statsBy {psych}R Documentation

Find statistics (including correlations) within and between groups for basic multilevel analyses


When examining data at two levels (e.g., the individual and by some set of grouping variables), it is useful to find basic descriptive statistics (means, sds, ns per group, within group correlations) as well as between group statistics (over all descriptive statistics, and overall between group correlations). Of particular use is the ability to decompose a matrix of correlations at the individual level into correlations within group and correlations between groups.


statsBy(data, group, cors = FALSE, method="pearson")



A matrix or dataframe with rows for subjects, columns for variables. One of these columns should be the values of a grouping variable.


The names or numbers of the variable in data to use as the grouping variables.


Should the results include the correlation matrix within each group? Default is FALSE.


What kind of correlations should be found (default is Pearson product moment)


The number of trials to run when bootstrapping statistics


Should the bootstrap be done by permuting the data (replace=FALSE) or sampling with replacement (replace=TRUE)


The results from statsBy.boot may be summarized using boot.stats


Name of the variable to be summarized from statsBy.boot


Multilevel data are endemic in psychological research. In multilevel data, observations are taken on subjects who are nested within some higher level grouping variable. The data might be experimental (participants are nested within experimental conditions) or observational (students are nested within classrooms, students are nested within college majors.) To analyze this type of data, one uses random effects models or mixed effect models, or more generally, multilevel models. There are at least two very powerful packages (nlme and multilevel) which allow for complex analysis of hierarchical (multilevel) data structures. statsBy is a much simpler function to give some of the basic descriptive statistics for two level models. It is meant to supplement true multilevel modeling.

For a group variable (group) for a data.frame or matrix (data), basic descriptive statistics (mean, sd, n) as well as within group correlations (cors=TRUE) are found for each group.

The amount of variance associated with the grouping variable compared to the total variance is the type 1 IntraClass Correlation (ICC1): ICC1 = (MSb-MSw)/(MSb + MSw*(npr-1)) where npr is the average number of cases within each group.

The reliability of the group differences may be found by the ICC2 which reflects how different the means are with respect to the within group variability. ICC2 = (MSb-MSw)/MSb. Because the mean square between is sensitive to sample size, this estimate will also reflect sample size.

Perhaps the most useful part of statsBy is that it decomposes the observed correlations between variables into two parts: the within group and the between group correlation. This follows the decomposition of an observed correlation into the pooled correlation within groups (rwg) and the weighted correlation of the means between groups discussed by Pedazur (1997) and by Bliese in the multilevel package.

r_{xy} = eta_{x_{wg}} * eta_{y_{wg}} * r_{xy_{wg}} + eta_{x_{bg}} * eta_{y_{bg}} * r_{xy_{bg}}

where r_{xy} is the normal correlation which may be decomposed into a within group and between group correlations r_{xy_{wg}} and r_{xy_{bg}} and eta is the correlation of the data with the within group values, or the group means.

It is important to realize that the within group and between group correlations are independent of each other. That is to say, inferring from the 'ecological correlation' (between groups) to the lower level (within group) correlation is inappropriate. However, these between group correlations are still very meaningful, if inferences are made at the higher level.

There are actually two ways of finding the within group correlations pooled across groups. We can find the correlations within every group, weight these by the sample size and then report this pooled value. This is found if the cors option is set to TRUE. It is logically equivalent to doing a sample size weighted meta-analytic correlation. The other way, rwg, considers the covariances, variances, and thus correlations when each subject's scores are given as deviation score from the group mean.

Confidence values and significance of r_{xy_{wg}}, pwg, reflect the pooled number of cases within groups, while r_{xy_{bg}} , pbg, the number of groups. These are not corrected for multiple comparisons.

withinBetween is an example data set of the mixture of within and between group correlations. sim.multilevel will generate simulated data with a multilevel structure.

The statsBy.boot function will randomize the grouping variable ntrials times and find the statsBy output. This can take a long time and will produce a great deal of output. This output can then be summarized for relevant variables using the statsBy.boot.summary function specifying the variable of interest. These two functions are useful in order to find if the mere act of grouping leads to large between group correlations.

Consider the case of the relationship between various tests of ability when the data are grouped by level of education (statsBy(sat.act,"education")) or when affect data are analyzed within and between an affect manipulation (statsBy(flat,group="Film") ). Note in this latter example, that because subjects were randomly assigned to Film condition for the pretest, that the pretest ICC1s cluster around 0.



The means for each group for each variable.


The standard deviations for each group for each variable.


The number of cases for each group and for each variable.


The intraclass correlation reflects the amount of total variance associated with the grouping variable.


The intraclass correlation (2) reflecting how much the groups means differ.


The F from a one-way anova of group means.


The pooled within group correlations.


The sample size weighted between group correlations.


The correlation of the data with the within group values.


The correlation of the data with the group means.


The probability of the between group correlation


The probability of the within group correlation


In the case that we want the correlations in each group, r is a list of the within group correlations for every group.


is just another way of displaying these correlations. within is a matrix which reports the lower off diagonal correlations as one row for each group.


The sample size weighted correlations. This is just within weighted by the sample sizes.


The statsBy.boot function will sometimes fail if sampling with replacement because if the group sizes differ drastically, some groups will be empty. In this case, sample without replacement.

The statsBy.boot function can take a long time. (As I am writing this, I am running 1000 replications of a problem with 64,000 cases and 84 groups. It is taking about 3 seconds per replication on a MacBook Pro.)


William Revelle


Pedhazur, E.J. (1997) Multiple regression in behavioral research: explanation and prediction. Harcourt Brace.

See Also

describeBy and the functions within the multilevel package.


#Taken from Pedhazur, 1997
pedhazur <- structure(list(Group = c(1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 
2L), X = c(5L, 2L, 4L, 6L, 3L, 8L, 5L, 7L, 9L, 6L), Y = 1:10), .Names = c("Group", 
"X", "Y"), class = "data.frame", row.names = c(NA, -10L))
ped.stats <- statsBy(pedhazur,"Group")

#Now do this for the sat.act data set
sat.stats <- statsBy(sat.act,c("education","gender"))   #group by two grouping variables
lowerMat(sat.stats$pbg)  #get the probability values

#show means by groups

[Package psych version 1.4.5 Index]