Analysis of the role of sex in the rate of accumulated error

Using the present data set, we have previously shown that that males had higher variance than females in CBT and LA within and across individuals [11]. These data (Fig. 1) were reanalyzed here to assess the contributions of different timescales and structures to variance observed.

Fig. 1

Core body temperature A, B and locomotor activity (C, D), for males (red) and females (blue), presented as mean ± SD. Some analyses use a data set in which females are aligned by days of estrous (“E”s; A, C), and some use a data set in which each individual (males and females) has been delayed 1 day relative to the previous individual in its group (B, D), thereby maximally de-aligning females to insure that interindividual structure is not conferred to the female group by way of predictable, 4-day ovulatory cycles

If variance is unaffected by sex, then one would expect that measurement errors would accumulate overtime at the same rate for males and females, or that sex does not impact precision of a given measurement. We did not find this to be the case. Here we assessed this using two kinds of error: 1) Static error; and 2) Dynamic error (see “Methods” section). Cumulative static error would be expected to differ from cumulative dynamic error if a portion of the variance were structured in time, leading to changes over time that reduce overall error (i.e., individual measures change over time in similar, non-random ways). Importantly, initially analyses used female data that had been staggered so that estrous days were maximally unaligned across animals, so that no 4-day pattern (i.e., the period of a mouse estrous cycle) is responsible for the error structure in the both-sexes mean and SD. A visual depiction of this strategy is presented in Fig. 1.

Raw data presented as distance from the overall mean revealed that males displayed higher variability than females for both CBT (Fig. 2A, B) and LA (Fig. 2C, D). Quantification of the cumulative error for CBT and LA (Fig. 2E, F, respectively) revealed that males accumulated significantly more error over time than did females (Static, CBT: χ2 = 401, p = 3 × 10–89; Static, LA: χ2 = 580, p = 4 × 10–128; Dynamic, CBT: χ2 = 282, p = 3 × 10–63; Dynamic, LA: χ2 = 791; p = 5 × 10–174). Cumulative static error rose more quickly than did cumulative dynamic error, confirming that some of the variance in the data was structured in time; comparison to the dynamic baseline was thereby confirmed to reduce measurement uncertainty. Note that error reduction was greater in CBT than in LA when a dynamic baseline was used (females p = 0.007, males p = 0.006), suggesting that CBT showed more structured variance across time than did LA. When these analyses were re-run on data in which females were staged (data aligned by days of estrus), the error was reduced further (3% for temperature, 1% for activity—Fig. 2E, F insets), confirming that estrous cycles contributed a small but structured amount of variance to the overall mean and SD of the population data.

Fig. 2

The estrous cycle does not contribute substantially to overall variance, of which males have more than females. Static error for all individual females (A, C, blue tones) and males (B, D, in red tones); female days arranged to provide minimum possible alignment across individuals’ estrous cycles; Y = 1 = Static SD; black line = dynamic SD. Males have higher variance than females in both temperature A, B and activity (C, D). Sex affects the accumulation rate of static and dynamic errors across the 14 day data window (E, F). Males have a higher cumulative error than do females, even when estrus is not aligned for females (E temperature, F activity); males in red, females in blue; line is intra-sex mean, filled areas are intra-sex SD of accumulated error; units in static SD of the population. Roughly one third of the error can be eliminated (blue and red arrow brackets) by comparison to a population dynamic baseline (solid lines) to a static baseline (dotted line). Insets: The estrous cycle adds additional structure, so that staging females (aligning individuals by estrous cycle) further reduces the accumulated errors for females, when comparing to a dynamic baseline, by 3% for temperature and 1% for activity; 0% change in males for the same realignment. *Indicates significant difference. See “Results” section for further details

Examination of amount of change within and across days, including ultradian structure, across individuals

Our previous findings using the present data set found that males exhibited larger variance across the day than did females, but the structure of that variance was not assessed [11]. As previously reported [11], males showed a higher amplitude of change in the CWT frequency power of their ultradian temperature rhythms (Fig. 3A). Further analysis revealed that males also showed a higher median power overall (Fig. 3B; p = 3 × 10–4), as well as a higher median power of the circadian modulation of ultradian power (Fig. 3C, D; p = 0.01). These findings indicate that males exhibited a greater change resulting from higher amplitude ultradian rhythms in body temperature, and also resulting from higher amplitude modulation of these ultradian rhythms across the day, than did females.

Fig. 3

Previously published temperature analyses A found that male mice (red lines—thick line is mean) show higher amplitude changes in ultradian rhythms (as determined by wavelet-based frequency band isolation—see “Methods” section) across time than females (blue). Here females are aligned by their 4-day estrous cycles (“E” marks days of estrus). Building on these analyses, males also have a higher median ultradian power, and inter-individual range of medians (B), than do females. Consistent with this finding, wavelet-based isolation of circadian modulation of ultradian power shown in A (C) reveals that males have greater circadian modulation of ultradian rhythms than do females. In addition to having higher median power D, males also show greater inter-individual variance than do females for both median ranges. *Indicates significant difference. See “Results” section for further details

Analysis of structure within individuals within a day

The analyses presented so far suggest that male mice exhibit more variability, less of which is structured, across days, than do female mice. However, it is possible that males show more structure within a day than females. Ultradian rhythms are not perfectly alignment day to day [11, 14, 15], so we chose two methods to assess self-similarity across days: comparison to a personal mean day, and day-to-day difference calculated by dynamic time warping (DTW).

We first examined within-a-day structure by constructing a mean daily profile for each individual, and calculating dynamic cumulative error in the same way as in the previous section, comparing each animal’s individual days to their own mean day, both for temperature and activity. By these analyses, males had a higher rate of cumulative error relative to their own mean day than did females (CBT: p = 0.003, Fig. 4C; LA: p = 7 × 10–4, Fig. 4G). However, given that males also had higher variance overall, it could still be argued that the proportional amount of structured variance was the same, even if the absolute structure was lower. To account for this possibility, we divided the cumulative error values for each individual by that individual’s SD. The result confirmed that once the individual’s SD was corrected for, the amount of cumulative error was not different between the sexes (temperature: p = 1, Fig. 4C; activity: p = 0.24, Fig. 4H).

Fig. 4

Males did not show more structure within days than females. Mean and SD (thick line with shaded surround) of temperature A, B and activity E, F of one male (red) and female (blue) across 24 h, overlaid on 14 days for the same individual (black lines underneath color). Any measurement more than 1 SD from the mean is defined as error. Error summed across 24 h allows comparison of structured variance by sex within a day. Males show significantly higher within individual, within-day error than females (C, G). Structure of the variance appeared similar across sexes; there was no difference between the sexes once each individual’s error was divided by that individual’s mean SD (D, H). This finding demonstrates that males did not have sufficient structure within a day to make them overall less variable if within-a-day structure is accounted for in this way. However, the amount of error was proportional to the SD in both sexes, suggesting that while males were more variable overall, they did not have less structure within a day than did females, by this approach. *Indicates significant difference. See “Results” section for further details

We next examined within-a-day structure by comparing the distance needed (in units of activity or temperature, respectively) to warp one day to best match the subsequent day (see “Methods” section). By these analyses, males showed higher between-day distance for body temperature (Fig. 5A, D; p = 1.6 × 10–5) and also for activity (Fig. 5E, G; p = 6 × 10–5). As with analysis of daily means, distance is in the same units as variance, and so greater variance should cause higher average distance for the same proportion of structured variance in a given data set. As with the preceding analysis of comparison to average day, DTW distance calculated for each individual was corrected by dividing by that individual’s SD. This correction resulted in less distance between males and females, but males still exhibited a higher distance on average than females (Fig. 5B, D; temperature: p = 1.6 × 10–5; Fig. 5F, H; activity: p = 6 × 10–4). Because DTW is better able to align ultradian cycles with small day-to-day phase differences than is the daily mean, this finding more strongly supports the hypothesis that males are not only more variable, but also show a lower proportion of structured variance within the day.

Fig. 5

Males exhibit less day-to-day similarity than females. Mean and SD (thick line with shaded surround) of CBT A, B and LA E, F for males (red) and females (blue) of DTW distance between successive days. Males show higher within individual, day-to-day distance than females (C, G). This distance is slightly reduced but remains significant between the sexes when each individual’s distances are divided by their mean SD (D, H). This finding suggests that males had higher day-to-day variability within individuals, and less proportional structure, than females. *Indicates significant difference. See “Results” section for further details