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Appendix: Calculation of linear and quadratic
contrasts
The following formulae were used
to calculate the contrasts (L) and their variance (V) and
confidence intervals (Cl) [22] used in estimating secular trends
in the length of the threeyearold Guatemalan children studied:
where
c = coefficients (see below)
X = mean length at three years of age
n = sample size
i = time of measurement (= 7)
s² = estimated variance
The coefficients were obtained by
the method proposed by Robson [15] for the construction of
polynomials for unequally spaced data (table A1).
TABLE A1. Coefficients for linear
and quadratic contrasts

Spacing 
Coefficients 
Linear 
Quadratic 
1968 
0 
0.43 
+0.56 
196970 
2 
0.31 
+0.20 
197172 
4 
0.19 
0.07 
197374 
6 
0.06 
0.28 
197576 
8 
+0.06 
0.41 
1977 
9 
+0.12 
0.44 
1988 
20 
+0.81 
+0.44 
As an example of the use of the
methodology, we show the calculation of the linear contrast and
its confidence interval for the atole group. Table A2 gives the
calculation of the summations and . From this:
Then, using s² = 19.78,
and
Cl = 0.61
Thus, the contrast = 1.85 ± 0.61
(1.24, 2.46).
TABLE A2. Calculation of and

c_{i}X_{i} 
c_{i}^{2}/n_{i} 
1968 
85.14
x 0.435801 
0.4358^{2}/196 
196970 
85.53
x 0.311286 
0.3113^{2}/82 
197172 
86.58
x 0.186772 
0.1868^{2}/100 
197374 
86.09
x 0.062257 
0.0622^{2}/117 
197576 
86.86
x +0.062257 
0.0622^{2}/117 
1977 
86.55
x +0.124515 
0.1245^{2}/35 
198889 
87.62
x +0.809345 
0.8093^{2}/344 

1.85 
0.0049 
Acknowledgements
Data collection and analyses were
supported by NIH grant HD22440. The study was a collaborative
effort involving investigators at several institutions: R.
Martorell (principal investigator, originally at Stanford
University, now at Cornell University), J. Rivera (INCAP,
Guatemala), E. Pollitt (University of California at Davis), and
J. Haas (Cornell University).
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