Designing Experiments and Analyzing Data: A Model Comparison Perspective (3rd edition) by Maxwell, Delaney, & Kelley
Information about the book is available at https://designingexperiments.com
R Code and Instructions to Accompany Chapter 7
library(AMCP)
data(chapter_7_table_1)
chapter_7_table_1$Group <- factor(chapter_7_table_1$Group)
chapter_7_table_1$Biofeedback <- as.numeric(chapter_7_table_1$Group %in% c(1, 2))
chapter_7_table_1$DrugTherapy <- as.numeric(chapter_7_table_1$Group %in% c(1, 3))
chapter_7_table_1## Group Score Biofeedback DrugTherapy
## 1 1 158 1 1
## 2 1 163 1 1
## 3 1 173 1 1
## 4 1 178 1 1
## 5 1 168 1 1
## 6 2 188 1 0
## 7 2 183 1 0
## 8 2 198 1 0
## 9 2 178 1 0
## 10 2 193 1 0
## 11 3 186 0 1
## 12 3 191 0 1
## 13 3 196 0 1
## 14 3 181 0 1
## 15 3 176 0 1
## 16 4 185 0 0
## 17 4 190 0 0
## 18 4 195 0 0
## 19 4 200 0 0
## 20 4 180 0 0This corresponds to Table 7.2. An ANOVA using the four Groups separately.
ANOVA.Object <- aov(Score~Group, data=chapter_7_table_1)
# Note the following produce the same summaries.
anova(ANOVA.Object)## Analysis of Variance Table
##
## Response: Score
## Df Sum Sq Mean Sq F value Pr(>F)
## Group 3 1540 513.33 8.2133 0.001553 **
## Residuals 16 1000 62.50
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(ANOVA.Object)## Df Sum Sq Mean Sq F value Pr(>F)
## Group 3 1540 513.3 8.213 0.00155 **
## Residuals 16 1000 62.5
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Table 7.3
aggregate(Score ~ Biofeedback+DrugTherapy, mean, data=chapter_7_table_1)## Biofeedback DrugTherapy Score
## 1 0 0 190
## 2 1 0 188
## 3 0 1 186
## 4 1 1 168Alternatively, for more than a single descriptive statistics, dplyr can be used.
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, uniongroup_by(chapter_7_table_1, Group) %>%
summarise(mean=mean(Score), sd=sd(Score))## # A tibble: 4 × 3
## Group mean sd
## <fctr> <dbl> <dbl>
## 1 1 168 7.905694
## 2 2 188 7.905694
## 3 3 186 7.905694
## 4 4 190 7.905694Interaction plot that compares to Figure 7.1, A. Note that the labels are in the opposite direction as compared to the book.
# Default names
interaction.plot(chapter_7_table_1$Biofeedback, chapter_7_table_1$DrugTherapy, chapter_7_table_1$Score)
# Using better names.
interaction.plot(chapter_7_table_1$Biofeedback, chapter_7_table_1$DrugTherapy, chapter_7_table_1$Score, xlab="Biofeedback", ylab="Score", trace.label="Drug Therepy", main="Interaction Plot")
Interaction plot that compares to Figure 7.1, B. Again, note that the labels are in the opposite direction as compared to the book.
# Default names
interaction.plot(chapter_7_table_1$DrugTherapy, chapter_7_table_1$Biofeedback, chapter_7_table_1$Score)
#Using better names.
interaction.plot(chapter_7_table_1$Biofeedback, chapter_7_table_1$DrugTherapy, chapter_7_table_1$Score, xlab="Drug Therepy", ylab="Score", trace.label="Biofeedback", main="Interaction Plot")
Moving to the data from Chapter 7, Table 5.
data("chapter_7_table_5")
chapter_7_table_5## Score Feedback Drug
## 1 170 1 1
## 2 175 1 1
## 3 165 1 1
## 4 180 1 1
## 5 160 1 1
## 6 186 1 2
## 7 194 1 2
## 8 201 1 2
## 9 215 1 2
## 10 219 1 2
## 11 180 1 3
## 12 187 1 3
## 13 199 1 3
## 14 170 1 3
## 15 204 1 3
## 16 173 2 1
## 17 194 2 1
## 18 197 2 1
## 19 190 2 1
## 20 176 2 1
## 21 189 2 2
## 22 194 2 2
## 23 217 2 2
## 24 206 2 2
## 25 199 2 2
## 26 202 2 3
## 27 228 2 3
## 28 190 2 3
## 29 206 2 3
## 30 224 2 3aggregate(Score ~ Feedback+Drug, mean, data=chapter_7_table_5)## Feedback Drug Score
## 1 1 1 170
## 2 2 1 186
## 3 1 2 203
## 4 2 2 201
## 5 1 3 188
## 6 2 3 210Here we perform a two-way ANOVA. Because the feedback and Drug data are numeric, we need to instruct R to treat it as a factor.
ANOVA.TwoWay_a <- aov(Score~as.factor(Feedback)*as.factor(Drug), data=chapter_7_table_5)
# Note the following produce the same summaries.
anova(ANOVA.TwoWay_a)## Analysis of Variance Table
##
## Response: Score
## Df Sum Sq Mean Sq F value Pr(>F)
## as.factor(Feedback) 1 1080 1080.00 6.9342 0.0145635
## as.factor(Drug) 2 3420 1710.00 10.9791 0.0004113
## as.factor(Feedback):as.factor(Drug) 2 780 390.00 2.5040 0.1028767
## Residuals 24 3738 155.75
##
## as.factor(Feedback) *
## as.factor(Drug) ***
## as.factor(Feedback):as.factor(Drug)
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(ANOVA.TwoWay_a)## Df Sum Sq Mean Sq F value Pr(>F)
## as.factor(Feedback) 1 1080 1080.0 6.934 0.014563 *
## as.factor(Drug) 2 3420 1710.0 10.979 0.000411 ***
## as.factor(Feedback):as.factor(Drug) 2 780 390.0 2.504 0.102877
## Residuals 24 3738 155.8
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1It is often better to define factors as factors in the dataset itself. Here we do that before replicating the analysis above.
chapter_7_table_5$Drug <- as.factor(chapter_7_table_5$Drug)
chapter_7_table_5$Biofeedback <- as.factor(chapter_7_table_5$Feedback)
# Although it appears nothing changes:
chapter_7_table_5## Score Feedback Drug Biofeedback
## 1 170 1 1 1
## 2 175 1 1 1
## 3 165 1 1 1
## 4 180 1 1 1
## 5 160 1 1 1
## 6 186 1 2 1
## 7 194 1 2 1
## 8 201 1 2 1
## 9 215 1 2 1
## 10 219 1 2 1
## 11 180 1 3 1
## 12 187 1 3 1
## 13 199 1 3 1
## 14 170 1 3 1
## 15 204 1 3 1
## 16 173 2 1 2
## 17 194 2 1 2
## 18 197 2 1 2
## 19 190 2 1 2
## 20 176 2 1 2
## 21 189 2 2 2
## 22 194 2 2 2
## 23 217 2 2 2
## 24 206 2 2 2
## 25 199 2 2 2
## 26 202 2 3 2
## 27 228 2 3 2
## 28 190 2 3 2
## 29 206 2 3 2
## 30 224 2 3 2# notice here that the structure of the data frame has changed (in that Drug and Biofeedback are Factors)
str(chapter_7_table_5)## 'data.frame': 30 obs. of 4 variables:
## $ Score : int 170 175 165 180 160 186 194 201 215 219 ...
## $ Feedback : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Drug : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 2 2 2 2 2 ...
## $ Biofeedback: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...# Replicate two-way ANOVA
ANOVA.TwoWay <- aov(Score~Biofeedback*Drug, data=chapter_7_table_5)
# Note the following produce the same summaries.
Summary_ANOVA.TwoWay <- summary(ANOVA.TwoWay)
Summary_ANOVA.TwoWay## Df Sum Sq Mean Sq F value Pr(>F)
## Biofeedback 1 1080 1080.0 6.934 0.014563 *
## Drug 2 3420 1710.0 10.979 0.000411 ***
## Biofeedback:Drug 2 780 390.0 2.504 0.102877
## Residuals 24 3738 155.8
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As in the code for Chapters 3 the relevant parts of the ANOVA table can be extracted for purposes of calculating the various effect sizes and follow-up tests (like in the Chapter 6 code), as discussed in the book (which can be used for further calculations).
# To see what the object contains.
str(Summary_ANOVA.TwoWay)## List of 1
## $ :Classes 'anova' and 'data.frame': 4 obs. of 5 variables:
## ..$ Df : num [1:4] 1 2 2 24
## ..$ Sum Sq : num [1:4] 1080 3420 780 3738
## ..$ Mean Sq: num [1:4] 1080 1710 390 156
## ..$ F value: num [1:4] 6.93 10.98 2.5 NA
## ..$ Pr(>F) : num [1:4] 0.014563 0.000411 0.102877 NA
## - attr(*, "class")= chr [1:2] "summary.aov" "listof"# Extract the sums of squares.
Summary_ANOVA.TwoWay[[1]]["Sum Sq"]## Sum Sq
## Biofeedback 1080
## Drug 3420
## Biofeedback:Drug 780
## Residuals 3738# Extract the sums of degrees of freedom.
Summary_ANOVA.TwoWay[[1]]["Df"]## Df
## Biofeedback 1
## Drug 2
## Biofeedback:Drug 2
## Residuals 24# Extract the mean squares.
Summary_ANOVA.TwoWay[[1]]["Mean Sq"]## Mean Sq
## Biofeedback 1080.00
## Drug 1710.00
## Biofeedback:Drug 390.00
## Residuals 155.75Form summary values of interest (which can be used for further calculations).
library(dplyr)
# First for Biofeedback
Summary_Score.Biofeedback <- chapter_7_table_5 %>%
group_by(Biofeedback) %>%
summarize(
mean.Score=mean(Score),
sd.Score=sd(Score),
n.Score=n()
)
Summary_Score.Biofeedback## # A tibble: 2 × 4
## Biofeedback mean.Score sd.Score n.Score
## <fctr> <dbl> <dbl> <int>
## 1 1 187 17.96823 15
## 2 2 199 15.62507 15# Now for Drug
Summary_Score.Drug <- chapter_7_table_5 %>%
group_by(Drug) %>%
summarize(
mean.Score=mean(Score),
sd.Score=sd(Score),
n.Score=n()
)
Summary_Score.Drug## # A tibble: 3 × 4
## Drug mean.Score sd.Score n.Score
## <fctr> <dbl> <dbl> <int>
## 1 1 178 12.29273 10
## 2 2 202 11.84155 10
## 3 3 199 18.18424 10Moving to the data from Table 15, Chapter 7.
data(chapter_7_table_15)
chapter_7_table_15## Sex Education Salary
## 1 1 1 24
## 2 1 1 26
## 3 1 1 25
## 4 1 1 24
## 5 1 1 27
## 6 1 1 24
## 7 1 1 27
## 8 1 1 23
## 9 1 2 15
## 10 1 2 17
## 11 1 2 20
## 12 1 2 16
## 13 2 1 25
## 14 2 1 29
## 15 2 1 27
## 16 2 2 19
## 17 2 2 18
## 18 2 2 21
## 19 2 2 20
## 20 2 2 21
## 21 2 2 22
## 22 2 2 19str(chapter_7_table_15)## 'data.frame': 22 obs. of 3 variables:
## $ Sex : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Education: int 1 1 1 1 1 1 1 1 2 2 ...
## $ Salary : int 24 26 25 24 27 24 27 23 15 17 ...chapter_7_table_15$Degree <- as.factor(as.numeric(chapter_7_table_15$Education==1))
chapter_7_table_15$Female <- as.factor(as.numeric(chapter_7_table_15$Sex==1))Unbalanced ANOVA designs are not straightforward, as discussed in the book. There are multiple ways in which the sums of squares can be calculated, and not everyone agrees with the best approach. The book generally recomends what is termed ``Type III’’ sums of squares. Whereas Type III sums of squares is the default in IBM SPSS Statistics and SAS, it is not in R. R uses what the book refers to as Type I sums of squares. Thus, the analysis of the same data in SAS or IBM SPSS vs. R, for example, will differ for unbalanced designs (e.g., when the sample sizes in the cells differ). To demonstrate, consider the “base” ANOVA in R, where Degree is entered first versus where Female is entered first. You will notice that the interaction and residual terms are the same, but that the main effects for the Degree and Female differ, based on the order of entry of the terms. To better understand some of the logic behind R’s appraoch, consider this presentation http://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf by W. N. Venables.
ANOVA_C7T15.Degree.1st <- aov(Salary~Degree*Female, data=chapter_7_table_15)
anova(ANOVA_C7T15.Degree.1st)## Analysis of Variance Table
##
## Response: Salary
## Df Sum Sq Mean Sq F value Pr(>F)
## Degree 1 242.227 242.227 87.2018 2.534e-08 ***
## Female 1 30.462 30.462 10.9662 0.003881 **
## Degree:Female 1 1.175 1.175 0.4229 0.523690
## Residuals 18 50.000 2.778
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ANOVA_C7T15.Female.1st <- aov(Salary~Female*Degree, data=chapter_7_table_15)
anova(ANOVA_C7T15.Female.1st)## Analysis of Variance Table
##
## Response: Salary
## Df Sum Sq Mean Sq F value Pr(>F)
## Female 1 0.297 0.297 0.1069 0.7475
## Degree 1 272.392 272.392 98.0611 1.038e-08 ***
## Female:Degree 1 1.175 1.175 0.4229 0.5237
## Residuals 18 50.000 2.778
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1One appraoch to obtaining the Type III sums of squares is to use the drop1() function for a fitted model (either order of entry is fine). See http://www.statmethods.net/stats/anova.html.
ANOVA_T3SS <- drop1(ANOVA_C7T15.Degree.1st, ~., test="F")
ANOVA_T3SS## Single term deletions
##
## Model:
## Salary ~ Degree * Female
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 50.000 26.062
## Degree 1 102.900 152.900 48.652 37.0440 9.447e-06 ***
## Female 1 22.909 72.909 32.360 8.2473 0.01014 *
## Degree:Female 1 1.175 51.175 24.573 0.4229 0.52369
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# The above is the same as the following.
drop1(ANOVA_C7T15.Female.1st, ~., test="F")## Single term deletions
##
## Model:
## Salary ~ Female * Degree
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 50.000 26.062
## Female 1 22.909 72.909 32.360 8.2473 0.01014 *
## Degree 1 102.900 152.900 48.652 37.0440 9.447e-06 ***
## Female:Degree 1 1.175 51.175 24.573 0.4229 0.52369
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1An alternative appraoch is to use the car package, which uses as it’s default Type II sums of squares. However, it is more straight-forward on switching between different types of sums of squares using its Anova() function (note the capital “A”) with the type parameter. Note that either of the previous (Type I) ANOVA objects can be used with the Anova() function to produce the Type III sums of squres. Also, an lm() object can be used, which we recomenmd (see below).
# install.packages("car")
library(car)##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodeAnova(ANOVA_C7T15.Degree.1st, type="III")## Anova Table (Type III tests)
##
## Response: Salary
## Sum Sq Df F value Pr(>F)
## (Intercept) 2800.00 1 1008.0000 < 2.2e-16 ***
## Degree 102.90 1 37.0440 9.447e-06 ***
## Female 22.91 1 8.2473 0.01014 *
## Degree:Female 1.17 1 0.4229 0.52369
## Residuals 50.00 18
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Anova(ANOVA_C7T15.Degree.1st, type="III")## Anova Table (Type III tests)
##
## Response: Salary
## Sum Sq Df F value Pr(>F)
## (Intercept) 2800.00 1 1008.0000 < 2.2e-16 ***
## Degree 102.90 1 37.0440 9.447e-06 ***
## Female 22.91 1 8.2473 0.01014 *
## Degree:Female 1.17 1 0.4229 0.52369
## Residuals 50.00 18
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Type III sums of squares using Anova() from the car package.
C7T15_lm <- lm(Salary~Female*Degree, data=chapter_7_table_15)
Anova(C7T15_lm, type="III")## Anova Table (Type III tests)
##
## Response: Salary
## Sum Sq Df F value Pr(>F)
## (Intercept) 2800.00 1 1008.0000 < 2.2e-16 ***
## Female 22.91 1 8.2473 0.01014 *
## Degree 102.90 1 37.0440 9.447e-06 ***
## Female:Degree 1.17 1 0.4229 0.52369
## Residuals 50.00 18
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Analysis for MMPI Depression data (Table 7.23).
data(chapter_7_table_23)
chapter_7_table_23## Therapy Severity Score
## 1 1 1 41
## 2 1 1 43
## 3 1 1 50
## 4 1 2 51
## 5 1 2 43
## 6 1 2 53
## 7 1 2 54
## 8 1 2 46
## 9 1 3 45
## 10 1 3 55
## 11 1 3 56
## 12 1 3 60
## 13 1 3 58
## 14 1 3 62
## 15 1 3 62
## 16 2 1 56
## 17 2 1 47
## 18 2 1 45
## 19 2 1 46
## 20 2 1 49
## 21 2 2 58
## 22 2 2 54
## 23 2 2 49
## 24 2 2 61
## 25 2 2 52
## 26 2 2 62
## 27 2 3 59
## 28 2 3 55
## 29 2 3 68
## 30 2 3 63
## 31 3 1 43
## 32 3 1 56
## 33 3 1 48
## 34 3 1 46
## 35 3 1 47
## 36 3 2 59
## 37 3 2 46
## 38 3 2 58
## 39 3 2 54
## 40 3 3 55
## 41 3 3 69
## 42 3 3 63
## 43 3 3 56
## 44 3 3 62
## 45 3 3 67str(chapter_7_table_23)## 'data.frame': 45 obs. of 3 variables:
## $ Therapy : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Severity: int 1 1 1 2 2 2 2 2 3 3 ...
## $ Score : int 41 43 50 51 43 53 54 46 45 55 ...chapter_7_table_23$Therapy <- chapter_7_table_23$Therapy
chapter_7_table_23$Therapy[chapter_7_table_23$Therapy==1] <- "Cognitive_Behavioral"
chapter_7_table_23$Therapy[chapter_7_table_23$Therapy==2] <- "Rogerian"
chapter_7_table_23$Therapy[chapter_7_table_23$Therapy==3] <- "Assertiveness_Training"
chapter_7_table_23$Severity <- chapter_7_table_23$Severity
chapter_7_table_23$Severity[chapter_7_table_23$Severity==1] <- "Mild"
chapter_7_table_23$Severity[chapter_7_table_23$Severity==2] <- "Moderate"
chapter_7_table_23$Severity[chapter_7_table_23$Severity==3] <- "Severe"
chapter_7_table_23## Therapy Severity Score
## 1 Cognitive_Behavioral Mild 41
## 2 Cognitive_Behavioral Mild 43
## 3 Cognitive_Behavioral Mild 50
## 4 Cognitive_Behavioral Moderate 51
## 5 Cognitive_Behavioral Moderate 43
## 6 Cognitive_Behavioral Moderate 53
## 7 Cognitive_Behavioral Moderate 54
## 8 Cognitive_Behavioral Moderate 46
## 9 Cognitive_Behavioral Severe 45
## 10 Cognitive_Behavioral Severe 55
## 11 Cognitive_Behavioral Severe 56
## 12 Cognitive_Behavioral Severe 60
## 13 Cognitive_Behavioral Severe 58
## 14 Cognitive_Behavioral Severe 62
## 15 Cognitive_Behavioral Severe 62
## 16 Rogerian Mild 56
## 17 Rogerian Mild 47
## 18 Rogerian Mild 45
## 19 Rogerian Mild 46
## 20 Rogerian Mild 49
## 21 Rogerian Moderate 58
## 22 Rogerian Moderate 54
## 23 Rogerian Moderate 49
## 24 Rogerian Moderate 61
## 25 Rogerian Moderate 52
## 26 Rogerian Moderate 62
## 27 Rogerian Severe 59
## 28 Rogerian Severe 55
## 29 Rogerian Severe 68
## 30 Rogerian Severe 63
## 31 Assertiveness_Training Mild 43
## 32 Assertiveness_Training Mild 56
## 33 Assertiveness_Training Mild 48
## 34 Assertiveness_Training Mild 46
## 35 Assertiveness_Training Mild 47
## 36 Assertiveness_Training Moderate 59
## 37 Assertiveness_Training Moderate 46
## 38 Assertiveness_Training Moderate 58
## 39 Assertiveness_Training Moderate 54
## 40 Assertiveness_Training Severe 55
## 41 Assertiveness_Training Severe 69
## 42 Assertiveness_Training Severe 63
## 43 Assertiveness_Training Severe 56
## 44 Assertiveness_Training Severe 62
## 45 Assertiveness_Training Severe 67str(chapter_7_table_23)## 'data.frame': 45 obs. of 3 variables:
## $ Therapy : chr "Cognitive_Behavioral" "Cognitive_Behavioral" "Cognitive_Behavioral" "Cognitive_Behavioral" ...
## $ Severity: chr "Mild" "Mild" "Mild" "Moderate" ...
## $ Score : int 41 43 50 51 43 53 54 46 45 55 ...Analysis of data from chapter_7_table_23
Descriptives of the cells usingdplyr can be used.
group_by(chapter_7_table_23, Therapy, Severity) %>%
summarise(mean=mean(Score), sd=sd(Score), n=n())## Source: local data frame [9 x 5]
## Groups: Therapy [?]
##
## Therapy Severity mean sd n
## <chr> <chr> <dbl> <dbl> <int>
## 1 Assertiveness_Training Mild 48.00000 4.847680 5
## 2 Assertiveness_Training Moderate 54.25000 5.909033 4
## 3 Assertiveness_Training Severe 62.00000 5.656854 6
## 4 Cognitive_Behavioral Mild 44.66667 4.725816 3
## 5 Cognitive_Behavioral Moderate 49.40000 4.722288 5
## 6 Cognitive_Behavioral Severe 56.85714 5.899960 7
## 7 Rogerian Mild 48.60000 4.393177 5
## 8 Rogerian Moderate 56.00000 5.176872 6
## 9 Rogerian Severe 61.25000 5.560276 4Descriptives of the marginal means for Therapy and Degree usingdplyr can be used.
group_by(chapter_7_table_23, Therapy) %>%
summarise(mean=mean(Score), sd=sd(Score), n=n())## # A tibble: 3 × 4
## Therapy mean sd n
## <chr> <dbl> <dbl> <int>
## 1 Assertiveness_Training 55.26667 8.013085 15
## 2 Cognitive_Behavioral 51.93333 7.085868 15
## 3 Rogerian 54.93333 6.922702 15group_by(chapter_7_table_23, Severity) %>%
summarise(mean=mean(Score), sd=sd(Score), n=n())## # A tibble: 3 × 4
## Severity mean sd n
## <chr> <dbl> <dbl> <int>
## 1 Mild 47.46154 4.539005 13
## 2 Moderate 53.33333 5.677860 15
## 3 Severe 59.70588 5.913594 17Interaction plot that compares to Figure 7.1, A. Note that the labels are in the opposite direction as compared to the book.
# Default names
interaction.plot(chapter_7_table_23$Therapy, chapter_7_table_23$Severity, chapter_7_table_23$Score, xlab="Type of Therepy", ylab="Score", trace.label="Degree of Severity", main="Interaction Plot for MMPI Depression Scores")
Type I sums of squares.
C7T23_SS1a <- aov(Score~Therapy*Severity, data=chapter_7_table_23)
anova(C7T23_SS1a)## Analysis of Variance Table
##
## Response: Score
## Df Sum Sq Mean Sq F value Pr(>F)
## Therapy 2 101.11 50.56 1.8102 0.1782
## Severity 2 1253.19 626.59 22.4357 4.711e-07 ***
## Therapy:Severity 4 14.19 3.55 0.1270 0.9717
## Residuals 36 1005.42 27.93
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1C7T23_SS1b <- aov(Score~Severity*Therapy, data=chapter_7_table_23)
anova(C7T23_SS1b)## Analysis of Variance Table
##
## Response: Score
## Df Sum Sq Mean Sq F value Pr(>F)
## Severity 2 1115.82 557.91 19.9764 1.458e-06 ***
## Therapy 2 238.48 119.24 4.2695 0.02168 *
## Severity:Therapy 4 14.19 3.55 0.1270 0.97170
## Residuals 36 1005.42 27.93
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Type II sums of squares.
C7T23_lm <- lm(Score~Therapy*Severity, data=chapter_7_table_23)
Anova(C7T23_lm, type="II")## Anova Table (Type II tests)
##
## Response: Score
## Sum Sq Df F value Pr(>F)
## Therapy 238.48 2 4.2695 0.02168 *
## Severity 1253.19 2 22.4357 4.711e-07 ***
## Therapy:Severity 14.19 4 0.1270 0.97170
## Residuals 1005.42 36
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Type III sums of squares.
Anova(C7T23_lm, type="III")## Anova Table (Type III tests)
##
## Response: Score
## Sum Sq Df F value Pr(>F)
## (Intercept) 11520.0 1 412.4828 < 2.2e-16 ***
## Therapy 31.4 2 0.5615 0.575263
## Severity 540.2 2 9.6708 0.000435 ***
## Therapy:Severity 14.2 4 0.1270 0.971701
## Residuals 1005.4 36
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1C7T23_lm <- lm(Score~Therapy*Severity, data=chapter_7_table_23) # Repeated from aboveFor an of the ANOVA models of interest, we can perform Tukey HSD tests with the TukeyHSD() function.
C7T23_ANOVA <- aov(Score~Therapy*Severity, data=chapter_7_table_23) # Repeated from above
TukeyHSD(C7T23_ANOVA)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Score ~ Therapy * Severity, data = chapter_7_table_23)
##
## $Therapy
## diff lwr upr
## Cognitive_Behavioral-Assertiveness_Training -3.3333333 -8.050125 1.383458
## Rogerian-Assertiveness_Training -0.3333333 -5.050125 4.383458
## Rogerian-Cognitive_Behavioral 3.0000000 -1.716792 7.716792
## p adj
## Cognitive_Behavioral-Assertiveness_Training 0.2090188
## Rogerian-Assertiveness_Training 0.9836924
## Rogerian-Cognitive_Behavioral 0.2781897
##
## $Severity
## diff lwr upr p adj
## Moderate-Mild 6.218803 1.323957 11.11365 0.0100799
## Severe-Mild 12.797888 8.038602 17.55717 0.0000004
## Severe-Moderate 6.579085 2.003125 11.15505 0.0033855
##
## $`Therapy:Severity`
## diff
## Cognitive_Behavioral:Mild-Assertiveness_Training:Mild -3.3333333
## Rogerian:Mild-Assertiveness_Training:Mild 0.6000000
## Assertiveness_Training:Moderate-Assertiveness_Training:Mild 6.2500000
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Mild 1.4000000
## Rogerian:Moderate-Assertiveness_Training:Mild 8.0000000
## Assertiveness_Training:Severe-Assertiveness_Training:Mild 14.0000000
## Cognitive_Behavioral:Severe-Assertiveness_Training:Mild 8.8571429
## Rogerian:Severe-Assertiveness_Training:Mild 13.2500000
## Rogerian:Mild-Cognitive_Behavioral:Mild 3.9333333
## Assertiveness_Training:Moderate-Cognitive_Behavioral:Mild 9.5833333
## Cognitive_Behavioral:Moderate-Cognitive_Behavioral:Mild 4.7333333
## Rogerian:Moderate-Cognitive_Behavioral:Mild 11.3333333
## Assertiveness_Training:Severe-Cognitive_Behavioral:Mild 17.3333333
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Mild 12.1904762
## Rogerian:Severe-Cognitive_Behavioral:Mild 16.5833333
## Assertiveness_Training:Moderate-Rogerian:Mild 5.6500000
## Cognitive_Behavioral:Moderate-Rogerian:Mild 0.8000000
## Rogerian:Moderate-Rogerian:Mild 7.4000000
## Assertiveness_Training:Severe-Rogerian:Mild 13.4000000
## Cognitive_Behavioral:Severe-Rogerian:Mild 8.2571429
## Rogerian:Severe-Rogerian:Mild 12.6500000
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Moderate -4.8500000
## Rogerian:Moderate-Assertiveness_Training:Moderate 1.7500000
## Assertiveness_Training:Severe-Assertiveness_Training:Moderate 7.7500000
## Cognitive_Behavioral:Severe-Assertiveness_Training:Moderate 2.6071429
## Rogerian:Severe-Assertiveness_Training:Moderate 7.0000000
## Rogerian:Moderate-Cognitive_Behavioral:Moderate 6.6000000
## Assertiveness_Training:Severe-Cognitive_Behavioral:Moderate 12.6000000
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Moderate 7.4571429
## Rogerian:Severe-Cognitive_Behavioral:Moderate 11.8500000
## Assertiveness_Training:Severe-Rogerian:Moderate 6.0000000
## Cognitive_Behavioral:Severe-Rogerian:Moderate 0.8571429
## Rogerian:Severe-Rogerian:Moderate 5.2500000
## Cognitive_Behavioral:Severe-Assertiveness_Training:Severe -5.1428571
## Rogerian:Severe-Assertiveness_Training:Severe -0.7500000
## Rogerian:Severe-Cognitive_Behavioral:Severe 4.3928571
## lwr
## Cognitive_Behavioral:Mild-Assertiveness_Training:Mild -16.0582074
## Rogerian:Mild-Assertiveness_Training:Mild -10.4200642
## Assertiveness_Training:Moderate-Assertiveness_Training:Mild -5.4385432
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Mild -9.6200642
## Rogerian:Moderate-Assertiveness_Training:Mild -2.5509082
## Assertiveness_Training:Severe-Assertiveness_Training:Mild 3.4490918
## Cognitive_Behavioral:Severe-Assertiveness_Training:Mild -1.3454541
## Rogerian:Severe-Assertiveness_Training:Mild 1.5614568
## Rogerian:Mild-Cognitive_Behavioral:Mild -8.7915407
## Assertiveness_Training:Moderate-Cognitive_Behavioral:Mild -3.7246585
## Cognitive_Behavioral:Moderate-Cognitive_Behavioral:Mild -7.9915407
## Rogerian:Moderate-Cognitive_Behavioral:Mild -0.9874730
## Assertiveness_Training:Severe-Cognitive_Behavioral:Mild 5.0125270
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Mild 0.1666004
## Rogerian:Severe-Cognitive_Behavioral:Mild 3.2753415
## Assertiveness_Training:Moderate-Rogerian:Mild -6.0385432
## Cognitive_Behavioral:Moderate-Rogerian:Mild -10.2200642
## Rogerian:Moderate-Rogerian:Mild -3.1509082
## Assertiveness_Training:Severe-Rogerian:Mild 2.8490918
## Cognitive_Behavioral:Severe-Rogerian:Mild -1.9454541
## Rogerian:Severe-Rogerian:Mild 0.9614568
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Moderate -16.5385432
## Rogerian:Moderate-Assertiveness_Training:Moderate -9.4973059
## Assertiveness_Training:Severe-Assertiveness_Training:Moderate -3.4973059
## Cognitive_Behavioral:Severe-Assertiveness_Training:Moderate -8.3140847
## Rogerian:Severe-Assertiveness_Training:Moderate -5.3208063
## Rogerian:Moderate-Cognitive_Behavioral:Moderate -3.9509082
## Assertiveness_Training:Severe-Cognitive_Behavioral:Moderate 2.0490918
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Moderate -2.7454541
## Rogerian:Severe-Cognitive_Behavioral:Moderate 0.1614568
## Assertiveness_Training:Severe-Rogerian:Moderate -4.0598962
## Cognitive_Behavioral:Severe-Rogerian:Moderate -8.8368157
## Rogerian:Severe-Rogerian:Moderate -5.9973059
## Cognitive_Behavioral:Severe-Assertiveness_Training:Severe -14.8368157
## Rogerian:Severe-Assertiveness_Training:Severe -11.9973059
## Rogerian:Severe-Cognitive_Behavioral:Severe -6.5283704
## upr
## Cognitive_Behavioral:Mild-Assertiveness_Training:Mild 9.391541
## Rogerian:Mild-Assertiveness_Training:Mild 11.620064
## Assertiveness_Training:Moderate-Assertiveness_Training:Mild 17.938543
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Mild 12.420064
## Rogerian:Moderate-Assertiveness_Training:Mild 18.550908
## Assertiveness_Training:Severe-Assertiveness_Training:Mild 24.550908
## Cognitive_Behavioral:Severe-Assertiveness_Training:Mild 19.059740
## Rogerian:Severe-Assertiveness_Training:Mild 24.938543
## Rogerian:Mild-Cognitive_Behavioral:Mild 16.658207
## Assertiveness_Training:Moderate-Cognitive_Behavioral:Mild 22.891325
## Cognitive_Behavioral:Moderate-Cognitive_Behavioral:Mild 17.458207
## Rogerian:Moderate-Cognitive_Behavioral:Mild 23.654140
## Assertiveness_Training:Severe-Cognitive_Behavioral:Mild 29.654140
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Mild 24.214352
## Rogerian:Severe-Cognitive_Behavioral:Mild 29.891325
## Assertiveness_Training:Moderate-Rogerian:Mild 17.338543
## Cognitive_Behavioral:Moderate-Rogerian:Mild 11.820064
## Rogerian:Moderate-Rogerian:Mild 17.950908
## Assertiveness_Training:Severe-Rogerian:Mild 23.950908
## Cognitive_Behavioral:Severe-Rogerian:Mild 18.459740
## Rogerian:Severe-Rogerian:Mild 24.338543
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Moderate 6.838543
## Rogerian:Moderate-Assertiveness_Training:Moderate 12.997306
## Assertiveness_Training:Severe-Assertiveness_Training:Moderate 18.997306
## Cognitive_Behavioral:Severe-Assertiveness_Training:Moderate 13.528370
## Rogerian:Severe-Assertiveness_Training:Moderate 19.320806
## Rogerian:Moderate-Cognitive_Behavioral:Moderate 17.150908
## Assertiveness_Training:Severe-Cognitive_Behavioral:Moderate 23.150908
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Moderate 17.659740
## Rogerian:Severe-Cognitive_Behavioral:Moderate 23.538543
## Assertiveness_Training:Severe-Rogerian:Moderate 16.059896
## Cognitive_Behavioral:Severe-Rogerian:Moderate 10.551101
## Rogerian:Severe-Rogerian:Moderate 16.497306
## Cognitive_Behavioral:Severe-Assertiveness_Training:Severe 4.551101
## Rogerian:Severe-Assertiveness_Training:Severe 10.497306
## Rogerian:Severe-Cognitive_Behavioral:Severe 15.314085
## p adj
## Cognitive_Behavioral:Mild-Assertiveness_Training:Mild 0.9935796
## Rogerian:Mild-Assertiveness_Training:Mild 1.0000000
## Assertiveness_Training:Moderate-Assertiveness_Training:Mild 0.7047196
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Mild 0.9999664
## Rogerian:Moderate-Assertiveness_Training:Mild 0.2656175
## Assertiveness_Training:Severe-Assertiveness_Training:Mild 0.0028675
## Cognitive_Behavioral:Severe-Assertiveness_Training:Mild 0.1325902
## Rogerian:Severe-Assertiveness_Training:Mild 0.0165147
## Rogerian:Mild-Cognitive_Behavioral:Mild 0.9814135
## Assertiveness_Training:Moderate-Cognitive_Behavioral:Mild 0.3279936
## Cognitive_Behavioral:Moderate-Cognitive_Behavioral:Mild 0.9450576
## Rogerian:Moderate-Cognitive_Behavioral:Mild 0.0919372
## Assertiveness_Training:Severe-Cognitive_Behavioral:Mild 0.0013422
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Mild 0.0447956
## Rogerian:Severe-Cognitive_Behavioral:Mild 0.0060621
## Assertiveness_Training:Moderate-Rogerian:Mild 0.8018175
## Cognitive_Behavioral:Moderate-Rogerian:Mild 0.9999996
## Rogerian:Moderate-Rogerian:Mild 0.3616114
## Assertiveness_Training:Severe-Rogerian:Mild 0.0048679
## Cognitive_Behavioral:Severe-Rogerian:Mild 0.1952294
## Rogerian:Severe-Rogerian:Mild 0.0255829
## Cognitive_Behavioral:Moderate-Assertiveness_Training:Moderate 0.9023328
## Rogerian:Moderate-Assertiveness_Training:Moderate 0.9998438
## Assertiveness_Training:Severe-Assertiveness_Training:Moderate 0.3846270
## Cognitive_Behavioral:Severe-Assertiveness_Training:Moderate 0.9965712
## Rogerian:Severe-Assertiveness_Training:Moderate 0.6353203
## Rogerian:Moderate-Cognitive_Behavioral:Moderate 0.5129491
## Assertiveness_Training:Severe-Cognitive_Behavioral:Moderate 0.0096887
## Cognitive_Behavioral:Severe-Cognitive_Behavioral:Moderate 0.3095077
## Rogerian:Severe-Cognitive_Behavioral:Moderate 0.0448108
## Assertiveness_Training:Severe-Rogerian:Moderate 0.5749751
## Cognitive_Behavioral:Severe-Rogerian:Moderate 0.9999980
## Rogerian:Severe-Rogerian:Moderate 0.8296447
## Cognitive_Behavioral:Severe-Assertiveness_Training:Severe 0.7131529
## Rogerian:Severe-Assertiveness_Training:Severe 0.9999998
## Rogerian:Severe-Cognitive_Behavioral:Severe 0.9166266