Plots effects from two-level designs with robust s.e. estimates (Eric D. Schoen & Enrico A.A. Kaul).
||What channel to use for anova and line-printer output; default
||Limit for factorial expansion of
||Error strata from which Yates effects are to be plotted; if unset, plots are made for all the strata|
||What type of graphics (
||Separate titles for each of the plots|
||Whether to make half-Normal or Normal plots (
||Robustness of scale estimators against contamination with active effects (
||Type I error (0.20, 0.15, 0.10, 0.05, 0.01); default 0.05|
||How many of the largest effects to withhold from each of the half-Normal plots; default 0|
||Data to be analysed|
||To save a variate for each error stratum containing the (sorted) Yates effects estimated there|
||To save a scalar with the standard error of the Yates effects for each error stratum|
||To save formulae containing the significant Yates effects in each stratum|
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
METHOD=* suppresses the plots. Options
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 (
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.
A2PLOT accesses the current
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
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
NED( (i-0.375) / (n+0.25) ) i=1…n
Half-Normal quantiles are calculated as
NED( 0.5 + 0.5 × (i-0.375) / (n+0.25) ) i=1…n
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
The authors thank Peter Lane for suggesting and sketching procedure
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.
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.
Commands for: Analysis of variance.
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