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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 three-year-old Guatemalan children studied:


c = coefficients (see below)
X = mean length at three years of age
n = sample size
i = time of measurement (= 7)
s = 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
1969-70 2 -0.31 +0.20
1971-72 4 -0.19 -0.07
1973-74 6 -0.06 -0.28
1975-76 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 s = 19.78,


Cl = 0.61

Thus, the contrast = 1.85 0.61 (1.24, 2.46).


TABLE A2. Calculation of and

  ciXi ci2/ni
1968 85.14 x -0.435801 -0.43582/196
1969-70 85.53 x -0.311286 -0.31132/82
1971-72 86.58 x -0.186772 -0.18682/100
1973-74 86.09 x -0.062257 -0.06222/117
1975-76 86.86 x +0.062257 0.06222/117
1977 86.55 x +0.124515 0.12452/35
1988-89 87.62 x +0.809345 0.80932/344
1.85 0.0049


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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