These notes build on the prior chapters’ R Code and Instructions, where we discuss some operational aspects of R and detail the use of many functions and how to install packages. Here, we skip details that have already been discussed in the prior R Code and Instructions.
Load the AMCP package.
library(AMCP)
data(chapter_13_table_1)
str(chapter_13_table_1)## 'data.frame': 5 obs. of 2 variables:
## $ Time1: int 2 4 6 8 10
## $ Time2: int 3 7 8 9 13chapter_13_table_1## Time1 Time2
## 1 2 3
## 2 4 7
## 3 6 8
## 4 8 9
## 5 10 13# Create difference score
chapter_13_table_1$Difference <- chapter_13_table_1$Time2 - chapter_13_table_1$Time1Use the Difference score on the left-hand-size of the “~” symbol.
RM.2DV <- lm(Difference ~ 1, data=chapter_13_table_1)
# Note summary() returns the individual models (for each dependent variable); this is not a MANOVA.
summary(RM.2DV)##
## Call:
## lm(formula = Difference ~ 1, data = chapter_13_table_1)
##
## Residuals:
## 1 2 3 4 5
## -1.000e+00 1.000e+00 -1.406e-17 -1.000e+00 1.000e+00
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.0000 0.4472 4.472 0.0111 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1 on 4 degrees of freedomThe above is equivilant to a paired samples t-test or a t-test on the difference score.
t.test(chapter_13_table_1$Difference, paired=FALSE)##
## One Sample t-test
##
## data: chapter_13_table_1$Difference
## t = 4.4721, df = 4, p-value = 0.01106
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 0.758336 3.241664
## sample estimates:
## mean of x
## 2#Or
t.test(chapter_13_table_1$Time2, chapter_13_table_1$Time1, paired=TRUE)##
## Paired t-test
##
## data: chapter_13_table_1$Time2 and chapter_13_table_1$Time1
## t = 4.4721, df = 4, p-value = 0.01106
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
## 0.758336 3.241664
## sample estimates:
## mean difference
## 2data(chapter_13_table_2)
str(chapter_13_table_2)## 'data.frame': 8 obs. of 3 variables:
## $ Time1: int 2 4 6 8 10 3 6 9
## $ Time2: int 3 7 8 9 13 4 9 11
## $ Time3: int 5 9 8 8 15 9 8 10chapter_13_table_2## Time1 Time2 Time3
## 1 2 3 5
## 2 4 7 9
## 3 6 8 8
## 4 8 9 8
## 5 10 13 15
## 6 3 4 9
## 7 6 9 8
## 8 9 11 10# Define difference scores to include in the multivariate analysis.
chapter_13_table_2$D1 <- chapter_13_table_2$Time2-chapter_13_table_2$Time1
chapter_13_table_2$D2 <- chapter_13_table_2$Time3-chapter_13_table_2$Time2
chapter_13_table_2## Time1 Time2 Time3 D1 D2
## 1 2 3 5 1 2
## 2 4 7 9 3 2
## 3 6 8 8 2 0
## 4 8 9 8 1 -1
## 5 10 13 15 3 2
## 6 3 4 9 1 5
## 7 6 9 8 3 -1
## 8 9 11 10 2 -1Until now only a single dependent variable was on the left-hand-side of the “~” symbol. Here, however, there will be a matrix that has as its columns the difference scores of interest.
MANOVA.Diff.3DV <- manova(cbind(D1, D2) ~ 1, data=chapter_13_table_2)
anova(MANOVA.Diff.3DV)## Analysis of Variance Table
##
## Df Pillai approx F num Df den Df Pr(>F)
## (Intercept) 1 0.86454 19.148 2 6 0.002485 **
## Residuals 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Alternative MANOVA tests.
anova(MANOVA.Diff.3DV, test="Wilks")## Analysis of Variance Table
##
## Df Wilks approx F num Df den Df Pr(>F)
## (Intercept) 1 0.13545 19.148 2 6 0.002485 **
## Residuals 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(MANOVA.Diff.3DV, test="Hotelling-Lawley")## Analysis of Variance Table
##
## Df Hotelling-Lawley approx F num Df den Df Pr(>F)
## (Intercept) 1 6.3825 19.148 2 6 0.002485 **
## Residuals 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(MANOVA.Diff.3DV, test="Roy")## Analysis of Variance Table
##
## Df Roy approx F num Df den Df Pr(>F)
## (Intercept) 1 6.3825 19.148 2 6 0.002485 **
## Residuals 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1data(chapter_13_table_6)
str(chapter_13_table_6)## 'data.frame': 12 obs. of 4 variables:
## $ Months30: int 108 103 96 84 118 110 129 90 84 96 ...
## $ Months36: int 96 117 107 85 125 107 128 84 104 100 ...
## $ Months42: int 110 127 106 92 125 96 123 101 100 103 ...
## $ Months48: int 122 133 107 99 116 91 128 113 88 105 ...chapter_13_table_6## Months30 Months36 Months42 Months48
## 1 108 96 110 122
## 2 103 117 127 133
## 3 96 107 106 107
## 4 84 85 92 99
## 5 118 125 125 116
## 6 110 107 96 91
## 7 129 128 123 128
## 8 90 84 101 113
## 9 84 104 100 88
## 10 96 100 103 105
## 11 105 114 105 112
## 12 113 117 132 130chapter_13_table_6$D1 <- chapter_13_table_6$Months36-chapter_13_table_6$Months30
chapter_13_table_6$D2 <- chapter_13_table_6$Months42-chapter_13_table_6$Months36
chapter_13_table_6$D3 <- chapter_13_table_6$Months48-chapter_13_table_6$Months42
chapter_13_table_6## Months30 Months36 Months42 Months48 D1 D2 D3
## 1 108 96 110 122 -12 14 12
## 2 103 117 127 133 14 10 6
## 3 96 107 106 107 11 -1 1
## 4 84 85 92 99 1 7 7
## 5 118 125 125 116 7 0 -9
## 6 110 107 96 91 -3 -11 -5
## 7 129 128 123 128 -1 -5 5
## 8 90 84 101 113 -6 17 12
## 9 84 104 100 88 20 -4 -12
## 10 96 100 103 105 4 3 2
## 11 105 114 105 112 9 -9 7
## 12 113 117 132 130 4 15 -2MANOVA.4DV <- manova(cbind(D1, D2, D3) ~ 1, data=chapter_13_table_6)
anova(MANOVA.4DV)## Analysis of Variance Table
##
## Df Pillai approx F num Df den Df Pr(>F)
## (Intercept) 1 0.42749 2.2401 3 9 0.1528
## Residuals 11anova(MANOVA.4DV, test="Pillai")## Analysis of Variance Table
##
## Df Pillai approx F num Df den Df Pr(>F)
## (Intercept) 1 0.42749 2.2401 3 9 0.1528
## Residuals 11anova(MANOVA.4DV, test="Wilks")## Analysis of Variance Table
##
## Df Wilks approx F num Df den Df Pr(>F)
## (Intercept) 1 0.57251 2.2401 3 9 0.1528
## Residuals 11anova(MANOVA.4DV, test="Hotelling-Lawley")## Analysis of Variance Table
##
## Df Hotelling-Lawley approx F num Df den Df Pr(>F)
## (Intercept) 1 0.7467 2.2401 3 9 0.1528
## Residuals 11anova(MANOVA.4DV, test="Roy")## Analysis of Variance Table
##
## Df Roy approx F num Df den Df Pr(>F)
## (Intercept) 1 0.7467 2.2401 3 9 0.1528
## Residuals 11Additionally, the Anova() function from the car can be used,
which also allows more options that we do not discuss. See https://socialsciences.mcmaster.ca/jfox/Books/Companion/appendices/Appendix-Multivariate-Linear-Models.pdf
for additional information on coding and capabilities of
Anova() and other functions in the car package. Of
particular relevance here is the Anova() and its option for
different types of sums of squares (see below). Note that the base R
appraoch (above) uses Type I sums of squares.
library(car)## Loading required package: carDataAnova(MANOVA.4DV, type=c("III"))##
## Type III MANOVA Tests: Pillai test statistic
## Df test stat approx F num Df den Df Pr(>F)
## (Intercept) 1 0.42749 2.2401 3 9 0.1528Anova(MANOVA.4DV, type=c("II"))## Note: model has only an intercept; equivalent type-III tests substituted.##
## Type III MANOVA Tests: Pillai test statistic
## Df test stat approx F num Df den Df Pr(>F)
## (Intercept) 1 0.42749 2.2401 3 9 0.1528