Response-surface regression

Fit a linear model with a response-surface component, and produce appropriate analyses and summaries.

Usage

rsm (formula, data, ...)

# S3 method for rsm
summary(object, adjust = rev(p.adjust.methods), ...)
# S3 method for summary.rsm
print(x, ...)

# S3 method for rsm
codings(object)

loftest (object)

canonical (object, threshold = 0.1*max.eigen)
xs (object, ...)

Arguments

formula: Formula to pass to lm. The model must include at least one FO(), SO(), TWI(), or PQ() term to define the response-surface portion of the model.
data: data argument to pass to lm.
...: In rsm, arguments that are passed to lm, summary.lm, or canonical, as appropriate. In summary, and print, additional arguments are passed to their generic methods.
object: An object of class rsm
adjust: Adjustment to apply to the P values in the coefficient matrix, chosen from among the available p.adjust methods in the stats package. The default is "none".
threshold: Threshold for canonical analysis -- see "Canonical analysis" below.
x: An object produced by summary

Details

In rsm, the model formula must contain at least an FO term; optionally, you can add one or more TWI() terms and/or a PQ() term. All variables that appear in TWI or PQ must be included in FO. For convenience, specifying SO() is the same as including FO(), TWI(), and PQ(), and is the safe, preferred way of specifying a full second-order model.

The variables in FO comprise the variables to consider in response-surface methods. They need not all appear in TWI and PQ terms; and more than one TWI term is allowed. For example, the following two model formulas are equivalent:


resp ~ Oper + FO(x1,x2,x3,x4) + TWI(x1,x2,x3) + TWI(x2,x3,x4) + PQ(x1,x3,x4)
resp ~ Oper + FO(x1,x2,x3,x4) + TWI(formula = ~x1*x2*x3 + x2*x3*x4) + PQ(x1,x3,x4)

The first version, however, creates duplicate x2:x3 terms -- which rsm can handle but there may be warning messages if it is subsequently used for predictions or plotted in contour.lm.

In summary.rsm, any ... arguments are passed to summary.lm, except for threshold, which is passed to canonical.

Value

rsm returns an rsm object, which is a lm object with additional members as follows:

order: The order of the model: 1 for first-order, 1.5 for first-order plus interactions, or 2 for a model that contains square terms.
b: The first-order response-surface coefficients.
B: The matrix of second-order response-surface coefficients, if present.
labels: Labels for the response-surface terms. These make the summary much more readable.
coding: Coding formulas, if provided in the codings argument or if the data argument passed to lm is a coded.data object.

Summary and print methods

The print method for rsm objects just shows the call and the regression coefficints.

The summarymethod for rsm objects returns an object of class summary.rsm, which is an extension of the summary.lm class with these additional list elements:

sa: Unit-length vector of the path of steepest ascent (first-order models only).
canonical: Canonical analysis (second-order models only) from canonical
lof: ANOVA table including lack-of-fit test.
coding: Coding formulas in parent rsm object.

Canonical analysis and stationary point

canonical returns a list with elements xs, the stationary point, and eigen, the eigenanalysis of the matrix B of second-order coefficients. Any eigenvalues less than threshold are taken to be zero, and a message is displayed. If this happens, the stationary point is determined using only the surviving eigenvectors, and stationary ridges or valleys are assumed to exist in their corresponding canonical directions. The default threshold is one tenth of the maximum eigenvalue, internally named max.eigen. Setting a small threshold may move the stationary point much farther from the origin.

When uncoded data are used, the canonical analysis and stationary point are not very meaningful and those results should probably be ignored. See vignette("rsm") for more details.

The function xs returns just the stationary point.

Other functions

loftest returns an anova object that tests the fitted model against a model that interpolates the means of the response-surface-variable combinations.

codings returns a list of coding formulas if the model was fitted to coded.data, or NULL otherwise.

emmeans support

Support is provided for the emmeans package: its emmeans and related functions work with special provisions for models fitted to coded data. The optional mode argument can have values of "asis" (the default), "coded", or "decoded". The first two are equivalent and simply return LS means based on the original model formula and the variables therein (raw or coded), without any conversion. When coded data were used and the user specifies mode = "decoded", the user must specify results in terms of the decoded variables rather than the coded ones. See the illustration in the Examples section.

References

Lenth RV (2009) ``Response-Surface Methods in R, Using rsm'', Journal of Statistical Software, 32(7), 1--17. doi:10.18637/jss.v032.i07

Author

Russell V. Lenth

Examples

library(rsm)
CR <- coded.data (ChemReact, x1~(Time-85)/5, x2~(Temp-175)/5)

### 1st-order model, using only the first block
CR.rs1 <- rsm (Yield ~ FO(x1,x2), data=CR, subset=1:7) 
summary(CR.rs1)
#> 
#> Call:
#> rsm(formula = Yield ~ FO(x1, x2), data = CR, subset = 1:7)
#> 
#>             Estimate Std. Error  t value  Pr(>|t|)    
#> (Intercept) 82.81429    0.54719 151.3456 1.143e-08 ***
#> x1           0.87500    0.72386   1.2088    0.2933    
#> x2           0.62500    0.72386   0.8634    0.4366    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Multiple R-squared:  0.3555,	Adjusted R-squared:  0.0333 
#> F-statistic: 1.103 on 2 and 4 DF,  p-value: 0.4153
#> 
#> Analysis of Variance Table
#> 
#> Response: Yield
#>             Df Sum Sq Mean Sq F value  Pr(>F)
#> FO(x1, x2)   2 4.6250  2.3125  1.1033 0.41534
#> Residuals    4 8.3836  2.0959                
#> Lack of fit  2 8.2969  4.1485 95.7335 0.01034
#> Pure error   2 0.0867  0.0433                
#> 
#> Direction of steepest ascent (at radius 1):
#>        x1        x2 
#> 0.8137335 0.5812382 
#> 
#> Corresponding increment in original units:
#>     Time     Temp 
#> 4.068667 2.906191 
#> 

### 2nd-order model, using both blocks
CR.rs2 <- rsm (Yield ~ Block + SO(x1,x2), data=CR) 
summary(CR.rs2)
#> 
#> Call:
#> rsm(formula = Yield ~ Block + SO(x1, x2), data = CR)
#> 
#>              Estimate Std. Error  t value  Pr(>|t|)    
#> (Intercept) 84.095427   0.079631 1056.067 < 2.2e-16 ***
#> BlockB2     -4.457530   0.087226  -51.103 2.877e-10 ***
#> x1           0.932541   0.057699   16.162 8.444e-07 ***
#> x2           0.577712   0.057699   10.012 2.122e-05 ***
#> x1:x2        0.125000   0.081592    1.532    0.1694    
#> x1^2        -1.308555   0.060064  -21.786 1.083e-07 ***
#> x2^2        -0.933442   0.060064  -15.541 1.104e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Multiple R-squared:  0.9981,	Adjusted R-squared:  0.9964 
#> F-statistic: 607.2 on 6 and 7 DF,  p-value: 3.811e-09
#> 
#> Analysis of Variance Table
#> 
#> Response: Yield
#>             Df Sum Sq Mean Sq   F value    Pr(>F)
#> Block        1 69.531  69.531 2611.0950 2.879e-10
#> FO(x1, x2)   2  9.626   4.813  180.7341 9.450e-07
#> TWI(x1, x2)  1  0.063   0.063    2.3470    0.1694
#> PQ(x1, x2)   2 17.791   8.896  334.0539 1.135e-07
#> Residuals    7  0.186   0.027                    
#> Lack of fit  3  0.053   0.018    0.5307    0.6851
#> Pure error   4  0.133   0.033                    
#> 
#> Stationary point of response surface:
#>        x1        x2 
#> 0.3722954 0.3343802 
#> 
#> Stationary point in original units:
#>      Time      Temp 
#>  86.86148 176.67190 
#> 
#> Eigenanalysis:
#> eigen() decomposition
#> $values
#> [1] -0.9233027 -1.3186949
#> 
#> $vectors
#>          [,1]       [,2]
#> x1 -0.1601375 -0.9870947
#> x2 -0.9870947  0.1601375
#> 
#> 

### Example of a rising-ridge situation from Montgomery et al, Table 6.2
RRex <- ccd(Response ~ A + B, n0 = c(0, 3), alpha = "face", 
            randomize = FALSE, oneblock = TRUE)
RRex$Response <- c(52.3, 5.3, 46.7, 44.2, 58.5, 33.5, 32.8, 49.2, 49.3, 50.2, 51.6)
RRex.rsm <- rsm(Response ~ SO(A,B), data = RRex)
canonical(RRex.rsm)  # rising ridge is detected
#> Near-stationary-ridge situation detected -- stationary point altered
#>  Change 'threshold' if this is not what you intend
#> $xs
#>          A          B 
#> -0.2928046  0.4526154 
#> 
#> $eigen
#> eigen() decomposition
#> $values
#> [1]   0.00000 -12.70637
#> 
#> $vectors
#>         [,1]       [,2]
#> A -0.8396245 -0.5431673
#> B -0.5431673  0.8396245
#> 
#> 
canonical(RRex.rsm, threshold = 0)  # xs is far outside of the experimental region
#> $xs
#>         A         B 
#> -5.176505 -2.706733 
#> 
#> $eigen
#> eigen() decomposition
#> $values
#> [1]  -0.509419 -12.706370
#> 
#> $vectors
#>         [,1]       [,2]
#> A -0.8396245 -0.5431673
#> B -0.5431673  0.8396245
#> 
#> 

if (FALSE) {
# Illustration of emmeans support
emmeans::emmeans(CR.rs2, ~ x1 * x2, mode = "coded", 
        at = list(x1 = c(-1, 0, 1), x2 = c(-2, 2)))
        
# The following will yield the same results, but based on the decoded data
emmeans::emmeans(CR.rs2, ~ Time * Temp, mode = "decoded", 
        at = list(Time = c(80, 85, 90), Temp = c(165, 185)))
}