Plots effects from two-level designs with robust s.e. estimates (Eric D. Schoen & Enrico A.A. Kaul).
Options
PRINT = string tokens |
Which ANOVA output to print, as in ADISPLAY; default aovt, effe |
|---|---|
CHANNEL = scalar |
What channel to use for anova and line-printer output; default * i.e. the current output channel |
FACTORIAL = scalar |
Limit for factorial expansion of TREATMENT formula; default 3 |
STRATUM = formula |
Error strata from which Yates effects are to be plotted; if unset, plots are made for all the strata |
GRAPHICS = string token |
What type of graphics (highresolution, lineprinter); default high |
TITLE = strings |
Separate titles for each of the plots |
METHOD = string token |
Whether to make half-Normal or Normal plots (halfnormal, normal); default half |
ROBUSTNESS = string token |
Robustness of scale estimators against contamination with active effects (low, medium, high); default medi |
ALPHALEVEL = scalar |
Type I error (0.20, 0.15, 0.10, 0.05, 0.01); default 0.05 |
EXCLUDE = scalars |
How many of the largest effects to withhold from each of the half-Normal plots; default 0 |
Parameters
Y = variates |
Data to be analysed |
|---|---|
EFFECTS = pointers |
To save a variate for each error stratum containing the (sorted) Yates effects estimated there |
SE = pointers |
To save a scalar with the standard error of the Yates effects for each error stratum |
SIGNIFICANT = pointers |
To save formulae containing the significant Yates effects in each stratum |
Description
Daniel (1959) shows how contrasts from two-level experiments in single or fractional replication can be evaluated through half-Normal plotting. Box et al. (1978) emphasize Normal plotting of the Yates effects. They suggest making separate plots for each error stratum. The Yates definition ensures that the effects from the same error stratum share a common variance. When there is sparsity of effects and Normality of error, most effects will come from a Normal distribution with zero mean and unknown variance. Inactive effects, plotted against quantiles of the Normal or half-Normal distribution, are roughly on a straight line through the origin. Effects not compatible with this line are designated active. Thus (half-)Normal plots will separate the few active effects from the inactive ones.
A well-known problem with the technique is the subjectivity as to which effects constitute the null-line. Many authors, therefore, have developed procedures for getting robust estimates of the standard errors of the Yates effects from unreplicated two-level experiments, see Haaland & O’Connell (1995) for an overview. Based on simulation results for 24 experiments (15 effects in the plot) the latter authors recommend three estimators according to a-priori ideas on the likely number of active effects (1-3, 4-6, and 7-8, respectively). The estimators are formed by (1) calculating an initial estimator of the standard error as a quantile of the full set of effects, multiplied with a consistency constant determined from the Normal distribution; (2) stripping of potential active effects by retaining only effects smaller than a constant times the initial scale estimate; (3) multiplying some function of the remaining effects with a simulated consistency constant. One of the three recommended estimators is based on the median of the full set and the sum of squares of the retained effects; it is called the Adaptive Standard Error (ASE). The other two estimators are based on the median and the 45th percentile, respectively, of the full set; these are Pseudo Standard Errors (PSE). Both use the median of the retained effects. In general, ASE is less robust against contamination with active effects than PSE, because it uses all the effects below the cut-off point. The median-based PSE is obviously less robust than the PSE based on the 45th percentile.
Haaland & O’Connell (1995) suggest judging t-values from the effects and the calculated scale estimate against critical values determined by simulation. They present consistency constants for two of the recommended estimators and critical values for one of them, each for 7, 11, 15, 17, 23 and 31 effects, respectively. We have extended their results to the whole range from 7 up to 127 effects and all three estimators.
The treatment effects to be studied should be specified using the TREATMENTSTRUCTURE directive before using A2PLOT. They are grouped according to the error strata as specified by a previous BLOCKSTRUCTURE statement. Normal or half-Normal plots, according to the METHOD option, are made in either lineprinter or high-resolution quality (option GRAPHICS). By default plots are made for each error stratum. Alternatively, option STRATUM can be set to a formula defining the strata from which the Yates effects are to be plotted. The EXCLUDE option specifies the number of largest effects to be exclude from half-Normal plots (the option does not work with Normal plots). The titles of the plots can be provided using option TITLE. Setting METHOD=* suppresses the plots. Options FACTORIAL, PRINT and CHANNEL, are as in ADISPLAY. Note, however, that effects are printed as Yates effects, and that CHANNEL also controls the line-printer graphics.
When the number of effects in the plot is in the range 7 to 127, robust estimators are calculated for the standard error of the effects. The robustness of the estimators against contamination with active effects is specified through option ROBUSTNESS. A vertical line in the plot indicates the least significant Yates effect (LSE). The type I error is controlled by option ALPHALEVEL. Effects larger than the LSE are labelled in the plot.
The data variates are specified using the Y parameter. The EFFECTS parameter can save a pointer holding a variate of effects, sorted from small to large, for each error stratum. Effects are either the usual Yates effects (METHOD=normal) or their absolute values (METHOD=halfnormal). Parameter SIGNIFICANT can save a formula with the joint significant effects of all the strata. Parameter SE holds scalars with the standard errors of the effects in the respective strata.
Options: PRINT, CHANNEL, FACTORIAL, STRATUM, GRAPHICS, TITLE, METHOD, ROBUSTNESS, ALPHALEVEL, EXCLUDE.
Parameters: Y, EFFECTS, SE, SIGNIFICANT.
Method
A2PLOT accesses the current BLOCKSTRUCTURE and TREATMENTSTRUCTURE settings using the GET directive. If the STRATUM option is unset, separate plots for each of the strata are to be produced. A2PLOT checks, therefore, whether all strata are set explicitly. If this is not the case it augments the current BLOCKSTRUCTURE with a bottom stratum using procedure AFUNITS. If no BLOCKSTRUCTURE is set, it generates an explicit Units stratum and sets the BLOCKSTRUCTURE and STRATUM options to this stratum.
Yates effects for each stratum are saved using AKEEP. They are ordered and plotted against either Normal or half-Normal quantiles. Normal quantiles are calculated as
qi = NED( (i-0.375) / (n+0.25) ) i=1…n
Half-Normal quantiles are calculated as
qi = NED( 0.5 + 0.5 × (i-0.375) / (n+0.25) ) i=1…n
For ROBUSTNESS=low, ASE based standard errors are calculated with the initial standard error calculated from the median of all effects, a cut-off of 2.5 times this value, and a final standard error from the sum of squares of the remaining effects. For ROBUSTNESS=medium, PSE based standard errors are calculated with the same cut-off as for ASE and a final standard error is calculated from the median of the remaining effects. For ROBUSTNESS=high, PSE based standard errors are calculated using the 45th percentile instead of the median for the initial estimate, and 1.25 instead of 2.5 as a multiplication factor to establish the cut-off. The final estimate uses the median of the retained effects.
Significant Yates effects are labelled in the half-Normal plots using the factor names from the TREATMENT statement.
Acknowledgements
The authors thank Peter Lane for suggesting and sketching procedure _A2PL_EXPAND.
Action with RESTRICT
AFUNITS (which may be called by A2PLOT if the STRATUM option is unset and no explicit bottom error stratum is specified in the current BLOCKSTRUCTURE setting) requires that none of the blocking factors be restricted.
References
Box, G.E.P., W.G. Hunter & J.S. Hunter (1978), Statistics for Experimenters. New York, Wiley.
Daniel, C. (1959), Use of half-normal plots in interpreting factorial two-level experiments. Technometrics, 1, 311-342.
Haaland, P.D. & M.A. O’Connell (1995), Inference for effect-saturated fractional factorials. Technometrics, 37, 82-93.
See also
Commands for: Analysis of variance.
Example
CAPTION 'A2PLOT example',\
!t('A half-fraction of a 2**5 design;',\
'data from Box, Hunter, and Hunter (1978):',\
'Statistics for Experimenters, p. 379',\
'(normal plot on p. 380).'); STYLE=meta,plain
FACTOR [NVALUES=16; LEVELS=!(-1,1)] feedrt,catalyst,agitrt,temp,conc
GENERATE temp,agitrt,catalyst,feedrt
CALCULATE conc=feedrt*catalyst*agitrt*temp
VARIATE [VALUES=56,53,63,65,53,55,67,61,69,45,78,93,49,60,95,82] %react
TREATMENT feedrt*catalyst*agitrt*temp*conc
A2PLOT [TITLE='% reacted'; METHOD=normal; PRINT=effects] Y=%react
CAPTION !T('To demonstrate handling of various error strata',\
'interactions temp x agitrt and temp x catalyst',\
'are used to define four blocks.')
FACTOR [NVALUES=16; LEVELS=!(-1,1)] BD,CD
CALCULATE BD,CD = catalyst,agitrt * temp
FACTOR [NVALUES=16; LEVELS=!(-3,-1,1,3)] Blocks
CALCULATE Blocks = 2 * BD + CD
BLOCKS Blocks
TREATMENT feedrt * catalyst * agitrt * temp * conc
A2PLOT [TITLE='between blocks','within blocks';PRINT=effects]\
Y=%react; EFFECT=eff
PRINT eff[]