Analysis of variance applies to the inferential analysis of means
across groups. The lessR function ANOVA(),
abbreviated av(), provides this analysis, based on the base
R function aov().
One-way Independent Groups
First, for illustrative purposes, ignore the type of wool and only
examine the impact of tension on breaks.
The output includes descriptive statistics, ANOVA table, effect size
indices, Tukey’s multiple comparisons of means, and residuals, as well
as the scatterplot of the response variable with the levels of the
independent variable, and a visualization of the mean comparisons.
ANOVA(breaks ~ tension, data=warpbreaks)
##
## BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable: tension
## Levels: L M H
##
## Number of cases (rows) of data: 54
## Number of cases retained for analysis: 54
##
##
## DESCRIPTIVE STATISTICS
##
## n mean sd min max
## L 18 36.39 16.45 14.00 70.00
## M 18 26.39 9.12 12.00 42.00
## H 18 21.67 8.35 10.00 43.00
##
## Grand Mean: 28.148
##
##
## ANOVA
##
## df Sum Sq Mean Sq F-value p-value
## tension 2 2034.26 1017.13 7.21 0.0018
## Residuals 51 7198.56 141.15
##
## R Squared: 0.220
## R Sq Adjusted: 0.190
## Omega Squared: 0.187
##
## Cohen's f: 0.479
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
## -------------------------------
## diff lwr upr p adj
## M-L -10.00 -19.56 -0.44 0.04
## H-L -14.72 -24.28 -5.16 0.00
## H-M -4.72 -14.28 4.84 0.46
##
##
## RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
## [sorted by Standardized Residuals, ignoring + or - sign]
## [res_rows = 20, out of 54 cases (rows) of data, or res_rows="all"]
## -------------------------------------------
## tension breaks fitted residual z-resid
## 5 L 70.00 36.39 33.61 2.91
## 9 L 67.00 36.39 30.61 2.65
## 29 L 14.00 36.39 -22.39 -1.94
## 24 H 43.00 21.67 21.33 1.85
## 3 L 54.00 36.39 17.61 1.53
## 31 L 19.00 36.39 -17.39 -1.51
## 35 L 20.00 36.39 -16.39 -1.42
## 37 M 42.00 26.39 15.61 1.35
## 6 L 52.00 36.39 15.61 1.35
## 7 L 51.00 36.39 14.61 1.27
## 14 M 12.00 26.39 -14.39 -1.25
## 19 H 36.00 21.67 14.33 1.24
## 41 M 39.00 26.39 12.61 1.09
## 44 M 39.00 26.39 12.61 1.09
## 23 H 10.00 21.67 -11.67 -1.01
## 4 L 25.00 36.39 -11.39 -0.99
## 8 L 26.00 36.39 -10.39 -0.90
## 40 M 16.00 26.39 -10.39 -0.90
## 1 L 26.00 36.39 -10.39 -0.90
## 18 M 36.00 26.39 9.61 0.83
##
## ----------------------------------------
## Plot 1: 95% family-wise confidence level
## Plot 2: Scatterplot with Cell Means
## ----------------------------------------
The brief version forgoes the multiple comparisons and the
residuals.
av_brief(breaks ~ tension, data=warpbreaks)
##
## BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable: tension
## Levels: L M H
##
## Number of cases (rows) of data: 54
## Number of cases retained for analysis: 54
##
##
## DESCRIPTIVE STATISTICS
##
## n mean sd min max
## L 18 36.39 16.45 14.00 70.00
## M 18 26.39 9.12 12.00 42.00
## H 18 21.67 8.35 10.00 43.00
##
## Grand Mean: 28.148
##
##
## ANOVA
##
## df Sum Sq Mean Sq F-value p-value
## tension 2 2034.26 1017.13 7.21 0.0018
## Residuals 51 7198.56 141.15
##
## R Squared: 0.220
## R Sq Adjusted: 0.190
## Omega Squared: 0.187
##
## Cohen's f: 0.479
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
##
## RESIDUALS
Two-way Independent Groups
Specify the second independent variable preceded by a *
sign. The plot of the cell means is generated automatically.
ANOVA(breaks ~ tension * wool, data=warpbreaks)
##
## BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable 1: tension
## Levels: L M H
##
## Factor Variable 2: wool
## Levels: A B
##
## Number of cases (rows) of data: 54
## Number of cases retained for analysis: 54
##
## Two-way Between Groups ANOVA
##
##
## DESCRIPTIVE STATISTICS
##
##
## Equal cell sizes, so balanced design
##
##
## tension
## wool L M H
## A 9 9 9
## B 9 9 9
##
##
## tension
## wool L M H
## A 44.56 24.00 24.56
## B 28.22 28.78 18.78
##
## tension
##
## L M H
## 1 36.39 26.39 21.67
##
## wool
##
## A B
## 1 31.04 25.26
##
## NA
##
##
##
## tension
## wool L M H
## A 18.10 8.66 10.27
## B 9.86 9.43 4.89
##
##
## ANOVA
##
##
## df Sum Sq Mean Sq F-value p-value
## tension 2 2034.26 1017.13 8.50 0.0007
## wool 1 450.67 450.67 3.77 0.0582
## tension:wool 2 1002.78 501.39 4.19 0.0210
## Residuals 48 5745.11 119.69
##
## Partial Omega Squared for tension: 0.217
## Partial Omega Squared for wool: 0.049
## Partial Omega Squared for tension & wool: 0.106
##
## Cohen's f for tension: 0.527
## Cohen's f for wool: 0.226
## Cohen's f for tension_&_wool: 0.344
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
##
## Factor: tension
## -------------------------------
## diff lwr upr p adj
## M-L -10.00 -18.82 -1.18 0.02
## H-L -14.72 -23.54 -5.90 0.00
## H-M -4.72 -13.54 4.10 0.40
##
## Factor: wool
## -----------------------------
## diff lwr upr p adj
## B-A -5.78 -11.76 0.21 0.06
##
## Cell Means
## ------------------------------------
## diff lwr upr p adj
## M:A-L:A -20.56 -35.86 -5.25 0.00
## H:A-L:A -20.00 -35.31 -4.69 0.00
## L:B-L:A -16.33 -31.64 -1.03 0.03
## M:B-L:A -15.78 -31.08 -0.47 0.04
## H:B-L:A -25.78 -41.08 -10.47 0.00
## H:A-M:A 0.56 -14.75 15.86 1.00
## L:B-M:A 4.22 -11.08 19.53 0.96
## M:B-M:A 4.78 -10.53 20.08 0.94
## H:B-M:A -5.22 -20.53 10.08 0.91
## L:B-H:A 3.67 -11.64 18.97 0.98
## M:B-H:A 4.22 -11.08 19.53 0.96
## H:B-H:A -5.78 -21.08 9.53 0.87
## M:B-L:B 0.56 -14.75 15.86 1.00
## H:B-L:B -9.44 -24.75 5.86 0.46
## H:B-M:B -10.00 -25.31 5.31 0.39
##
##
## RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
## [sorted by Standardized Residuals, ignoring + or - sign]
## [res_rows = 20, out of 54 cases (rows) of data, or res_rows="all"]
## ------------------------------------------------
## tension wool breaks fitted residual z-resid
## 5 L A 70.00 44.56 25.44 2.47
## 9 L A 67.00 44.56 22.44 2.18
## 4 L A 25.00 44.56 -19.56 -1.90
## 1 L A 26.00 44.56 -18.56 -1.80
## 8 L A 26.00 44.56 -18.56 -1.80
## 24 H A 43.00 24.56 18.44 1.79
## 36 L B 44.00 28.22 15.78 1.53
## 23 H A 10.00 24.56 -14.56 -1.41
## 2 L A 30.00 44.56 -14.56 -1.41
## 29 L B 14.00 28.22 -14.22 -1.38
## 37 M B 42.00 28.78 13.22 1.28
## 34 L B 41.00 28.22 12.78 1.24
## 40 M B 16.00 28.78 -12.78 -1.24
## 14 M A 12.00 24.00 -12.00 -1.16
## 18 M A 36.00 24.00 12.00 1.16
## 19 H A 36.00 24.56 11.44 1.11
## 16 M A 35.00 24.00 11.00 1.07
## 41 M B 39.00 28.78 10.22 0.99
## 44 M B 39.00 28.78 10.22 0.99
## 39 M B 19.00 28.78 -9.78 -0.95
Can also obtain the cell mean plot directly from the means. Here use
lessR pivot() to compute the cell means of
breaks across tension and wool.
data(warpbreaks)
dm <- pivot(warpbreaks, mean, breaks, c(tension, wool))
dm
## tension wool breaks_n breaks_na breaks_mean
## 1 L A 9 0 44.556
## 2 M A 9 0 24.000
## 3 H A 9 0 24.556
## 4 L B 9 0 28.222
## 5 M B 9 0 28.778
## 6 H B 9 0 18.778
Store the aggregated data in the data frame named dm, so
explicitly identify with the data parameter. The computed
mean of breaks variable in the dm data frame from the
previous call to pivot() is named breaks_mean by
default.
Plot(tension, breaks_mean, by=wool, segments=TRUE, size=2, data=dm, main="Cell Means")
## >>> Suggestions
## Plot(tension, breaks_mean, data=dm, by=wool, size=2, segments=TRUE, main="Cell Means", means=FALSE) # do not plot means
## Plot(tension, breaks_mean, data=dm, by=wool, size=2, segments=TRUE, main="Cell Means", stat="mean") # only plot means
## ANOVA(breaks_mean ~ tension) # inferential analysis
##
## breaks_mean
## - by levels of -
## tension
##
## n miss mean sd min mdn max
## L 2 0 36.3890 11.5499 28.2220 36.3890 44.5560
## M 2 0 26.3890 3.3786 24.0000 26.3890 28.7780
## H 2 0 21.6670 4.0857 18.7780 21.6670 24.5560
##
## Some Parameter values (can be manually set)
## -------------------------------------------------------
## fill: #4398D0 #B28B2A filled color of the points
## color: #4398D0 #B28B2A edge color of the points
## size: 1.60 size of plotted points
## jitter_y: 0.00 random vertical movement of points
## jitter_x: 0.00 random horizontal movement of points
Randomized Block Design
The randomized block design has a treatment variable, usually
administered over time, and a blocking variable. The values of the
treatment variable are measured across each instance of the blocking
variable. In this example, repetitions are measured across four
different workout sessions. The person takes one of four supplements
before each session. Person is the blocking variable, and Supplement is
the treatment variable. Repetitions is the response variable.
The data are presented in a wide-form data table, a single row for
each person.
d <- read.csv(header=TRUE, text="
Person,sup1,sup2,sup3,sup4
p1,2,4,4,3
p2,2,5,4,6
p3,8,6,7,9
p4,4,3,5,7
p5,2,1,2,3
p6,5,5,6,8
p7,2,3,2,4")
The ANOVA, however, requires data to be in long-form. Reshape data
from wide form to long form with base R reshape() according
to the following parameters. With each parameter, either
identify existing variables in the given wide-form data, or
name newly created variables in the long-form. This R function
refers to a time-variable, which in the context of ANOVA is the
treatment variable, of which the values occur over time: first
treatment, second treatment, etc.
The reshaping from a wide-form to a long-form data table creates two
new variables: the variable whose values are collected over time, here
the blocking variable, Supplement, and the response variable, here
Reps.
idvar: Identify the existing blocking (within) variable
in the wide-form datavarying: Identify the wide-form variables, which occur
over time, to be gathered into a single variable in long-formattimevar: Name the corresponding treatment (factor)
variable in the created long-formv.names: Name the response variable in the created
long-form
There are many ways to identify the names of the wide-form variables
to be gathered into a single time-oriented long-form variable. The most
general is to specify a vector of the names, here
c("sup1", "sup2" "sup3", "sup4")
In this example use the lessR to
function to create that vector without needed to individually list each
variable.
## [1] "sup1" "sup2" "sup3" "sup4"
d <- reshape(d, direction="long",
idvar="Person", varying=list(to("sup", 4)),
timevar="Supplement", v.names="Reps")
Do not need the row names, so remove before displaying new long-form
data.
row.names(d) <- NULL
d[1:10,]
## Person Supplement Reps
## 1 p1 1 2
## 2 p2 1 2
## 3 p3 1 8
## 4 p4 1 4
## 5 p5 1 2
## 6 p6 1 5
## 7 p7 1 2
## 8 p1 2 4
## 9 p2 2 5
## 10 p3 2 6
To run the ANOVA, specify the blocking variable preceded by a
+ sign.
ANOVA(Reps ~ Supplement + Person)
##
## BACKGROUND
##
## Response Variable: Reps
##
## Factor Variable 1: Supplement
## Levels: 1 2 3 4
##
## Factor Variable 2: Person
## Levels: p1 p2 p3 p4 p5 p6 p7
##
## Number of cases (rows) of data: 28
## Number of cases retained for analysis: 28
##
## Randomized Blocks ANOVA
## Factor of Interest: Supplement
## Blocking Factor: Person
##
## Note: For the resulting F statistic for Supplement to be distributed as F,
## the population covariances of Reps must be spherical.
##
##
## DESCRIPTIVE STATISTICS
##
## Supplement
##
## X1 X2 X3 X4
## 1 3.57 3.86 4.29 5.71
##
## Person
##
## p1 p2 p3 p4 p5 p6 p7
## 1 3.25 4.25 7.50 4.75 2.00 6.00 2.75
##
## NA
##
##
##
## ANOVA
##
##
## df Sum Sq Mean Sq F-value p-value
## Supplement 3 19.00 6.33 6.71 0.0031
## Person 6 88.43 14.74 15.61 0.0000
## Residuals 18 17.00 0.94
##
## Partial Omega Squared for Supplement: 0.379
## Partial Intraclass Correlation for Person: 0.785
##
## Cohen's f for Supplement: 0.782
## Cohen's f for Person: 1.911
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
##
## Factor: Supplement
## ---------------------------
## diff lwr upr p adj
## 2-1 0.29 -1.18 1.75 0.95
## 3-1 0.71 -0.75 2.18 0.53
## 4-1 2.14 0.67 3.61 0.00
## 3-2 0.43 -1.04 1.90 0.84
## 4-2 1.86 0.39 3.33 0.01
## 4-3 1.43 -0.04 2.90 0.06
##
##
## RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
## [sorted by Standardized Residuals, ignoring + or - sign]
## [res_rows = 20, out of 28 cases (rows) of data, or res_rows="all"]
## ---------------------------------------------------
## Supplement Person Reps fitted residual z-resid
## 22 4 p1 3 4.61 -1.61 -2.06
## 2 1 p2 2 3.46 -1.46 -1.88
## 3 1 p3 8 6.71 1.29 1.65
## 8 2 p1 4 2.75 1.25 1.60
## 11 2 p4 3 4.25 -1.25 -1.60
## 9 2 p2 5 3.75 1.25 1.60
## 10 2 p3 6 7.00 -1.00 -1.28
## 25 4 p4 7 6.11 0.89 1.15
## 15 3 p1 4 3.18 0.82 1.05
## 5 1 p5 2 1.21 0.79 1.01
## 14 2 p7 3 2.25 0.75 0.96
## 21 3 p7 2 2.68 -0.68 -0.87
## 27 4 p6 8 7.36 0.64 0.83
## 12 2 p5 1 1.50 -0.50 -0.64
## 13 2 p6 5 5.50 -0.50 -0.64
## 1 1 p1 2 2.46 -0.46 -0.60
## 17 3 p3 7 7.43 -0.43 -0.55
## 23 4 p2 6 5.61 0.39 0.50
## 26 4 p5 3 3.36 -0.36 -0.46
## 18 3 p4 5 4.68 0.32 0.41
##
## ---------------------
## Plot 1: Data Values
## Plot 2: Fitted Values
## ---------------------
