mantelhaen.test package:stats R Documentation _C_o_c_h_r_a_n-_M_a_n_t_e_l-_H_a_e_n_s_z_e_l _C_h_i-_S_q_u_a_r_e_d _T_e_s_t _f_o_r _C_o_u_n_t _D_a_t_a _D_e_s_c_r_i_p_t_i_o_n: Performs a Cochran-Mantel-Haenszel chi-squared test of the null that two nominal variables are conditionally independent in each stratum, assuming that there is no three-way interaction. _U_s_a_g_e: mantelhaen.test(x, y = NULL, z = NULL, alternative = c("two.sided", "less", "greater"), correct = TRUE, exact = FALSE, conf.level = 0.95) _A_r_g_u_m_e_n_t_s: x: either a 3-dimensional contingency table in array form where each dimension is at least 2 and the last dimension corresponds to the strata, or a factor object with at least 2 levels. y: a factor object with at least 2 levels; ignored if 'x' is an array. z: a factor object with at least 2 levels identifying to which stratum the corresponding elements in 'x' and 'y' belong; ignored if 'x' is an array. alternative: indicates the alternative hypothesis and must be one of '"two.sided"', '"greater"' or '"less"'. You can specify just the initial letter. Only used in the 2 by 2 by K case. correct: a logical indicating whether to apply continuity correction when computing the test statistic. Only used in the 2 by 2 by K case. exact: a logical indicating whether the Mantel-Haenszel test or the exact conditional test (given the strata margins) should be computed. Only used in the 2 by 2 by K case. conf.level: confidence level for the returned confidence interval. Only used in the 2 by 2 by K case. _D_e_t_a_i_l_s: If 'x' is an array, each dimension must be at least 2, and the entries should be nonnegative integers. 'NA''s are not allowed. Otherwise, 'x', 'y' and 'z' must have the same length. Triples containing 'NA''s are removed. All variables must take at least two different values. _V_a_l_u_e: A list with class '"htest"' containing the following components: statistic: Only present if no exact test is performed. In the classical case of a 2 by 2 by K table (i.e., of dichotomous underlying variables), the Mantel-Haenszel chi-squared statistic; otherwise, the generalized Cochran-Mantel-Haenszel statistic. parameter: the degrees of freedom of the approximate chi-squared distribution of the test statistic (1 in the classical case). Only present if no exact test is performed. p.value: the p-value of the test. conf.int: a confidence interval for the common odds ratio. Only present in the 2 by 2 by K case. estimate: an estimate of the common odds ratio. If an exact test is performed, the conditional Maximum Likelihood Estimate is given; otherwise, the Mantel-Haenszel estimate. Only present in the 2 by 2 by K case. null.value: the common odds ratio under the null of independence, '1'. Only present in the 2 by 2 by K case. alternative: a character string describing the alternative hypothesis. Only present in the 2 by 2 by K case. method: a character string indicating the method employed, and whether or not continuity correction was used. data.name: a character string giving the names of the data. _N_o_t_e: The asymptotic distribution is only valid if there is no three-way interaction. In the classical 2 by 2 by K case, this is equivalent to the conditional odds ratios in each stratum being identical. Currently, no inference on homogeneity of the odds ratios is performed. See also the example below. _R_e_f_e_r_e_n_c_e_s: Alan Agresti (1990). _Categorical data analysis_. New York: Wiley. Pages 230-235. Alan Agresti (2002). _Categorical data analysis_ (second edition). New York: Wiley. _E_x_a_m_p_l_e_s: ## Agresti (1990), pages 231--237, Penicillin and Rabbits ## Investigation of the effectiveness of immediately injected or 1.5 ## hours delayed penicillin in protecting rabbits against a lethal ## injection with beta-hemolytic streptococci. Rabbits <- array(c(0, 0, 6, 5, 3, 0, 3, 6, 6, 2, 0, 4, 5, 6, 1, 0, 2, 5, 0, 0), dim = c(2, 2, 5), dimnames = list( Delay = c("None", "1.5h"), Response = c("Cured", "Died"), Penicillin.Level = c("1/8", "1/4", "1/2", "1", "4"))) Rabbits ## Classical Mantel-Haenszel test mantelhaen.test(Rabbits) ## => p = 0.047, some evidence for higher cure rate of immediate ## injection ## Exact conditional test mantelhaen.test(Rabbits, exact = TRUE) ## => p - 0.040 ## Exact conditional test for one-sided alternative of a higher ## cure rate for immediate injection mantelhaen.test(Rabbits, exact = TRUE, alternative = "greater") ## => p = 0.020 ## UC Berkeley Student Admissions mantelhaen.test(UCBAdmissions) ## No evidence for association between admission and gender ## when adjusted for department. However, apply(UCBAdmissions, 3, function(x) (x[1,1]*x[2,2])/(x[1,2]*x[2,1])) ## This suggests that the assumption of homogeneous (conditional) ## odds ratios may be violated. The traditional approach would be ## using the Woolf test for interaction: woolf <- function(x) { x <- x + 1 / 2 k <- dim(x)[3] or <- apply(x, 3, function(x) (x[1,1]*x[2,2])/(x[1,2]*x[2,1])) w <- apply(x, 3, function(x) 1 / sum(1 / x)) 1 - pchisq(sum(w * (log(or) - weighted.mean(log(or), w)) ^ 2), k - 1) } woolf(UCBAdmissions) ## => p = 0.003, indicating that there is significant heterogeneity. ## (And hence the Mantel-Haenszel test cannot be used.) ## Agresti (2002), p. 287f and p. 297. ## Job Satisfaction example. Satisfaction <- as.table(array(c(1, 2, 0, 0, 3, 3, 1, 2, 11, 17, 8, 4, 2, 3, 5, 2, 1, 0, 0, 0, 1, 3, 0, 1, 2, 5, 7, 9, 1, 1, 3, 6), dim = c(4, 4, 2), dimnames = list(Income = c("<5000", "5000-15000", "15000-25000", ">25000"), "Job Satisfaction" = c("V_D", "L_S", "M_S", "V_S"), Gender = c("Female", "Male")))) ## (Satisfaction categories abbreviated for convenience.) ftable(. ~ Gender + Income, Satisfaction) ## Table 7.8 in Agresti (2002), p. 288. mantelhaen.test(Satisfaction) ## See Table 7.12 in Agresti (2002), p. 297.