station, we used the data of the stations in the closest cities. The prefecture-specific mortality rates were calculated using the person-time method presented in Rothman and Greenland.19) In brief, for one prefecture, the number of decedents (males or females) aged 65 or older were divided by their corresponding person-days (the risk set population size multiplied by 1 day, which is considered to be approximately the same as the actual person-days), and, for convenience, the mortality rates were multiplied by 100000000. Twenty-three-year-average of daily mean temperatures was used as an average temperature for each prefecture. Data Analysis —— For our analysis we used R statistical software, version 2.1.1,20) available through the Comprehensive R Archive Network (CRAN) mirror site (http://cran.r-project.org/). Among the many different nonparametric curve-fitting estimates that are provided by the software, we selected smoothing spline and LOESS, which are well known as better tools to capture complex behavior than polynomials, in contrast to the parametric regression. We assumed that we have a 47 × n data set matrix, in which the component mi j represents a mortality rate for the ith prefecture on the jth day. The data were smoothed to estimate the regression curves using the following model mi j = si (temp j , k) + εi j , i = 1, . . . , 47, j = 1, . . . , n

(1)

where si (·) is a smooth function of a maximum temperature with k degree of smoothing, which is the degree of freedom (df ) for the smoothing spline

function [Appendix A] or the span for the LOESS function [Appendix B], n = 8401 days for all prefectures, except for Okinawa (n = 8035), and εi j ∼ N (0, σ2i ). Firstly, we specified the optimal degree of smoothing (ds) for the smoothers using “a priori” approach. We used a wide range of the ds for each model. For the smoothing splines we examined degrees of freedom from 2 df to 20 df (with the step size of 0.1). Within the range the degrees of freedom yielded curves from too rough to too smooth for all prefectures. Similarly for the LOESS, we searched the spans (degrees of smoothness) ranged from 1.0 to 0.1 with the step size of 0.01. Then, we exhaustively fitted all 47 datasets to obtain all the best fits of the relation between daily maximum temperature and sex-specific mortality rate for each prefecture. After observing all the graphs, we concluded that the step sizes of 0.5 df and 0.05 span were sufficient. Following this, we selected the ranges of ds for the smoothers, which yielded OT. When there was no OT, as exemplified by Figs. 1, 2, we did not assign it. Since some ds yielded OT for some prefectures but did not yield OT for the other prefectures, we selected the “best” range of ds such that all the prefectures (except for those with no OT for any of the ds) had OT as long as ds was within the “best” range. Secondly, to illustrate the performance of GCV method and to compare it to the “a priori” approach, we plotted OTs obtained by default and by the “a priori” methods against the long-term average temperatures (AVT) for all prefectures. For the “a priori” plots, we used the lower limit of the “best” ranges of ds and, only for comparison, retained OT candidates that were actually non-OTs, i.e., the tem-

Fig. 1. Difference in Temperature-mortality Relations for Hokkaido (Left, V-shaped) and Hiroshima (Right, Non-V-shaped) Prefectures, Males, 1972–1994. Results were Obtained Using Smoothing Splines with Degrees of Freedom, Selected by GCV Method. mr all: all-cause mortality rate (day−1 ×100000000); MXT: daily maximum temperature (◦ C).

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Fig. 2. Example of a Non-V-shape Temperature-mortality Relation Obtained Using Smoothing Spline Model, with the Range of Degrees of Freedom from 2 to 20 for Hiroshima Prefecture, Males, 1972–1994.

peratures that yielded minimum mortality rates, but were not between the two extreme temperatures, or the slopes were not monotonous. In addition, in order to illustrate the performance of two nonparametric models fitted to our data, we plotted OTs obtained with the smoothing splines against OTs obtained with the LOESS model using the “best” ds.

RESULTS Shape of the Relation between Temperature and Mortality in Japan For most prefectures, the relation between daily

maximum temperature and mortality rates was Vshaped (as shown in the left panel of Fig. 1) with some exceptions, which were Yamanashi, Nagano, Hiroshima, Saga, Kumamoto and Okinawa (the right panel of Fig. 1). For southern prefectures OTs tended to be higher (Table 1). Optimal ds for Selected Nonparametric Models Table 1 represents GCV and “a priori” selected ds for smoothing splines and LOESS models for males and females of 47 prefectures in Japan. With GCV we obtained the range of ds for smoothing splines from 5 df to 12 df for males and from 4 df to 17 df for females, and the span for LOESS, which was 0.67 for all prefectures. However, although

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Table 1. Degrees of Smoothing and the Corresponding OTs Determined by GCV and “a Priori” Methods for Smoothing Splines and LOESS Models for 47 Prefectures in Japan

Prefecture name Hokkaido Aomori Iwate Miyagi Akita Yamagata Fukushima Ibaraki Tochigi Gunma Saitama Chiba Tokyo Kanagawa Niigata Toyama Ishikawa Fukui Yamanashi Nagano Gifu Shizuoka Aichi Mie Shiga Kyoto Osaka Hyogo Nara Wakayama Tottori Shimane Okayama Hiroshima Yamaguchi Tokushima Kagawa Ehime Kochi Fukuoka Saga Nagasaki Kumamoto Oita Miyazaki Kagoshima Okinawa

GCVa) , ss males females df OT df OT 7.0 9.3 8.3 6.9 8.3 7.7 5.6 6.0 5.8 9.1 6.9 9.1 12.4 6.7 7.2 5.7 6.1 5.4 4.9 5.7 5.9 8.7 8.9 6.2 6.4 6.9 10.9 6.6 7.2 5.7 5.7 5.4 6.2 11.2 6.3 4.9 5.4 6.6 5.5 7.7 4.7 5.7 4.9 7.3 4.8 8.5 5.5

24.3 25.1 27.0 26.3 27.0 28.9 30.0 27.3 28.1 29.5 28.4 26.8 28.8 27.8 26.9 26.5 28.2 29.3 33.3 38.7 29.8 28.4 29.5 29.0 28.8 30.1 29.7 29.2 28.3 29.2 28.8 29.2 29.2 38.7 29.6 38.4 29.2 28.7 30.7 30.2 39.6 37.7 38.8 29.3 37.6 31.0 34.9

9.5 8.4 5.8 16.4 7.2 11.0 8.1 6.4 13.6 12.1 7.9 6.5 10.9 7.1 8.5 9.1 5.3 6.3 5.0 7.3 7.1 5.7 9.3 8.1 7.8 14.1 7.4 6.2 5.9 6.4 7.6 5.3 5.5 13.1 5.5 4.4 5.9 11.6 5.5 12.4 17.5 6.6 5.1 7.1 7.1 8.3 5.8

23.6 24.3 24.2 25.7 25.5 29.2 28.8 26.4 27.9 28.3 27.9 27.2 27.4 26.6 27.2 25.6 27.0 27.8 39.8 27.4 28.8 27.1 29.5 27.5 26.9 30.0 28.4 27.7 27.3 28.0 28.3 28.6 29.9 38.7 28.5 30.6 27.3 28.9 29.8 37.7 39.6 28.8 38.8 27.7 30.1 30.3 31.6

GCV, ls males females span OT span OT (%) (%) 67 36.2 67 26.7 67 28.6 67 23.3 67 36.6 67 31.2 67 36.2 67 24.2 67 38.2 67 30.3 67 38.9 67 31.7 67 38.8 67 38.8 67 36.4 67 25.8 67 36.9 67 32.3 67 39.1 67 26.3 67 28.0 67 26.4 67 37.6 67 37.6 67 31.3 67 26.6 67 36.3 67 26.8 67 31.9 67 30.3 67 30.8 67 23.8 67 38.0 67 33.1 67 37.1 67 32.3 67 39.8 67 39.8 67 38.7 67 38.7 67 39.7 67 33.1 67 38.4 67 33.2 67 39.8 67 28.4 67 39.5 67 27.0 67 36.0 67 31.4 67 39.8 67 34.2 67 34.5 67 28.3 67 38.8 67 31.4 67 39.3 67 32.0 67 38.1 67 31.8 67 39.1 67 39.1 67 38.5 67 38.5 67 39.3 67 39.3 67 38.7 67 38.7 67 38.4 67 38.4 67 38.4 67 38.4 67 38.2 67 28.0 67 37.0 67 37.0 67 37.3 67 37.3 67 37.7 67 37.7 67 39.6 67 39.6 67 37.7 67 37.7 67 38.8 67 38.8 67 36.7 67 32.2 67 37.6 67 37.6 67 36.7 67 36.7 67 34.9 67 34.9

males df 3.5 – 7.0 3.5 – 8.5 4.0 – 7.5 3.5 – 12.5 4.0 – 8.5 4.0 – 8.0 4.5 – 11.0 4.0 – 10.0 4.0 – 8.0 3.5 – 13.5 3.5 – 10.5 4.5 – 7.5 3.5 – 13.5 3.5 – 14.5 3.5 – 14.5 3.5 – 8.0 3.5 – 12.5 4.5 – 7.5 4.5 – 8.0 X 4.0 – 15.5 4.0 – 11.0 3.5 – 9.0 4.0 – 9.5 4.5 – 10.0 3.5 – 10.0 3.5 – 10.5 4.0 – 9.0 4.0 – 9.5 4.0 – 11.5 4.0 – 7.5 4.5 – 12.5 4.0 – 9.5 X 5.0 – 8.5 5.5 – 6.0 4.0 – 11.0 4.0 – 10.5 4.5 – 11.0 5.0 – 11.5 5.0 – 9.5 6.0 – 14.0 X 4.0 – 12.5 5.5 – 10.0 4.5 – 20.0 X

“a priorib )”, ss females avrOT df avrOT 25.0 25.2 27.3 26.7 27.0 29.1 30.2 27.5 28.3 29.4 28.5 27.4 28.3 28.5 27.3 26.8 28.6 29.2 33.2 — 29.9 28.7 29.5 29.3 28.8 30.5 29.5 29.6 28.6 29.7 29.1 28.5 29.4 — 29.6 29.9 28.9 28.9 30.1 30.3 31.8 31.9 — 29.4 30.9 31.2 —

3.0 – 5.5 3.0 – 9.5 3.5 – 11.5 3.5 – 9.5 3.5 – 10.0 3.5 – 12.0 3.5 – 9.5 3.5 – 7.5 3.5 – 8.5 3.0 – 9.0 3.0 – 10.0 3.5 – 12.0 3.0 – 9.5 3.5 – 8.5 3.0 – 12.0 3.0 – 10.0 3.5 – 9.5 3.5 – 11.5 Xc) 4.0 – 9.5 3.5 – 14.0 3.5 – 6.5 3.0 – 10.5 3.0 – 10.0 3.5 – 11.0 3.5 – 13.0 3.0 – 11.5 3.5 – 15.0 3.5 – 8.0 3.5 – 12.5 5.0 – 6.0 4.0 – 6.5 4.0 – 12.5 X 4.0 – 9.0 4.0 – 8.5 3.0 – 12.0 4.0 – 7.5 4.5 – 9.5 4.5 – 7.0 X 4.5 – 7.5 X 3.5 – 13.5 4.0 – 17.5 4.5 – 15.0 5.5 – 10.0

23.0 23.8 23.9 25.4 25.6 28.2 28.1 26.5 27.1 27.8 27.2 27.4 26.5 26.5 27.3 25.4 27.2 28.0 — 27.7 28.7 27.6 28.8 27.6 26.9 29.6 28.4 28.3 27.5 28.3 29.3 28.9 29.9 — 28.7 29.5 27.5 28.8 29.1 29.3 — 29.2 — 27.7 30.2 30.5 31.6

“a priori”, ls males females span (%) avrOT span (%) avrOT 60 – 25 70 – 30 50 – 25 65 – 25 55 – 25 50 – 20 45 – 20 55 – 15 50 – 20 60 – 20 70 – 20 55 – 20 70 – 20 55 – 20 65 – 20 70 – 25 60 – 25 50 – 20 40 – 25 40 – 30 55 – 20 55 – 15 55 – 20 55 – 20 45 – 20 55 – 20 60 – 20 50 – 20 55 – 20 50 – 20 55 – 25 45 – 20 50 – 20 X 45 – 25 45 – 25 55 – 20 55 – 15 45 – 25 40 – 15 35 – 20 30 – 20 X 50 – 15 30 – 25 45 – 15 X

24.7 25.2 27.1 26.0 27.5 29.2 29.9 26.9 28.7 29.1 27.9 27.7 27.9 27.8 26.8 26.4 28.4 28.3 30.1 27.3 29.6 28.9 29.2 29.3 28.0 30.2 29.0 29.0 28.5 29.6 28.5 28.2 29.2 — 28.9 29.0 28.7 28.8 31.0 29.7 31.4 31.8 — 29.1 30.4 31.0 —

70 – 30 85 – 25 60 – 25 70 – 35 65 – 20 65 – 20 60 – 30 70 – 20 60 – 20 75 – 35 85 – 30 60 – 15 80 – 20 70 – 25 70 – 15 85 – 20 60 – 15 65 – 25 40 – 35 55 – 20 65 – 20 60 – 25 70 – 15 75 – 20 60 – 20 60 – 20 75 – 20 70 – 20 70 – 20 70 – 20 50 – 30 55 – 25 45 – 20 X 55 – 25 45 – 25 70 – 20 55 – 25 50 – 20 50 – 25 X 50 – 25 X 65 – 20 50 – 20 55 – 15 50 – 30

22.8 24.4 24.3 24.9 25.6 27.2 27.1 25.5 26.4 27.1 26.8 27.5 27.2 26.7 26.9 25.4 27.2 27.9 29.2 27.5 28.8 27.0 28.8 27.3 26.6 28.9 28.5 28.2 28.0 28.5 28.3 28.4 29.3 — 28.0 28.9 27.4 28.7 29.2 29.3 — 28.8 — 27.3 30.2 30.4 31.1

a) GCV : generalized cross-validation method; b) “a priori” based method to select a range of the degrees of smoothing; ss : smoothing splines; ls : LOESS; avrOT : average value of the optimal temperature within the range; c) X : non-V-shaped relation; bold-italic font : non-OT by “a priori” approach.

we obtained OTs by default method for all prefectures, some of the computed OT candidates (temperatures with the lowest mortality rates) were not considered to be OT, since the curves were non-Vshaped, based on a priori knowledge. The OT candidates captured by the GCV method, but non-OTs by a priori method, were shown only for comparison with the “a priori” approach (Table 1 bold-italic

font). All the “a priori” selected OTs were obtained from the V-shaped regression curves, estimated by smoothing the data with ds varied among prefectures from 3 df to 20 df for smoothing splines, and from 0.80 span to 0.15 span for LOESS. In fact, there were only two prefectures, Hiroshima and Kumamoto, for which we did not assign any OT when we used the “a priori” approach with

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Table 2. “A priori” Selected “best” Ranges of Degrees of Smoothing, Within Which the Minimum Number of Non-OTs was Observed

Males

Females

Smoothing Splines “best” range of ds non-OTs 6.0 – 7.5 df Nagano Hiroshima Kumamoto Okinawa 5.0 – 8.0 df Yamanashi Hiroshima Saga Kumamoto

both, smoothing splines and LOESS, models. There was one more prefecture without minimum mortality rate between two extreme temperatures for males, Nagano, was for the smoothing splines, and one more for females, Yamanashi, for the LOESS. In addition, for both models, there was one more non-OT for males and one more for females, Okinawa and Saga, respectively (Table 2). With an exhaustive search based on the a priori knowledge, we determined the ranges of smoothing for all prefectures within which less non-OTs were obtained. Following this, with smoothing splines, the degree of freedom ranged from 6.0 to 7.5 df for males and from 5.0 to 8.0 df for females. With LOESS, the span ranged from 0.35 to 0.25 for males and from 0.40 to 0.25 for females. Thus, we regarded the range of df from 6 to 7.5 as the “best” range for a smoothing spline model for both males and females, and the range of span from 0.35 to 0.25 as the “best” range for LOESS. Table 3 represents OT candidates and their maximum differences within the “best” ranges. It shows that an average difference within the range for the smoothing spline model was 0.3◦ C for both males and females, and for the LOESS model it was 0.5◦ C for males and females. GCV Performance by the “a Priori” Approach Figure 3 represents the plots of OTs for males using smoothing spline (top two panels) and LOESS (bottom two panels) models with ds selected by the GCV (left) and “a priori” (right) methods. With GCV method, for smoothing splines there were eight non-OTs, which were Nagano, Hiroshima, Tokushima, Saga, Nagasaki, Kumamoto, Miyazaki and Okinawa. The results of using LOESS yielded only six prefectures with well pronounced V-shape relations, which were Aomori, Saitama, Tokyo,

LOESS “best” range of ds non-OTs 0.35 – 0.25 spans Hiroshima Kumamoto Okinawa 0.40 – 0.25 spans

Hiroshima Saga Kumamoto

Niigata, Toyama and Osaka. With the “a priori” approach, smoothing spline model with 6 df and LOESS model with 0.35 span showed similar results with only four non-OTs for smoothing splines, Nagano, Hiroshima, Kumamoto and Okinawa, and three non-OTs for LOESS, Hiroshima, Kumamoto and Okinawa. Difference in OT for Males and Females OTs for males and females obtained with the “a priori” approach were similar within the “best” range for both models. We used a smoothing spline model with 6 df to illustrate a shift in OTs for males and females (Fig. 4). The shift, a, was estimated as an average difference between OTs for males and OTs for females: 47
(OT i(males) − OT i(females) )

a=

i=1

47

The results show that females have on average 1.3◦ C lower OT levels than males, which is consistent with the previous study by Honda et al. (1998).21) Difference in OT between Smoothing Spline and LOESS Models In order to investigate the difference in OTs, regardless of the model used, we plotted OTs obtained with smoothing splines with degrees of freedom from 6 to 7.5 df against OTs obtained with LOESS with spans from 0.35 to 0.25. Among the plots we examined, the plot of OTs with 6 df vs. OTs with 0.30 span showed the relation that was closest to linear (Fig. 5). Also, there was a tendency that smoother curves yielded higher OTs.

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Table 3. OT Levels for Smoothing Spline (ss) and LOESS (ls) Models by the Range of Degrees of Smoothing and their Maximum Differences (max ∆) within the Range Prefecture Hokkaido Aomori Iwate Miyagi Akita Yamagata Fukushima Ibaraki Tochigi Gunma Saitama Chiba Tokyo Kanagawa Niigata Toyama Ishikawa Fukui Yamanashi Nagano Gifu Shizuoka Aichi Mie Shiga Kyoto Osaka Hyogo Nara Wakayama Tottori Shimane Okayama Hiroshima Yamaguchi Tokushima Kagawa Ehime Kochi Fukuoka Saga Nagasaki Kumamoto Oita Miyazaki Kagoshima Okinawa AVR max∆

6 24.3 25.0 26.9 26 26.8 28.7 29.9 27.3 28.0 29.1 28.2 27.2 27.7 27.5 26.9 26.4 28.2 29.0 32.7 X 29.8 28.6 29.2 29.0 28.9 30.0 29.2 29.3 28.6 29.2 28.8 28.9 29.3 X 29.7 29.8 29.0 28.8 30.3 30.7 32.0 33.7 X 29.5 31.7 31.2 X

ss (males), df 6.5 7 7.5 24.3 24.3 24.3 25.0 25.1 25.1 26.9 26.9 26.9 26.2 26.3 26.5 26.8 26.9 26.9 28.7 28.8 28.9 29.9 30.0 30.1 27.3 27.4 27.4 28.0 27.9 27.9 29.2 29.3 29.4 28.3 28.5 28.6 27.1 27.0 26.9 27.9 28.0 28.2 27.7 27.8 28.0 26.9 26.9 27.0 26.4 26.3 26.2 28.3 28.3 28.4 28.9 28.8 28.7 32.8 33.0 33.1 X X X 29.8 29.8 29.8 28.5 28.5 28.4 29.2 29.3 29.4 29.0 29.1 29.1 28.8 28.6 28.6 30.1 30.2 30.3 29.3 29.4 29.5 29.2 29.2 29.2 28.5 28.3 28.2 29.3 29.4 29.5 28.9 29.0 29.2 28.7 28.5 28.4 29.2 29.1 29.0 X X X 29.5 29.3 29.2 29.5 29.4 29.3 28.9 28.8 28.7 28.7 28.6 28.6 30.1 30.0 29.9 30.5 30.3 30.2 31.8 31.6 31.5 32.8 32.4 32.2 X X X 29.4 29.4 29.3 31.2 30.9 30.7 31.1 31.0 31.0 X X X

max∆ 0 0.1 0 0.5 0.1 0.2 0.2 0.1 0.1 0.3 0.4 0.3 0.5 0.5 0.1 0.2 0.2 0.3 0.4 — 0 0.2 0.2 0.1 0.3 0.3 0.3 0.1 0.4 0.3 0.4 0.5 0.3 — 0.5 0.5 0.3 0.2 0.4 0.5 0.5 1.5 — 0.2 1 0.2 — 0.3

6 22.8 23.5 24.2 25.1 25.3 27.4 27.1 26.3 26.8 27.6 26.6 27.2 26.2 26.3 26.7 25.2 27.0 27.8 Xa) 27.5 28.6 27.1 28.4 27.3 26.6 29.2 28.0 27.6 27.3 28.0 28.9 28.4 29.8 X 28.4 29.3 27.3 28.6 29.4 29.1 X 29.0 X 27.7 30.1 30.3 31.5

ss (females), df 6.5 7 7.5 22.9 23.1 23.3 23.7 23.9 24.0 24.1 24.0 23.8 25.4 25.6 25.7 25.4 25.5 25.6 27.7 28.0 28.2 27.3 27.7 28.3 26.5 26.7 26.9 26.9 27.0 27.1 27.9 28.0 28.2 27.0 27.4 27.7 27.2 27.2 27.2 26.4 26.6 26.7 26.4 26.6 26.7 26.8 26.9 27.0 25.3 25.4 25.4 27.0 27.0 27.1 27.8 27.9 28.0 X X X 27.5 27.4 27.4 28.7 28.8 28.8 27.0 27.0 27.1 28.6 28.8 29.0 27.4 27.5 27.5 26.7 26.8 26.9 29.3 29.4 29.6 28.1 28.3 28.4 27.7 27.8 27.9 27.3 27.3 27.3 28.0 28.1 28.1 28.6 28.5 28.4 28.4 28.5 28.5 29.7 29.6 29.6 X X X 28.4 28.4 28.5 29.1 29.1 29.0 27.3 27.3 27.4 28.6 28.7 28.7 29.1 28.9 28.7 29.0 29.0 28.9 X X X 28.8 28.8 28.7 X X X 27.7 27.7 27.7 30.1 30.1 30.1 30.3 30.3 30.3 31.3 31.3 31.4

max∆ 0.5 0.5 0.4 0.6 0.3 0.8 1.2 0.6 0.3 0.6 1.1 0 0.5 0.4 0.3 0.2 0.1 0.2 — 0.1 0.2 0.1 0.6 0.2 0.3 0.4 0.4 0.3 0 0.1 0.5 0.1 0.2 — 0.1 0.3 0.1 0.1 0.7 0.2 — 0.3 — 0 0 0 0.2 0.3

0.35 23.8 24.2 26.8 26.6 26.8 29.0 29.3 26.5 28.4 28.7 28.0 27.0 27.3 27.5 26.2 25.5 28.3 28.3 30.8 27.2 29.8 28.1 29.2 28.6 28.4 30.1 29.0 29.0 28.8 29.4 28.8 27.7 29.1 X 28.9 28.7 28.0 28.4 30.7 29.7 32.1 33.7 X 29.1 34.1 30.8 X

ls (males), span 0.30 0.25 23.8 23.8 25.1 25.1 26.8 26.8 26.6 26.6 26.8 26.8 29.0 29.0 29.3 29.3 27.3 27.3 28.4 28.4 28.7 29.5 28.0 28.0 27.0 27.0 28.1 28.1 28.2 28.2 26.2 27.0 25.5 25.5 29.1 29.1 28.3 28.3 30.8 27.6 27.2 27.2 29.0 29.0 28.1 28.8 29.2 29.2 28.6 29.3 27.6 27.6 30.1 30.1 29.0 29.8 29.0 29.0 28.0 28.0 29.4 30.2 28.8 29.6 27.7 27.7 29.1 29.1 X X 28.9 28.9 28.7 28.7 28.0 28.0 28.4 28.4 30.7 31.5 29.7 29.7 31.3 31.3 33.7 31.4 X X 29.1 29.1 32.0 29.9 30.8 30.8 X X

max∆ 0 0.9 0 0 0 0 0 0.8 0 0.8 0 0 0.8 0.7 0.8 0 0.8 0 3.2 0 0.8 0.7 0 0.7 0.8 0 0.8 0 0.8 0.8 0.8 0 0 — 0 0 0 0 0.8 0 0.8 2.3 — 0 4.2 0 — 0.5

0.35 21.9 24.2 23.2 25.0 25.0 27.3 26.7 25.0 26.8 27.9 26.4 27.0 27.3 26.8 26.2 24.7 27.5 28.3 29.2 27.2 28.2 27.3 29.2 27.0 26.9 29.3 28.3 28.1 28.0 27.9 27.9 27.7 29.1 X 28.1 28.7 27.3 28.4 29.3 28.9 X 28.2 X 27.6 29.9 30.1 31.0

ls (females), span 0.30 0.25 21.9 21.9 25.1 25.1 23.2 22.3 25.8 25.8 25.9 25.9 28.1 28.1 28.5 29.3 25.8 25.8 26.8 26.8 27.9 27.9 27.2 28.0 27.0 27.0 27.3 28.1 26.8 26.8 27.0 27.0 25.5 25.5 27.5 27.5 28.3 28.3 30.0 30.0 27.2 27.2 28.2 28.2 27.3 27.3 29.2 29.2 27.0 27.0 26.9 27.6 29.3 29.3 29.0 29.0 28.1 29.0 28.0 28.0 28.6 28.6 27.9 27.9 28.6 29.4 29.1 29.1 X X 28.1 28.9 29.5 29.5 27.3 27.3 28.4 29.2 29.3 27.8 28.9 28.9 X X 28.2 29.0 X X 27.6 26.9 29.9 30.6 30.1 30.1 30.0 30.0

max∆ 0 0.9 0 0.8 0.9 0.8 2.6 0.8 0 0 1.6 0 0.8 0 0.8 0.8 0 0 0.8 0 0 0 0 0 0.7 0 0.7 0.9 0 0.7 0 1.7 0 — 0.8 0.8 0 0.8 1.5 0 — 0.8 — 0.7 0.7 0 1.0 0.5

ss : smoothing splines; ls : LOESS; max∆ : maximum difference within a given range of the degrees of smoothing; a) X : non-V-shaped relations.

DISCUSSION Refinement of the model by incorporating as many confounders as possible in it is important. However, as noted in the INTRODUCTION section, our intention is to assess the impact of global warming in the world, including developing countries. In developing countries, it is rarely possible to obtain accurate mortality data. Even when vital statistics are available, it is usually unreliable in terms of age

and cause of death. Thus, we considered gender, only, as a confounder. Our results showed that, in relation to temperature, the use of smoothing spline model with the a priori selected degrees of freedom ranged from 6 df to 7.5 df is the most appropriate method for analyzing the Japanese data. An average difference in OT levels within the range of spans (0.35–0.25) was 0.5◦ C for the LOESS, while for the smoothing splines it was 0.3◦ C (Table 3), which showed

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Fig. 3. Plots of OT Levels for Males vs. Long-term Average Temperatures for 47 Prefectures of Japan Obtained with the Smoothing Spline and LOESS Models Using GCV and “a Priori” Approaches The circles surrounded by the dashed lines are non-OTs, determined by the “a priori”, i.e., the temperatures with minimum mortality rates represent the last values of the curves. The non-OTs for the smoothing spline model were Nagano, Hiroshima, Kumamoto and Okinawa, and the non-OTs for the LOESS model were Hiroshima, Kumamoto and Okinawa. Instead of exclusion the prefectures from the analysis we plotted the non-OTs for comparison with the GCV method, with which all prefectures were included in the analysis.

Fig. 4. Plot of OTs for Males against OTs for Females Obtained with Smoothing Spline (df = 6). Prefectures without OTs were excluded, which where Yamanashi, Nagano, Hiroshima, Saga, Kumamoto, Okinawa.

that LOESS model was more sensitive to the degree of smoothing than the smoothing splines. In either model, however, the difference was not large, compared with the OT difference among the prefectures (the range of OT is about 5◦ C). Thus, as long as a fixed degree of smoothing is used within an analysis, the choice of the degree of smoothing may not substantially affect the results of further analyses on the relation between OT and climate change in the world, in which we evaluate the relation between OT and climatic indices. The important implication of this study is that, by the exhaustive search, we found some OTs which were not identified by GCV method. For instance, for Chiba prefecture, in Table 1 with LOESS model exhaustive search has identified OTs (with extreme temperatures on both sides), whereas GCV yielded a degree of smoothing, by which a non-OT was captured. This implies that even careful researchers who eliminate OT candidates because they are not

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Fig. 5. Plots of OTs Obtained with LOESS (Span = 0.30) Against OTs Obtained with Smoothing Spline (df = 6). Prefectures without OTs were excluded: males — Nagano, Hiroshima, Kumamoto, Okinawa; females — Yamanashi, Hiroshima, Saga, Kumamoto.

between the two extreme temperatures may miss some of the OTs when they use GCV method in drawing curves. Judging from Fig. 3, it seems that the problem is very serious in the case of LOESS. Unlike LOESS model, smoothing spline yielded OTs in most of the cases. Hence, as far as the dataset used for the analysis is concerned, blind use of GCV method may not be very problematic in the case of smoothing spline. Another problem in using GCV method is as follows: By the GCV method we obtained more non-OTs than by the “a priori” method we concluded that GCV method was not suitable for our study and, hence, cannot be used for our further analyses. The sex-specific analysis of the relation between temperature and mortality showed a 1.3◦ C shift in OTs to colder temperatures for females, which was consistent with a previous study.21) In Honda and coworkers’ report, daily maximum temperature was categorized by a 5◦ C interval, and the difference was not directly calculated as temperature difference. The relation between temperature and mortality showed a V-shape with identified OTs for most prefectures. However, there were prefectures with no OTs even when we exhaustively examined the degrees of smoothing for the selected nonparametric smoothers. This phenomenon is puzzling. One possible clue is that most of these prefectures are located in the southern region of Japan, but to explore the mechanism of this phenomenon is beyond the scope of this paper and will be analyzed in a future study. In conclusion, although a further validation study is necessary, this nonparametric regression approach may be a better alternative to widely cate-

gorized analyses. Acknowledgements This study was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Global Research Fund S-4 by the Ministry of Environment, Japan. Appendix A. Estimation Using Smoothing Splines The method described in Hastie and Tibshirani22) obtains the smoother estimates of the true but unknown function among all functions f (x) with two continuous derivatives by minimizing the penalized residual sum of squares RSS( f, λ) =

 N
{yi − f (xi )}2 + λ j=1

xn

f  (x)2 dx,

x1

(2) a criterion that consists of the first term that measures the closeness to the data and the second term that penalizes curvature in the function. Here, λ is a fixed smoothing parameter (constant) with respect to the unknown regression function f (x) that is found on the basis of the data (xi , yi ) and (xi ≤ · · · ≤ xn ) represent the knots at the values of xi . The rate of change of the slope of the function f is given by f  . In other words, λ (which must be positive) establishes a trade-off between the closeness of fit and the roughness of penalty. If λ = 0, f is the function that interpolates the data; if λ → ∞, the second derivative is constrained to 0, and thus we have a constant slope. It has been shown that Eq. (2) has an explicit, unique minimizer and that the minimizer is a natural cubic spline with knots of the unique values of xi . The approximate number of parameters for the smoothing spline is specified by degrees of freedom

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dfλ = trace(S λ ). Since dfλ is monotonous in λ, we can invert the relationship and specify λ by fixing the degrees of freedom. A large df results in a small penalty λ, resulting in a jagged curve. The function smooth.spline in the splines (stats) package of R software was used to fit the smoothing splines. By default, this function uses GCV method to choose λ. Since dfλ = trace(S λ ) we can specify either λ directly, or invert the relationship and specify the degrees of freedom instead. The latter method is much easier and somewhat more intuitive. Appendix B. Estimation Using Locally Weighted Regression (LOESS) The locally-weighted regression smoother was introduced by Cleveland.23) This method locally (in neighbourhoods) adjusts polynomials of low degree, whose smoother estimates are obtained by the regression of mortality rates over maximum temperatures by means of weighted least squares in each neighbourhood N(x). It can be computed in a number of steps:23) (i) Identification of the h nearest neighbours of focal x0 , which can be denoted as nearest neighbourhood or window width N(x). The parameter h controls the appearance of the running-line smooth. Large values of h tend to produce smoother curves while small values tend to produce more jagged curves. The bandwidth h can either be fixed or it can vary as a function of the focal x0 . It is more convenient to think not in terms of h but instead in terms of the proportion of points in each neighbour2h + 1 that is called span. The size of s hood s = n has an important effect on the curve. Once s is chosen, each local neighbourhood around the focal x0 always contains a specified proportion of observations. In other words, span allows you to adjust the smoothness of the results. (ii) Assignment of the weights ωi to each point in N(x), using the Tricube Weight Function (WT ), which gives the greatest weight to observations that are closest to the focal x0 observation: WT (z) = (1− |z|3 )3 , for 0 ≤ z < 1 or WT (z) = 0 otherwise. zi = xi − x0 is a distance between the predictor xi and h the focal x0 . (iii) The Weighted Least-Squares Regression fits the equation yi = a + b1 (xi − x0 ) + b2 (xi − x0 )2 + · · · +b p (xi − x0 ) p + εi to minimize the weighted residual sum of squares

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x − x  i 0 is a local kerh i=1 nel weight. Selecting p = 1 produces a local linear fit that promises reduced bias, that is apparent at the boundaries. The values p = 2 or p = 3, local quadratic or cubic fits, produce more flexible regressions. Greater flexibility has the potential to reduce bias further, but flexibility also entails the cost of greater variation. (iv) The fitted values at focal x0 from the weighted least-squares fit of y to x confined to N(x) using the weights computed in (ii), (iii) carried out for each observation in the data are connected, producing the local polynomial nonparametric regression curve. N 

ωi ε2i , where ωi = K

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