MANOVA
fangya
10/25/2021
knitr::opts_chunk$set(echo = TRUE)
data(mtcars)
summary(mtcars)## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000MANOVA
Iris data
yvars<- cbind(mtcars$mpg, mtcars$cyl, mtcars$disp, mtcars$hp, mtcars$drat,mtcars$wt, mtcars$qsec, mtcars$gear, mtcars$vs)
colnames(yvars) <-c("mpg","cyl","disp","hp","drat","wt","qsec","gear","vs")
factor_c <-factor(mtcars$carb)
factor_am <-factor(mtcars$am)
mma<-manova(yvars~ factor_am +factor_c, data=mtcars)
mma## Call:
## manova(yvars ~ factor_am + factor_c, data = mtcars)
##
## Terms:
## factor_am factor_c Residuals
## mpg 405.15 407.78 313.12
## cyl 27.00 35.18 36.69
## disp 166450.1 106702.0 203032.7
## hp 8619.50 90571.44 46535.94
## drat 4.50 0.81 3.55
## wt 14.23 7.48 7.97
## qsec 5.23 45.10 48.65
## gear 10.64 1.78 4.46
## vs 0.22 3.58 4.07
## Deg. of Freedom 1 5 25
##
## Residual standard errors: 3.539016 1.21144 90.1183 43.14438 0.3769924 0.5646695 1.395061 0.4223634 0.4036956
## Estimated effects may be unbalancedNULL hypthoesis
Means are equal to each other
#wilks test
summary(mma,test="Wilks", intercept=TRUE)## Df Wilks approx F num Df den Df Pr(>F)
## (Intercept) 1 0.000501 3765.2 9 17.000 < 2.2e-16 ***
## factor_am 1 0.153018 10.5 9 17.000 2.427e-05 ***
## factor_c 5 0.009093 3.3 45 79.148 1.984e-06 ***
## Residuals 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Roys Test
summary(mma, test="Roy", intercept=TRUE)## Df Roy approx F num Df den Df Pr(>F)
## (Intercept) 1 1993.33 3765.2 9 17 < 2.2e-16 ***
## factor_am 1 5.54 10.5 9 17 2.427e-05 ***
## factor_c 5 11.02 25.7 9 21 2.190e-09 ***
## Residuals 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Hotelling lawyley test
summary(mma, test="Hotelling-Lawley", intercept=TRUE)## Df Hotelling-Lawley approx F num Df den Df Pr(>F)
## (Intercept) 1 1993.33 3765.2 9 17 < 2.2e-16 ***
## factor_am 1 5.54 10.5 9 17 2.427e-05 ***
## factor_c 5 14.50 5.0 45 77 3.969e-10 ***
## Residuals 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Pillai Test
summary(mma, test="Pillai", intercept=TRUE)## Df Pillai approx F num Df den Df Pr(>F)
## (Intercept) 1 0.99950 3765.2 9 17 < 2.2e-16 ***
## factor_am 1 0.84698 10.5 9 17 2.427e-05 ***
## factor_c 5 2.46882 2.3 45 105 0.000304 ***
## Residuals 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Relation between IV and DV
yvars## mpg cyl disp hp drat wt qsec gear vs
## [1,] 21.0 6 160.0 110 3.90 2.620 16.46 4 0
## [2,] 21.0 6 160.0 110 3.90 2.875 17.02 4 0
## [3,] 22.8 4 108.0 93 3.85 2.320 18.61 4 1
## [4,] 21.4 6 258.0 110 3.08 3.215 19.44 3 1
## [5,] 18.7 8 360.0 175 3.15 3.440 17.02 3 0
## [6,] 18.1 6 225.0 105 2.76 3.460 20.22 3 1
## [7,] 14.3 8 360.0 245 3.21 3.570 15.84 3 0
## [8,] 24.4 4 146.7 62 3.69 3.190 20.00 4 1
## [9,] 22.8 4 140.8 95 3.92 3.150 22.90 4 1
## [10,] 19.2 6 167.6 123 3.92 3.440 18.30 4 1
## [11,] 17.8 6 167.6 123 3.92 3.440 18.90 4 1
## [12,] 16.4 8 275.8 180 3.07 4.070 17.40 3 0
## [13,] 17.3 8 275.8 180 3.07 3.730 17.60 3 0
## [14,] 15.2 8 275.8 180 3.07 3.780 18.00 3 0
## [15,] 10.4 8 472.0 205 2.93 5.250 17.98 3 0
## [16,] 10.4 8 460.0 215 3.00 5.424 17.82 3 0
## [17,] 14.7 8 440.0 230 3.23 5.345 17.42 3 0
## [18,] 32.4 4 78.7 66 4.08 2.200 19.47 4 1
## [19,] 30.4 4 75.7 52 4.93 1.615 18.52 4 1
## [20,] 33.9 4 71.1 65 4.22 1.835 19.90 4 1
## [21,] 21.5 4 120.1 97 3.70 2.465 20.01 3 1
## [22,] 15.5 8 318.0 150 2.76 3.520 16.87 3 0
## [23,] 15.2 8 304.0 150 3.15 3.435 17.30 3 0
## [24,] 13.3 8 350.0 245 3.73 3.840 15.41 3 0
## [25,] 19.2 8 400.0 175 3.08 3.845 17.05 3 0
## [26,] 27.3 4 79.0 66 4.08 1.935 18.90 4 1
## [27,] 26.0 4 120.3 91 4.43 2.140 16.70 5 0
## [28,] 30.4 4 95.1 113 3.77 1.513 16.90 5 1
## [29,] 15.8 8 351.0 264 4.22 3.170 14.50 5 0
## [30,] 19.7 6 145.0 175 3.62 2.770 15.50 5 0
## [31,] 15.0 8 301.0 335 3.54 3.570 14.60 5 0
## [32,] 21.4 4 121.0 109 4.11 2.780 18.60 4 1summary.aov(mma)## Response mpg :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 405.15 405.15 32.3483 6.368e-06 ***
## factor_c 5 407.78 81.56 6.5117 0.000527 ***
## Residuals 25 313.12 12.52
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response cyl :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 27.005 27.0046 18.4007 0.0002348 ***
## factor_c 5 35.181 7.0362 4.7944 0.0032913 **
## Residuals 25 36.690 1.4676
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response disp :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 166450 166450 20.4955 0.0001269 ***
## factor_c 5 106702 21340 2.6277 0.0483721 *
## Residuals 25 203033 8121
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response hp :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 8619 8619.5 4.6306 0.04126 *
## factor_c 5 90571 18114.3 9.7313 2.966e-05 ***
## Residuals 25 46536 1861.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response drat :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 4.5017 4.5017 31.6745 7.412e-06 ***
## factor_c 5 0.8076 0.1615 1.1364 0.3673
## Residuals 25 3.5531 0.1421
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response wt :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 14.2324 14.2324 44.6365 5.313e-07 ***
## factor_c 5 7.4750 1.4950 4.6887 0.003714 **
## Residuals 25 7.9713 0.3189
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response qsec :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 5.230 5.2301 2.6874 0.113672
## factor_c 5 45.103 9.0206 4.6350 0.003951 **
## Residuals 25 48.655 1.9462
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response gear :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 10.6402 10.6402 59.6454 4.44e-08 ***
## factor_c 5 1.7750 0.3550 1.9901 0.115
## Residuals 25 4.4598 0.1784
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Response vs :
## Df Sum Sq Mean Sq F value Pr(>F)
## factor_am 1 0.2232 0.22318 1.3694 0.252943
## factor_c 5 3.5776 0.71551 4.3905 0.005254 **
## Residuals 25 4.0743 0.16297
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Boxplot
To check the mean differences
ggboxplot(mtcars, x="carb", y=c("mpg", "cyl","qsec"), merge=TRUE, palette="jco")
ggboxplot(mtcars, x="carb", y=c("disp","hp"), merge=TRUE, palette="jco")
ggboxplot(mtcars, x="carb", y=c("drat","wt","am"), merge=TRUE, palette="jco")
MANOVA Linearity
Key :MANOVA consider the relationship of dependent variable together !
m2<-lm(yvars~ am, data=mtcars)
lin_comb <- matrix(c(1,-1), nrow=1, ncol=2)
linearHypothesis(m2,lin_comb)##
## Sum of squares and products for the hypothesis:
## mpg cyl disp hp drat wt
## mpg 341.131711 303.767474 15062.1666 6672.1452 86.902881 176.614287
## cyl 303.767474 270.495752 13412.4039 5941.3436 77.384388 157.269683
## disp 15062.166567 13412.403935 665047.7062 294598.7107 3837.068305 7798.142844
## hp 6672.145184 5941.343559 294598.7107 130499.5108 1699.720734 3454.372981
## drat 86.902881 77.384388 3837.0683 1699.7207 22.138402 44.992271
## wt 176.614287 157.269683 7798.1428 3454.3730 44.992271 91.438601
## qsec 654.754194 583.038814 28909.7039 12806.2414 166.797821 338.986208
## gear 70.153829 62.469864 3097.5386 1372.1285 17.871601 36.320776
## vs 6.834071 6.085533 301.7483 133.6666 1.740971 3.538207
## qsec gear vs
## mpg 654.75419 70.153829 6.8340708
## cyl 583.03881 62.469864 6.0855334
## disp 28909.70386 3097.538650 301.7482979
## hp 12806.24142 1372.128471 133.6665906
## drat 166.79782 17.871601 1.7409711
## wt 338.98621 36.320776 3.5382068
## qsec 1256.70831 134.650378 13.1170348
## gear 134.65038 14.427154 1.4054285
## vs 13.11703 1.405429 0.1369105
##
## Sum of squares and products for error:
## mpg cyl disp hp drat wt
## mpg 7.208966e+02 -179.744939 -11413.9880 -8073.9522 25.340316 -82.6812053
## cyl -1.797449e+02 71.870445 4069.3482 2677.4170 -9.693684 22.7838947
## disp -1.141399e+04 4069.348178 309734.6793 170478.2668 -593.360474 1799.0578579
## hp -8.073952e+03 2677.417004 170478.2668 137107.3765 -313.001579 1019.7205263
## drat 2.534032e+01 -9.693684 -593.3605 -313.0016 4.360642 -3.5499774
## wt -8.268121e+01 22.783895 1799.0579 1019.7205 -3.549977 15.4463138
## qsec 1.858162e+02 -70.376842 -3910.6387 -2902.1958 7.553621 -18.0976537
## gear 5.489879e-01 -3.174089 -244.0696 105.7166 1.634737 -0.7475789
## vs 5.302227e+01 -20.170040 -1182.9680 -730.7652 2.675789 -6.7012632
## qsec gear vs
## mpg 185.816158 0.5489879 53.0222672
## cyl -70.376842 -3.1740891 -20.1700405
## disp -3910.638737 -244.0696356 -1182.9680162
## hp -2902.195789 105.7165992 -730.7651822
## drat 7.553621 1.6347368 2.6757895
## wt -18.097654 -0.7475789 -6.7012632
## qsec 93.758011 -1.2326316 21.8678947
## gear -1.232632 6.2348178 0.8340081
## vs 21.867895 0.8340081 7.6518219
##
## Multivariate Tests:
## Df test stat approx F num Df den Df Pr(>F)
## Pillai 1 0.99459 449.3124 9 22 < 2.22e-16 ***
## Wilks 1 0.00541 449.3124 9 22 < 2.22e-16 ***
## Hotelling-Lawley 1 183.80961 449.3124 9 22 < 2.22e-16 ***
## Roy 1 183.80961 449.3124 9 22 < 2.22e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ANOVA
an <- aov(hp ~ am, data=mtcars)
summary(an)## Df Sum Sq Mean Sq F value Pr(>F)
## am 1 8619 8619 1.886 0.18
## Residuals 30 137107 4570ggboxplot(mtcars, x="hp", y="am", palette="jco")
an1 <- aov(am ~ mpg, data=mtcars)
summary(an1)## Df Sum Sq Mean Sq F value Pr(>F)
## mpg 1 2.777 2.7772 16.86 0.000285 ***
## Residuals 30 4.942 0.1647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ggboxplot(mtcars, x="am", y="mpg", palette="jco")
ref:ANOVA and MANOVA Analysis in R - Spencer Pao