| Anova {car} | R Documentation |
Calculates type-II or type-III analysis-of-variance tables for
model objects produced by lm, glm, multinom
(in the nnet package), and polr (in the MASS
package). 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.
Anova(mod, ...)
Manova(mod, ...)
## S3 method for class 'lm':
Anova(mod, error, type=c("II","III", 2, 3), ...)
## 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")), ...)
mod |
lm, aov, glm, multinom, polr
or mlm model object. |
error |
for a linear model, an 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 estimated. If not specified, mod is used. |
type |
type of test, "II", "III", 2, or 3. |
test.statistic |
for a generalized linear model, whether to calculate
"LR" (likelihood-ratio), "Wald", or "F" tests.
For a multivariate linear model, the multivariate test statistic to compute — one of
"Pillai", "Wilks", "Hotelling-Lawley", or "Roy",
with "Pillai" as the default. The summary method for Anova.mlm
objects permits the specification of more than one multivariate
test statistic, and the default is to report all four. |
error.estimate |
for F-tests for a generalized linear model, base the
dispersion estimate on the Pearson residuals (pearson, the default); use the
dispersion estimate in the model object (dispersion), which, e.g., is
fixed to 1 for binomial and Poisson models; or base the dispersion estimate on
the residual deviance (deviance). |
SSPE |
The error sum-of-squares-and-products matrix; if missing, will be computed from the residuals of the model. |
error.df |
The degrees of freedom for error; if missing, will be taken from the model. |
idata |
an optional data frame giving a factor or factors defining the intra-subject model for multivariate repeated-measures data. See Details for an explanation of the intra-subject design and for further explanation of the other arguments relating to intra-subject factors. |
idesign |
a one-sided model formula using the ``data'' in idata and
specifying the intra-subject design. |
icontrasts |
names of contrast-generating functions to be applied by default
to factors and ordered factors, respectively, in the within-subject
``data''; the contrasts must produce an intra-subject model
matrix in which different terms are orthogonal. The default is
c("contr.sum", "contr.poly"). |
x, object |
object of class "Anova.mlm" to print or summarize. |
multivariate, univariate |
print multivariate and univariate tests for a repeated-measures
ANOVA; the default is TRUE for both. |
digits |
minimum number of significant digits to print. |
... |
arguments to be passed to linear.hypothesis; only use
white.adjust for a linear model. |
The designations "type-II" and "type-III" are borrowed from SAS, but the definitions used here do not correspond precisely to those employed by SAS. Type-II tests are calculated according to the principle of marginality, testing each term after all others, except ignoring the term's higher-order relatives; so-called type-III tests violate marginality, testing each term in the model after all of the others. This definition of Type-II tests corresponds to the tests produced by SAS for analysis-of-variance models, where all of the predictors are factors, but not more generally (i.e., when there are quantitative predictors). Be very careful in formulating the model for type-III tests, or the hypotheses tested will not make sense.
As implemented here, type-II Wald tests for generalized linear models are actually differences of Wald statistics.
For tests for linear models, multivariate linear models, and Wald tests for generalized linear models,
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.
An object of class "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.
Be careful of type-III tests.
John Fox jfox@mcmaster.ca
Fox, J. (1997) Applied Regression, Linear Models, and Related Methods. Sage.
Hand, D. J., and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Behavioural Scientists. Chapman and Hall.
O'Brien, R. G., and Kaiser, M. K. (1985) MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin 97, 316–333.
linear.hypothesis, anova
anova.lm, anova.glm,
anova.mlm
## 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 Compound Symmetry
##
## 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
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Greenhouse-Geisser and Huynh-Feldt Corrections
## for Departure from Compound Symmetry
##
## 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