Goodman Kruskal Lambda

Calculate symmetric and asymmetric Goodman Kruskal lambda and their confidence intervals. Lambda is a measure of proportional reduction in error in cross tabulation analysis. For any sample with a nominal independent variable and dependent variable (or ones that can be treated nominally), it indicates the extent to which the modal categories and frequencies for each value of the independent variable differ from the overall modal category and frequency, i.e. for all values of the independent variable together

Usage

lambda(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  direction = c("symmetric", "row", "column"),
  ...
)

Arguments

x: either a contingency table, a two-column object (matrix, data.frame or list), or a vector of observations (together with y)
y: optional second vector. If x is not a vector, y must be NULL.
conf.level: confidence level of the interval. If set to NA (the default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See details in ConfidenceIntervals().
direction: type of lambda. Can be one out of "symmetric" (default), "row", "column" (abbreviations are allowed). If direction is set to "row" then lambda(R|C) (column dependent) will be reported. See Details.
...: further arguments, passed on to normalizeToConfusion() and table() for building the table - useNA is the usual one.

Value

if conf.level = NA, a numeric scalar. Otherwise a named numeric vector with elements:

est: point estimate of Goodman–Kruskal lambda
lci: lower confidence interval bound
uci: upper confidence interval bound

Details

Asymmetric lambda is interpreted as the probable improvement in predicting the column variable Y given knowledge of the row variable X.
The nondirectional lambda is the average of the two asymmetric lambdas, lambda(C|R) and lambda(R|C). lambda (asymmetric and symmetric) has a scale ranging from 0 to 1.

Note

Based on code by Antti Arppe and Nanina Anderegg (confidence interval symmetric lambda), adapted to conform to package standards.

References

Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons

Goodman, L. A., Kruskal W. H. (1979) Measures of Association for Cross Classifications. New York: Springer-Verlag (contains articles appearing in J. Amer. Statist. Assoc. in 1954, 1959, 1963, 1972).
http://www.nssl.noaa.gov/users/brooks/public_html/feda/papers/goodmankruskal1.pdf (might be outdated)

Liebetrau, A. M. (1983) Measures of Association, Sage University Papers Series on Quantitative Applications in the Social Sciences, 07-004. Newbury Park, CA: Sage, pp. 17–24

Examples


# example from Goodman Kruskal (1954)
m <- as.table(cbind(c(1768,946,115), c(807,1387,438), c(189,746,288), c(47,53,16)))
dimnames(m) <- list(paste("A", 1:3), paste("B", 1:4))
m
#>      B 1  B 2  B 3  B 4
#> A 1 1768  807  189   47
#> A 2  946 1387  746   53
#> A 3  115  438  288   16

# direction default is "symmetric"
lambda(m)
#> [1] 0.2076188
lambda(m, conf.level=0.95)
#>       est       lci       uci 
#> 0.2076188 0.1871747 0.2280629 

lambda(m, direction="row")
#> [1] 0.2241003
lambda(m, direction="column")
#> [1] 0.1923949