Fits nonlinear contrasts to quantitative factors in
ANOVA (R.C. Butler).
|Printed output required (
||Curve (as in
||Printing of probabilities for variance ratios (
||Standard errors to print with means tables (
||Variate of weights for each unit; default
||Data to be analysed|
||Factor with quantitative levels for which contrasts are to be found|
||Variate of values to use for the levels of
||Factor whose interaction with
||Structures to hold the estimates of the fitted contrasts:
||Structures to save the standard errors for the contrast estimates, including
||Structures to save the degrees of freedom for the contrast estimates; the pointer has the same form as the
ANOVA directive allows linear contrasts to be fitted and incorporated into the analysis-of-variance table.
NLCONTRASTS extends this to enable nonlinear contrasts to be fitted to the effects of a quantitative factor and its interaction with another factor. The analysis should include both main effects and the interaction between the factors. The procedure will work for any block structure providing each treatment term is estimated entirely within one stratum. The result is similar to
ANOVA with a polynomial contrast, but with slightly different partitions of the treatment sums of squares. The main effect is partitioned into the sum of squares for the “Curve” and the remainder or “Deviations”. The interaction sum of squares is partitioned into the sum of squares due to curves with “Common Nonlinear” parameters for the levels of the non-quantitative factor, and the extra sum of squares due to having “Separate Curves” for each level of that factor, and the remaining sum of squares which again represents “Deviations”.
TREATMENTSTRUCTURE directives must be used in the normal way before the procedure is called, and any
COVARIATES should also be defined first. The structure of the analysis-of-variance table is then accessed from inside the procedure. The
Y parameter defines the variate to be analysed, and the form of nonlinear contrast is defined using the
CURVE option of the procedure. The same choices of curves are available as for
FITCURVE. There are four other options,
WEIGHT, which are exactly as for
XFACTOR parameter defines the factor to which the contrasts are to be fitted, and the
XLEVELS parameter may be used to define x values for the regressions if the levels already defined for the factor are unsuitable. The
GROUPFACTOR parameter defines the factor whose interaction with
XFACTOR is to be assessed. The final three parameters
DFCONTRASTS can be used to save the parameter estimates for the contrasts, their standard errors and degrees of freedom respectively.
ANOVA is used to obtain the basic analysis-of-variance table and the sums of squares for the treatment terms.
FITCURVE is then used with the treatment means to fit three sets of curves: a single curve, curves with common nonlinear parameters, and entirely separate curves. The deviances and degrees of freedom obtained from these are used in conjunction with the treatment sums of squares to calculate the contrast sums of squares and degrees of freedom. Further details are given by Butler & Brain (1992). New lines for the analysis-of-variance table are then constructed using
EDIT, and these lines are then inserted into the table (saved in a text with
EDIT. The standard errors for the parameter estimates and deviances are based on the Residual Mean Square for the appropriate stratum. Standard errors for deviations are calculated using the method in the Guide to the Genstat Command Language, Part 2, Section 4.5.
Y variate is restricted, the procedure will use only the units not excluded by the restriction.
Butler, R.C. & Brain, P. (1993). Nonlinear Contrasts in
ANOVA. Genstat Newsletter, 29, 20-27.
CAPTION 'NLCONTRAST example',\ !t('Doses of a herbicide were applied in two volumes of liquid.',\ 'The analysis asseses whether the dose-response of weed Fresh',\ 'Weight changes with Volume. The dose-reponse is fitted as a',\ 'logistic curve using log(Dose).'); STYLE=meta,plain VARIATE [NVALUES=120] Fwt READ Fwt 4.33 4.94 5.26 4.15 6.64 3.67 4.26 4.93 7.00 5.37 3.79 3.82 3.75 3.11 4.80 5.24 6.37 6.89 3.38 4.80 3.69 5.65 4.92 2.57 3.77 6.19 3.31 6.39 2.28 5.23 3.29 4.08 6.06 5.04 2.66 3.95 2.21 0.50 5.14 3.34 0.71 3.32 2.61 3.74 2.41 2.90 3.32 2.77 2.66 4.48 2.56 2.64 3.57 2.11 2.29 2.26 1.36 2.92 2.77 2.40 6.52 5.51 4.58 5.91 5.85 5.33 4.88 7.01 5.54 6.21 6.27 5.29 5.67 5.23 4.80 4.07 5.09 3.14 7.63 2.84 3.59 2.92 2.38 4.02 2.92 3.53 3.42 3.12 3.20 2.77 3.39 4.11 3.49 2.50 3.11 3.15 2.03 2.85 1.65 3.13 1.37 1.25 0.62 2.11 1.87 3.81 0.55 2.29 2.62 1.44 1.60 1.15 2.27 2.46 1.91 2.16 1.84 2.99 2.56 1.39 : FACTOR [NVALUES=120; LABELS=!t(Small, Large); values=60(1,2)] Volume FACTOR [NVALUES=120; LEVEL=6; VALUES=10(1...6)2] Doses FACTOR [NVALUES=120; LEVEL=10; VALUES=(1...10)12] Block VARIATE [VALUES=10,20,40,100,160,340] Dose CALC Ldose=log(Dose) BLOCK Block TREAT Doses*Volume NLCONTRASTS [CURVE=logistic] Fwt; XFACTOR=Doses; XLEVELS=Ldose;\ GROUPFACTOR=Volume