Anova                  package:car                  R Documentation

_A_n_o_v_a _T_a_b_l_e_s _f_o_r _V_a_r_i_o_u_s _S_t_a_t_i_s_t_i_c_a_l _M_o_d_e_l_s

_D_e_s_c_r_i_p_t_i_o_n:

     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'.

_U_s_a_g_e:

     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")), ...)

_A_r_g_u_m_e_n_t_s:

     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.

_D_e_t_a_i_l_s:

     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.

_V_a_l_u_e:

     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.

_W_a_r_n_i_n_g:

     Be careful of type-III tests.

_A_u_t_h_o_r(_s):

     John Fox jfox@mcmaster.ca

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     'linear.hypothesis', 'anova' 'anova.lm', 'anova.glm',  'anova.mlm'

_E_x_a_m_p_l_e_s:

     ## 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

