2.3. Data analysis

To test whether there was a temporal pattern or cyclicity in the breeding number time series (i.e., whether a breeding numbers occurred regularly over the years), a wavelet analysis was applied. Wavelet analysis decomposes the time‐series into time and frequency domains and determines the dominant (significant) mode of variability (Cazelles et al., 2008). We used wavelet decomposition analysis from the package “biwavelet” in R (version 4.0.2) R Core Team (2020). The analysis can be displayed as 3D contour plots with breeding frequency along time (year) and at which frequency (cycle period in number of years) a fluctuation is significant (cf. Figure 3). The analysis was run for total males, polygonous males, and monogamous males separately.

Wavelets analyses (using R package “biwavelet”) of the dipper breeding population size between 1978 and 2019 [(a) total breeding males, (b) only polygynous, (c) only monogamous]. The plots are the bias‐corrected power normalized by the variance (using “type = power.corr.norm”). The cone of influence (white lines) illustrates the loss in statistical power near the start and the end of the series and must be interpreted with caution. The red color, indicates that over time exists a dominant ca 8‐year periodicity covering the studied period. While not significant over the whole period (black line, 1991–2018), partly due to border effect, it is consistent for all categories of males.

The relationship between the number of polygynous male number to the total number of males (Figure 4a) and that of monogamous males (Figure 4b) was estimated using generalized linear models with the function glm in R. To take into account the overdispersion of the residuals, we used a quasibinomial error distribution. We did not detect any significant autocorrelation in the residuals (using autocorrelation function acf).

The number of polygynous dippers versus total population size (left) and number of monogamous males (right) over 42 years. Linear regression is the dotted line. p < .001 for both, with R ² = .38 and .31, respectively.

The description of the effect of age on polygamy/monogamy was done using the function boxplot. The significance difference between groups was estimated using a rank test (Siegel & Castellan, 1988) followed when necessary by a Dunn's test (1964) with the Benjamini–Hochberg p‐value adjustment (Benjamini & Hochberg, 1995), to report the differences among multiple pairwise comparisons, using the package dunn.test in R. The statistical tests are two‐tailed with an α‐level of .05.