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Longitudinal Development of Secondary Sexual
Characteristics in Girls and Boys Between
Ages 91⁄2 and 151⁄2 Years
Elizabeth J. Susman, PhD; Renate M. Houts, PhD; Laurence Steinberg, PhD; Jay Belsky, PhD; Elizabeth Cauffman, PhD;
Ganie DeHart, PhD; Sarah L. Friedman, PhD; Glenn I. Roisman, PhD; Bonnie L. Halpern-Felsher, PhD;
for the Eunice Kennedy Shriver NICHD Early Child Care Research Network
Arch Pediatr Adolesc Med. 2010;164(2):166-173
eAppendix. Details of Analysis Methods
The ages when girls and boys were in sexual maturity stage 2
through 5 were estimated in separate logistic regressions for
each sexual maturity stage (2, 3, 4, or 5) for each secondary
sexual characteristic: breast (girls), genital (boys), and pubic
hair (girls and boys) development. For this analysis, whether
an adolescent was in a particular sexual maturity stage at each
assessment was dichotomized (0 vs 1). The probability of being
in the stage being estimated at a given age was modeled using
a random-effects logistic regression with the dichotomized indicator as the dependent variable and age at each assessment
(in years) as the independent variable.
PSM = q (yti = 1|Ageti) =
1
,
1 + e−(β0i + β1 [ageti] + u0i)
the probability that individual i had reached sexual maturity
stage a at assessment t. Other parameters were defined as follows:
yti =Dummy coded (0/1) variable indicating whether individual i had reached sexual maturity stage q at assessment t
Ageti =Individual i’s age (in years) at assessment t
β0i =Intercept
β1 = Slope of ageti on the probability that individual i had
reached sexual maturity stage q at ageti
u0i =Random within-person error
The righthand side of this equation is the cumulative distribution function for the logistic distribution with variance
␴2 =␲2/3 and mean –β0 /β1. Thus, if age is the time at which an
individual was in a given sexual maturity stage, then this expression provides the probability of the individual being in the
specified sexual maturity stage by the time ageti. As in traditional logistic regression, β0i moves an individual’s curve to the
left or right, whereas β1 changes the steepness of the estimated
curve. This model is the simplest model for each component
of pubertal development. In this case, β0i is equal to the grand
mean plus a random individual effect, u0i; β1 is fixed at the grand
mean. Although we attempted to fit models that also allowed
β1 to be random (ie, include random individual variance for
the slope), these models failed to converge.
Using a logit transformation, we reformulated the above expression to estimate the age at which an adolescent was expected to be in the specified sexual maturity stage:
logit(Pi) = β0 + β1Ageti + u0i.
From this, we estimate the age at which an adolescent was
expected to be in the specified sexual maturity stage as the mean
of the distribution:
−β0i −(β0 + u0i),
=
β1i
β1
where q=the sexual maturity stage under consideration (ie, 2,
3, 4, or 5) and β0, β1, and u0i are as defined above.
Once the ages of being in the various sexual maturity stages
for the different secondary sexual characteristics were estimated, differences across time and across characteristics were
modeled using a doubly repeated-measures analysis of variance. For these analyses, the estimated ages were the dependent variables and sexual maturity stage (2-5) and secondary
sexual characteristics (breast or pubic hair for girls; genital or
pubic hair for boys) were repeated factors. Additional models
added between-subjects factors for menarche (girls only), race
(black vs white), years of maternal education, and family income to needs ratio.
Finally, the length of time (in years) that girls and boys remained in puberty was calculated as the difference between the
ages that individuals were estimated to be in sexual maturity
stage 5 (full development) and sexual maturity stage 2 (the beginning of puberty).
(REPRINTED) ARCH PEDIATR ADOLESC MED/ VOL 164 (NO. 2), FEB 2010
E1
AgeSM = q =
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100
% of Participants
80
SMS 1
SMS 2
SMS 3
White Girls: Breast Development
White Girls: Pubic Hair Development
Black Girls: Breast Development
Black Girls: Pubic Hair Development
White Boys: Genital Development
White Boys: Pubic Hair Development
Black Boys: Genital Development
Black Boys: Pubic Hair Development
SMS 4
SMS 5
60
40
20
0
100
% of Participants
80
60
40
20
0
100
% of Participants
80
60
40
20
0
100
% of Participants
80
60
40
20
0
9½
10½
11½
12½
13½
14½
15½
9½
10½
11½
Age, y
12½
13½
14½
Age, y
eFigure. Percentage of adolescents in each sexual maturity stage (SMS) (1-5) by breast, genital, and pubic hair development and by age and race.
(REPRINTED) ARCH PEDIATR ADOLESC MED/ VOL 164 (NO. 2), FEB 2010
E2
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15½