Freeman's theta

Calculates Freeman's theta for a table with one ordinal variable and one nominal variable; confidence intervals by bootstrap.

Usage

freemanTheta(
  x,
  g = NULL,
  group = "row",
  verbose = FALSE,
  progress = FALSE,
  ci = FALSE,
  conf = 0.95,
  type = "perc",
  R = 1000,
  histogram = FALSE,
  digits = 3,
  reportIncomplete = FALSE
)

Arguments

x: Either a two-way table or a two-way matrix. Can also be a vector of observations of an ordinal variable.
g: If x is a vector, g is the vector of observations for the grouping, nominal variable.
group: If x is a table or matrix, group indicates whether the "row" or the "column" variable is the nominal, grouping variable.
verbose: If TRUE, prints statistics for each comparison.
progress: If TRUE, prints a message as each comparison is conducted.
ci: If TRUE, returns confidence intervals by bootstrap. May be slow.
conf: The level for the confidence interval.
type: The type of confidence interval to use. Can be any of "norm", "basic", "perc", or "bca". Passed to boot.ci.
R: The number of replications to use for bootstrap.
histogram: If TRUE, produces a histogram of bootstrapped values.
digits: The number of significant digits in the output.
reportIncomplete: If FALSE (the default), NA will be reported in cases where there are instances of the calculation of the statistic failing during the bootstrap procedure.

Value

A single statistic, Freeman's theta. Or a small data frame consisting of Freeman's theta, and the lower and upper confidence limits.

Details

Freeman's coefficent of differentiation (theta) is used as a measure of association for a two-way table with one ordinal and one nominal variable. See Freeman (1965).

Currently, the function makes no provisions for NA values in the data. It is recommended that NAs be removed beforehand.

Because theta is always positive, if type="perc", the confidence interval will never cross zero, and should not be used for statistical inference. However, if type="norm", the confidence interval may cross zero.

When theta is close to 0 or very large, or with small counts in some cells, the confidence intervals determined by this method may not be reliable, or the procedure may fail.

References

Freeman, L.C. 1965. Elementary Applied Statistics for Students in Behavioral Science. Wiley.

https://rcompanion.org/handbook/H_11.html

Author

Salvatore Mangiafico, mangiafico@njaes.rutgers.edu

Examples

data(Breakfast)
library(coin)
chisq_test(Breakfast, scores = list("Breakfast" = c(-2, -1, 0, 1, 2)))
#> 
#> 	Asymptotic Generalized Pearson Chi-Squared Test
#> 
#> data:  Breakfast (ordered) by Travel (Walk, Bus, Drive)
#> chi-squared = 8.6739, df = 2, p-value = 0.01308
#> 
freemanTheta(Breakfast)
#> Freeman.theta 
#>         0.312 

### Example from Freeman (1965), Table 10.6
Counts = c(1, 2, 5, 2, 0, 10, 5, 5, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 3)
Matrix = matrix(Counts, byrow=TRUE, ncol=5,
                dimnames = list(Marital.status = c("Single", "Married",
                                                   "Widowed", "Divorced"),
                                Social.adjustment = c("5","4","3","2","1")))
Matrix
#>               Social.adjustment
#> Marital.status  5 4 3 2 1
#>       Single    1 2 5 2 0
#>       Married  10 5 5 0 0
#>       Widowed   0 0 2 2 1
#>       Divorced  0 0 0 2 3
freemanTheta(Matrix)
#> Freeman.theta 
#>         0.749 

### Example after Kruskal Wallis test
data(PoohPiglet)
kruskal.test(Likert ~ Speaker, data = PoohPiglet)
#> 
#> 	Kruskal-Wallis rank sum test
#> 
#> data:  Likert by Speaker
#> Kruskal-Wallis chi-squared = 16.842, df = 2, p-value = 0.0002202
#> 
freemanTheta(x = PoohPiglet$Likert, g = PoohPiglet$Speaker)
#> Freeman.theta 
#>          0.64 

### Same data, as table of counts
data(PoohPiglet)
XT = xtabs( ~ Speaker + Likert , data = PoohPiglet)
freemanTheta(XT)
#> Freeman.theta 
#>          0.64 

### Example from Freeman (1965), Table 10.7
Counts = c(52, 28, 40, 34, 7, 9, 16, 10, 8, 4, 10, 9, 12,6, 7, 5)
Matrix = matrix(Counts, byrow=TRUE, ncol=4,
                dimnames = list(Preferred.trait = c("Companionability",
                                                    "PhysicalAppearance",
                                                    "SocialGrace",
                                                    "Intelligence"),
                                Family.income = c("4", "3", "2", "1")))
Matrix
#>                     Family.income
#> Preferred.trait       4  3  2  1
#>   Companionability   52 28 40 34
#>   PhysicalAppearance  7  9 16 10
#>   SocialGrace         8  4 10  9
#>   Intelligence       12  6  7  5
freemanTheta(Matrix, verbose=TRUE)
#> 
#>   Comparison   Di   Ti
#> 1          1 1024 6468
#> 2          2  636 4774
#> 3          3  434 4620
#> 4          4    8 1302
#> 5          5  332 1260
#> 6          6  211  930
#> 
#> Sum Di =  2645 
#> T2     =  19354 
#>  
#> Freeman.theta 
#>         0.137