There has been considerable debate on the existence of trends in climate in the highlands of East Africa and hypotheses about their potential effect on the trends in malaria in the region. We apply a new robust trend test to mean temperature time series data from three editions of the University of East Anglia's Climatic Research Unit database (CRU TS) for several relevant locations. We find significant trends in the data extracted from newer editions of the database but not in the older version for periods ending in 1996. The trends in the newer data are even more significant when post-1996 data are added to the samples. We also test for trends in the data from the Kericho meteorological station prepared by Omumbo et al. We find no significant trend in the 1979-1995 period but a highly significant trend in the full 1979-2009 sample. However, although the malaria cases observed at Kericho, Kenya rose during a period of resurgent epidemics (1994-2002) they have since returned to a low level. A large assembly of parasite rate surveys from the region, stratified by altitude, show that this decrease in malaria prevalence is not limited to Kericho.
Citation: Stern DI, Gething PW, Kabaria CW, Temperley WH, Noor AM, Okiro EA, et al. (2011) Temperature and Malaria Trends in Highland East Africa. PLoS ONE6(9): e24524. https://doi.org/10.1371/journal.pone.0024524
Editor: Matthew Baylis, University of Liverpool, United Kingdom
Received: April 17, 2011; Accepted: August 12, 2011; Published: September 15, 2011
Copyright: © 2011 Stern et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Wellcome Trust. SH is funded by a Senior Research Fellowship from the Wellcome Trust (# 079091) which also supports CK and PG. EO is supported by the Wellcome Trust as a Research Training Fellow (# 086166). AN is supported by the Wellcome Trust as a Research Training Fellow (# 081829). RS is supported by the Wellcome Trust as Principal Research Fellow (# 079080). This work forms part of the output of the Malaria Atlas Project (http://www.map.ox.ac.uk), principally funded by the Wellcome Trust, United Kingdom. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Controversy over the cause of malaria resurgences reported in the late 1990s in some areas of highland East Africa continues despite reports of an overall global reduction in prevalence of the disease , marked declines across many well studied communities in East Africa , and at the majority of seventeen non-lakeside hospitals across Kenya . Using the database developed by the Climate Research Unit at the University of East Anglia (CRU), Hay et al.  tested for trends in the time series of a range of climate variables from various locations in East Africa where increases in the prevalence of malaria had been indicated. They found no evidence of a significant increase in temperature at any of the locations and some evidence of an increase in rainfall at some locations. The inference was made that, as there was no significant warming in these East African locations, global warming was unlikely to be responsible for the increases in malaria admissions seen at facilities in these locations. Rather, a host of other plausible causes could have been responsible, among which the rise of anti-malarial drug resistance was viewed as most plausible . This has sparked considerable debate.
Chaves and Koenraadt  claim (incorrectly) that, because only one test - the Q test  - was used to test for serial correlation in the residuals, the procedures used by Hay et al.  were not robust. They applied three different time series methods to CRU data interpolated for Kericho and find increases in temperature in the period from 1966 to 1996. We will show in this paper that the differences between their results and our earlier work are due to differences in data rather than differences in methods. Pascual et al.  applied two time series methods to data for the four locations analyzed by Hay et al.  for the period 1950 to 2002, finding positive and significant trend components. They suggest that the differing results of Hay et al.  may stem from the differencing used in the Dickey-Fuller regression or from the method of treating seasonality with dummy variables. As the Dickey-Fuller procedure does not actually difference the data, this cannot explain the differences in findings.
By contrast, Omumbo et al.  argue that this debate is due to the inappropriate use of regionally interpolated data in place of actual observations from individual meteorological stations. Omumbo et al. show that the number of stations used to construct the CRU temperature database has fallen very strongly in the Kericho, Kenya region since the mid 1990s, calling into question the usefulness of the database for recent years. In collaboration with the Kenyan meteorological service, Omumbo et al. prepared quality-controlled series for temperature and rainfall at Kericho that takes into account the shift in the location of the weather station in 1986. For the period 1979-2009, they find that mean temperature increased by 0.21 K per decade, which they claim is significant at the 1% level.
Given the different, but formally untested, arguments made against the Hay et al.  results and the different methodologies used, it is still unclear why there is a difference between the results of these more recent studies and our earlier work. Because of the importance of this debate , we apply a uniform methodology to test for trends in the various datasets used to date and time periods previously considered in the literature to determine whether differences in data or in methods are responsible for the variant findings. We find that it is differences in data examined, not methods, that is responsible.
We apply a relatively new test for temporal trends that is robust to autocorrelation of unknown form including a possible random walk in the regression residuals , . We test the original monthly time series from the Climate Research Unit Time Series (CRU TS) 1.0 data set  used by Hay et al.  for four locations in highland East Africa, as well as the more recently published CRU TS 2.1 data  used by Chaves and Koenraadt  and Pascual et al. , and the newest data set, CRU TS 3.0. For these newer datasets we also test for trends in the regional average temperature across East Africa. Additionally, we test for trends in the data from the Kericho meteorological station prepared by Omumbo et al. .
We also apply the new trend test to various sub-samples of an updated time series of malaria cases at a tea estate hospital in Kericho, Kenya, that has formed the centerpiece of much of the debate regarding climate change and malaria in the highlands of East Africa , , . Finally, through the data collection initiatives of the Malaria Atlas Project (MAP, http://www.map.ox.ac.uk)  we are able to present information on more than 5000 geo-positioned, post-1985, malaria prevalence surveys that allow insights into the changing endemicity of malaria in several East African countries during the last 25 years and if these show any difference by altitude.
Time Series Data and Samples
We test mean monthly temperature data for the four locations (Kericho, Kenya; Kabale, Uganda; Gikongoro, Rwanda; and Muhanga, Burundi) examined by Hay et al. (2002a) (Figure 1). We also test the CRU TS 2.1 and 3.0 data for an area of East Africa defined by Patz et al  (Figure 1) to investigate whether similar results are also manifest across the region. Though much debate has focused on Kericho in Kenya, broader trends across highland East Africa should really be of interest in the debate about the role of climate change in malaria incidence. All sample sizes can be computed from the number of months in the period considered.
Sites at which monthly temperature data were analyzed are shown in red. Countries for which P. falciparum parasite rate data were analyzed are labeled. The map extent exactly matches the area defined by Patz et al.  across which CRU TS 2.1 and 3.0 data were tested: 4°S, 4°N, 28°E, 38°E.
Since the data and periods examined in the literature have become somewhat numerous, a series of tests are required to explore the reasons for differences between results obtained in the various studies. The data samples considered are: (A) the actual data (CRU TS 1.0) used by Hay et al.  for 1970:1-1995:12; (B) the newer dataset (CRU TS 2.1) used by Chaves and Koenraadt  for 1966:1-1996:12, which provides a direct test of Chaves and Koenraadt's claims using our methods; (C) these same data (CRU TS 2.1) for 1950:1-2002:12 to provide a direct test of the claims of Pascual et al.  using our methods; (D) the CRU TS 2.1 data for the period 1970:1-1995:12 as a direct test of the effect of the change in data source on Hay et al.'s  results; (E) the new dataset (CRU TS 3.0) for 1970:1-1995:12 to test if there are any differences in moving from CRU TS 2.1 to CRU TS 3.0; (F) the CRU TS 3.0 series for 1966:1-1995:12 in order to test if adding earlier years influenced the results; (G) the CRU TS 3.0 for 1966:1-2006:6 to test whether the trend in temperature persists or accelerates after 1995; (H) Omumbo et al.'s data for 1979:1-1995:12; and (I) Omumbo et al.'s data for 1979:1-2009:12.
The CRU TS 2.1 and TS 3.0 data are similar to the data we used previously for the period 1970-1995 but there are differences. In particular, the new series trend more. Figure 2 compares CRU TS 2.1 to CRU TS 1.0 and Figure 3 compares CRU TS 3.0 to CRU TS 1.0 for the Kericho location. During the 1970s, the cold season temperatures tend to be lower in the new series, while in the 1980s the warm season temperatures tend to be higher. An increase in temperature after 1995 is also apparent from both figures.
We also test for trends in a series of the monthly number of malaria cases from the Unilever Tea Kenya Ltd (UTKL) tea estate hospital in Kericho using the robust trend tests (Figure 4). We test this data for the following periods: (a) 1966:1-1995:1, the sample period covered by Hay et al.  who tested for trends in the same series using the Dickey Fuller regression and suite of test statistics; (b) 1966:1-2006:6 to correspond to the temperature sample from test “G” above and (c) 1966:1-2010:5, a period that includes the most recent malaria data.
Community Parasite Rate Surveys
As an additional indication of wider temporal patterns in malaria endemicity across East Africa, data for the region were extracted from the MAP database of community parasite rate surveys spanning the years 1985 to 2010 . A total of 2,443 such surveys were available for Kenya, 2,084 for Tanzania, and a total of 721 for Burundi, Rwanda, and Uganda which were pooled for analysis. The observed parasite rate in each survey was first standardized to the epidemiologically informative two-up-to-ten year old age group using a previously described age-adjustment algorithm , but no adjustments were made for sample size or whether the surveys refer to urban or rural areas. In each of the three groupings, temporal changes in observed parasite prevalence were summarized graphically by plotting the median and inter-quartile range of surveyed values in each year, accompanied by a moving-average line generated with the LOWESS smoother . Importantly, these summaries were calculated separately for low (<1500 m) and high (>1500 m) altitude areas.
Time Series Methods
The tDAN  or Dan-J test  is based on a modified t-test on the slope parameter of the simple linear trend regression model:(1)where t is a linear time trend, u, is a stochastic process that may or may not be stationary and and are regression parameters to be estimated. Then the trend test statistic is given by:(2)where is the estimate of the slope parameter in (1) and its standard error. b is a parameter derived by Bunzel and Vogelsang  and:(3)where RSS1 is the sum of squared residuals from (1), and RSS4 is the sum of squared residuals from the following regression:(4)
The recommended value of b and the critical values of tDAN for a two-tailed test are as follows: b = 2.466, tDAN = 2.462 at 1%; b = 1.795, tDAN = 2.052 at 2.5%; b = 1.322, tDAN = 1.710 at 5% and b = 0.965, tDAN = 1.329 at the 10% significance level. Further values can be derived from the formulae in . J can also be used in a left-tailed test of the null hypothesis that the errors in (1) contain a unit root autoregressive process or random walk. The null hypothesis is rejected for small values of the statistic. The critical values are 0.488 at 1%, 0.678 at 2.5%, and 0.908 at the 5% significance levels. We also carried out the tDAN test on deseasonalized data constructed by removing monthly means from the data.
Table 1 presents the results of the robust trend tests for the unadjusted temperature series. For the CRU TS 1.0 data used by Hay et al. , the actual temperature changes estimated by the trend component vary from 0.21 K for Kericho to 0.27 K for Gikongoro and Muhanga. None of these changes are statistically significant at the 5% level as determined by the tDAN statistics and only Muhanga and Gikongoro have a statistically significant trend at the 10% level. By contrast, the CRU TS 2.1 data used by Chaves and Koenraadt  show highly significant trends for all four sites and the East Africa region as a whole for the 1966 to 1996 period that they used. The increases in temperature over the period are much higher than we found for the 1970-1995 period using the CRU TS 1.0 series. Moreover, when we restrict the CRU TS 2.1 sample to the latter narrower window the temperature increases are still higher for the four locations, though not for East Africa as a whole. The test statistics are also highly significant when we consider the CRU TS 2.1 data for the period 1950-2002 used by Pascual et al .
Compared to the data we used previously, the CRU TS 3.0 data also show uniformly greater temperature increases and more significant t-statistics, especially for Kericho, though the changes are smaller than in the CRU TS 2.1 data. In the 1966 to 1995 period we find a 0.76 K temperature increase at Kericho, which is significant at the 1% level. The temperature increase in Kabale is significant at the 5% level and in the other two locations and across rhe region at the 1% level. Increasing the start date to 1970 results in similar but slightly steeper slope coefficients. Trends are significant at the 5% level in all four locations but at the 1% level only in Kericho and Muhanga. For 1966 to 2006, the temperature increases are larger still, ranging from 1.38 K to 1.69 K. The increase is significant at the 1% level at all locations.
All of the J statistics used to test for stochastic trends in the data are highly significant at the 1% level allowing us to reject the null hypothesis of a unit root in the residuals of the trend regression. The results for deseasonalized data are almost identical to those for the unadjusted data for most of the samples (Table 2) showing that the seasonal cycle does not have a strong influence on our findings for the CRU datasets.
Table 3 presents results for Omumbo et al.'s mean temperature series. For the unadjusted data there is no significant trend in the 1979-1995 period but there is a highly significant trend for 1979-2009. For the deseasonalized data there is a significant trend in both periods. So here, deseasonalization has an effect on the results.
Results for the Kericho hospital malaria inpatient series are shown in Table 4. The tDAN trend test finds a significant trend in malaria cases for 1966-1995 at either the 1% or 2.5% level depending on whether the data were deseasonalized or not. The slope coefficient is positive and implies an increase in monthly incidence over the period of 29 cases per month. The maximum number of monthly cases seen in this period was 269 in August 1994 with an average of 23 per month. The major upsurge in cases came towards the end of the period. The trend coefficient for 1966-2006 is about the same as that for 1966-1995 with an implied increase over the full period of 40 cases per month but is significant at the 5% level. The average number of cases over the period was 29 per month. The maximum number of monthly cases of 376 was recorded in August 1997 soon after the end of the earlier sample. Again, the deseasonalized results are very similar. Adding the four most recent years of data, however, reduces the estimated trend coefficient to 0.0436, which is only significant at the 10% level. The overall increase in malaria cases is reduced to 23 but in the last four years only July 2009 saw that many cases and the average number of cases over the last four years has been just eight.
Summary time series plots for community parasite rate surveys in Kenya, Tanzania, Burundi, Rwanda, and Uganda are shown in Figure 5. In Kenya, the moving average lines for both low and high altitude regions display an overall decreasing trend across the period 1985-2008, although in low altitude regions this decline slows or reverses slightly during the period 1995-2000. In Tanzania, the high altitude trend line displays a similar overall decreasing pattern interrupted by a phase of increasing parasite rates in the middle years of the period. For low altitude areas an increasing trend is displayed up to 1996 but this is based on very few data points. After 1996 a clear decreasing trend is displayed. For the combined Burundi, Rwanda, and Uganda region, parasite rates in both high and low altitude regions are seen to generally increase until the late 1990s/early 2000s after which a marked decrease is displayed. Although based on an opportunistic assembly of available community parasite rate surveys, and despite substantial within- and between- year variation not explained by the smoothed trends, these simple summary plots point to a consistent and substantial decline in P. falciparum prevalence since 2002 or earlier across both low and high altitude regions of East Africa. These findings suggest that the recent decreases in case incidence described above for Kericho may be indicative of wider regional trends not limited to specific high-altitude sites. Such findings are also consistent with continental and global scale changes in P. falciparum endemicity described recently across much longer time periods .
Time series plots summarizing available community P. falciparum parasite rate survey data for (top) Kenya, (middle) Tanzania and (bottom) the combined region of Burundi, Rwanda, and Uganda. Data are stratified into low and high altitude regions and summarized by the median (dots) and interquartile range (bars) of surveys available in each year. Smoothed lines show the LOWESS-smoothed moving average. PfPR2-10: Plasmodium falciparum parasite rate in the age-standardized two-up-to-ten year old age range (Smith et al. 2007).
We have re-examined temperature time series from locations in highland East Africa using the tDAN robust test for trends , . Applying simple regression models evaluated using the tDAN robust t-test to the data from CRU that we previously used  finds positive but insignificant trends. However, the new CRU TS 2.1 and CRU TS 3.0 datasets shows a highly significant temperature increase in Kericho in both the 1966-1996 period examined by Chavez and Koenraadt , in the shorter 1970-1995 period examined by Hay et al.  and in the longer 1966-2006 period. The trend in Kericho is not, however, significant for the 1950-2002 period used by Pascual et al . The trend is significant for East Africa as a whole at least the 5% level for both the TS 2.1 and TS 3.0 datasets for all the periods that we test.
Therefore, Chavez and Koenraadt's  general claim that, contrary to Hay et al. , temperature did increase in Kericho is correct but this is due to changes in the data used rather than any inadequacies of our statistical methods. The trend of increasing temperature has continued or accelerated further in the 1996-2006 period. These findings also explain the results of Pascual et al.  for the 1950-2002 period. But our results in this paper again show that there is no significant trend in the 1970-1995 period in the temperature data previously published by CRU. It is crucial when making claims for the empirical superiority of one statistical method over another to test both methods on the same datasets.
Our examination of the mean temperature series for Kericho prepared by Omumbo et al.  shows that again, adding post-1995 data results in a steeper trend. The 1979-1995 sample does not have a significant time trend. Hence, while high quality data is of value, use of the data from a single met station vs. use of the interpolated CRU database is not the reason that Hay et al.  failed to find a significant temperature trend for Kericho in the 1970 to 1995 period.
Finally, regardless of its etiology, malaria in Kericho and many other areas of East Africa has decreased during periods of unambiguous warming.
We wish to thank Charles Godfray, Dave L. Smith, and several anonymous referees for their comments on the manuscript. The malaria inpatient data (1995-2006) were kindly supplied by the chief physician Dr Walter Odonde, Unilever Tea Kenya Ltd and updated to 2010 by Mr. Geoffrey Kores, the records officer.
Conceived and designed the experiments: DS SH. Analyzed the data: DS PG AN EO RS. Contributed reagents/materials/analysis tools: PG AN EO RS GDS CK WT. Wrote the paper: DS SH PG AN EO RS WT GDS CK.
- 1. Gething PW, Smith DL, Patil AP, Tatem AJ, Snow RW, et al. (2010) Climate change and the global malaria recession. Nature 465(7296): 342–346.PW GethingDL SmithAP PatilAJ TatemRW Snow2010Climate change and the global malaria recession.Nature4657296342346
- 2. Okiro EA, Al-Taiar A, Reyburn H, Idro R, Berkley JA, et al. (2009) Age patterns of severe paediatric malaria and their relationship to Plasmodium falciparum transmission intensity. Malaria Journal 8: 4.EA OkiroA. Al-TaiarH. ReyburnR. IdroJA Berkley2009Age patterns of severe paediatric malaria and their relationship to Plasmodium falciparum transmission intensity.Malaria Journal84
- 3. Okiro EA, Alegana VA, Noor AM, Mutheu JJ, Juma E, et al. (2009) Malaria paediatric hospitalization between 1999 and 2008 across Kenya. BMC Medicine 7: 75.EA OkiroVA AleganaAM NoorJJ MutheuE. Juma2009Malaria paediatric hospitalization between 1999 and 2008 across Kenya.BMC Medicine775
- 4. Hay SI, Cox J, Rogers DJ, Randolph SE, Stern DI, et al. (2002) Climate change and the resurgence of malaria in the East African highlands. Nature 415(6874): 905–909.SI HayJ. CoxDJ RogersSE RandolphDI Stern2002Climate change and the resurgence of malaria in the East African highlands.Nature4156874905909
- 5. Hay SI, Rogers DJ, Randolph SE, Stern DI, Cox J, et al. (2002) Hot topic or hot air? Climate change and malaria resurgence in East African highlands. Trends in Parasitology 18(12): 530–534.SI HayDJ RogersSE RandolphDI SternJ. Cox2002Hot topic or hot air? Climate change and malaria resurgence in East African highlands.Trends in Parasitology1812530534
- 6. Chaves LF, Koenraadt CJM (2010) Climate change and highland malaria: fresh air for a hot debate. Quarterly Review of Biology 85(1): 27–55.LF ChavesCJM Koenraadt2010Climate change and highland malaria: fresh air for a hot debate.Quarterly Review of Biology8512755
- 7. Box GEP, Pierce DA (1970) Distribution of the autocorrelations in autoregressive moving average time series models. Journal of American Statistical Association 65: 1509–1526.GEP BoxDA Pierce1970Distribution of the autocorrelations in autoregressive moving average time series models.Journal of American Statistical Association6515091526
- 8. Pascual M, Ahumada JA, Chaves LF, Rodo X, Bouma M (2006) Malaria resurgence in the East African highlands: temperature trends revisited. Proceedings of the National Academy of Sciences 103(15): 5829–5834.M. PascualJA AhumadaLF ChavesX. RodoM. Bouma2006Malaria resurgence in the East African highlands: temperature trends revisited.Proceedings of the National Academy of Sciences1031558295834
- 9. Omumbo JA, Lyon B, Waweru SM, Connor SJ, Thomson MC (2011) Raised temperatures over the Kericho tea estates: revisiting the climate in the East African highlands malaria debate. Malaria Journal 10: 12.JA OmumboB. LyonSM WaweruSJ ConnorMC Thomson2011Raised temperatures over the Kericho tea estates: revisiting the climate in the East African highlands malaria debate.Malaria Journal1012
- 10. Confalonieri U, Menne B, Akhtar R, Ebi KL, Hauengue M, et al. (2007) Human health. In: Parry ML, Canziani OF, Palutikof JP, van der Linden PJ, Hanson CE, editors. Climate Change 2007: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK. pp. 391–431.U. ConfalonieriB. MenneR. AkhtarKL EbiM. Hauengue2007Human health.ML ParryOF CanzianiJP PalutikofPJ van der LindenCE HansonClimate Change 2007: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK391431
- 11. Fomby TB, Vogelsang TJ (2003) Test of common deterministic trend slopes applied to quarterly global temperature data. In: Fomby TB, Hill RC, editors. Maximum likelihood estimation of misspecified models: twenty years later. Advances in Econometrics 17. Elsevier. TB FombyTJ Vogelsang2003Test of common deterministic trend slopes applied to quarterly global temperature data.TB FombyRC HillMaximum likelihood estimation of misspecified models: twenty years laterAdvances in Econometrics 17. Elsevier
- 12. Bunzel H, Vogelsang TJ (2005) Powerful trend function tests that are robust to strong serial correlation, with an application to the Prebisch-Singer hypothesis. Journal of Business and Economic Statistics 23: 381–394.H. BunzelTJ Vogelsang2005Powerful trend function tests that are robust to strong serial correlation, with an application to the Prebisch-Singer hypothesis.Journal of Business and Economic Statistics23381394
- 13. New M, Hulme M, Jones PD (2000) Representing twentieth century space-time climate variability. Part 2: development of 1901-96 monthly grids of terrestrial surface climate. Journal of Climate 13: 2217–2238.M. NewM. HulmePD Jones2000Representing twentieth century space-time climate variability. Part 2: development of 1901-96 monthly grids of terrestrial surface climate.Journal of Climate1322172238
- 14. Mitchell TD, Jones PD (2005) An improved method of constructing a database of monthly climate observations and associated high-resolution grids. Int. J. Climatology 25: 693–712.TD MitchellPD Jones2005An improved method of constructing a database of monthly climate observations and associated high-resolution grids. Int.J. Climatology25693712
- 15. Alonso D, Bouma MJ, Pascual M (2011) Epidemic malaria and warmer temperatures in recent decades in an East African highland. Proceedings of the Royal Society B 278(1712): 1661–1669.D. AlonsoMJ BoumaM. Pascual2011Epidemic malaria and warmer temperatures in recent decades in an East African highland.Proceedings of the Royal Society B278171216611669
- 16. Hay SI, Snow RW (2006) The Malaria Atlas Project: developing global maps of malaria risk. PLoS Medicine 3(12): e473.SI HayRW Snow2006The Malaria Atlas Project: developing global maps of malaria risk.PLoS Medicine312e473
- 17. Patz JA, Hulme M, Rosenzweig C, Mitchell TD, Goldberg RA, et al. (2002) Climate change - Regional warming and malaria resurgence. Nature 420(6916): 627–628.JA PatzM. HulmeC. RosenzweigTD MitchellRA Goldberg2002Climate change - Regional warming and malaria resurgence.Nature4206916627628
- 18. Hay SI, Guerra CA, Gething PW, Patil AP, Tatem AJ, et al. (2009) A world malaria map: Plasmodium falciparum endemicity in 2007. PLoS Medicine 6(3): e1000048.SI HayCA GuerraPW GethingAP PatilAJ Tatem2009A world malaria map: Plasmodium falciparum endemicity in 2007.PLoS Medicine63e1000048
- 19. Smith DL, Guerra CA, Snow RW, Hay SI (2007) Standardizing estimates of the Plasmodium falciparum parasite rate. Malaria Journal 6: 131.DL SmithCA GuerraRW SnowSI Hay2007Standardizing estimates of the Plasmodium falciparum parasite rate.Malaria Journal6131
- 20. Cleveland WS (1981) LOWESS: A program for smoothing scatterplots by robust locally weighted regression. The American Statistician 35(1): 54.WS Cleveland1981LOWESS: A program for smoothing scatterplots by robust locally weighted regression.The American Statistician35154