Finds the replication to detect a treatment effect or contrast (R.W. Payne & P. Brain).

### Options

`PRINT` = string tokens |
Prints the replication or produces a printed summary of the power etc for the various amounts of replication (`power` , `replication` ); default `powe` , `repl` |
---|---|

`TERM` = formula |
Treatment term to be assessed in the analysis |

`REPLICATES` = factor |
Factor identifying the replication in the design |

`MINREPLICATION` = scalar |
Minimum number of replicates to try; default 2 |

`MAXREPLICATION` = scalar |
Maximum feasible number of replicates; default `*` i.e. no limit |

`TREATMENTSTRUCTURE` = formula |
Treatment structure of the design; determined automatically from an `ANOVA` save structure if `TREATMENTSTRUCTURE` is unset or if `SAVE` is set |

`BLOCKSTRUCTURE` = formula |
Block structure of the design; determined automatically from an `ANOVA` save structure if `BLOCKSTRUCTURE` is unset or if `SAVE` is set |

`COMPONENTS` = variate or scalar |
Variate of variance components of all the terms in the block structure or, if `TERM` is estimated in the final stratum of the design, scalar containing only the variance component of the final stratum itself; determined automatically (if possible) from an `ANOVA` save structure if unset |

`FACTORIAL` = scalar |
Limit on the number of factors in treatment terms; default 3 |

`PROBABILITY` = scalar |
Significance level at which the response is required to be detected (assuming a one-sided test); default 0.05 |

`POWER` = scalar |
The required power (i.e. probability of detection) of the test; default 0.9 |

`TMETHOD` = string token |
Type of test to be made (`onesided` , `twosided` , `equivalence` , `noninferiority` , `fratio` ); default `ones` |

`XCONTRASTS` = variate |
X-variate defining a a contrast to be detected |

`CONTRASTTYPE` = string token |
Type of contrast (`regression` , `comparison` ) default `rege` |

`SAVE` = asave |
`ANOVA` save structure to provide the information about the design |

### Parameters

`RESPONSE` = scalars |
Size of the difference or contrast between `TERM` effects that is to be detected |
---|---|

`NREPLICATES` = scalars |
Number of replicates required to detect `RESPONSE` |

### Description

When designing an experiment, it is often possible to vary the replication of the treatments. For example, in a randomized block design you can adjust the number of blocks, or in a design with no blocking structure you can choose how many units to allocate to each of the treatments.

To decide how many replicates to include, you need to specify the size of difference between treatment effects that you would like the design to be able to detect. The treatment term of interest is specified using the `TERM`

option of `ASAMPLESIZE`

, and the difference that you want to detect between its effects is given by the `RESPONSE`

parameter. As an alternative to detecting a difference between treatment effects, you can ask to detect a contrast, but here the treatment term must be a main effect (that is, `TERM`

must involve just one factor). The `XCONTRASTS`

option then specifies a variate containing the coefficients defining the contrast, and the `CONTRASTTYPE`

option indicates whether this is a regression contrast (as specified by the `REG`

function) or a comparison (as specified by `COMPARISON`

).

The `PROBABILITY`

option specifies the significance level that you will be using in the future analysis to detect the treatment difference (default 0.05, i.e. 5%). The `POWER`

option specifies the probability with which you want the experiment to be able to detect the difference (that is, the *power* of the test); by default this is 0.9 i.e. 90%. In the language of hypothesis testing, `PROBABILITY`

specifies the type I error rate, and `POWER`

specifies one minus the type II error rate. By default, `ASAMPLESIZE`

assumes a one-sided t-test is to be used, but you can set option `TMETHOD=twosided`

to take a two-sided t-test instead. Alternatively, if you set `TMETHOD=fratio`

, `ASAMPLESIZE`

takes `RESPONSE`

as the maximum difference between the effects of `TERM`

, and uses

an F-test.

Other settings of `TMETHOD`

enable you to test for equivalence or for non-inferiority. With equivalence (`TMETHOD=equivalence`

), `RESPONSE`

provides a threshold below which the treatments can be assumed to be equivalent. If the treatments have effects *e*_{1} and *e*_{2}, the null hypothesis that the treatments are not equivalent is that either

(*e*_{1} – *e*_{2}) ≤ –`RESPONSE`

or

(*e*_{1} – *e*_{2}) ≥ `RESPONSE`

with the alternative hypothesis that they are equivalent, i.e.

–`RESPONSE`

< (*e*_{1} – *e*_{2}) < `RESPONSE`

With non-inferiority (`TMETHOD=noninferiority`

), `RESPONSE`

again specifies the threshold for the effect of one treatment to be superior to another. So, for example, to demonstrate non-inferiority of treatment 1 compared to treatment 2, the null hypothesis becomes

(*e*_{1} – *e*_{2}) ≥ –`RESPONSE`

(which, in fact, represents a simple one-sided t-test).

To determine the replication, `ASAMPLESIZE`

needs to know the about the structure of the design, and the likely amount of variability. This is most easily obtained by taking the analysis of a design with similar units and the same block and treatment structures as those that are to be used in the new design. To do this, you should analyse the earlier set of data with the `ANOVA`

directive in the usual way. First define the strata (or error terms) for the design using the `BLOCKSTRUCTURE`

directive, and the treatment model to be fitted using the `TREATMENTSTRUCTURE`

directive. Then analyse the y-variate using the `ANOVA`

. Provided you do not give any other `ANOVA`

commands in the interim, `ASAMPLESIZE`

will pick up the information automatically from the save information held within Genstat about that analysis. Alternatively, you can save the information explicitly in an `ANOVA`

save structure, using the `SAVE`

parameter of `ANOVA`

, and then use this same save structure as the setting of the `SAVE`

option of `ASAMPLESIZE`

.

If you do not have a suitable earlier set of data, you should set up the design factors to contain the values required to define the units of the design for any convenient number of replicates. (It does not matter how many replicates you choose, as the form of the design should be the same in every replicate.) Then use the `TREATMENTSTRUCTURE`

and `BLOCKSTRUCTURE`

options of `ASAMPLESIZE`

to define the treatment model and the block model, and the `COMPONENTS`

option to specify the variance components of the strata. Note: if `TERM`

is estimated in the bottom (or final) stratum of the design, `COMPONENTS`

can be set to a scalar to specify only the variance component of this stratum – which is then equal to its residual mean square.

There is also the compromise possibility that you can take the information about the design and the block and treatment model from an `ANOVA`

save structure (generated for example by the analysis of an artificial data set), but use the `COMPONENTS`

option to specify different variance components from those in the analysis in the save structure.

The treatment terms to be included are controlled by the `FACTORIAL`

option. This sets a limit (by default 3) on the number of factors in a treatment term. Treatment terms containing more than that number are deleted.

Finally, you must set the `REPLICATES`

option to the factor in the block formula whose number of levels is to be increased or decreased to change the replication of the treatments. You can set the `MINREPLICATION`

option to indicate the minimum number of replicates to try; by default this is 2. You can use the `MAXREPLICATION`

option to define a maximum feasible number of replicates; by default this is no limit. The number of replicates that is required can be saved using the `NREPLICATES`

parameter.

The `PRINT`

option controls the printed output, with settings:

`power` |
prints a table summarising the situation for a range of numbers of replicates (defined by `MINREPLICATION` and `MAXREPLICATION` if set, otherwise set automatically to a range covering the required number of replicates) – the table contains the residual degrees of freedom, the residual mean square, the standard error of difference (sed), `RESPONSE` divided by the sed, the t-value for a difference of `RESPONSE` , and the detection probability (i.e. power) at the level defined by the `PROBABILITY` option; |
---|---|

`replication` |
prints the required replication. |

By default both are printed.

For example, the following program would determine the number of blocks required to detect a treatment difference of 3 in a randomized block design with an anticipated residual mean square of 2.5 in the final stratum `Block.Plot`

(i.e. within blocks); there is a single treatment factor `Treat`

with 3 levels. We first use `AGHIERARCHICAL`

to define the design for one replicate (or block), and then call `ASAMPLESIZE`

to discover how many blocks are actually needed.

`AGHIERARCHICAL [PRINT=*; ANALYSE=no; SEED=-1] Block,Plot;\`

` TREATMENTFACTORS=*,Treat; LEVELS=1,3`

`ASAMPLESIZE [PRINT=power,rep; TERM=Treat;\`

` REPLICATES=Block; TREATMENTSTRUCTURE=Treat;\`

` BLOCKSTRUCTURE=Block/Plot; COMPONENT=2.5]\`

` 1; NREPLICATES=Nrep`

As another example, suppose we wish to have a split-plot design, with block structure `Rep/Wplot/Subplot`

. The factor `Variety`

with 3 levels is applied to whole plots (and is thus estimated in the `Rep.Wplot`

stratum) and the factor `Nitrogen`

with 4 levels is applied to the sub-plots (and is thus estimated in the `Rep.Wplot.Subplot`

stratum). The variance components for `Rep`

, `Rep.Wplot`

and `Rep.Wplot.Subplot`

are anticipated to be 6, 3 and 5 respectively, and we wish to detect varietal differences of 3. Again we first define a split-plot with a single replicate, and then use `ASAMPLESIZE`

to find out how many reps we need.

`AGHIERARCHICAL [PRINT=*; ANALYSE=no; SEED=-1]\`

` Rep,Wplot,Subplot;\`

` TREATMENTFACTORS=*,Variety,Nitrogen;\`

` LEVELS=1,3,4`

`ASAMPLESIZE [PRINT=power,rep; TERM=Variety;\`

` REPLICATES=Rep;\`

` TREATMENTSTRUCTURE=Variety*Nitrogen;\`

` BLOCKSTRUCTURE=Rep/Wplot/Subplot;\`

` COMPONENTS=!(6,3,5)] 3; NREPLICATES=Nrep`

Options: `PRINT`

, `TERM`

, `REPLICATES`

, `MINREPLICATION`

, `MAXREPLICATION`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `COMPONENTS`

, `FACTORIAL`

, `PROBABILITY`

, `POWER`

, `TMETHOD`

, `XCONTRASTS`

, `CONTRASTTYPE`

, `SAVE`

.

Parameters: `RESPONSE`

, `NREPLICATES`

.

### Method

The standard error of difference between two treatment effects is

√( *s*^{2} × 2 / (*r* × *e*))

where *s*^{2} is the stratum variance of the stratum where the treatment term is estimated, *e* is the efficiency factor, and *r* is the replication of each effect. For a regression contrast the standard error is

√( *s*^{2} × 2 / (*r* × *sdiv* × *e*))

where *sdiv* is the sum of squares of the `XCONTRASTS`

variate, and for a comparison contrast the standard error is

√( *s*^{2} × *sdiv* / (*r* × *e*))

`ASAMPLESIZE`

assumes that the treatment effects have equal replication, and also that all the effects (or residuals) of each block term have equal replication.

The stratum variance can be calculated as the variance component of the stratum `S`

where the treatment term is estimated multiplied by the replication of its effects (residuals), plus the variance component of each stratum to which the stratum `S`

is marginal, again multiplied by the replication of its effects (residuals). See for example Payne & Tobias (1992).

Comparing the null hypothesis that the treatments are not equivalent, i.e.

(*m*_{1} – *m*_{2}) ≤ –*d*

or

(*m*_{1} – *m*_{2}) ≥ *d*

with the alternative hypothesis that they are equivalent, i.e.

–*d* < (*m*_{1} – *m*_{2}) < *d*

defines an *intersection-union* test, in which each component of the null hypothesis must be rejected separately. Here this implies performing two one-sided t-tests (this is known as a *TOST* procedure). If the significance level for the full test is to be α, each t-test must have significance level α (see Berger & Hsu 1996). To obtain a detection probability (or power) of (1 – β), each of the t-tests must have detection probabilities of (1 – β/2).

To demonstrate non-inferiority of treatment 1 compared to treatment 2, the null hypothesis is

(*m*_{1} – *m*_{2}) ≥ –*d*

This is equivalent to a one-sided t-test.

For the F-test, it is assumed that one effect will be -0.5 × `RESPONSE`

, another will be 0.5 × `RESPONSE`

, and the others will be zero. This gives the smallest sum of squares for any table of effects with a maximum pair-wise difference of `RESPONSE`

, which represents the most difficult case that needs to be detected.

### References

Berger, M.L. & Hsu, J.C. (1996). Bioequivalence trials, intersection-union tests and equivalence confidence sets. *Statistical Science*, 11, 283-319.

Payne, R.W. & Tobias, R.D. (1992). General balance, combination of information and the analysis of covariance. *Scandinavian Journal of Statistics*, 19, 3-23.

### See also

Directive: `ANOVA`

.

Procedures: `ADETECTION`

, `APOWER`

, `STTEST`

, `VSAMPLESIZE`

.

Commands for: Design of experiments, Analysis of variance.

### Example

CAPTION 'ASAMPLESIZE example',\ !t('1) determine the number of blocks required to detect a treatment',\ 'difference of 3 in a randomized block design with anticipated',\ 'residual mean square of 2.5 in the final stratum Block.Plot;',\ 'there is a single treatment factor Treat with 3 levels.');\ STYLE=meta,plain AGHIERARCHICAL [PRINT=*; ANALYSE=no; SEED=-1] Block,Plot;\ TREATMENTFACTORS=*,Treat; LEVELS=1,3 ASAMPLESIZE [PRINT=power,rep; TERM=Treat; REPLICATES=Block;\ TREATMENTSTRUCTURE=Treat; BLOCKSTRUCTURE=Block/Plot;\ COMPONENT=2.5] 3; NREPLICATES=Nrep PRINT Nrep; DECIMALS=0 CAPTION !t('2) Suppose we have a split-plot design with block structure',\ 'Rep/Wplot/Subplot. The factor Variety with 3 levels is applied',\ 'to whole plots (and so is estimated in the Rep.Wplot stratum)',\ 'and factor Nitrogen with 4 levels is applied to the sub-plots',\ '(and is thus estimated in the Rep.Wplot.Subplot stratum).',\ 'Variance components for Rep, Rep.Wplot and Rep.Wplot.Subplot',\ 'are anticipated to be 6, 3 and 5 respectively, and we wish to',\ 'detect varietal differences of 3.') AGHIERARCHICAL [PRINT=*; ANALYSE=no; SEED=-1] Rep,Wplot,Subplot;\ TREATMENTFACTORS=*,Variety,Nitrogen; LEVELS=1,3,4 ASAMPLESIZE [PRINT=power,rep; TERM=Variety; REPLICATES=Rep;\ TREATMENTSTRUCTURE=Variety*Nitrogen;\ BLOCKSTRUCTURE=Rep/Wplot/Subplot; COMPONENTS=!(6,3,5)]\ 3; NREPLICATES=Nrep PRINT Nrep; DECIMALS=0 CAPTION !t('3) Suppose we have a split-plot design as in (2) and we wish', 'to detect differences of 3 between the nitrogen levels.') ASAMPLESIZE [PRINT=power,rep; TERM=Nitrogen; REPLICATES=Rep; TREATMENTSTRUCTURE=Variety*Nitrogen; BLOCKSTRUCTURE=Rep/Wplot/Subplot; COMPONENTS=!(6,3,5)] 3; NREPLICATES=Nrep PRINT Nrep; DECIMALS=0 CAPTION !t('4) Suppose we have a split-plot design as in (2) and we wish',\ 'to detect a comparison of size 3 between the mean of the',\ 'first two varieties and the third variety.') ASAMPLESIZE [PRINT=power,rep; TERM=Variety; REPLICATES=Rep;\ TREATMENTSTRUCTURE=Variety*Nitrogen;\ BLOCKSTRUCTURE=Rep/Wplot/Subplot; COMPONENTS=!(6,3,5);\ XCONTRASTS=!(0.5,0.5,-1); CONTRASTTYPE=comparison]\ 3; NREPLICATES=Nrep PRINT Nrep; DECIMALS=0 CAPTION !t('5) Suppose we have a split-plot design as in (2) and we wish',\ 'to detect a regression contrast of size 5 between nitrogen',\ 'levels 0.1, 0.2, 0.3 and 0.4.') ASAMPLESIZE [PRINT=power,rep; TERM=Nitrogen; REPLICATES=Rep;\ TREATMENTSTRUCTURE=Variety*Nitrogen;\ BLOCKSTRUCTURE=Rep/Wplot/Subplot; COMPONENTS=!(6,3,5);\ XCONTRASTS=!(0.1,0.2...0.4)] 5; NREPLICATES=Nrep PRINT Nrep; DECIMALS=0