Forms equations for single-factor polynomial contrasts fitted by `ANOVA`

(R.W. Payne).

### Options

`PRINT` = string token |
Whether to print the equation of the polynomial (`equation` ); default `equa` |
---|---|

`SAVE` = `ANOVA` save structure |
Save structure (from `ANOVA` ) to provide details of the analysis from which the equations are to be formed; default uses the save structure from the most recent `ANOVA` |

### Parameters

`TERMS` = formula |
Model terms whose polynomial equations are required |
---|---|

`COEFFICIENTS` = pointers |
Saves the coefficients of each polynomial |

### Description

The `ANOVA`

directive fits polynomial contrasts of the effects of a factor by forming orthogonal polynomials (see Section 4.5 of the *Guide to the Genstat Command Language, Part 2 Statistics*). This allows the sums of squares for the factor to be partitioned into the amount that can be explained by a linear relationship, then the extra amount that can be explained if the relationship is quadratic, then the extra amount given by a cubic relationship, and so on. As a result, though, the estimates that are produced by `ANOVA`

are the regression coefficients of the orthogonal polynomials, not the coefficients of the polynomial equation. `ANOVA`

can also estimate interactions between the (orthogonal) polynomial contrasts and other factors.

The polynomial coefficients can, however, be obtained using procedure `APOLYNOMIAL`

. The `TERMS`

parameter specifies the treatment terms whose equations are required. Each term must contain no more than one factor with a polynomial function (`POL`

or `POLND`

), and no factors with regression or comparison functions (`REG`

, `REGND`

or `COMPARISON`

); otherwise it is ignored. If `TERMS`

is not set, `APOLYNOMIAL`

takes the full treatment model (see `TREATMENTSTRUCTURE`

).

`APOLYNOMIAL`

usually prints the equation, but you can set option `PRINT=*`

to suppress this. The `COEFFICIENTS`

parameter can supply a pointer to save the coefficients of the equations. The pointer will contain a pointer for each term. These are given suffixes 0 upwards, corresponding to the powers of the factor in each polynomial.

By default, the equation is formed for the contrasts estimated in the most recent analysis performed by `ANOVA`

, but the `SAVE`

option can be used to supply the save structure from an earlier analysis to use instead.

Option: `PRINT`

.

Parameters: `FACTOR`

, `LEVELS`

, `GROUPS`

, `COEFFICIENTS`

, `SAVE`

.

### Method

`APOLYNOMIAL`

first needs to duplicate the process of forming the orthogonal polynomials, regressing each power of the factor levels on the lower powers. Suppose, for example, a fourth-order polynomial was fitted, and the orthogonal polynomials were given by

p_{1} = *y*

p_{2} = *y*^{2} – *b*_{21} × *y*

p_{3} = *y*^{3} – *b*_{31} × *y* – *b*_{32} × *y*^{2}

p_{4} = *y*^{4} – *b*_{41} × *y* – *b*_{42} × *y*^{2} – *b*_{41} × *y*^{3}

and that the estimated coefficients of the orthogonal polynomials were *e*_{1}, *e*_{2}, *e*_{3} and *e*_{4}. The coefficients of the polynomial equation are then calculated as

*c*_{1} = *e*_{1} – *b*_{21} × *e*_{2} – *b*_{31} × *e*_{3} – *b*_{41} × *e*_{4}

*c*_{2} = *e*_{2} – *b*_{32} × *e*_{3} – *b*_{42} × *e*_{4}

*c*_{3} = *e*_{3} – *b*_{43} × *e*_{4}

*c*_{4} = *e*_{4}

### See also

Directives: `ANOVA`

, `TREATMENTSTRUCTURE`

.

Procedure: `ADPOLYNOMIAL`

.

Commands for: Analysis of variance.

### Example

CAPTION 'APOLYNOMIAL example'; STYLE=meta FACTOR [NVALUES=72; LEVELS=6] Blocks & [LEVELS=3] Wplots & [LEVELS=4] Subplots GENERATE Blocks,Wplots,Subplots FACTOR [LABELS=!T('0 cwt','0.2 cwt','0.4 cwt','0.6 cwt')] Nitrogen & [LABELS=!T(Victory,'Golden rain',Marvellous)] Variety VARIATE Yield; DECIMALS=2; EXTRA=' of oats in cwt. per acre' READ [SERIAL=yes] Nitrogen,Variety,Yield 4 3 2 1 1 2 4 3 1 2 3 4 3 1 2 4 4 1 2 3 2 1 3 4 2 3 4 1 4 2 3 1 1 4 2 3 3 4 1 2 1 3 4 2 2 3 4 1 4 1 3 2 3 4 1 2 3 4 2 1 3 1 4 2 4 3 1 2 1 2 3 4 : 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 : 156 118 140 105 111 130 174 157 117 114 161 141 104 70 89 117 122 74 89 81 103 64 132 133 108 126 149 70 144 124 121 96 61 100 91 97 109 99 63 70 80 94 126 82 90 100 116 62 96 60 89 102 112 86 68 64 132 124 129 89 118 53 113 74 104 86 89 82 97 99 119 121 : " Convert yields to cwt per acre." CALCULATE Yield=(Yield*80)/(112*4) " Define the treatment structure: factorial effects of V and N." TREATMENTS Variety*Nitrogen " Subplots nested within whole-plots nested within blocks." BLOCK Blocks/Wplots/Subplots VARIATE [VALUES=0,0.2,0.4,0.6] Nitlev " Polynomial effects of nitrogen to order 2 (lin. and quad.)." TREATMENTS POL(Nitrogen;2;Nitlev) * Variety " Analysis printing contrasts to order 4 and the aov table." ANOVA [PRINT=aov,contrasts,effects] Yield APOLYNOMIAL Nitrogen + Variety.Nitrogen