See pairwise for further description.
# S3 method for pairwise
summary(
object,
stat.table = TRUE,
test.type = c("dist", "stdist", "mdist", "VC", "DL", "var"),
angle.type = c("rad", "deg"),
confidence = 0.95,
show.vectors = FALSE,
...
)Object from pairwise
Logical argument for whether results should be returned in one table (if TRUE) or separate pairwise tables (if FALSE)
the type of statistic to test. See below should be used in the test.
If test.type = "VC", whether angle results are expressed in radians or degrees.
Confidence level to use for upper confidence limit; default = 0.95 (alpha = 0.05)
Logical value to indicate whether vectors should be printed.
Other arguments passed onto pairwise
Michael Collyer
The following summarize the test that can be performed:
Vectors for LS means or slopes originate at the origin and point to some location, having both a magnitude and direction. A distance between two vectors is the inner-product of of the vector difference, i.e., the distance between their endpoints. For LS means, this distance is the difference between means. For multivariate slope vectors, this is the difference in location between estimated change for the dependent variables, per one-unit change of the covariate considered. For univariate slopes, this is the absolute difference between slopes.
Same as the distance between vectors, but distances are divided by the model standard error (square-root of the trace of the residual covariance matrix). Pairwise tests with this statistic should be consistent with ANOVA results.
Similar to the standardized distance between vectors but the inverse of the residual covariance matrix is used in calculation of the distance, rather than dividing the Euclidean distance between means and dividing by the standard error. If the residual covariance matrix is singular, Mahalanobis distances will not be calculated. Pairwise tests with this statistic should be consistent with MANOVA results.
If LS mean or slope vectors are scaled to unit size, the vector correlation is the inner-product of the scaled vectors. The arccosine (acos) of this value is the angle between vectors, which can be expressed in radians or degrees. Vector correlation indicates the similarity of vector orientation, independent of vector length.
If the length of a vector is an important attribute -- e.g., the amount of multivariate change per one-unit change in a covariate -- then the absolute value of the difference in vector lengths is a practical statistic to compare vector lengths, rather than the estimates the vectors make. Let d1 and d2 be the distances (length) of vectors. Then |d1 - d2| is a statistic that compares their lengths. For slope vectors, this is a comparison of rates. For comparison, if vectors are rates, "dist" finds the difference between estimates per unit change of, e.g., time, size, etc., which could be large, even for small rates of change, if vectors point in dissimilar directions. "DL" is a comparison of rates, irrespective of direction.
Vectors of residuals from a linear model indicate can express the distances of observed values from fitted values. Mean squared distances of values (variance), by group, can be used to measure the amount of dispersion around estimated values for groups. Absolute differences between variances are used as test statistics to compare mean dispersion of values among groups. Variance degrees of freedom equal n, the group size, rather than n-1, as the purpose is to compare mean dispersion in the sample. (Additionally, tests with one subject in a group are possible, or at least not a hindrance to the analysis.)
The argument, test.type is used to select one of the tests
above. See pairwise for examples.
In previous versions of pairwise, summary.pairwise had three
test types: "dist", "VC", and "var". When one chose "dist", for LS mean
vectors, the statistic was the inner-product of the vector difference.
For slope vectors, "dist" returned the absolute value of the difference
between vector lengths, which is "DL" in 0.6.2 and subsequent versions. This
update uses the same calculation, irrespective of vector types. Generally,
"DL" is the same as a contrast in rates for slope vectors, but might not have
much meaning for LS means. Likewise, "dist" is the distance between vector
endpoints, which might make more sense for LS means than slope vectors.
Nevertheless, the user has more control over these decisions with version 0.6.2
and subsequent versions.
The test types, standardized distance between vectors, "stdist", and Mahalanobis distances between vectors were added. The former simply divides the distance between vectors by model standard error (square-root of the trace of the residual covariance matrix). This is a multivariate generalization of a t-statistic for the difference between means. It is not the same as Hotelling squared-T-statistic, which requires incorporating the inverse of the residual covariance matrix in the calculation of the distance, a calculation that also requires a non-singular covariance matrix. However, the Mahalanobis distances are similar (and proportional) to the Hotelling squared-T-statistic. Pairwise tests using Mahalanobis distances represent a non-parametric analog to the parametric Hotelling squared-T test. Both tests should be better for GLS model fits compared to Euclidean distances between means, as the total sums of squares are more likely to vary across random permutations. In general, if ANOVA is performed a pairwise test with "stdist" is a good association; if MANOVA is performed, a pairwise test with "mdist" is a good association.