Plotting means and error bars (ggplot2)

Table of contents

Plotting means and error bars (ggplot2)
- Problem
- Solution

Problem

You want to plot means and error bars for a dataset.

Solution

To make graphs with ggplot2, the data must be in a data frame, and in "long" (as opposed to wide) format. If your data needs to be restructured, see this page for more information.

Sample data

The examples below will the ToothGrowth dataset. Note that dose is a numeric column here; in some situations it may be useful to convert it to a factor.

df <- ToothGrowth
#  len supp dose
#  4.2   VC  0.5
# 11.5   VC  0.5
#  ...
# 29.4   OJ    2
# 23.0   OJ    2

library(ggplot2)

First, it is necessary to summarize the data. This can be done in a number of ways, as described on this page. In this case, we'll use the summarySE() function defined on that page, and also at the bottom of this page. (The code for the summarySE function must be entered before it is called here).

# summarySE provides the standard deviation, standard error of the mean, and a (default 95%) confidence interval
dfc <- summarySE(df, measurevar="len", groupvars=c("supp","dose"))
# supp dose  N   len       sd        se       ci
#   OJ  0.5 10 13.23 4.459709 1.4102837 3.190283
#   OJ  1.0 10 22.70 3.910953 1.2367520 2.797727
#   OJ  2.0 10 26.06 2.655058 0.8396031 1.899314
#   VC  0.5 10  7.98 2.746634 0.8685620 1.964824
#   VC  1.0 10 16.77 2.515309 0.7954104 1.799343
#   VC  2.0 10 26.14 4.797731 1.5171757 3.432090

Line graphs

After the data is summarized, we can make the graph. These are basic line and point graph with error bars representing either the standard error of the mean, or 95% confidence interval.

# Standard error of the mean
ggplot(dfc, aes(x=dose, y=len, colour=supp, group=supp)) + 
    geom_point() +
    geom_line() +
    geom_errorbar(aes(ymin=len-se, ymax=len+se),
                  width=.1)

# The errorbars overlapped, so use position_dodge to move them horizontally
pd <- position_dodge(.1) # move them .05 to the left and right

ggplot(dfc, aes(x=dose, y=len, colour=supp, group=supp)) + 
    geom_point(position=pd) +
    geom_line(position=pd) +
    geom_errorbar(aes(ymin=len-se, ymax=len+se),
                  width=.1, position=pd)

# Use 95% confidence interval instead of SEM
ggplot(dfc, aes(x=dose, y=len, colour=supp, group=supp)) + 
    geom_point(position=pd) +
    geom_line(position=pd) +
    geom_errorbar(aes(ymin=len-ci, ymax=len+ci),
                  width=.1, position=pd)

A finished graph with error bars representing the standard error of the mean might look like this. The points are drawn last so that the white fill goes on top of the lines and error bars.

ggplot(dfc, aes(x=dose, y=len, colour=supp, group=supp)) + 
    geom_errorbar(aes(ymin=len-se, ymax=len+se),
                  width=.15, position=pd) +
    geom_line(position=pd) +
    geom_point(position=pd, shape=21, fill="white") + # 21 is filled circle
    xlab("Dose (mg)") +
    ylab("Tooth length") +
    scale_colour_hue(name="Supplement type", # Legend label, use darker colors
                     breaks=c("OJ", "VC"),
                     labels=c("Orange juice", "Ascorbic acid"),
                     l=40) +                  # Use darker colors, lightness=40
    opts(title="The Effect of Vitamin C on\nTooth Growth in Guinea Pigs") +
    scale_y_continuous(limits=c(0, max(dfc$len + dfc$se)), # Set y range
                       breaks=0:20*4) +              # Set tick every 4
    theme_bw() +
    opts(legend.position=c(.7, .3)) # Position legend inside
                                    # This must go after theme_bw

Bar graphs

The procedure is similar for bar graphs. Note that dfc$size must be a factor. If it is a numeric vector, then it will not work.

# Use dose as a factor rather than numeric
dfc2 <- dfc
dfc2$dose <- factor(dfc2$dose)

# Error bars represent standard error of the mean
ggplot(dfc2, aes(x=dose, y=len, fill=supp)) + 
    geom_bar(position=position_dodge()) +
    geom_errorbar(aes(ymin=len-se, ymax=len+se),
                  width=.2,                    # Width of the error bars
                  position=position_dodge(.9))

# Use 95% confidence intervals instead of SEM
ggplot(dfc2, aes(x=dose, y=len, fill=supp)) + 
    geom_bar(position=position_dodge()) +
    geom_errorbar(aes(ymin=len-ci, ymax=len+ci),
                  width=.2,                    # Width of the error bars
                  position=position_dodge(.9))

A finished graph might look like this.

ggplot(dfc2, aes(x=dose, y=len, fill=supp)) + 
    geom_bar(position=position_dodge(),
             colour="black", # Use black outlines,
             size=.3) +      # Thinner lines
    geom_errorbar(aes(ymin=len-se, ymax=len+se),
                  size=.3,    # Thinner lines
                  width=.2,
                  position=position_dodge(.9)) +
    xlab("Dose (mg)") +
    ylab("Tooth length") +
    scale_fill_hue(name="Supplement type", # Legend label, use darker colors
                   breaks=c("OJ", "VC"),
                   labels=c("Orange juice", "Ascorbic acid")) +
    opts(title="The Effect of Vitamin C on\nTooth Growth in Guinea Pigs") +
    scale_y_continuous(breaks=0:20*4) +
    theme_bw()

Error bars for within-subjects variables

When all variables are between-subjects, it is straightforward to plot standard error or confidence intervals. However, when there are within-subjects variables (repeated measures), plotting the standard error or regular confidence intervals may be misleading for making inferences about differences between conditions.

The method below is from Morey (2008), which is a correction to Cousineau (2005), which in turn is meant to be a simpler method of that in Loftus and Masson (1994). See these papers for a more detailed treatment of the issues involved in error bars with within-subjects variables.

One within-subjects variable

Here is a data set (from Morey 2008) with one within-subjects variable: pre/post-test.

dfw <- read.table(header=T, con <- textConnection('
 subject pretest posttest
       1    59.4     64.5
       2    46.4     52.4
       3    46.0     49.7
       4    49.0     48.7
       5    32.5     37.4
       6    45.2     49.5
       7    60.3     59.9
       8    54.3     54.1
       9    45.4     49.6
      10    38.9     48.5
 '))
close(con)

# Treat subject ID as a factor
dfw$subject <- factor(dfw$subject)

The first step is to convert it to long format. See this page for more information about the conversion.

# Convert to long format
library(reshape)
dfw.long <- melt(dfw,
                 id.vars = "subject",
                 measure.vars = c("pretest","posttest"),
                 variable_name = "condition")
# subject condition value
#       1   pretest  59.4
#       2   pretest  46.4
#       3   pretest  46.0
#       4   pretest  49.0
#       5   pretest  32.5
#       6   pretest  45.2
#       7   pretest  60.3
#       8   pretest  54.3
#       9   pretest  45.4
#      10   pretest  38.9
#       1  posttest  64.5
#       2  posttest  52.4
#       3  posttest  49.7
#       4  posttest  48.7
#       5  posttest  37.4
#       6  posttest  49.5
#       7  posttest  59.9
#       8  posttest  54.1
#       9  posttest  49.6
#      10  posttest  48.5

Collapse the data using summarySEwithin (defined at the bottom of this page; both of the helper functions below must be entered before the function is called here).

dfwc <- summarySEwithin(dfw.long, measurevar="value", withinvars="condition", idvar="subject", na.rm=FALSE, conf.interval=.95)
# condition  N value       sd        se       ci
#   pretest 10 47.74 2.262361 0.7154214 1.618396
#  posttest 10 51.43 2.262361 0.7154214 1.618396

library(ggplot2)
# Make the graph with the 95% confidence interval
ggplot(dfwc, aes(x=condition, y=value, group=1)) +
    geom_line() +
    geom_errorbar(width=.1, aes(ymin=value-ci, ymax=value+ci)) +
    geom_point(shape=21, size=3, fill="white") +
    ylim(40,60)

Understanding within-subjects error bars

This section explains how the within-subjects error bar values are calculated. The steps here are for explanation purposes only; they are not necessary for making the error bars.

The graph of individual data shows that there is a consistent trend for the within-subjects variable condition, but this would not necessarily be revealed by taking the regular standard errors (or confidence intervals) for each group. The method in Morey (2008) and Cousineau (2005) essentially normalizes the data to remove the between-subject variability and calculates the variance from this normalized data.

# Use a consistent y range
ymax <- max(dfw.long$value)
ymin <- min(dfw.long$value)

# Plot the individuals
ggplot(dfw.long, aes(x=condition, y=value, colour=subject, group=subject)) +
    geom_line() + geom_point(shape=21, fill="white") + 
    ylim(ymin,ymax)

# Create the normed version of the data
dfwNorm.long <- normDataWithin(data=dfw.long, idvar="subject", measurevar="value")

# Plot the normed individuals
ggplot(dfwNorm.long, aes(x=condition, y=valueNormed, colour=subject, group=subject)) +
    geom_line() + geom_point(shape=21, fill="white") + 
    ylim(ymin,ymax)

The differences in the error bars for the regular (between-subject) method and the within-subject method are shown here. The regular error bars are in red, and the within-subject error bars are in black.

# Instead of summarySEwithin, use summarySE, which treats condition as though it were a between-subject variable
dfwc.between <- summarySE(data=dfw.long, measurevar="value", groupvars="condition", na.rm=FALSE, conf.interval=.95)
# condition  N value       sd       se       ci
#   pretest 10 47.74 8.598992 2.719240 6.151348
#  posttest 10 51.43 7.253972 2.293907 5.189179

# Show the between-S CI's in red, and the within-S CI's in black
ggplot(dfwc.between, aes(x=condition, y=value, group=1)) +
    geom_line() +
    geom_errorbar(width=.1, aes(ymin=value-ci, ymax=value+ci), colour="red") +
    geom_errorbar(width=.1, aes(ymin=value-ci, ymax=value+ci), data=dfwc) +
    geom_point(shape=21, size=3, fill="white") +
    ylim(ymin,ymax)

Two within-subjects variables

If there is more than one within-subjects variable, the same function, summarySEwithin, can be used. This data set is taken from Hays (1994), and used for making this type of within-subject error bar in Rouder and Morey (2005).

data <- read.table(header=T, con <- textConnection('
 Subject RoundMono SquareMono RoundColor SquareColor
       1        41         40         41          37
       2        57         56         56          53
       3        52         53         53          50
       4        49         47         47          47
       5        47         48         48          47
       6        37         34         35          36
       7        47         50         47          46
       8        41         40         38          40
       9        48         47         49          45
      10        37         35         36          35
      11        32         31         31          33
      12        47         42         42          42
'))
close(con)

The data must first be converted to long format. In this case, the column names indicate two variables, shape (round/square) and color scheme (monochromatic/colored).

# Convert it to long format
library(reshape)
data.long <- melt(data=data, id.var="Subject", measure.vars=c("RoundMono", "SquareMono", "RoundColor", "SquareColor"), variable_name="Condition")
names(data.long)[names(data.long)=="value"] <- "Time"

# Split Condition column into Shape and ColorScheme
data.long$Shape <- NA
data.long$Shape[grepl("^Round",  data.long$Condition)] <- "Round"
data.long$Shape[grepl("^Square", data.long$Condition)] <- "Square"
data.long$Shape <- factor(data.long$Shape)

data.long$ColorScheme <- NA
data.long$ColorScheme[grepl("Mono$",  data.long$Condition)] <- "Monochromatic"
data.long$ColorScheme[grepl("Color$", data.long$Condition)] <- "Colored"
data.long$ColorScheme <- factor(data.long$ColorScheme, levels=c("Monochromatic","Colored"))

# Remove the Condition column now
data.long$Condition <- NULL

data.long
# Subject Time  Shape   ColorScheme
#       1   41  Round Monochromatic
#       2   57  Round Monochromatic
#       3   52  Round Monochromatic
#   ...
#       1   37 Square       Colored
#       2   53 Square       Colored
#   ...
#      11   33 Square       Colored
#      12   42 Square       Colored

Now it can be summarized and graphed.

datac <- summarySEwithin(data.long, measurevar="Time", withinvars=c("Shape","ColorScheme"), idvar="Subject")
#  Shape   ColorScheme  N     Time       sd        se        ci
#  Round Monochromatic 12 44.58333 1.331438 0.3843531 0.8459554
#  Round       Colored 12 43.58333 1.212311 0.3499639 0.7702654
# Square Monochromatic 12 43.58333 1.261312 0.3641095 0.8013997
# Square       Colored 12 42.58333 1.461630 0.4219364 0.9286757

library(ggplot2)
ggplot(datac, aes(x=Shape, y=Time, fill=ColorScheme)) +
    geom_bar(position=position_dodge(.9), colour="black") +
    geom_errorbar(position=position_dodge(.9), width=.25, aes(ymin=Time-ci, ymax=Time+ci)) +
    coord_cartesian(ylim=c(40,46)) +
    scale_colour_hue(l=40) +
    scale_fill_manual(values=c("#CCCCCC","#FFFFFF")) +
    scale_y_continuous(breaks=seq(1:100)) +
    theme_bw() +
    geom_hline(yintercept=38)

Helper functions

The summarySE function is also defined on this page. If you only are working with between-subjects variables, that is the only function you will need in your code. If you have within-subjects variables and want to adjust the error bars so that inter-subject variability is removed as in Loftus and Masson (1994), then the other two functions, normDataWithin and summarySEwithin must also be added to your code; summarySEwithin will then be the function that you call.

## Summarizes data.
## Gives count, mean, standard deviation, standard error of the mean, and confidence interval (default 95%).
## If there are within-subject variables, calculate adjusted values using method from Morey (2008).
##   data: a data frame.
##   measurevar: the name of a column that contains the variable to be summariezed
##   groupvars: a vector containing names of columns that contain grouping variables
##   na.rm: a boolean that indicates whether to ignore NA's
##   conf.interval: the percent range of the confidence interval (default is 95%)
summarySE <- function(data=NULL, measurevar, groupvars=NULL, na.rm=FALSE, conf.interval=.95) {
    require(doBy)

    # New version of length which can handle NA's: if na.rm==T, don't count them
    length2 <- function (x, na.rm=FALSE) {
        if (na.rm) sum(!is.na(x))
        else       length(x)
    }

    # Collapse the data
    formula <- as.formula(paste(measurevar, paste(groupvars, collapse=" + "), sep=" ~ "))
    datac <- summaryBy(formula, data=data, FUN=c(length2,mean,sd), na.rm=na.rm)

    # Rename columns
    names(datac)[ names(datac) == paste(measurevar, ".mean",    sep="") ] <- measurevar
    names(datac)[ names(datac) == paste(measurevar, ".sd",      sep="") ] <- "sd"
    names(datac)[ names(datac) == paste(measurevar, ".length2", sep="") ] <- "N"

    datac$se <- datac$sd / sqrt(datac$N)  # Calculate standard error of the mean

    # Confidence interval multiplier for standard error
    # Calculate t-statistic for confidence interval: 
    # e.g., if conf.interval is .95, use .975 (above/below), and use df=N-1
    ciMult <- qt(conf.interval/2 + .5, datac$N-1)
    datac$ci <- datac$se * ciMult

    return(datac)
}

## Norms the data within specified groups in a data frame; it normalizes each
## subject (identified by idvar) so that they have the same mean, within each group
## specified by betweenvars.
##   data: a data frame.
##   idvar: the name of a column that identifies each subject (or matched subjects)
##   measurevar: the name of a column that contains the variable to be summariezed
##   betweenvars: a vector containing names of columns that are between-subjects variables
##   na.rm: a boolean that indicates whether to ignore NA's
normDataWithin <- function(data=NULL, idvar, measurevar, betweenvars=NULL, na.rm=FALSE) {
    require(doBy)

    # Measure var on left, idvar + between vars on right of formula.
    subjMeanFormula <- as.formula(paste(measurevar, paste(c(idvar, betweenvars), collapse=" + "), sep=" ~ "))
    data.subjMean   <- summaryBy(subjMeanFormula, data=data, keep.names=TRUE, na.rm=na.rm)
    names(data.subjMean)[names(data.subjMean) == measurevar] <- "subjMean"

    # Put the subject means with original data
    data <- merge(data, data.subjMean)

    # Get the normalized data in a new column
    measureNormedVar <- paste(measurevar, "Normed", sep="")
    data[,measureNormedVar] <- data[,measurevar] - data[,"subjMean"] + mean(data[,measurevar])

    # Remove this subject mean column
    data$subjMean <- NULL

    return(data)
}

## Summarizes data, handling within-subjects variables by removing inter-subject variability.
## It will still work if there are no within-S variables.
## Gives count, mean, standard deviation, standard error of the mean, and confidence interval (default 95%).
## If there are within-subject variables, calculate adjusted values using method from Morey (2008).
##   data: a data frame.
##   measurevar: the name of a column that contains the variable to be summariezed
##   betweenvars: a vector containing names of columns that are between-subjects variables
##   withinvars: a vector containing names of columns that are within-subjects variables
##   idvar: the name of a column that identifies each subject (or matched subjects)
##   na.rm: a boolean that indicates whether to ignore NA's
##   conf.interval: the percent range of the confidence interval (default is 95%)
summarySEwithin <- function(data=NULL, measurevar, betweenvars=NULL, withinvars=NULL, idvar=NULL, na.rm=FALSE, conf.interval=.95) {
    # Norm each subject's data    
    data <- normDataWithin(data, idvar, measurevar, betweenvars, na.rm)

    # This is the name of the new column
    measureNormedVar <- paste(measurevar, "Normed", sep="")

    # Replace the original data column with the normed one
    data[,measurevar] <- data[,measureNormedVar]

    # Collapse the normed data - now we can treat between and within vars the same
    datac <- summarySE(data, measurevar, groupvars=c(betweenvars, withinvars), na.rm=na.rm, conf.interval=conf.interval)

    # Apply correction from Morey (2008) to the standard error and confidence interval
    #  Get the product of the number of conditions of within-S variables
    nWithinGroups    <- prod(sapply(datac[,withinvars, drop=FALSE], FUN=nlevels))
    correctionFactor <- sqrt( nWithinGroups / (nWithinGroups-1) )

    # Apply the correction factor
    datac$sd <- datac$sd * correctionFactor
    datac$se <- datac$se * correctionFactor
    datac$ci <- datac$ci * correctionFactor

    return(datac)
}