lodds: Calculate Generalized Log Odds for Frequency Tables

Description

Computes (log) odds and their asymptotic variance covariance matrix for R (by strata) tables. Odds are calculated for pairs of levels of one array dimension (typically a response or focal variable) separately for each level of all stratifying dimensions. See Friendly et al. (2011) for a sketch of a general theory.

Usage

lodds(x, ...)
# S3 method for default
lodds(x, response = NULL, strata = NULL, log = TRUE, 
  ref = NULL, correct = any(x == 0), ...)
# S3 method for formula
lodds(formula, data = NULL, ...,
subset = NULL, na.action = NULL)
odds(x, log = FALSE, ...)
# S3 method for lodds
coef(object, log = object$log, ...)
# S3 method for lodds
vcov(object, log = object$log, ...)
# S3 method for lodds
print(x, log = x$log, ...)
# S3 method for lodds
confint(object, parm, level = 0.95, log = object$log, ...)
# S3 method for lodds
dim(x, ...)
# S3 method for lodds
dimnames(x, ...)

# S3 method for lodds
as.array(x, log=x$log, ...)
# S3 method for lodds
t(x)
# S3 method for lodds
aperm(a, perm, ...)

Value

An object of class lodds, with the following components:

coefficients: A named vector, of length (R-1) x (C-1) x prod(dim(x)[strata]) containing the log odds. Use the coef method to extract these from the object, and the confint method for confidence intervals. For a two-way table, the names for the log oddsare constructed in the form Ri:Rj using the table names for rows and columns. For a stratified table, the names are constructed in the form Ri:Rj|Lk.
vcov: Variance covariance matrix of the log odds.
dimnames: Dimension names for the log odds, considered as a table of size (R-1, C-1, dim(x)[strata]). Use the dim and dimnames methods to extract these and manipulate the log odds in relation to the original table.
dim: Corresponding dimension vector.
contrasts: A matrix C, such that C %*% as.vector(log(x)) gives the log odds ratios. Each row corresponds to one log odds, and is all zero, except for 2 elements of c(1, -1) for a given 2 x 1 subtable.
log: A logical, indicating the value of log in the original call.

Arguments

x: an object. For the default method a k-way matrix/table/array of frequencies. The number of margins has to be at least 2.
response: Numeric or character indicating the margin of a $k$-way table x (with $k$ greater than 2) that should be employed as the response variable. By default the first dimension is used.
strata: Numeric or character indicating the margins of a $k$-way table x (with $k$ greater than 2) that should be employed as strata. Ignored if response is specified. By default all dimensions except the first are used as strata.
ref: numeric or character. Reference categories for the (non-stratum) row and column dimensions that should be employed for computing the odds. By default, odds for profile contrasts (or sequential contrasts, i.e., successive differences of adjacent categories) are used. See details below.
formula: a formula specifying the variables used to create a contingency table from data. A conditioning formula can be specified; the conditioning variables will then be used as strata variables.
data: either a data frame, or an object of class "table" or "ftable".
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Ignored if data is a contingency table.
log: logical. Should the results be displayed on a log scale or not? All internal computations are always on the log-scale but the results are transformed by default if log = TRUE.
correct: logical or numeric. Should a continuity correction be applied before computing odds? If TRUE, 0.5 is added to all cells; if numeric (or an array conforming to the data) that value is added to all cells. By default, this not employed unless there are any zero cells in the table, but this correction is often recommended to reduce bias when some frequencies are small (Fleiss, 1981).
a, object: an object of class lodds as computed by lodds.
perm: numeric or character vector specifying a permutation of strata.
...: arguments passed to methods.
parm: a specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.
level: the confidence level required for the confint method.

Author

Achim Zeileis, Michael Friendly and David Meyer.

Details

For an n-way table with the response variable containing R levels, (log) odds are formed (by default) for the set of (R-1) contrasts among the response levels. The ref argument allows these to be specified in a general way.

ref = NULL (default) corresponds to “profile contrasts” (or sequential contrasts or successive differences) for ordered categories, i.e., R1--R2, R2--R3, R3--R4, etc., and similarly for the column categories. These are sometimes called “local odds” or “adjacent odds”.

ref = 1 gives contrasts with the first category; ref = dim(x) gives contrasts with the last category.

Note that all such parameterizations are equivalent, in that one can derive all other possible odds from any non-redundant set, but the interpretation of these values depends on the parameterization.

See the help page of plot.loddsratio for related visualization methods. There is as yet no plot method for lodds objects.

References

A. Agresti (2013), Categorical Data Analysis, 3rd Ed. New York: Wiley.

Fleiss, J. L. (1981). Statistical Methods for Rates and Proportions. 2nd Edition. New York: Wiley.

M. Friendly (2000), Visualizing Categorical Data. SAS Institute, Cary, NC.

Friendly, M., Turner, H,, Firth, D., Zeileis, A. (2011). Advances in Visualizing Categorical Data Using the vcd, gnm and vcdExtra Packages in R. Correspondence Analysis and Related Methods (CARME 2011). http://www.datavis.ca/papers/adv-vcd-4up.pdf

Examples

Run this code

## artificial example
set.seed(1)
x <- matrix(rpois(5 * 3, 7), ncol = 5, nrow = 3)
dimnames(x) <- list(Row = head(letters, 3), Col = tail(letters, 5))

x_lodds <- lodds(x)
coef(x_lodds)
x_lodds
confint(x_lodds)
summary(x_lodds)

### 2 x 2 x k cases
##data(CoalMiners, package = "vcd")
#lor_CM <- loddsratio(CoalMiners)
#lor_CM
#coef(lor_CM)
#confint(lor_CM)
#confint(lor_CM, log = FALSE)
#
### 2 x k x 2
#lor_Emp <-loddsratio(Employment)
#lor_Emp
#confint(lor_Emp)
#
### 4 way tables 
#data(Punishment, package = "vcd")
#lor_pun <- loddsratio(Freq ~ memory + attitude | age + education, data = Punishment)
#lor_pun
#confint(lor_pun)
#summary(lor_pun)
#
## fit linear model using WLS
#lor_pun_df <- as.data.frame(lor_pun)
#pun_mod1 <- lm(LOR ~ as.numeric(age) * as.numeric(education),
#               data = lor_pun_df, weights = 1 / ASE^2)
#anova(pun_mod1)
#
### illustrate ref levels
#VA.fem <- xtabs(Freq ~ left + right, subset=gender=="female", data=VisualAcuity)
#VA.fem
#loddsratio(VA.fem)                  # profile contrasts
#loddsratio(VA.fem, ref=1)           # contrasts against level 1
#loddsratio(VA.fem, ref=dim(VA.fem)) # contrasts against level 4
#

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