halfnormal: Creation of half normal effects plots and numeric methods for...
In DoE.base: Full Factorials, Orthogonal Arrays and Base Utilities for DoE Packages

halfnormal

R Documentation

Creation of half normal effects plots and numeric methods for significance assessment

Description

Generic function and methods for creating half normal effects plots

Usage

halfnormal(x, ...)
## Default S3 method:
halfnormal(x, labs=names(x), codes = NULL, pch = 1, cex.text = 1,
   alpha = 0.05, xlab = "absolute effects", large.omit = 0, plot=TRUE, 
   crit=NULL, ...)
## S3 method for class 'lm'
halfnormal(x, labs = NULL, code = FALSE, pch = NULL, cex.text = 1, 
   alpha = 0.05, xlab = "absolute coefficients", large.omit = 0, plot=TRUE, 
   keep.colons = !code, ME.partial = FALSE, 
   external.pe = NULL, external.center = FALSE, contr.center = "contr.poly", 
   pch.set = c(1, 16, 8), scl = NULL, method="Lenth", 
   legend=code, err.points=TRUE, err.line=TRUE, linecol="darkgray", linelwd=2, 
   ...)
## S3 method for class 'design'
halfnormal(x, response = NULL, labs = NULL, code = FALSE, pch = NULL, 
   cex.text = 1, 
   alpha = 0.05, xlab = "absolute coefficients", large.omit = 0, plot=TRUE, 
   keep.colons = !code, ME.partial = FALSE, 
   external.pe = NULL, external.center = FALSE, contr.center = "contr.poly", 
   pch.set = c(1, 16, 8), scl = NULL, method="Lenth", 
   legend=code, err.points=TRUE, err.line=TRUE, linecol="darkgray", linelwd=2, 
   ...)

ME.Lenth(b, simulated=TRUE, alpha=NULL)
CME.LW98(b, sterr, dfe, simulated=TRUE, alpha=NULL)
CME.EM08(b, sterr, dfe, simulated=TRUE, weight0=5, alpha=NULL)

Arguments

`x`	a numeric vector of effects, a linear model from experimental data, or an experimental design of class `design`
`labs`	effect labels; default labels: for the default method, names of the vector x, or `b1`, `b2`, ... for unnamed vectors; for classes `design` or `lm` taken from the linear model
`codes`	a vector with a code for each effect; the default `NULL` uses the `labs` values
`code`	a logical; `TRUE` implies that factor letters are used instead of factor codes, and that the default for default for `keep.colons` is changed to `FALSE`
`pch`	plot symbol; `NULL`, a number or a vector of plot symbol numbers or the same length as the effects in x; in the default method, a single number (default 1) implies that the given plotting symbol is used for for all points; for the other methods, the default `NULL` or a single number implies that pch.set is employed for lack of fit or pure error contrast points; for the non-default methods, a vector-valued `pch` will only rarely be useful (see Details section)
`cex.text`	factor to hand to `cex` argument for point labeling with function `text` and margin annotations with function `mtext`; for `mtext`, it is multiplied with `par("cex")`, in order to obtain the same size for point labels and the margin annotations.
`alpha`	number between 0 and 1: the significance level for labelling effects; for functions `ME.Lenth`, `CME.LW98` and `CME.EM08`, `alpha` can also be `NULL` or a numeric vector; for using the simulated critical bounds, all elements of alpha must be in 0.01,0.02,...,0.25
`xlab`	character string: the x axis label
`plot`	logical; if `FALSE`, plotting is suppressed
`large.omit`	integer number of largest effects to be omitted from plot and calculations in order to concentrate on the smaller effects; (note that the significance is also re-assessed; if that is undesirable, an explicit `crit` value can be specified from all coefficients, or `alpha` can be adjusted to reflect the same significant effects as with all coefficients)
`crit`	default `NULL`; not meant for the end user; allows the `method` option for linear models and experimental designs to choose alternatives to Lenth's method
`keep.colons`	if `TRUE`, the automatic effect labels contain colons for interactions
`ME.partial`	if `TRUE`, partial aliasing among main effects is permitted and will be orthogonalized away
`external.pe`	numeric vector with values from outside the experimental data for use in estimating the error variance
`external.center`	if `TRUE`, external values from `external.pe` are taken as center point values, and a nonlinearity check contrast is estimated from them
`contr.center`	contrasts used for external center points; `contr.poly` or `contr.XuWu`
`pch.set`	plot symbols used for experimental effects, automatically determined lack of fit contrasts or pure error effects
`scl`	squared column length to which the model matrix is normalized; default: number of experimental runs
`method`	the default `"Lenth"` applies Lenth's method to the combined set of effects including error contrasts (if any); the alternatives `"LW98"` or `"EM08"` apply the methods proposed by Larntz and Whitcomb (1998) or Edwards and Mee (2008) with weight0=5; if there is no pure error, method `"Lenth"` is always used, with a warning
`legend`	squared column length to which the model matrix is normalized; default: number of experimental runs
`err.points`	logical, default `TRUE`; determines, whether pure error points are added to the plot (lack-of-fit points are always added)
`err.line`	logical, default `TRUE`; determines, whether null line is added to the plot in case pure error points are available
`linecol`	specifies the color for the null line, if applicable
`linelwd`	specifies the width of the null line, if applicable
`response`	response for which the plot is to be created
`...`	further options to be handed to the `plot` function; among these, if options `col` and/or `cex` have an element for each effect, these are used in the expected order (first color refers to first element of x and so forth); this change was introduced in version 0.26-2 and causes an appropriate reordering in the actual plot function.
`b`	vector of coefficients
`simulated`	logical; if `FALSE`, the original critical values from Lenth 1989 are used, otherwise the methods use stored simulated values from a million simulation runs for significance levels of 0.01, 0.02, ..., to 0.25
`sterr`	a standard error for `b`, obtained from (a few, `dfe`) pure error degrees of freedom; the methods by Larntz and Whitcomb (1998) and Edwards and Mee (2008) combine this with Lenth's method
`dfe`	the number of pure error degrees of freedom on which `sterr` was based
`weight0`	a tuning parameter for the method by Edwards and Mee 2008; Edwards and Mee recommend to set this to 5

Details

Function halfnormal creates half normal effects plots with automatic effect labelling according to significance. It also prints the significant effects and creates an output object that contains only the vector if signifcant effects (for the default method) or in addition several further components (see section "Value"). Note: The methods for linear models and experimental designs plot absolute coefficients from a linear model (i.e. in case of 2-level factors with the usual -1/+1 coding, half of the absolute effects).

The methods for linear models and experimental designs allow to automatically create lack of fit and pure error contrasts to also be included in the plot, following an orthogonalization strategy similar to Section 5 in Langsrud (2001). Furthermore, they handle factors with more than two levels, and they handle partially aliased effects by orthogonalizing out previous effects from later effects in the model order (similar to what Langsrud 2001 proposed for multiple response variables); thus, the plots are order dependent in case of partial aliasing. The more severe the partial aliasing, the more drastic the difference between the different effect orders. Per default, main effects are required to be orthogonal; this can be changed via option ME.partial.

The functions ME.Lenth, CME.LW98 and CME.EM08 yield standard error estimates and critical values. For alpha in 0.01, 0.02, ..., 0.25, function ME.Lenth uses simulated critical values from a large number of simulations (1000000), if the number of effects is in 7 to 143. Functions CME.LW98 and CME.EM08 currently simulate critical values from 10000 simulation runs on the fly. If no simulated values are available or simulation has been switched off, the half-normal plotting routines will use the conservative t-values proposed by Lenth (1989) (ME.Lenth) or Larntz and Whitcomb (CME.LW98 and CME.EM08).

Vector valued entries for pch, col and cex are handled very specifically for the class lm and class design methods: They make the most sense if the model is already saturated: If no pure error effects have been automatically calculated, effects whose pch is identical to the third element of pch.set will be treated as pure error effects; this allows to manually code these effects.
Generally, vector-valued pch (and col and cex) must have as many elements as the final coefficients vector after augmenting the coefficients; the coefficient vector carries first the experimental coefficients, then the automatically calculated lack-of-fit coefficients, then the automatically calculated pure error coefficients, then lack-of fit coefficients from external replications, and finally the pure error coefficients from external replications. Even for err.points=FALSE, entries for all these elements are needed. The value for pch determines, which coefficients are considered pure error.

Value

The default method for halfnormal visibly returns a character vector of significant effects only. The methods for linear models and experimental designs invisibly return lists of nine elements:

`coef`	contains the estimated coefficients
`mm`	contains the model matrix
	after adjustment to equally scaled independent effects
`mod.effs`	the effects that are part of the model
`res`	list that indicates the effects (named vector of position numbers)
	that were projected out from any particular model effect (element name)
`LCs`	contains the coefficients of the linear combinations
	taken from the residuals after projecting out the effects
	listed in `res` from the original model matrix columns.
	Where `LCs` elements are `NULL`,
	the original effect completely disappeared
	because of complete confounding with previous effects.
`alpha`	contains the significance level
`method`	contains the method of significance assessment
`signif`	is a character vector of significant effects
`pchs`	is a numeric vector of plot character identifiers

The functions ME.Lenth, CME.LW98 and CME.EM08 each return lists of length 4 with an estimate for s0, PSE, ME and SME for Lenth's method or their respective modifications for the other two methods (called s0, CPSE, CME and CSME for CME.LW98 and Cs0, CPSE, CME and CSME for CME.EM08). The length of the (C)ME and (C)SME components depends on the length of alpha (default: 25 critical values for alphas from 0.25 to 0.01).

Note

If someone worked out how to modify symbol colors (option col) and/or sizes (option cex) for a version before 0.26-2, version 0.26-2 will mess up the order of the symbol colors and/or sizes. The benefit: colors and symbol sizes can now be specified in the natural order, see description of the ... argument.

Author(s)

Ulrike Groemping, Berliner Hochschule fuer Technik

References

Daniel, C. (1959) Use of Half Normal Plots in Interpreting Two Level Experiments. Technometrics 1, 311–340.

Daniel, C. (1976) Application of Statistics to Industrial Experimentation. New York: Wiley.

Edwards, D. and Mee, R. (2008) Empirically Determined p-Values for Lenth t Statistics. Journal of Quality Technology 40, 368–380.

Langsrud, O. (2001) Identifying Significant Effects in Fractional Factorial Multiresponse Experiments. Technometrics 43, 415–424.

Larntz, K. and Whitcomb, P. (1998) Use of replication in almost unreplicated factorials. Manuscript of a presentation given at the 42nd ASQ Fall Technical conference in Corning, New York. Downloaded 4/26/2013 at https://cdnm.statease.com/pubs/use-of-rep.pdf.

Lenth, R.V. (1989) Quick and easy analysis of unreplicated factorials. Technometrics 31, 469–473.

Examples

### critical values
b <- rnorm(12)
ME.Lenth(b)
ME.Lenth(b)$ME
ME.Lenth(b, alpha=0.22)
ME.Lenth(b, alpha=0.123)
ME.Lenth(b, alpha=0.12)
ME.Lenth(rnorm(144), alpha=0.1)
(mel <- ME.Lenth(b, alpha=0.1))
## assuming an external effect standard error based on 3df
## Not run: CME.EM08(b, 0.1, 3, alpha=0.1)    
         ## does not run for saving CRAN check time 
         ## much smaller than Lenth, if external 
         ## standard error much smaller than s0 (see mel)

### Half normal plots
## the default method
halfnormal(rnorm(15), labs=paste("b",1:15,sep=""))
b <- c(250, 8,7,6, rnorm(11))
halfnormal(b, labs=paste("b",1:15,sep=""))
halfnormal(b, labs=paste("b",1:15,sep=""), large.omit=1)

## the design method, saturated main effects design
plan <- oa.design(L12.2.11)
halfnormal(add.response(plan,rnorm(12)))

## the design method, saturated main effects design, 
## partial aliasing due to a missing value
y <- c(NA, rnorm(11))
## the following line would yield an error, because there is even 
## complete aliasing among main effects: 
## Not run: halfnormal(lm(y~., add.response(plan, y)), ME.partial=TRUE)
## this can only be helped by omitting a main effect from the model;
## afterwards, there is still partial aliasing,
## which must be explicitly permitted by the ME.partial option:
halfnormal(lm(y~.-D, add.response(plan, y)), ME.partial=TRUE)

## the linear model method
yc <- rnorm(12)
## partial aliasing only
halfnormal(lm(yc~A+B+C+D+E+F+G+H+J+A:B, plan))
## both partial (A:B) and complete (E:F) aliasing are present
halfnormal(lm(yc~A+B+C+D+E+F+G+H+J+A:B+E:F, plan))
## complete aliasing only because of the missing value in the response
halfnormal(lm(y~A+B+C+D+E+F+G+H+J+A:B+E:F, plan),ME.partial=TRUE)
## omit a large dominating effect

halfnormal(lm(y~A+B+C+D+E+F+G+H+J+A:B+E:F, plan),ME.partial=TRUE)


## a regular fractional factorial design with center points
y20 <- rnorm(20)
## Not run: halfnormal(lm(y20~.^2, FrF2(16,7,ncenter=4)))

DoE.base documentation built on Nov. 15, 2023, 1:06 a.m.