線形統計モデル構築と当てはめ
> LinearModel.1 <- lm(DATA ~ REP + N + V + N:V, data=Dataset)
> summary(LinearModel.1)
Call:
lm(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Residuals:
Min 1Q Median 3Q Max
-0.87830 -0.17743 0.00708 0.24239 0.57413
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.3003 0.2130 15.493 < 2e-16 ***
REP2 -0.2203 0.1420 -1.551 0.12840
REP3 0.0140 0.1420 0.099 0.92194
REP4 -0.4989 0.1420 -3.513 0.00107 **
NN1 1.5995 0.2750 5.816 7.30e-07 ***
NN2 1.3355 0.2750 4.856 1.70e-05 ***
NN3 2.5930 0.2750 9.429 6.30e-12 ***
NN4 2.6990 0.2750 9.814 1.96e-12 ***
VV2 0.4240 0.2750 1.542 0.13063
VV3 0.6540 0.2750 2.378 0.02203 *
NN1:VV2 -0.3415 0.3889 -0.878 0.38490
NN2:VV2 0.5525 0.3889 1.421 0.16281
NN3:VV2 -0.4015 0.3889 -1.032 0.30782
NN4:VV2 -0.5665 0.3889 -1.457 0.15266
NN1:VV3 -0.6240 0.3889 -1.604 0.11611
NN2:VV3 0.2065 0.3889 0.531 0.59824
NN3:VV3 -0.8185 0.3889 -2.105 0.04135 *
NN4:VV3 -0.5905 0.3889 -1.518 0.13643
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3889 on 42 degrees of freedom
Multiple R-squared: 0.8813, Adjusted R-squared: 0.8333
F-statistic: 18.35 on 17 and 42 DF, p-value: 3.375e-14
Tukey HSD 検定
> library(mvtnorm, pos=17)
> library(survival, pos=17)
> library(MASS, pos=17)
> library(TH.data, pos=17)
Attaching package: 'TH.data'
The following object is masked from 'package:MASS':
geyser
> library(multcomp, pos=17)
> library(abind, pos=22)
窒素要因(N)の多重比較
> AnovaModel.2 <- aov(DATA ~ REP + N + V + N:V, data=Dataset)
> summary(AnovaModel.2)
Df Sum Sq Mean Sq F value Pr(>F)
REP 3 2.60 0.867 5.729 0.00222 **
N 4 41.23 10.309 68.153 < 2e-16 ***
V 2 1.05 0.526 3.480 0.03995 *
N:V 8 2.29 0.286 1.893 0.08671 .
Residuals 42 6.35 0.151
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> with(Dataset, numSummary(DATA, groups=N, statistics=c("mean", "sd")))
mean sd data:n
N0 3.483333 0.5139808 12
N1 4.761000 0.3313680 12
N2 5.071833 0.6897728 12
N3 5.669667 0.4411381 12
N4 5.796667 0.2710335 12
> local({
+ .Pairs <- glht(AnovaModel.2, linfct = mcp(N = "Tukey"))
+ print(summary(.Pairs)) # pairwise tests
+ print(confint(.Pairs)) # confidence intervals
+ print(cld(.Pairs)) # compact letter display
+ old.oma <- par(oma=c(0,5,0,0))
+ plot(confint(.Pairs))
+ par(old.oma)
+ })
Warning in mcp2matrix(model, linfct = linfct): covariate interactions found
-- default contrast might be inappropriate
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
N1 - N0 == 0 1.5995 0.2750 5.816 < 1e-04 ***
N2 - N0 == 0 1.3355 0.2750 4.856 0.000175 ***
N3 - N0 == 0 2.5930 0.2750 9.429 < 1e-04 ***
N4 - N0 == 0 2.6990 0.2750 9.814 < 1e-04 ***
N2 - N1 == 0 -0.2640 0.2750 -0.960 0.871311
N3 - N1 == 0 0.9935 0.2750 3.613 0.006836 **
N4 - N1 == 0 1.0995 0.2750 3.998 0.002217 **
N3 - N2 == 0 1.2575 0.2750 4.573 0.000431 ***
N4 - N2 == 0 1.3635 0.2750 4.958 0.000115 ***
N4 - N3 == 0 0.1060 0.2750 0.385 0.995140
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)
Simultaneous Confidence Intervals
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Quantile = 2.8496
95% family-wise confidence level
Linear Hypotheses:
Estimate lwr upr
N1 - N0 == 0 1.5995 0.8158 2.3832
N2 - N0 == 0 1.3355 0.5518 2.1192
N3 - N0 == 0 2.5930 1.8093 3.3767
N4 - N0 == 0 2.6990 1.9153 3.4827
N2 - N1 == 0 -0.2640 -1.0477 0.5197
N3 - N1 == 0 0.9935 0.2098 1.7772
N4 - N1 == 0 1.0995 0.3158 1.8832
N3 - N2 == 0 1.2575 0.4738 2.0412
N4 - N2 == 0 1.3635 0.5798 2.1472
N4 - N3 == 0 0.1060 -0.6777 0.8897
N0 N1 N2 N3 N4
"a" "b" "b" "c" "c"
品種要因(V)の多重比較
> AnovaModel.3 <- aov(DATA ~ REP + N + V + N:V, data=Dataset)
> summary(AnovaModel.3)
Df Sum Sq Mean Sq F value Pr(>F)
REP 3 2.60 0.867 5.729 0.00222 **
N 4 41.23 10.309 68.153 < 2e-16 ***
V 2 1.05 0.526 3.480 0.03995 *
N:V 8 2.29 0.286 1.893 0.08671 .
Residuals 42 6.35 0.151
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> with(Dataset, numSummary(DATA, groups=V, statistics=c("mean", "sd")))
mean sd data:n
V1 4.7694 1.0548145 20
V2 5.0420 0.9574259 20
V3 5.0581 0.8559827 20
> local({
+ .Pairs <- glht(AnovaModel.3, linfct = mcp(V = "Tukey"))
+ print(summary(.Pairs)) # pairwise tests
+ print(confint(.Pairs)) # confidence intervals
+ print(cld(.Pairs)) # compact letter display
+ old.oma <- par(oma=c(0,5,0,0))
+ plot(confint(.Pairs))
+ par(old.oma)
+ })
Warning in mcp2matrix(model, linfct = linfct): covariate interactions found
-- default contrast might be inappropriate
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
V2 - V1 == 0 0.424 0.275 1.542 0.2821
V3 - V1 == 0 0.654 0.275 2.378 0.0561 .
V3 - V2 == 0 0.230 0.275 0.836 0.6828
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)
Simultaneous Confidence Intervals
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Quantile = 2.4298
95% family-wise confidence level
Linear Hypotheses:
Estimate lwr upr
V2 - V1 == 0 0.4240 -0.2442 1.0922
V3 - V1 == 0 0.6540 -0.0142 1.3222
V3 - V2 == 0 0.2300 -0.4382 0.8982
V1 V2 V3
"a" "a" "a"
ブロック要因(REP)の多重比較
> AnovaModel.4 <- aov(DATA ~ REP + N + V + N:V, data=Dataset)
> summary(AnovaModel.4)
Df Sum Sq Mean Sq F value Pr(>F)
REP 3 2.60 0.867 5.729 0.00222 **
N 4 41.23 10.309 68.153 < 2e-16 ***
V 2 1.05 0.526 3.480 0.03995 *
N:V 8 2.29 0.286 1.893 0.08671 .
Residuals 42 6.35 0.151
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> with(Dataset, numSummary(DATA, groups=REP, statistics=c("mean", "sd")))
mean sd data:n
1 5.132800 0.9331476 15
2 4.912533 0.9578146 15
3 5.146800 1.0105777 15
4 4.633867 0.9102152 15
> local({
+ .Pairs <- glht(AnovaModel.4, linfct = mcp(REP = "Tukey"))
+ print(summary(.Pairs)) # pairwise tests
+ print(confint(.Pairs)) # confidence intervals
+ print(cld(.Pairs)) # compact letter display
+ old.oma <- par(oma=c(0,5,0,0))
+ plot(confint(.Pairs))
+ par(old.oma)
+ })
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
2 - 1 == 0 -0.2203 0.1420 -1.551 0.41712
3 - 1 == 0 0.0140 0.1420 0.099 0.99965
4 - 1 == 0 -0.4989 0.1420 -3.513 0.00579 **
3 - 2 == 0 0.2343 0.1420 1.650 0.36275
4 - 2 == 0 -0.2787 0.1420 -1.962 0.21847
4 - 3 == 0 -0.5129 0.1420 -3.612 0.00415 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)
Simultaneous Confidence Intervals
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DATA ~ REP + N + V + N:V, data = Dataset)
Quantile = 2.6761
95% family-wise confidence level
Linear Hypotheses:
Estimate lwr upr
2 - 1 == 0 -0.2203 -0.6003 0.1598
3 - 1 == 0 0.0140 -0.3660 0.3940
4 - 1 == 0 -0.4989 -0.8790 -0.1189
3 - 2 == 0 0.2343 -0.1458 0.6143
4 - 2 == 0 -0.2787 -0.6587 0.1014
4 - 3 == 0 -0.5129 -0.8930 -0.1329
1 2 3 4
"b" "ab" "b" "a"