This routine performs the covariance theorem decomposition discussed by Robinson (1950) and Dansereau, Alutto and Yammarino (1984). Dansereau et al. have labeled the variance decomposition Within-And-Between-Analysis II or WABA II. The program decomposes a raw correlation from a two-level nested design into 6 components. These components are (1) eta-between value for X, (2) eta-between value for Y, (3) the group-size weighted group-mean correlation, (4) the within-eta value for X, (5) the within-eta value for Y, and (6) the within-group correlation between X and Y. The last value represents the correlation between X and Y after each variable has been group-mean centered.

The program is designed to automatically perform listwise deletion on missing values; consequently, users should pay attention to the diagnostic information (Number of Groups and Number of Observations) provided as part of the output.

Note that Within-And-Between-Analysis proposed by Dansereau et al. involves more than covariance theorem decomposition of correlations. Specifically, WABA involves decision rules based on eta-values. These are not replicated in the R multilevel library because the eta based decision rules have been shown to be highly related to group size (Bliese, 2000; Bliese & Halverson, 1998), a factor not accounted for in the complete Within-And-Between-Analysis methodology.

1 | ```
waba(x, y, grpid)
``` |

`x` |
A vector representing one variable in the correlation. |

`y` |
A vector representing the other variable in the correlation. |

`grpid` |
A vector identifying the groups from which x and y originated. |

Returns a list with three elements.

`Cov.Theorem` |
A 1 row dataframe with all of the elements of the covariance theorem. |

`n.obs` |
The number of observations used to calculate the covariance theorem. |

`n.grps` |
The number of groups in the data set. |

Paul Bliese paul.bliese@moore.sc.edu

Bliese, P. D. (2000). Within-group agreement, non-independence, and reliability: Implications for data aggregation and Analysis. In K. J. Klein & S. W. Kozlowski (Eds.), Multilevel Theory, Research, and Methods in Organizations (pp. 349-381). San Francisco, CA: Jossey-Bass, Inc.

Bliese, P. D., & Halverson, R. R. (1998). Group size and measures of group-level properties: An examination of eta-squared and ICC values. Journal of Management, 24, 157-172.

Dansereau, F., Alutto, J. A., & Yammarino, F. J. (1984). Theory testing in organizational behavior: The varient approach. Englewood Cliffs, NJ: Prentice-Hall.

Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15, 351-357.

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