Description Usage Arguments Details Value Summary and print methods Canonical analysis and stationary point Other functions emmeans support Author(s) References See Also Examples
Fit a linear model with a responsesurface component, and produce appropriate analyses and summaries.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 
formula 
Formula to pass to 
data 

... 
In 
object 
An object of class 
adjust 
Adjustment to apply to the P values in the coefficient matrix, chosen from among the available 
threshold 
Threshold for canonical analysis – see "Canonical analysis" below. 
x 
An object produced by 
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 secondorder model.
The variables in FO
comprise the variables to consider in responsesurface 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:
1 2 
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
.
rsm
returns an rsm
object, which is a lm
object with
additional members as follows:
order 
The order of the model: 1 for firstorder, 1.5 for firstorder plus interactions, or 2 for a model that contains square terms. 
b 
The firstorder responsesurface coefficients. 
B 
The matrix of secondorder responsesurface coefficients, if present. 
labels 
Labels for the responsesurface terms. These make the summary much more readable. 
coding 
Coding formulas, if provided in the 
The print
method for rsm
objects just shows the call and the regression
coefficints.
The summary
method for rsm
objects returns an object of class
summary.rsm
, which is an extension of the summary.lm
class with these additional list elements:
Unitlength vector of the path of steepest ascent (firstorder models only).
Canonical analysis (secondorder models only) from canonical
ANOVA table including lackoffit test.
Coding formulas in parent rsm
object.
Its print
method shows the regression summary,
followed by an ANOVA and lackoffit test.
For firstorder models, it shows the direction of
steepest ascent (see steepest
), and for secondorder models, it shows the canonical analysis of the
response surface.
canonical
returns a list with elements xs
, the stationary point, and eigen
, the eigenanalysis of the matrix B of secondorder 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.
loftest
returns an anova
object that tests the fitted model against a model
that interpolates the means of the responsesurfacevariable combinations.
codings
returns a list
of coding formulas if the model was fitted to
coded.data
, or NULL
otherwise.
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.
Russell V. Lenth
Lenth RV (2009) “ResponseSurface Methods in R, Using rsm”, Journal of Statistical Software, 32(7), 1–17. doi: 10.18637/jss.v032.i07
FO
, SO
,
lm
, summary
, coded.data
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  library(rsm)
CR < coded.data (ChemReact, x1~(Time85)/5, x2~(Temp175)/5)
### 1storder model, using only the first block
CR.rs1 < rsm (Yield ~ FO(x1,x2), data=CR, subset=1:7)
summary(CR.rs1)
### 2ndorder model, using both blocks
CR.rs2 < rsm (Yield ~ Block + SO(x1,x2), data=CR)
summary(CR.rs2)
### Example of a risingridge 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
canonical(RRex.rsm, threshold = 0) # xs is far outside of the experimental region
## Not run:
# 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)))
## End(Not run)

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 1.143e08 ***
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 Rsquared: 0.3555, Adjusted Rsquared: 0.0333
Fstatistic: 1.103 on 2 and 4 DF, pvalue: 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
Call:
rsm(formula = Yield ~ Block + SO(x1, x2), data = CR)
Estimate Std. Error t value Pr(>t)
(Intercept) 84.095427 0.079631 1056.067 < 2.2e16 ***
BlockB2 4.457530 0.087226 51.103 2.877e10 ***
x1 0.932541 0.057699 16.162 8.444e07 ***
x2 0.577712 0.057699 10.012 2.122e05 ***
x1:x2 0.125000 0.081592 1.532 0.1694
x1^2 1.308555 0.060064 21.786 1.083e07 ***
x2^2 0.933442 0.060064 15.541 1.104e06 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Multiple Rsquared: 0.9981, Adjusted Rsquared: 0.9964
Fstatistic: 607.2 on 6 and 7 DF, pvalue: 3.811e09
Analysis of Variance Table
Response: Yield
Df Sum Sq Mean Sq F value Pr(>F)
Block 1 69.531 69.531 2611.0950 2.879e10
FO(x1, x2) 2 9.626 4.813 180.7341 9.450e07
TWI(x1, x2) 1 0.063 0.063 2.3470 0.1694
PQ(x1, x2) 2 17.791 8.896 334.0539 1.135e07
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
$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
$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
x1 x2 emmean SE df lower.CL upper.CL
1 2 75.0 0.298 7 74.3 75.7
0 2 77.0 0.240 7 76.4 77.5
1 2 76.4 0.298 7 75.6 77.1
1 2 76.8 0.298 7 76.1 77.5
0 2 79.3 0.240 7 78.7 79.9
1 2 79.2 0.298 7 78.5 79.9
Results are averaged over the levels of: Block
Confidence level used: 0.95
Time Temp emmean SE df lower.CL upper.CL
80 165 75.0 0.298 7 74.3 75.7
85 165 77.0 0.240 7 76.4 77.5
90 165 76.4 0.298 7 75.6 77.1
80 185 76.8 0.298 7 76.1 77.5
85 185 79.3 0.240 7 78.7 79.9
90 185 79.2 0.298 7 78.5 79.9
Results are averaged over the levels of: Block
Confidence level used: 0.95
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