Data formats for proportions"
In ANOPA: Analyses of Proportions using Anscombe Transform

cat("this is hidden; general initializations.\n")
library(ANOPA)
w1 <- anopa( {success;total} ~ Class * Difficulty, twoWayExample)
dataWide1     <- toWide(w1)
dataCompiled1 <-toCompiled(w1)
dataLong1     <- toLong(w1)
rownames(dataLong1) <- NULL

w2 <- anopa( cbind(bpre, bpost, b1week, b5week) ~ Status, minimalMxExample, WSFactors = "Moment(4)")
dataWide2     <- toWide(w2)
dataCompiled2 <-toCompiled(w2)
dataLong2     <- toLong(w2)
rownames(dataLong2) <- NULL

Data formats for proportions

proportions are actually not raw data: they are the proportion of one response (typically called a success) over all the responses (the other responses being called collectively a failure). As such, a proportion is a summary statistic, a bit like the mean is a summary statistic of continuous data.

Very often, the success are coded using the digit 1 and the failure, with the digit 0. When this is the case, computing the mean is actually the same as computing the proportion of successes. However, it is a conceptual mistake to think of proportions as means, because they must the processed completely differently from averages. For example, standard error and confidence intervals for proportions are obtained using very different procedures than standard error and confidence intervals for the mean.

In this vignette, we review various ways that data can be coded in a data frame. In a nutshell, there are three ways to represent success or failures, Wide, Long, and Compiled. The first two shows raw scores whereas the last shows a summary of the data.

Before we begin, we load the package ANOPA (if is not present on your computer, first upload it to your computer from CRAN or from the source repository devtools::install_github("dcousin3/ANOPA")):

library(ANOPA)

First format: Wide data format

In this format, there is one line per subject and one column for each measurements. The columns contain only 1s (success) or 0s (`failure).

If the participant was measured multiple times, there is one (or some) within-subject factor(s) resulting in multiple columns of measurements. In between-group design, there is only a single column of scores.

As an example, consider the following data for a between-subject factor design with two factors: Class (2 levels) and Difficulty (3 levels) for 6 groups. There is an identical number of participants in each, 12, for a total of 72 participants.

dataWide1

When the data are in a wide format, the formula in anopa() must provide the columns where the success/failure are stored, and the conditions after the usual ~, as in

w1 <- anopa( success ~ Class * Difficulty, dataWide1)

(how dataWide1 was obtained is shown below in the Section Converting between formats below.)

As another example, consider the following example obtained in a mixed, within- and between- subject design. It has a factor Status with 8, 9 and 7 participants per group respectively. It also has four repeated measures, bpre, bpost, b1week and b5week which represent four different Moments of measurements. The data frame is

dataWide2

The formula for analyzing these data in this format is

w2 <- anopa( cbind(bpre, bpost, b1week, b5week) ~ Status, dataWide2, WSFactors = "Moment(4)" )

It is necessary to (a) group all the measurement columns using cbind(); (b) indicate the within-subject factor(s) using the argument WSFactors along with the number of levels each in a string.

Alternatively, cbind() can be replaced by crange() with the first and last variable to be binded. The in-between variables will be taken from the data.frame().

w2bis <- anopa( crange(bpre, b5week) ~ Status, dataWide2, WSFactors = "Moment(4)" )

Second format: Long data format

This format may be preferred for linear modelers (but it may rapidly becomes very long!). There is always at least these columns: One Id column, one column to indicate a within-subject level, and one column to indicate the observed score. On the other hand, this format has fewer columns in repeated measure designs.

This example shows the first 6 lines of the 2-factor between design data above, stored in the long format.

head(dataLong1)

To analyse such data format within anopa(), use

w1Long <- anopa( Value ~ Class * Difficulty * Variable  | Id, dataLong1 )

The vertical line symbol indicates that the observations are nested within Id (i.e., all the lines with the same Id are actually the same subject).

With the mixed design described above, the data begin as:

head(dataLong2)

and are analyzed with the formula:

w2Long <- anopa( Value ~ Status * Variable  | Id, dataLong2, WSFactors="Moment(4)" )

Third format: Compiled data format

This format is compiled, in the sense that the 0s and 1s have been replaced by a single count of success for each cell of the design. Hence, we no longer have access to the raw data. This format however has the advantage of being very compact, requiring few lines. Here is the data for the 2 between-subject factors example

dataCompiled1

To use a compiled format in anopa(), use

w1Compiled <- anopa( {success; Count} ~ Class * Difficulty, dataCompiled1 )

where succes identifies in which column the total number of successes are stored. The column Count indicates the total number of observations in that cell. The notation {s;n} is read s over n (note the curly braces and semicolon).

For the mixed design presented earlier, the data looks like:

dataCompiled2

where there are columns for the number of success for each repeated measures. A new columns appear uAlpha. This column (called unitary alpha) is a measure of correlation (between -1 and +1). In this fictitious example, the correlations are near zero (negative actually) by chance as the data were generated randomly.

To run an ANOPA on compiled data having repeated measures, use

w2Compiled <- anopa( {cbind(bpre, bpost, b1week, b5week); Count; uAlpha} ~ Status, 
                    dataCompiled2, WSFactors = "Week(4)")
summary(w2Compiled)

where cbind() lists all the within-subject success count columns, Count is the column in the data.frame with the total number of observations, and uAlpha is the column containing the mean pairwise correlation measured with the unitary alpha.

Here again, crange() can be used in place of cbind() with

w2Compiledbis <- anopa( {crange(bpre, b5week); Count; uAlpha} ~ Status, 
                    dataCompiled2, WSFactors = "Week(4)")
summary(w2Compiledbis)

Converting between formats

Once entered in an anopa() structure, it is possible to convert to any format using toWide(), toCompiled() and toLong(). For example:

toCompiled(w1)
toCompiled(w2)

The compiled format is probably the most compact format, but the wide format is the most explicit format (as we see all the subjects and their scores on a single line, one subject per line).

Getting the example data frame

Above, we used two examples. They are available in this package under the names twoWayExample and minimalMxExample. The first is available in compiled form, the second in wide form.

We converted these data set in other formats using:

w1 <- anopa( {success;total} ~ Class * Difficulty, twoWayExample)
dataWide1     <- toWide(w1)
dataCompiled1 <-toCompiled(w1)
dataLong1     <- toLong(w1)

w2 <- anopa( cbind(bpre, bpost, b1week, b5week) ~ Status, minimalMxExample, WSFactors = "Moment(4)")
dataWide2     <- toWide(w2)
dataCompiled2 <-toCompiled(w2)
dataLong2     <- toLong(w2)

Multiple repeated-measure factors

One limitation is with regards to repeated measures: It is not possible to guess the name of the within-subject factors from the names of the columns. This is why, as soon as there are more than one measurement, the argument WSFactors must be added.

Suppose a two-way within-subject design with 2 x 3 levels. The data set twoWayWithinExample has 6 columns; the first three are for the factor A, level 1, and the last three are for factor A, level 2. Within each triplet of column, the factor B goes from 1 to 3.

w3 <- anopa( cbind(r11,r12,r13,r21,r22,r23) ~ . , 
             twoWayWithinExample, 
             WSFactors = c("B(3)","A(2)") 
            )
toCompiled(w3)

A "fyi" message is shown which lets you see how the variables are interpreted. Take the time to verify that the order of the variables within cbind() does match the expected order from anopa(). Note that FYI messages can be inhibited by changing the option