effect: Functions For Constructing Effect Displays

Description

Effect and effect construct an "eff" object for a term (usually a high-order term) in a linear model (fit by lm or gls) or generalized linear model (fit by glm), or an "effpoly" object for a term in a multinomial or proportional-odds logit model (fit respectively by multinom or polr), absorbing the lower-order terms marginal to the term in question, and averaging over other terms in the model. For multivariate linear models (of class "mlm", fit by lm), the function constructs a list of "eff" objects separately for the various response variables.

effect builds the required object by specifying explicity a focal term like "a:b" for an a by b interaction. Effect specifies the predictors in the term, for example c("a", "b"), rather than the term itself. Effect is consequently more flexible and robust than effect, and will succeed with some models for which effect fails. The effect function works by constructing a call to Effect.

The Effect and effect functions can also be used with many other models; see Effect.default and the Defining Effect Methods for Other Models vignette.

allEffects identifies all of the high-order terms in a model and returns a list of "eff" or "effpoly" objects (i.e., an object of type "efflist").

For information on computing and displaying predictor effects, see predictorEffect and plot.predictoreff.

For further information about plotting effects, see plot.eff.

Usage

effect(term, mod, vcov.=vcov, ...)
# S3 method for default
effect(term, mod, vcov.=vcov, ...)
Effect(focal.predictors, mod, ...)
# S3 method for lm
Effect(focal.predictors, mod, xlevels=list(),
    fixed.predictors, vcov. = vcov, se=TRUE,
    residuals=FALSE, quantiles=seq(0.2, 0.8, by=0.2),
    x.var=NULL,  ...,
    #legacy arguments:
    given.values, typical, offset, confint, confidence.level,
    partial.residuals, transformation)
# S3 method for multinom
Effect(focal.predictors, mod,
    xlevels=list(), fixed.predictors,
    vcov. = vcov, se=TRUE, ...,
    #legacy arguments:
    confint, confidence.level, given.values, typical)
# S3 method for polr
Effect(focal.predictors, mod,
    xlevels=list(), fixed.predictors,
    vcov.=vcov, se=TRUE, latent=FALSE, ...,
    #legacy arguments:
    confint, confidence.level, given.values, typical)
# S3 method for svyglm
Effect(focal.predictors, mod, fixed.predictors, ...)
allEffects(mod, ...)
# S3 method for default
allEffects(mod, ...)

Arguments

term

the quoted name of a term, usually, but not necessarily, a high-order term in the model. The term must be given exactly as it appears in the printed model, although either colons (:) or asterisks (*) may be used for interactions. If term is NULL, the function returns the formula for the linear predictor.

focal.predictors

a character vector of one or more predictors in the model in any order.

mod

an object of the appropriate class. If no method exists for that class, Effect.default will be called.

xlevels

this argument is used to set the number of levels for any focal predictor that is not a factor (where character and logical predictors are treated as factors). If xlevels=NULL, then each numeric predictor is represented by five values equally spaced over its range and then rounded to 'nice' numbers. If xlevels=n is an integer, then each numeric predictor is represented by n equally spaced values rounded to 'nice' numbers. More generally, xlevels can be a named list of values at which to set each numeric predictor. For example, xlevels=list(x1=c(2, 4, 7), x2=5) would use the values 2, 4 and 7 for the levels of x1, use 5 equally spaced levels for the levels of x2, and use the default for any other numeric predictors. If partial residuals are computed, then the focal predictor that is to appear on the horizontal axis of an effect plot is evaluated at 100 equally spaced values along its full range, and, by default, other numeric predictors are evaluated at the quantiles specified in the quantiles argument, unless their values are given explicitly in xlevels.

fixed.predictors

an optional list of specifications affecting the values at which fixed predictors for an effect are set, potentially including:

given.values: given.values="default" specifies averaging over levels of a non-focal factor using the default that weights levels of the factor by sample size. given.values="equal" uses unweighted averages over factor levels for non-focal factors. For finer control, the user can also provide a named numeric vector of weights for particular columns of the model matrix that correspond to regressors for the factor. Character and logical predictors are treated as factors. For example, for a factor X with three levels a, b and c, the regressors generated using the default parameterization for a factor will be named Xb and Xc as the regressor for level a is usually excluded. The specification given.values=c(Xb=1/2, Xc=1/4) would average over the levels of X with weight 1/2 for level b, 1/4 for c, and weight 1 = 1/2 - 1/4 = 1/4 for the baseline level a. Setting given.values=c(Xb=1) will fix X and level b.
typical: a function to be applied to the columns of the model matrix over which the effect is "averaged"; with the exception of the "svyglm" method, the default is mean. For"svyglm" objects, the default is to use the survey-design weighted mean.
apply.typical.to.factors: It generally doesn't make sense to apply typical values that aren't means (e.g., medians) to the columns of the model-matrix representing contrasts for factors. This value generally defaults to FALSE except for "svyglm" objects, for which the default is TRUE, using the the survey-design weighted mean.
offset: a function to be applied to the offset values (if there is an offset) in a linear or generalized linear model, or a mixed-effects model fit by lmer or glmer; or a numeric value, to which the offset will be set. The default is the mean function, and thus the offset will be set to its mean; in the case of "svyglm" objects, the default is to use the survey-design weighted mean. Note: Only offsets defined by the offset argument to lm, glm, svyglm, lmer, or glmer will be handled correctly; use of the offset function in the model formula is not supported.

vcov.

A function or the name of a function that will be used to get the estimated variance-covariance matrix of the estimated coefficients. This will ordinarily be the default, vcov, which will result in the function call vcov(mod) to get the variance-covariance matrix. You can use the name of any function that takes the model object as its first argument and returns an estimated sample covariance matrix, such as the hccm function in the car package, which returns a heteroscedasticity corrected estimate for a linear model.

TRUE (the default), FALSE, or a list with any or all of the following elements, controlling whether and how standard errors and confidence limits are computed for the effects: compute (default TRUE), whether or not to compute standard errors and confidence limits; level (default 0.95), confidence level for confidence limits; type, one of "pointwise" (the default), "Scheffe", or "scheffe", whether to compute confidence limits with specified coverage at each point for an effect or to compute limits for a Scheffe-type confidence envelope. For mer, merMod, and lme objects, the normal distribution is used to get confidence limits.

residuals

if TRUE, residuals for a linear or generalized linear model will be computed and saved; if FALSE (the default), residuals are suppressed. If residuals are saved, partial residuals are computed when the effect is plotted: see plot.eff and the vignette Effect Displays with Partial Residuals. This argument may also be used for mixed-effects and some other models.

quantiles

quantiles at which to evaluate numeric focal predictors not on the horizontal axis, used only when partial residuals are displayed; superceded if the xlevels argument gives specific values for a predictor.

x.var

the name or index of the numeric predictor to define the horizontal axis of an effect plot for a linear or generalized linear model; the default is NULL, in which case the first numeric predictor in the effect will be used if partial residuals are to be computed. This argument is intended to be used when residuals is TRUE; otherwise, the variable on the horizontal axis can be chosen when the effect object is plotted: see plot.eff.

latent

if TRUE, effects in a proportional-odds logit model are computed on the scale of the latent response; if FALSE (the default) effects are computed as individual-level probabilities and logits.

an object of class "eff", "effpoly", or "efflatent".

...

arguments to be passed down.

confint, confidence.level, given.values, typical, offset, partial.residuals, transformation

legacy arguments retained for backwards compatability; if present, these arguments take precedence over level element of the confint list argument and the given.values, typical, and offset elements of the fixed.predictors list argument; confint may be used in place of the se argument; partial.residuals may be used in place of the residuals argument. See LegacyArguments for details.

Value

For lm, glm,svyglm, mer and lme, effect and Effect return an "eff" object, and for multinom, polr, clm, clmm and clm2, an "effpoly" object, with the components listed below. For an "mlm" object with one response specified, an "eff" object is returned, otherwise an "efflist" object is returned, containing one "eff" object for each response.

term

the term to which the effect pertains.

formula

the complete model formula.

response

a character string giving the name of the response variable.

y.levels

(for "effpoly" objects) levels of the polytomous response variable.

variables

a list with information about each predictor, including its name, whether it is a factor, and its levels or values.

fit

(for "eff" objects) a one-column matrix of fitted values, representing the effect on the scale of the linear predictor; this is a ravelled table, representing all combinations of predictor values.

prob

(for "effpoly" objects) a matrix giving fitted probabilities for the effect for the various levels of the the response (columns) and combinations of the focal predictors (rows).

logit

(for "effpoly" objects) a matrix giving fitted logits for the effect for the various levels of the the response (columns) and combinations of the focal predictors (rows).

a data frame, the columns of which are the predictors in the effect, and the rows of which give all combinations of values of these predictors.

model.matrix

the model matrix from which the effect was calculated.

data

a data frame with the data on which the fitted model was based.

discrepancy

the percentage discrepancy for the `safe' predictions of the original fit; should be very close to 0. Note: except for gls models, this is now necessarily 0.

offset

value to which the offset is fixed; 0 if there is no offset.

model

(for "effpoly" objects) "multinom" or "polr", as appropriate.

vcov

(for "eff" objects) a covariance matrix for the effect, on the scale of the linear predictor.

(for "eff" objects) a vector of standard errors for the effect, on the scale of the linear predictor.

se.prob, se.logit

(for "effpoly" objects) matrices of standard errors for the effect, on the probability and logit scales.

lower, upper

(for "eff" objects) one-column matrices of confidence limits, on the scale of the linear predictor.

lower.prob, upper.prob, lower.logit, upper.logit

(for "effpoly" objects) matrices of confidence limits for the fitted logits and probabilities; the latter are computed by transforming the former.

confidence.level

for the confidence limits.

transformation

(for "eff" objects) a two-element list, with element link giving the link function, and element inverse giving the inverse-link (mean) function.

residuals

(working) residuals for linear or generalized linear models, to be used by plot.eff to plot partial residuals.

x.var

the name of the predictor to appear on the horizontal axis of an effect plot made from the returned object; will usually be NULL if partial residuals aren't computed.

family

for a "glm" model, the name of the distributional family of the model; for an "lm" model, this is "gaussian"; otherwise NULL. The family controls how partial residuals are smoothed in plots.

link

the value returned by family(mod). Down-stream methods may need the link, inverse link and derivative functions.

allEffects returns an "efflist" object, a list of "eff" or "effpoly" objects corresponding to the high-order terms of the model.

If mod is of class "poLCA" (from the poLCA package), representing a polytomous latent class model, effects are computed for the predictors given the estimated latent classes. The result is of class "eff" if the latent class model has 2 categories and of class "effpoly" with more than 2 categories.

Warnings and Limitations

The Effect function handles factors and covariates differently, and is likely to be confused if one is changed to the other in a model formula. Consequently, formulas that include calls to as.factor, factor, or numeric (as, e.g., in y ~ as.factor(income)) will cause errors. Instead, create the modified variables outside of the model formula (e.g., fincome <- as.factor(income)) and use these in the model formula.

Factors cannot have colons in level names (e.g., "level:A"); the effect function will confuse the colons with interactions; rename levels to remove or replace the colons (e.g., "level.A").

The functions in the effects package work properly with predictors that are numeric variables, factors, character variables, or logical variables; consequently, e.g., convert dates to numeric. Character predictors and logical predictors are treated as factors, the latter with "levels" "FALSE" and "TRUE".

Empty cells in crossed-factors are now permitted for "lm", "glm", and "multinom" models. For "multinom" models with two or more crossed factors with an empty cell, stacked area plots apparently do not work because of a bug in the barchart function in the lattice package. However, the default line plots do work.

Offsets in linear and generalized linear models are supported, as are offsets in mixed models fit by lmer or glmer, but must be supplied through the offset argument to lm, glm, lmer or glmer; offsets supplied via calls to the offset function on the right-hand side of the model formula are not supported.

Fitting ordinal mixed-models using clmm or clmm2 permits many options, including a variety of link functions, scale functions, nominal regressors, and various methods for setting thresholds. Effects are currently generated only for the default values of the arguments scale, nominal, link and threshold, which is equivalent to fitting an ordinal response mixed effects model with a logit link. The effect methods can also be used with objects created using clm or clm2 fitting ordinal response models with the same links permitted by polr with no random effects, with results similar to those from polr in the MASS package.

Calling any of these functions from within a user-written function may result in errors due to R's scoping rules. See the vignette embedding.pdf for the car package for a solution to this problem.

Details

Normally, the functions to be used directly are allEffects, to return a list of high-order effects, and the generic plot function to plot the effects. (see plot.efflist, plot.eff, and plot.effpoly). Alternatively, Effect can be used to vary a subset of predictors over their ranges, while other predictors are held to typical values. Plots are drawn using the xyplot (or in some cases, the densityplot) function in the lattice package. Effects may also be printed (implicitly or explicitly via print) or summarized (using summary) (see print.efflist, summary.efflist, print.eff, summary.eff, print.effpoly, and summary.effpoly).

If asked, the effect function will compute effects for terms that have higher-order relatives in the model, averaging over those terms (which rarely makes sense), or for terms that do not appear in the model but are higher-order relatives of terms that do. For example, for the model Y ~ A*B + A*C + B*C, one could compute the effect corresponding to the absent term A:B:C, which absorbs the constant, the A, B, and C main effects, and the three two-way interactions. In either of these cases, a warning is printed.

See predictorEffects for an alternative paradigm for getting effects.

References

Fox, J. (1987). Effect displays for generalized linear models. Sociological Methodology 17, 347--361.

Fox, J. (2003) Effect displays in R for generalised linear models. Journal of Statistical Software 8:15, 1--27, <http://www.jstatsoft.org/v08/i15/>.

Fox, J. and R. Andersen (2006). Effect displays for multinomial and proportional-odds logit models. Sociological Methodology 36, 225--255.

Fox, J. and J. Hong (2009). Effect displays in R for multinomial and proportional-odds logit models:? Extensions to the effects package. Journal of Statistical Software 32:1, 1--24, <http://www.jstatsoft.org/v32/i01/>.

Fox, J. and S. Weisberg (2018). Visualizing Fit and Lack of Fit in Complex Regression Models with Predictor Effect Plots with Partial Residuals. Journal of Statistical Software 87:9, 1--27, <http://www.jstatsoft.org/v87/i09/>

Hastie, T. J. (1992). Generalized additive models. In Chambers, J. M., and Hastie, T. J. (eds.) Statistical Models in S, Wadsworth.

Weisberg, S. (2014). Applied Linear Regression, 4th edition, Wiley, http://z.umn.edu/alr4ed.

Examples

Run this code

# NOT RUN {
mod.cowles <- glm(volunteer ~ sex + neuroticism*extraversion,
                  data=Cowles, family=binomial)
eff.cowles <- allEffects(mod.cowles, xlevels=list(extraversion=seq(0, 24, 6)),
                   fixed.predictors=list(given.values=c(sexmale=0.5)))
eff.cowles
as.data.frame(eff.cowles[[2]])

# }
# NOT RUN {
# the following are equivalent:
eff.ne <- effect("neuroticism*extraversion", mod.cowles)
Eff.ne <- Effect(c("neuroticism", "extraversion"), mod.cowles)
all.equal(eff.ne$fit, Eff.ne$fit)

plot(eff.cowles, 'sex', axes=list(y=list(lab="Prob(Volunteer)")))

plot(eff.cowles, 'neuroticism:extraversion',
     axes=list(y=list(lab="Prob(Volunteer)",
        ticks=list(at=c(.1,.25,.5,.75,.9)))))

plot(Effect(c("neuroticism", "extraversion"), mod.cowles,
            se=list(type="Scheffe"),
            xlevels=list(extraversion=seq(0, 24, 6)),
            fixed.predictors=list(given.values=c(sexmale=0.5))),
     axes=list(y=list(lab="Prob(Volunteer)",
        ticks=list(at=c(.1,.25,.5,.75,.9)))))

plot(eff.cowles, 'neuroticism:extraversion', lines=list(multiline=TRUE),
     axes=list(y=list(lab="Prob(Volunteer)")))

plot(effect('sex:neuroticism:extraversion', mod.cowles,
            xlevels=list(extraversion=seq(0, 24, 6))),
     lines=list(multiline=TRUE))
# }
# NOT RUN {
# a nested model:

mod <- lm(log(prestige) ~ income:type + education, data=Prestige)

plot(Effect(c("income", "type"), mod, transformation=list(link=log, inverse=exp)),
     axes=list(y=list(lab="prestige")))


if (require(nnet)){
    mod.beps <- multinom(vote ~ age + gender + economic.cond.national +
                             economic.cond.household + Blair + Hague + Kennedy +
                             Europe*political.knowledge, data=BEPS)
    
# }
# NOT RUN {
    plot(effect("Europe*political.knowledge", mod.beps,
                xlevels=list(political.knowledge=0:3)))
    
# }
# NOT RUN {
    plot(Effect(c("Europe", "political.knowledge"), mod.beps,
                xlevels=list(Europe=1:11, political.knowledge=0:3),
                fixed.predictors=list(given.values=c(gendermale=0.5))),
         lines=list(col=c("blue", "red", "orange")),
         axes=list(x=list(rug=FALSE), y=list(style="stacked")))

    
# }
# NOT RUN {
    plot(effect("Europe*political.knowledge", mod.beps, # equivalent
                xlevels=list(Europe=1:11, political.knowledge=0:3),
                fixed.predictors=list(given.values=c(gendermale=0.5))),
         lines=list(col=c("blue", "red", "orange")),
         axes=list(x=list(rug=FALSE), y=list(style="stacked")))
    
# }
# NOT RUN {
}

if (require(MASS)){
    mod.wvs <- polr(poverty ~ gender + religion + degree + country*poly(age,3),
                    data=WVS)
    
# }
# NOT RUN {
    plot(effect("country*poly(age, 3)", mod.wvs))
    
# }
# NOT RUN {
    plot(Effect(c("country", "age"), mod.wvs),
         axes=list(y=list(style="stacked")))

    
# }
# NOT RUN {
    plot(effect("country*poly(age, 3)", mod.wvs),
         axes=list(y=list(style="stacked"))) # equivalent

    plot(effect("country*poly(age, 3)", latent=TRUE, mod.wvs))
    plot(effect("country*poly(age, 3)", latent=TRUE, mod.wvs,
         se=list(type="scheffe"))) # Scheffe-type confidence envelopes
    
# }
# NOT RUN {
}


mod.pres <- lm(prestige ~ log(income, 10) + poly(education, 3) + poly(women, 2),
               data=Prestige)
eff.pres <- allEffects(mod.pres, xlevels=50)
plot(eff.pres)
plot(eff.pres[1],
     axes=list(x=list(income=list(
             transform=list(trans=log10, inverse=function(x) 10^x),
             ticks=list(at=c(1000, 2000, 5000, 10000, 20000))
    ))))
# }
# NOT RUN {
# linear model with log-response and log-predictor
# to illustrate transforming axes and setting tick labels
mod.pres1 <- lm(log(prestige) ~ log(income) + poly(education, 3) + poly(women, 2),
                data=Prestige)
# effect of the log-predictor
eff.log <- Effect("income", mod.pres1)
# effect of the log-predictor transformed to the arithmetic scale
eff.trans <- Effect("income", mod.pres1, transformation=list(link=log, inverse=exp))
#variations:
# y-axis:  scale is log, tick labels are log
# x-axis:  scale is arithmetic, tick labels are arithmetic
plot(eff.log)

# y-axis:  scale is log, tick labels are log
# x-axis:  scale is log, tick labels are arithmetic
plot(eff.log, axes=list(x=list(income=list(
    transform=list(trans=log, inverse=exp),
    ticks=list(at=c(5000, 10000, 20000)),
    lab="income, log-scale"))))

# y-axis:  scale is log, tick labels are arithmetic
# x-axis:  scale is arithmetic, tick labels are arithmetic
plot(eff.trans, axes=list(y=list(lab="prestige")))

# y-axis:  scale is arithmetic, tick labels are arithmetic
# x-axis:  scale is arithmetic, tick labels are arithmetic
plot(eff.trans, axes=list(y=list(type="response", lab="prestige")))

# y-axis:  scale is log, tick labels are arithmetic
# x-axis:  scale is log, tick labels are arithmetic
plot(eff.trans, axes=list(
       x=list(income=list(
            transform=list(trans=log, inverse=exp),
            ticks=list(at=c(1000, 2000, 5000, 10000, 20000)),
            lab="income, log-scale")),
        y=list(lab="prestige, log-scale")),
     main="Both response and X in log-scale")

# y-axis:  scale is arithmetic, tick labels are arithmetic
# x-axis:  scale is log, tick labels are arithmetic
plot(eff.trans, axes=list(
        x=list(
            income=list(transform=list(trans=log, inverse=exp),
                        ticks=list(at=c(1000, 2000, 5000, 10000, 20000)),
                        lab="income, log-scale")),
        y=list(type="response", lab="prestige")))
# }
# NOT RUN {
if (require(nlme)){ # for gls()
    mod.hart <- gls(fconvict ~ mconvict + tfr + partic + degrees, data=Hartnagel,
                    correlation=corARMA(p=2, q=0), method="ML")
    plot(allEffects(mod.hart))
    detach(package:nlme)
}

if (require(lme4)){
    data(cake, package="lme4")
    fm1 <- lmer(angle ~ recipe * temperature + (1|recipe:replicate), cake,
                REML = FALSE)
    plot(Effect(c("recipe", "temperature"), fm1))
    
# }
# NOT RUN {
    plot(effect("recipe:temperature", fm1),
         axes=list(grid=TRUE)) # equivalent (plus grid)
    
# }
# NOT RUN {
    if (any(grepl("pbkrtest", search()))) detach(package:pbkrtest)
    detach(package:lme4)
}

# }
# NOT RUN {
if (require(nlme) && length(find.package("lme4", quiet=TRUE)) > 0){
    data(cake, package="lme4")
    cake$rep <- with(cake, paste( as.character(recipe), as.character(replicate), sep=""))
    fm2 <- lme(angle ~ recipe * temperature, data=cake,
               random = ~ 1 | rep, method="ML")
    plot(Effect(c("recipe", "temperature"), fm2))
    plot(effect("recipe:temperature", fm2),
         axes=list(grid=TRUE))  # equivalent (plus grid)
    }
    detach(package:nlme)
# }
# NOT RUN {
# }
# NOT RUN {
if (require(poLCA)){
    data(election)
    f2a <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG,
                 MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~PARTY*AGE
    nes2a <- poLCA(f2a,election,nclass=3,nrep=5)
    plot(Effect(c("PARTY", "AGE"), nes2a),
         axes=list(y=list(style="stacked")))
}
# }
# NOT RUN {
# mlm example
if (require(heplots)) {
    data(NLSY, package="heplots")
    mod <- lm(cbind(read,math) ~ income+educ, data=NLSY)
    eff.inc <- Effect("income", mod)
    plot(eff.inc)
    eff.edu <- Effect("educ", mod)
    plot(eff.edu, axes=list(x=list(rug=FALSE), grid=TRUE))
    
# }
# NOT RUN {
    plot(Effect("educ", mod, response="read"))
    
# }
# NOT RUN {
    detach(package:heplots)
}

# svyglm() example (adapting an example from the survey package)

# }
# NOT RUN {
if (require(survey)){
  data("api")
  dstrat<-svydesign(id=~1, strata=~stype, weights=~pw,
    data=apistrat, fpc=~fpc)
  mod <- svyglm(sch.wide ~ ell + meals + mobility, design=dstrat,
    family=quasibinomial())
  plot(allEffects(mod),
    axes=list(y=list(lim=log(c(0.4, 0.99)/c(0.6, 0.01)),
      ticks=list(at=c(0.4, 0.75, 0.9, 0.95, 0.99)))))
}
# }
# NOT RUN {
# component + residual plot examples

# }
# NOT RUN {
Prestige$type <- factor(Prestige$type, levels=c("bc", "wc", "prof"))

mod.prestige.1 <- lm(prestige ~ income + education, data=Prestige)
plot(allEffects(mod.prestige.1, residuals=TRUE)) # standard C+R plots
plot(allEffects(mod.prestige.1, residuals=TRUE,
    se=list(type="scheffe"))) # with Scheffe-type confidence bands

mod.prestige.2 <- lm(prestige ~ type*(income + education), data=Prestige)
plot(allEffects(mod.prestige.2, residuals=TRUE))

mod.prestige.3 <- lm(prestige ~ type + income*education, data=Prestige)
plot(Effect(c("income", "education"), mod.prestige.3, residuals=TRUE),
     partial.residuals=list(span=1))
# }
# NOT RUN {
#  artificial data

set.seed(12345)
x1 <- runif(500, -75, 100)
x2 <- runif(500, -75, 100)
y <- 10 + 5*x1 + 5*x2 + x1^2 + x2^2 + x1*x2 + rnorm(500, 0, 1e3)
Data <- data.frame(y, x1, x2)
mod.1 <- lm(y ~ poly(x1, x2, degree=2, raw=TRUE), data=Data)
# raw=TRUE necessary for safe prediction
mod.2 <- lm(y ~ x1*x2, data=Data)
mod.3 <- lm(y ~ x1 + x2, data=Data)

plot(Effect(c("x1", "x2"), mod.1, residuals=TRUE)) # correct model
plot(Effect(c("x1", "x2"), mod.2, residuals=TRUE)) # wrong model
plot(Effect(c("x1", "x2"), mod.3, residuals=TRUE)) # wrong model
# }

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