halfnormal: 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, ...)
## S3 method for class 'default':
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

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 o

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 ob

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 sim

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 ca

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 (199

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

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

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: ll{ 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).

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.

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 http://www.statease.com/pubs/use-of-rep.pdf. Lenth, R.V. (1989) Quick and easy analysis of unreplicated factorials. Technometrics 31, 469--473.

Examples

Run this code

### 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
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: 
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)
halfnormal(lm(y20~.^2, FrF2(16,7,ncenter=4)))

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