lm
, glm
, multinom
(in the polr
(in the coxph
(in the vcov
and coef
functions. For linear
models, F-tests are calculated; for generalized linear models,
likelihood-ratio chisquare, Wald chisquare, or F-tests are calculated;
for multinomial logit and proportional-odds logit models, likelihood-ratio
tests are calculated. Various test statistics are provided for multivariate
linear models produced by lm
or manova
. Partial-ikelihood-ratio tests
or Wald tests are provided for Cox models. Wald chi-square or F tests are provided
in the default case.Anova(mod, ...)
Manova(mod, ...)
## S3 method for class 'lm':
Anova(mod, error, type=c("II","III", 2, 3),
white.adjust=c("hc3", "hc0", "hc1", "hc2", "hc4"), ...)
## S3 method for class 'aov':
Anova(mod, ...)
## S3 method for class 'glm':
Anova(mod, type=c("II","III", 2, 3),
test.statistic=c("LR", "Wald", "F"),
error, error.estimate=c("pearson", "dispersion", "deviance"), ...)
## S3 method for class 'multinom':
Anova(mod, type = c("II","III", 2, 3), ...)
## S3 method for class 'polr':
Anova(mod, type = c("II","III", 2, 3), ...)
## S3 method for class 'mlm':
Anova(mod, type=c("II","III", 2, 3), SSPE, error.df,
idata, idesign, icontrasts=c("contr.sum", "contr.poly"),
test.statistic=c("Pillai", "Wilks", "Hotelling-Lawley", "Roy"),...)
## S3 method for class 'manova':
Anova(mod, ...)
## S3 method for class 'mlm':
Manova(mod, ...)
## S3 method for class 'Anova.mlm':
print(x, ...)
## S3 method for class 'Anova.mlm':
summary(object, test.statistic, multivariate=TRUE,
univariate=TRUE, digits=unlist(options("digits")), ...)
## S3 method for class 'coxph':
Anova(mod, type=c("II","III", 2, 3),
test.statistic=c("LR", "Wald"), ...)
## S3 method for class 'default':
Anova(mod, type=c("II","III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod), ...)
lm
, aov
, glm
, multinom
, polr
or mlm
model object.lm
model object from which the
error sum of squares and degrees of freedom are to be calculated. For
F-tests for a generalized linear model, a glm
object from which the
dispersion is to be e"II"
, "III"
, 2
, or 3
."LR"
(likelihood-ratio), "Wald"
, or "F"
tests; for a Cox
model, whether to calculate "LR"
(partial-likelihood ratio) or
"Wald"
pearson
, the default); use the
dispersion estimate in the model object (dispersion
), which, e.g., is
fixed to 1 forhccm
for details. If idata
and
specifying the intra-subject design."Anova.mlm"
to print or summarize.TRUE
for both.vcov
function to the model object."anova"
, or "Anova.mlm"
, which usually is printed.
For objects of class "Anova.mlm"
, there is also a summary
method,
which provides much more detail than the print
method about the MANOVA, including
traditional mixed-model univariate F-tests with Greenhouse-Geisser and Hunyh-Feldt
corrections.Anova
finds the test statistics without refitting the model.
The standard R anova
function calculates sequential ("type-I") tests.
These rarely test interesting hypotheses.
A MANOVA for a multivariate linear model (i.e., an object of
class "mlm"
or "manova"
) can optionally include an
intra-subject repeated-measures design.
If the intra-subject design is absent (the default), the multivariate
tests concern all of the response variables.
To specify a repeated-measures design, a data frame is provided defining the repeated-measures factor or
factors
via idata
, with default contrasts given by the icontrasts
argument. An intra-subject model-matrix is generated from the formula
specified by the idesign
argument; columns of the model matrix
corresponding to different terms in the intra-subject model must be orthogonal
(as is insured by the default contrasts). Note that the contrasts given in
icontrasts
can be overridden by assigning specific contrasts to the
factors in idata
. Manova
is essentially a synonym for Anova
for multivariate linear models.linear.hypothesis
, anova
anova.lm
, anova.glm
,
anova.mlm
, anova.coxph
.## Two-Way Anova
mod <- lm(conformity ~ fcategory*partner.status, data=Moore,
contrasts=list(fcategory=contr.sum, partner.status=contr.sum))
Anova(mod)
## Anova Table (Type II tests)
##
## Response: conformity
## Sum Sq Df F value Pr(>F)
## fcategory 11.61 2 0.2770 0.759564
## partner.status 212.21 1 10.1207 0.002874
## fcategory:partner.status 175.49 2 4.1846 0.022572
## Residuals 817.76 39
Anova(mod, type="III")
## Anova Table (Type III tests)
##
## Response: conformity
## Sum Sq Df F value Pr(>F)
## (Intercept) 5752.8 1 274.3592 < 2.2e-16
## fcategory 36.0 2 0.8589 0.431492
## partner.status 239.6 1 11.4250 0.001657
## fcategory:partner.status 175.5 2 4.1846 0.022572
## Residuals 817.8 39
## One-Way MANOVA
## See ?Pottery for a description of the data set used in this example.
summary(Anova(lm(cbind(Al, Fe, Mg, Ca, Na) ~ Site, data=Pottery)))
## Type II MANOVA Tests:
##
## Sum of squares and products for error:
## Al Fe Mg Ca Na
## Al 48.2881429 7.08007143 0.60801429 0.10647143 0.58895714
## Fe 7.0800714 10.95084571 0.52705714 -0.15519429 0.06675857
## Mg 0.6080143 0.52705714 15.42961143 0.43537714 0.02761571
## Ca 0.1064714 -0.15519429 0.43537714 0.05148571 0.01007857
## Na 0.5889571 0.06675857 0.02761571 0.01007857 0.19929286
##
## ------------------------------------------
##
## Term: Site
##
## Sum of squares and products for the hypothesis:
## Al Fe Mg Ca Na
## Al 175.610319 -149.295533 -130.809707 -5.8891637 -5.3722648
## Fe -149.295533 134.221616 117.745035 4.8217866 5.3259491
## Mg -130.809707 117.745035 103.350527 4.2091613 4.7105458
## Ca -5.889164 4.821787 4.209161 0.2047027 0.1547830
## Na -5.372265 5.325949 4.710546 0.1547830 0.2582456
##
## Multivariate Tests: Site
## Df test stat approx F num Df den Df Pr(>F)
## Pillai 3.00000 1.55394 4.29839 15.00000 60.00000 2.4129e-05 ***
## Wilks 3.00000 0.01230 13.08854 15.00000 50.09147 1.8404e-12 ***
## Hotelling-Lawley 3.00000 35.43875 39.37639 15.00000 50.00000 < 2.22e-16 ***
## Roy 3.00000 34.16111 136.64446 5.00000 20.00000 9.4435e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## MANOVA for a randomized block design (example courtesy of Michael Friendly:
## See ?Soils for description of the data set)
soils.mod <- lm(cbind(pH,N,Dens,P,Ca,Mg,K,Na,Conduc) ~ Block + Contour*Depth,
data=Soils)
Manova(soils.mod)
## Type II MANOVA Tests: Pillai test statistic
## Df test stat approx F num Df den Df Pr(>F)
## Block 3 1.6758 3.7965 27 81 1.777e-06 ***
## Contour 2 1.3386 5.8468 18 52 2.730e-07 ***
## Depth 3 1.7951 4.4697 27 81 8.777e-08 ***
## Contour:Depth 6 1.2351 0.8640 54 180 0.7311
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## a multivariate linear model for repeated-measures data
## See ?OBrienKaiser for a description of the data set used in this example.
phase <- factor(rep(c("pretest", "posttest", "followup"), c(5, 5, 5)),
levels=c("pretest", "posttest", "followup"))
hour <- ordered(rep(1:5, 3))
idata <- data.frame(phase, hour)
idata
## phase hour
## 1 pretest 1
## 2 pretest 2
## 3 pretest 3
## 4 pretest 4
## 5 pretest 5
## 6 posttest 1
## 7 posttest 2
## 8 posttest 3
## 9 posttest 4
## 10 posttest 5
## 11 followup 1
## 12 followup 2
## 13 followup 3
## 14 followup 4
## 15 followup 5
mod.ok <- lm(cbind(pre.1, pre.2, pre.3, pre.4, pre.5,
post.1, post.2, post.3, post.4, post.5,
fup.1, fup.2, fup.3, fup.4, fup.5) ~ treatment*gender,
data=OBrienKaiser)
(av.ok <- Anova(mod.ok, idata=idata, idesign=~phase*hour))
## Type II Repeated Measures MANOVA Tests: Pillai test statistic
## Df test stat approx F num Df den Df Pr(>F)
## treatment 2 0.4809 4.6323 2 10 0.0376868 *
## gender 1 0.2036 2.5558 1 10 0.1409735
## treatment:gender 2 0.3635 2.8555 2 10 0.1044692
## phase 1 0.8505 25.6053 2 9 0.0001930 ***
## treatment:phase 2 0.6852 2.6056 4 20 0.0667354 .
## gender:phase 1 0.0431 0.2029 2 9 0.8199968
## treatment:gender:phase 2 0.3106 0.9193 4 20 0.4721498
## hour 1 0.9347 25.0401 4 7 0.0003043 ***
## treatment:hour 2 0.3014 0.3549 8 16 0.9295212
## gender:hour 1 0.2927 0.7243 4 7 0.6023742
## treatment:gender:hour 2 0.5702 0.7976 8 16 0.6131884
## phase:hour 1 0.5496 0.4576 8 3 0.8324517
## treatment:phase:hour 2 0.6637 0.2483 16 8 0.9914415
## gender:phase:hour 1 0.6950 0.8547 8 3 0.6202076
## treatment:gender:phase:hour 2 0.7928 0.3283 16 8 0.9723693
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(av.ok, multivariate=FALSE)
## Univariate Type II Repeated-Measures ANOVA Assuming Sphericity
##
## SS num Df Error SS den Df F Pr(>F)
## treatment 211.286 2 228.056 10 4.6323 0.037687
## gender 58.286 1 228.056 10 2.5558 0.140974
## treatment:gender 130.241 2 228.056 10 2.8555 0.104469
## phase 167.500 2 80.278 20 20.8651 1.274e-05
## treatment:phase 78.668 4 80.278 20 4.8997 0.006426
## gender:phase 1.668 2 80.278 20 0.2078 0.814130
## treatment:gender:phase 10.221 4 80.278 20 0.6366 0.642369
## hour 106.292 4 62.500 40 17.0067 3.191e-08
## treatment:hour 1.161 8 62.500 40 0.0929 0.999257
## gender:hour 2.559 4 62.500 40 0.4094 0.800772
## treatment:gender:hour 7.755 8 62.500 40 0.6204 0.755484
## phase:hour 11.083 8 96.167 80 1.1525 0.338317
## treatment:phase:hour 6.262 16 96.167 80 0.3256 0.992814
## gender:phase:hour 6.636 8 96.167 80 0.6900 0.699124
## treatment:gender:phase:hour 14.155 16 96.167 80 0.7359 0.749562
##
## treatment *
## gender
## treatment:gender
## phase ***
## treatment:phase **
## gender:phase
## treatment:gender:phase
## hour ***
## treatment:hour
## gender:hour
## treatment:gender:hour
## phase:hour
## treatment:phase:hour
## gender:phase:hour
## treatment:gender:phase:hour
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Mauchly Tests for Sphericity
##
## Test statistic p-value
## phase 0.74927 0.27282
## treatment:phase 0.74927 0.27282
## gender:phase 0.74927 0.27282
## treatment:gender:phase 0.74927 0.27282
## hour 0.06607 0.00760
## treatment:hour 0.06607 0.00760
## gender:hour 0.06607 0.00760
## treatment:gender:hour 0.06607 0.00760
## phase:hour 0.00478 0.44939
## treatment:phase:hour 0.00478 0.44939
## gender:phase:hour 0.00478 0.44939
## treatment:gender:phase:hour 0.00478 0.44939
##
##
## Greenhouse-Geisser and Huynh-Feldt Corrections
## for Departure from Sphericity
##
## GG eps Pr(>F[GG])
## phase 0.79953 7.323e-05 ***
## treatment:phase 0.79953 0.01223 *
## gender:phase 0.79953 0.76616
## treatment:gender:phase 0.79953 0.61162
## hour 0.46028 8.741e-05 ***
## treatment:hour 0.46028 0.97879
## gender:hour 0.46028 0.65346
## treatment:gender:hour 0.46028 0.64136
## phase:hour 0.44950 0.34573
## treatment:phase:hour 0.44950 0.94019
## gender:phase:hour 0.44950 0.58903
## treatment:gender:phase:hour 0.44950 0.64634
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## HF eps Pr(>F[HF])
## phase 0.92786 2.388e-05 ***
## treatment:phase 0.92786 0.00809 **
## gender:phase 0.92786 0.79845
## treatment:gender:phase 0.92786 0.63200
## hour 0.55928 2.014e-05 ***
## treatment:hour 0.55928 0.98877
## gender:hour 0.55928 0.69115
## treatment:gender:hour 0.55928 0.66930
## phase:hour 0.73306 0.34405
## treatment:phase:hour 0.73306 0.98047
## gender:phase:hour 0.73306 0.65524
## treatment:gender:phase:hour 0.73306 0.70801
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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