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Comparison of WABA and DETECT to the one-way analysis of variance (ANOVA)

Note: No misunderstandings and misconceptions about WABA and DETECT were identified as coming from individuals who advocate the ANOVA model.

Description of Traditional ANOVA approach

In the traditional ANOVA, the independent variable (x) is aligned with groups so that it does not vary within groups. In contrast, the dependent variable (y) is free to vary within groups and between groups. The variation of the dependent variable (y) within groups is not attributable the independent variable (x). Thus, the ANOVA correctly asserts that:

Within-group variance is error
Between-group variance is valid.

The variance between groups is represented by averages y' for each group and the variance within groups is represented by the within-group deviations (a score minus the group average y-y')). Thus a total score y = (y-y') + y' or total score = within-group score + between-group score. If the grand mean (m) for the dependent variable is subtracted from both sides of the equation, the following equation results:

(y-m) = (y - y') + (y'-m)

If both sides are squared the equation remains the same but each term in parentheses is squared or

Total deviations squared = Within-group deviations squared + Between-group deviations squared

The percent of variation-between groups, which is the valid variation in the dependent variable (y) associated with the independent variable (x), is Eta squared which is calculated as follows

Eta Squared (Between ) = the sum of between-group deviations squared divided by the total deviations squared

The statistical significance of eta squared is tested by the traditional F-ratio from Fisher (1925). The practical significance can be tested using the "f" value described by Cohen (1988), which equals the squared between-groups eta divided by 1 minus the squared between group eta correlation. If one assumes that between-group variance is valid and within-group variance is error, the intraclass correlation (ICC(1)) can be used to estimate the degree of association between individual scores in the same group. The test of significance for the intraclass correlation is the same F-ratio (Fisher, 1925) as used for testing the significance of the eta correlation. In the multivariate ANOVA (Finn, 1974), when there are multiple dependent variables the correlations among the variables based on the within-group deviation scores is an indication of the correlations among the variables not due to the independent variable (pooled within-group correlations).

This gives rise to the following indicators available in any ANOVA using virtually any statistical package (e.g., SPSS or SAS)

Between group eta squared = Sum of squares between divided by total sum of squares.
F-ratio whose statistical significance is taken to indicate a statistically significant between-group effect.
f which indicates the strength of the effect. (In WABA, the square root of f is called an E ratio)
Intraclass correlation which indicates correlation among individuals in the groups.
Pooled within-group correlation.

Inferences are drawn that there is a significant between effect (x and y are related) based on the values of the F-ratio and occasionally on the value of eta squared, the f value, or the intraclass correlation. The pooled within-group correlations estimate the correlations among the dependent variables holding the independent variable constant.

Compatibility of Within and Between Analysis (WABA/DETECT) with the ANOVA

As pointed out by Hays (1985), WABA involves straightforward extensions of bivariate correlations among variables (Dansereau, Alutto, & Yammarino, 1984) and more recently of multiple correlations (Schriesheim, 1995). The purpose of WABA is very different from an analysis of variance. Some of the statistics and mathematical calculations are identical to those performed in an ANOVA. Nevertheless, the purpose is to examine correlations (bivariate or multiple) in order to assess the level or levels of analysis that may underlie an observed bivariate or multiple correlation.

WABA begins with a very different question than the ANOVA. Groups are not formed based on the values of the independent variables to assess the relationship of the independent and dependent variables. Instead, naturally occurring groups are identified and the question of interest is is whether one or some combination of groups at higher levels of may or may not underlie an observed relationship. Thus, in WABA all variables are included in groupings not just the dependent variables as in the ANOVA. Although there may be many levels: individual, group, department, organization, etc., each level is considered one at a time.

Unlike the ANOVA, WABA does not assume that variation within-groups is always error. Nor does it assume that within-group variation always represents a nongroup effect. Under certain defined conditions a WABA will lead to the inference that the within-group variance represents a valid frog-pond group based effect (see Firebaugh (1978), Dansereau, Alutto, &Yammarino, 1984). This condition is called the parts condition (see the single-level analysis tutorial). Under other conditions, WABA will lead to the inference that within-group variance represents a nongroup based effect. This is called an equivocal condition (see the single-level tutorial). WABA may also lead to an inference of error for the within-group variation. This is called the null condition (see the single-level tutorial)

Why the above difference between WABA and the ANOVA? In the ANOVA, individuals in groups are typically forced to be independent. Thus within-group scores can not represent a frog pond effect. In a naturally occurring group, it is not possible to assume that such a group can not occur. Thus, WABA allows for frog pond effects. Moreover, in the ANOVA, any variation within groups can not be due to the independent variable, so individual responses that can not be associated with the independent variable represent error. In a naturally occurring group, independent and dependent variables can vary between and within-groups. Thus, different conditions must be allowed for error and individual-level effects.

Likewise unlike the ANOVA, WABA does not assume that between-group variation is always valid. Nor does it assume that between-group variation always represents a group effect. Under certain defined conditions a WABA will lead to the inference that the between-group variance represents a valid between-group effect (Dansereau, Alutto, &Yammarino, 1984). This condition is called the wholes condition (see the single-level analysis tutorial). Under other conditions, WABA will lead to the inference that between-group variance represents a nongroup based individual effect. This is called an equivocal condition (see the single-level tutorial). WABA may also lead to an inference of error for the between-group variation. This is called the null condition (see the single-level tutorial).

Despite these differences WABA is totally compatible with the ANOVA. It uses the following indicators from the ANOVA

Between group eta squared = Sum of squares between divided by total sum of squares.
F-ratio whose statistical significance is taken to indicate a statistically significant between-group effect.
f which indicates the strength of the between-group effect.(In WABA, the square root of f is called an E ratio)

But, the following are added because within-group variation is not assumed to be error:

Within-group eta squared = 1 minus the between-group eta squared.
A within-group F-ratio to test the statistical significance of a within-group effect.
f is used to indicate the strength of within-group effects.(In WABA, the square root of f is called an E-ratio)

Because the pooled within-group correlations may be valid it is necessary to examine the between-group correlations.

Thus, the solution to decomposing a correlation then becomes exactly what was shown by Robinson (1950):

A total correlation between x and y equals the between-groups component plus the within-groups component:

The between component equals the multiplication of :

The between-groups eta for x
The between-groups eta for y
The between group correlation of x and y

The within groups component equals the multiplication of:

The within-groups eta for x
The within-groups eta for y
The within-groups correlation for x and y

Responding to misconceptions and misunderstandings about WABA and DETECT

At this time, no published criticisms of the mathematics for above partitioning process were found. To do so would require the rejection of the work not only of the statisticians listed above but also the work of Duncan (1961), Pedhazur (1982) and Maddala (1971). Misconceptions and misunderstandings are described in all other sections. If you know of published material that raises questions about WABA from an ANOVA perspective, please send the citation by e-mail to conceptualizers@levelsofanalysis.com.

Occasionally, reviewers will make comments about the extensive number of tests that are used . It is sometimes helpful to be sure to refer to the references for these tests since they are very well-established.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum.

Dansereau, F., Alutto, J., and Yammarino, F. (1984) Theory Testing in Organizational Behavior. Englewood Cliffs NJ: Prentice Hall

Duncan, O.D., Cuzort, R., Duncan, R. (1961). Statistical geography. Glencoe, IL : Free Press.

Finn, J. (1974). A general model for multivariate analysis. New York: Holt, Rinehart, and Winston.

Firebaugh, G. (1978). A rule for inferring individual-level relationships from aggregate data. American Sociological Review, 43, 557-72.

Fisher, R. A. (1925). Statistical methods for research workers. New York: Hafner.

Hays, J. (1985). Review of Theory testing in Organizational behavior. Contemporary Psychology 30, 160-161.

Maddala, G. (1971). The use of variance components in pooling cross section and time series data. Economtrica, 39, 341-358.

Pedhazur, E. (1982). Multiple regression in behavior research. New York: Holt, Rinehart and Winston.

Robinson, W. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15, 351-57.

Schriesheim, C. (1995). Multivariate and moderated within-and between-entity analysis (WABA) using hierarchical linear multiple regression. Leadership Quarterly, 6, 1-18.

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