# Rank difference test Performs Kornbrot's rank difference test. It's a modified Wilcoxon signed-rank test which produces consistent and meaningful results for ordinal or monotonically transformed data. ```r rdt( data, formula, zero.method = c("Pratt", "Wilcoxon"), distribution = c("asymptotic", "approximate", "exact"), alternative = c("two.sided", "greater", "less"), return = c("data.frame", "coin"), ... ) ``` ## Arguments - `data`: A data.frame - `formula`: A formula of either form: - **y ~ x | block**: For use when `data` is in long format. `y` is the numeric outcome, `x` is the binary grouping factor, and `block` is the observation-level grouping factor. `x` and `block` must be factors. - **y ~ x**: For use when `data` is in wide format. `y` is the first measurement and `x` is the second measurement on the same observation. The differences in ranks are calculated as `y - x` or, when `x` is a factor, the first factor level minus the second. - `zero.method`: A string for the method used to handle differences equal to zero: `"Pratt"` (default) or `"Wilcoxon"`. - `distribution`: A string for the method used to calculate the conditional null distribution of the test statistic: Asymptotic distribution `"asymptotic"` (default), Monte Carlo resampling `"approximate"`, or the exact distribution `"exact"`. - `alternative`: A string for the alternative hypothesis: `"two.sided"` (default), `"greater"`, or `"less"`. - `return`: A string for the return object: `"data.frame"` (default) or `"coin"`. - `...`: Further arguments passed to `coin::wilcoxsign_test`. ## Returns If `return = "data.frame"`, a data.frame with columns `p.value`, `z.statistic`, `formula`, `alternative`, and `method`. If `return = "coin"`, an object inheriting from class `"IndependenceTest"`. ## Details For ordinal scale data, the Wilcoxon signed-rank test results in subtraction of those ordinal scale values. However, this subtraction is not meaningful on the ordinal scale. In addition, any monotone transformation of the data will result in different signed ranks, thus different p-values. However, ranking the original data allows for meaningful addition and subtraction of ranks and preserves ranks over monotonic transformation. Kornbrot described the rank difference test for this reason. Kornbrot recommends that the rank difference procedure be used in preference to the Wilcoxon signed-rank test in all paired comparison designs where the data are not both of interval scale type and of known distribution type. The rank difference test preserves good power compared to Wilcoxon's signed-rank test, is more powerful than the sign test, and has the benefit of being a true distribution-free test. The procedure for Wilcoxon's signed-rank test is as follows: 1. Calculate differences for each paired observation. 2. Remove differences equal to zero. 3. Order the absolute differences from smallest to largest. 4. Assign ranks $1, \dots, n$ with average rank for ties. 5. Calculate W+ = sum of the ranks for positive differences. The sum of W+ and W- is $n(n+1)/2$, so either can be calculated from the other. 6. Choose the smaller of W+ and W- as the test statistic W. 7. Since the test statistic is the smaller of W+ and W-, the critical region is the left tail of the distribution. W is distributed approximately normal with mean $mu = (n(n+1))/4$ and variance $sigma^2 = (Tn(n+1)(2n+1))/24$, where T is a correction for ties and $T = 1-(sum(t^3-t)/(N^3-N))$, summed over all ties, where t is the length of a tie. The continuity corrected mean $mu = ((n(n+1))/4)+0.5$. The procedure for Kornbrot's rank difference test is as follows: 1. Combine all 2n observations. 2. Assign ranks $1, \dots, 2n$ with average rank for ties. 3. Perform the Wilcoxon signed-rank test using the paired ranks. The test statistic for the rank difference test (D) is not exactly equal to the test statistic (W) of the naive rank-transformed Wilcoxon signed-rank test (the latter being implemented in `rdt()`). Using W should result in a conservative estimate for D, and they approach in distribution as the sample size increases. if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="kornbrot1990;textual",package="rankdifferencetest",cached_env=.Rdpack.currefs) discusses methods for calculating D when n<7 and 8

rdt function