Fits nonlinear contrasts to quantitative factors in `ANOVA`

(R.C. Butler).

### Options

`PRINT` = string tokens |
Printed output required (`aovtable` , `information` , `covariates` , `effects` , `residuals` , `contrasts` , `means` , `%cv` , `missingvalues` ); default `aovt` , `info` , `cova` , `mean` , `miss` |
---|---|

`CURVE` = string token |
Curve (as in `FITCURVE` ) to use for nonlinear regression (`exponential` , `dexponential` , `cexponential` , `lexponential` , `logistic` , `glogistic` , `gompertz` , `ldl` , `qdl` , `qdq` ); default `expo` |

`FPROBABILITY` = string token |
Printing of probabilities for variance ratios (`yes` , `no` ); default `no` |

`PSE` = string token |
Standard errors to print with means tables (`differences` , `means` ); default `diff` |

`WEIGHT` = variate |
Variate of weights for each unit; default `*` (no weights) |

### Parameters

`Y` = variates |
Data to be analysed |
---|---|

`XFACTOR` = factors |
Factor with quantitative levels for which contrasts are to be found |

`XLEVELS` = variates |
Variate of values to use for the levels of `XFACTOR` ; if unset, the factor levels themselves are used |

`GROUPFACTOR` = factors |
Factor whose interaction with `XFACTOR` is to be assessed |

`CONTRASTS` = pointers |
Structures to hold the estimates of the fitted contrasts: `CONTRASTS[1]` is a pointer with two values, labelled `'Curve'` (parameter estimates for a single fitted curve) and `'Deviations'` (the differences between this curve and the means for `XFACTOR` ); `CONTRASTS[2]` has three values, labelled `'Common NonLin'` (parameter estimates for curves fitted with common nonlinear parameters for all levels of `GROUPFACTOR` ), `'Separate Curves'` (parameter estimates for curves fitted with all parameters varying with the levels of `GROUPFACTOR` ) and `'Deviations'` (differences between the treatment means and the Separate Curves); the order of the parameters is as in the output of the procedure, the variates of estimated contrasts are labelled by the parameter names as used in the printed output, while the `'Deviations'` are both tables, labelled by the relevant factors |

`SECONTRASTS` = pointers |
Structures to save the standard errors for the contrast estimates, including `'deviations'` ; the pointer has the same form as the `CONTRASTS` pointer |

`DFCONTRASTS` = pointers |
Structures to save the degrees of freedom for the contrast estimates; the pointer has the same form as the `CONTRASTS` pointer, except that the variates and tables are replaced by scalars |

### Description

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”.

The `BLOCKSTRUCTURE`

and `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, `PRINT`

, `FPROBABILITY`

, `PSE`

, and `WEIGHT`

, which are exactly as for `ANOVA`

. The `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 `CONTRASTS`

, `SECONTRASTS`

and `DFCONTRASTS`

can be used to save the parameter estimates for the contrasts, their standard errors and degrees of freedom respectively.

Options: `PRINT`

, `CURVE`

, `FPROBABILITY`

, `PSE`

, `WEIGHT`

.

Parameters: `Y`

, `XFACTOR`

, `XLEVELS`

, `GROUPFACTOR`

, `CONTRASTS`

, `SECONTRASTS`

, `DFCONTRASTS`

.

### Method

`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 `PRINT`

and `EDIT`

, and these lines are then inserted into the table (saved in a text with `ADISPLAY`

) using `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.

### Action with `RESTRICT`

If the `Y`

variate is restricted, the procedure will use only the units not excluded by the restriction.

### Reference

Butler, R.C. & Brain, P. (1993). Nonlinear Contrasts in `ANOVA`

. *Genstat Newsletter*, 29, 20-27.

### See also

Directives: `ANOVA`

, `FITCURVE`

, `FITNONLINEAR`

.

Commands for: Analysis of variance, Regression analysis.

### Example

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