### Results

#### Interepidemic Periods

Four distinct patterns of periodicity were detected in the 1938–1955 data (Figure 2), with states classified as follows. Group 1 consisted of 5 states in which epidemics were annual throughout the period (see Figure 2A and 2C for an example of incidence data and the associated dominant period of a group 1 state). Group 2 was comprised of 17 states in which a clear transition from annual epidemics to multiennial epidemics was observed (Figure 2B and 2D). Twenty states with consistent multiennial epidemics were categorized in group 3 (Figure 2E and 2G), while group 4 was populated by Oregon and Vermont, in which multiyear outbreaks gave way to annual epidemics (Figure 2F and 2H).

Figure 2.

Weekly pertussis incidence data (1938–1955) for 4 states: A) Arizona (group 1), B) California (group 2), E) New Jersey (group 3), and F) Vermont (group 4). The associated dominant periods are plotted below the corresponding incidence data for C) Arizona, D) California, G) New Jersey, and H) Vermont. The dominant periods represent the periods with the largest power in the wavelet spectrum. Black and gray lines distinguish the significance (or otherwise) of the detected interepidemic periods (see Methods).

In Figure 3, we dissect the diversity of observed patterns in periodicity across states. Epidemics in states belonging to group 1 (Figure 3A) or group 2 before the transition (Figure 3C) were characterized by a significant dominant annual component alone. After the transition, however, group 2 states exhibited a range of periodicities from approximately 2 years to 6 years (Figure 3D). Similarly, interepidemic periods in states that were characterized by multiennial outbreaks throughout this interval (group 3) were found to vary from approximately 2 years to 5 years (Figure 3B). We mapped the classification of each state in Figure 4. This figure is visually suggestive of a latitudinal gradient in pertussis epidemiology, a suggestion that was confirmed by our statistical analyses (Figure 5). For comparison, we also present the periodicity of incidence data from 1951 to 1970 in Web Appendix 6 (refer to Web Figure 2) http://aje.oxfordjournals.org/content/181/12/921/suppl/DC1. During this era, throughout the United States, pertussis epidemics were characterized by a strong 4-year signal, in addition to an annual component in group 1 and many group 2 states.

Figure 3.

Normalized global power spectrum derived from the wavelet decomposition of 1938–1955 pertussis time series. Darker shades represent higher powers in the wavelet spectrum. The dominant period is the 1-year period for group 1 states (A) and a multiyear period for group 3 states (B). For the group 2 states, the 1-year period is originally dominant (C) but is replaced by a multiyear period after some transition time (D). The estimated timing of the transition in group 2 is given in Web Table 2.

Figure 4.

Classification of US states according to periodicity of pertussis outbreaks from 1938 to 1955. Oregon and Vermont are the only states in group 4 where the period displayed a transition from multiyear outbreaks to annual outbreaks—the reverse of the group 2 transition. Mississippi, Nevada, North Dakota, South Dakota, and Wyoming were not classified (refer to Web Appendix 3 for details).

Figure 5.

Comparison of the distribution of values for the characteristics listed in Table 2, according to state pattern of pertussis periodicity (groups 1–3) during the early vaccine era (1938–1955). In each panel, the first *P* value presented is from analysis of variance and the second is from the Kruskal-Wallis test. The whiskers represent the most extreme data points—no greater than 1.5 times the interquartile range. Variation in population (panel C) was calculated as change in population size from 1938 to 1955 relative to the mean population size. Variation in birth rate (panel D) was calculated as change in the monthly birth rate from 1938 to 1955 relative to the mean birth rate. For residual phase of the annual period (panel M), it was inappropriate to compare arithmetic mean values because phases varied from −π to π; hence, we calculated circular means instead. (This involves conversion of the phase from an angle to a point on the unit circle, from which the arithmetic mean is computed.) The circular means for phase are shown with heavy dashed lines.

#### Comparison Among Groups

Results of the comparison of state-specific characteristics (outlined in Table 2) across groups are summarized in Figure 5. We found the following characteristics to be different between groups according to both ANOVA and the Kruskal-Wallis test: per capita birth rate, household crowding, the fraction of families with no children, the fraction of families with more than 1 child, latitude of the state population centroid, and recent pertussis incidence (from 2001–2010). In particular, group 1 states had higher per capita birth rates, more household crowding, and higher fractions of families with more than 1 child than states in group 3. The reverse relationship held for the fraction of families with no children. We found that group 3 states were mainly concentrated in the northern part of the country. Our results also indicated contemporary pertussis incidence to be highest among group 1 states and lowest in group 2 states.

ANOVA and the Kruskal-Wallis test generated dissimilar *P* values concerning group differences in mean population size, population size variation, school attendance, and the slope of resurgence. In these instances, the disagreement resulted from the presence of outliers (shown in Figure 5), which is inconsistent with the assumptions of ANOVA. Thus, conclusions based on the Kruskal-Wallis test are appropriate. For instance, consider school attendance. The outlier in group 3 is due to the very low (53%) school attendance in Kentucky. Removing this point from the analysis yields *P* values of 0.001 and 0.003 for ANOVA and the Kruskal-Wallis test, respectively, suggesting that school attendance is different between groups, with the values for group 3 being higher than those in group 1. Regarding population size and variation in population size, the Kruskal-Wallis test does not indicate any significant difference among groups.

Similarly, our statistical tests lead to contradictory conclusions concerning differences in 1938–1955 mean incidence among groups. Here, because none of the assumptions of either test are clearly violated, we cannot confidently conclude that differences in mean incidence exist among groups.

There were no differences among groups in population size, per capita birth rate, variation in population size, variation in birth rates, proportion of the population living in urban regions, per capita health spending, timing of the annual peaks in pertussis incidence, or the timing of the breakpoint.

Although group 4 was not included in the analysis, we note that the historical mean weekly incidence and recent monthly incidence of the group 4 states were 4.2 per 100,000 population and 0.9 per 100,000 population, respectively, both of which are higher than the group mean values of groups 1–3. Vermont contributed the higher values in both cases.

Am J Epidemiol. 2015;181(12):921-931. © 2015 Oxford University Press