Fractals, Vigilance, and Adolescent Diabetes Management: A Case for When Regulation May Be Difficult to Measure With the Current Medical Standards

Abstract

Temporal patterning in blood glucose (BG) consistent with fractals—how BG follows a repetitive pattern through resolutions of time—was used to examine 2 different samples of adolescents with Type 1 diabetes (10–14 years). Sample 1 contained 10 adolescents with longtime series for accurate estimations of long-term dependencies associated with fractals. The second contained 94 adolescents measured multiple times daily over a 2-week period corresponding to psychosocial measures. In both samples, temporal dependencies in BG showed patterns consistent with fractals. In the second sample, temporal dependencies were associated with indicators of vigilant regulation including adolescents' higher anxiety, mothers' higher monitoring, and intrusive support. The existence of temporal dependencies in BG moderated the relationship between glycosylated hemoglobin (Hb_A1c) and indicators of low BG risk but not the relationship between Hb_A1c and high BG risk. These results show how a biomedical indicator may be susceptible to metric issues associated with fractals.

Notes

^a Differences in AIC between an ARIMA (1 0 1) model and an ARFIMA (1 d 1) model.

*Significant at alpha = .05. All terms were entered centered at their mean. All standard errors are adjusted for nonnormality.

¹We used the equations from Aiken and West (1991) to calculate the simple slopes. Note that these equations are not quite correct for this circumstance for two reasons. First, the equations are correct under regression logic, which is not exactly the case here. There are equivalent versions for multilevel modeling, but these equations are for the circumstance where the estimates are treated as the criterion rather than predictors. Thus, the regression versions are closer approximations. Second, the maximum likelihood estimation method used in multilevel modeling produces asymptotically correct estimates and are therefore only considered good enough rather than correct in finite samples.