Multiyear dengue incidence patterns in Puerto Rico, Mexico, and Thailand were not explicitly periodic. Though we found high power at multiyear scales in wavelet spectra of all three, the power did not reach significance relative to randomly generated autocorrelated time series. The high degree of interannual variation in dengue incidence is often described as periodic, but our analysis suggests that this oscillation lacks a regular periodicity. This does not mean that dengue transmission does not cycle on multiyear scales, but that there is not enough data to support explicit determination of stationary or nonstationary cycles. It is possible that this seemingly chaotic behavior is a result of serotype-specific dynamics of dengue transmission in human populations 
. In contrast, significant periodicity was present on the annual scale for both dengue and weather variables and on the interannual scale for ENSO.
Using coherence analysis to compare these time series in the frequency domain, we found some associations between climate and dengue incidence. In Puerto Rico, Mexico, and Thailand we found strong coherence between temperature, precipitation and dengue incidence at a periodicity of approximately 1 y. This finding is expected due to the regular seasonality observed in all three. Of greater interest are the relationships on multiyear scales. In Puerto Rico, we found significant association between ENSO and dengue incidence between 1995 and 2002. The biological basis for this relationship is that ENSO drives local changes in weather, and local changes in weather affect dengue transmission. Analyzing this pathway, we found that ENSO was associated with temperature but not precipitation, and that precipitation but not temperature was associated with dengue incidence. As a result, we must treat this link cautiously. The observed time lag of the effect of rainfall on dengue incidence is also problematic. Dengue incidence influencing precipitation is a biologically implausible relationship. It is more plausible that decreased precipitation increases subsequent dengue transmission given the observation that decreased rainfall can lead to increased water storage and thus, increased Ae. aegypti
breeding habitat 
. However, the observed lag of 7 mo is also suspicious, because it would require the effect to occur over many Ae. aegypti
In Thailand, ENSO was associated with changes in local temperature and precipitation, but only precipitation cohered with dengue incidence. The lag of the positive effect of ENSO on precipitation was 14 mo and the negative effect of precipitation on dengue incidence was observed a further 2 mo later. This is biologically plausible as decreasing ENSO could result in decreased rain leading to increased water storage, increased Ae. aegypti breeding habitat, and, later, increased dengue transmission. However, there is reason for skepticism. The two associations occur on slightly different frequency modes (2.6–4 y for ENSO-precipitation and 2.3–2.6 y for precipitation-dengue), and direct coherence between ENSO and dengue incidence was not significant. Furthermore, the observed relationships are nonstationary implying that sometimes precipitation plays a role and at other times, it does not. Biologically, the nonstationarity is difficult to explain as breeding habitat is always necessary for the mosquito vector.
These results combined with the complete lack of multiyear coherence with any of the datasets for Mexico suggest that neither ENSO nor temperature or precipitation are the most important determinants of multiyear variability in dengue incidence in these endemic settings. The tenuous relationships demonstrated on the multiyear scale are clearly different from coherence on the seasonal scale where the case for the effect of weather is much stronger. There are several plausible explanations for our findings. One is that ENSO has no effect on dengue transmission. While this is possible, the biology of transmission suggests that temperature and precipitation, and thus the effects of ENSO, are important determinants of transmission efficiency. If these effects do exist, we may lack sufficient long-term datasets with which to observe them. Another possibility is that local effects of ENSO are obscured by summarizing weather and dengue incidence to large political boundaries. Although Puerto Rico is a relatively small geographical area, the association of temperature and precipitation with dengue incidence varies geographically 
. On the scale of Mexico, the extent of spatial heterogeneity is likely much larger, possibly explaining the lack of any significant associations in the current analysis. Finally, the effects may be present but obscured by other more prominent factors. In particular, a theoretical basis for complex multiyear oscillations in dengue incidence based solely on intrinsic factors has been hypothesized by several groups 
. These factors may overshadow any extrinsic effects of ENSO.
Indeed, many of the observed associations may be the result of independent, coincident El Niño episodes and major dengue epidemics. At the scale of highest coherence in Puerto Rico, ENSO is periodic throughout the observed time period while dengue incidence fluctuates in the pattern of a single Morlet wavelet (). The similarity between the reconstruction and the wavelet used for transformation, suggests that transformation is capturing a single event rather than a periodic pattern. This means that the observed coherence may simply be the result of a single dengue outbreak occurring on the same scale as ENSO variation. The reported association in Thailand appears to follow this same pattern 
. Unfortunately, the very nature of multi-annual variation makes it difficult to find relationships that are more than coincidental because of the vast amount of data required. In Mexico, for example, there appears to be two or three multi-annual peaks in dengue incidence over 22 y (Figure S2A
). Though this may reflect an 8-y periodic, at least one more event is required to assess its significance (observe the shaded region of Figure S2B
). Even then, it may be hard to differentiate coherence from coincidence.
Wavelet analysis, because of its ability to decompose and compare frequency specific components of time series, is a powerful tool for the analysis of long-term epidemiological data. While particularly well-suited to comparing periodic variations at different time scales, wavelets also can be used to assess other types of temporal changes such as those related to vaccine introduction 
. Integral to any analysis is the testing of significance. Unlike previous analysis of the potential effect of ENSO and weather on dengue incidence, we consider the effects of autocorrelation on frequency-specific decomposition. The autocorrelation of epidemiological data over time leads to higher spectral power at low frequencies than would be expected from independent observations. We allow for this by using a statistical test incorporating autocorrelation in the null hypothesis. In the coherence analysis we assessed the role of autocorrelation and scale selection. Autocorrelation was found to have little effect and scale resolution was selected to balance computational efficiency and sensitivity to low resolution. Because random coherence still occurs at a high rate, we also developed a test for random coherence based on the duration over which it occurs.
With these considerations, the associations between temperature, precipitation, and dengue incidence on the annual scale in Puerto Rico, Mexico, and Thailand are clear. Although these associations are indistinguishable by wavelet analysis, they demonstrate how a strong temporal relationship can be characterized by coherence analysis. Both ENSO and dengue incidence vary on multiyear scales, but they do not exhibit similarly strong coherence. It is possible that there is a nonstationary relationship between climate and dengue incidence, but further evidence explaining the nonstationarity and demonstrating its occurrence at more than one time point is necessary to effectively support this hypothesis. Moreover, given the magnitude of interannual variation in dengue transmission, it is unlikely that a weakly supported nonstationary effect is the dominant driver of this important component of dengue transmission dynamics. Further elucidation of these dynamics may require explicit modeling of intrinsic factors. In particular, though difficult to do, there is a need to go beyond theory to the application and assessment of biologically reasonable theories using empirical data.