Mantel Linear-by-Linear Association Test for Detecting Linear Association Between Ordinal Variables

A chi-squared test for linear association between two ordinal variables in a two-way contingency table, using row and column scores.

Usage

mantelTrendTest(
  x,
  srow = scores(x, MARGIN = 1L, method = "table"),
  scol = scores(x, MARGIN = 2L, method = "table")
)

Arguments

x: a numeric matrix of counts ($r \times c$)
srow: numeric vector of row scores; length must equal nrow(x). Defaults to the numeric row dimnames of x if present, otherwise 1:nrow(x). See the Details.
scol: numeric vector of column scores; length must equal ncol(x). Defaults to the numeric column dimnames of x if present, otherwise 1:ncol(x). See the Details.

Value

A list of class "htest" containing:

statistic: the Mantel linear association chi-squared statistic
parameter: degrees of freedom (always 1)
p.value: the p-value
estimate: the Pearson correlation coefficient $r$
method: a character string describing the test
data.name: a character string giving the name of the data

Details

Tests for linear trend in a two-way $r \times c$ contingency table by computing the Mantel linear-by-linear association statistic $$Q_{MH} = (n - 1) \cdot r^2$$ where $r$ is the Pearson correlation between the row and column variables using the supplied scores. Under the null hypothesis of no linear association, $Q$ has an asymptotic chi-squared distribution with one degree of freedom.

This test is sometimes called the Mantel-Haenszel chi-squared test for trend, but it is not the stratified Cochran-Mantel-Haenszel test for $2 \times 2 \times k$ tables (see mantelhaen.test() for that). It is a score test for ordinal association, also known as the linear-by-linear association test.

Both variables should be measured on an ordinal scale. If x has numeric row and/or column dimnames (e.g. income brackets or dose levels stored as label strings that parse as numbers), those values are used as the default scores; otherwise the default is 1:nrow(x) resp. 1:ncol(x), i.e. the categories are assumed equally spaced. This mirrors the scoring convention used by cochranArmitageTest(). The choice of scores affects the result: any monotone scores are permitted; non-monotone scores (neither strictly increasing nor strictly decreasing) produce a warning, since $r$ would then no longer reflect a consistent ordinal trend.

References

Agresti, A. (2002) Categorical Data Analysis, John Wiley & Sons, pp. 57, 86.

Mantel, N. (1963) Chi-square tests with one degree of freedom: extensions of the Mantel-Haenszel procedure. Journal of the American Statistical Association, 58, 690-700.

Examples

## Agresti (2002, p. 57) Job Satisfaction
Job <- matrix(c(1,2,1,0, 3,3,6,1, 10,10,14,9, 6,7,12,11), 4, 4,
              dimnames = list(
                income       = c("< 15k","15-25k","25-40k","> 40k"),
                satisfaction = c("VeryD","LittleD","ModerateS","VeryS")))

mantelTrendTest(Job)
#> 
#> 	Mantel linear-by-linear association test
#> 
#> data:  Job
#> X-squared = 2.983, df = 1, p-value = 0.08414
#> sample estimates:
#>         r 
#> 0.1772001 
#> 
mantelTrendTest(Job, srow = c(7.5, 20, 32.5, 60))
#> 
#> 	Mantel linear-by-linear association test
#> 
#> data:  Job
#> X-squared = 3.8075, df = 1, p-value = 0.05102
#> sample estimates:
#>         r 
#> 0.2001962 
#> 

## Automatic scores from numeric dimnames
dose <- matrix(c(10, 9, 10, 7, 0, 1, 0, 3), nrow = 4,
               dimnames = list(dose = c("0", "1", "2", "3"),
                               resp = c("no", "yes")))
mantelTrendTest(dose)  # srow taken as c(0, 1, 2, 3), not 1:4
#> 
#> 	Mantel linear-by-linear association test
#> 
#> data:  dose
#> X-squared = 3.4667, df = 1, p-value = 0.06262
#> sample estimates:
#>         r 
#> 0.2981424 
#>