Comparing two discrete independent variables

Question

I would like to perform something like two-sample Wilcoxon tests/T-test (paired = FALSE) to compare two method's expected scores.

Here is the table of counts:

table(data$method_type, data$score)

   Score:  3  4  5  6
  method1  2  3 12 13
  method2  1  6 15  8

Experiment was designed, such that score could be from set {0,1,2,3,4,5,6}, which represents points from exam test (0 point worst, 6 point best)

I have another dataset, where score (let's named it score_2), where:

table(data$method_type, data$score_2)

  Score_2  0  1  2  3  4  5  6
  method1  1  2  1  5  5  3 13
  method2  1  0  4  7  5  3 10

Problem: there is just 4 unique values of score (3,4,5 and 6). I definitely can't use t-test, but what about wilcoxon (Mann-Whitney)? Can I use it? Or which test would you advice me?

Dataset:

   method_type score
1      method1     5
2      method1     6
3      method1     5
4      method1     4
5      method1     6
6      method1     5
7      method1     6
8      method1     3
9      method1     5
10     method1     6
11     method1     5
12     method1     5
13     method1     6
14     method1     4
15     method1     6
16     method1     5
17     method1     5
18     method1     6
19     method1     6
20     method1     5
21     method1     3
22     method1     6
23     method1     6
24     method1     5
25     method1     6
26     method1     5
27     method1     6
28     method1     5
29     method1     4
30     method1     6
31     method2     4
32     method2     5
33     method2     5
34     method2     6
35     method2     4
36     method2     6
37     method2     5
38     method2     5
39     method2     5
40     method2     3
41     method2     5
42     method2     6
43     method2     4
44     method2     5
45     method2     6
46     method2     5
47     method2     5
48     method2     6
49     method2     5
50     method2     4
51     method2     5
52     method2     6
53     method2     5
54     method2     5
55     method2     5
56     method2     6
57     method2     6
58     method2     5
59     method2     4
60     method2     4

Answer 1

Or which test would you advice me?

If your question is whether the score distribution with method 2 differs from the distribution with method 1 then you can use Fisher's exact test:

x <- matrix(
  c(1L, 2L, 1L, 5L, 5L, 3L, 13L,
    1L, 0L, 4L, 7L, 5L, 3L, 10L), 
  nrow = 2, byrow = TRUE, 
  dimnames = list(c("method1", "method2"), 0:6))

set.seed(18548777)
fisher.test(x, simulate.p.value = TRUE, B = 10000)
#R 
#R Fisher's Exact Test for Count Data with simulated p-value (based on 10000 replicates)
#R 
#R data:  x
#R p-value = 0.6969
#R alternative hypothesis: two.sided

The null hypothesis is that the rows and columns are independent. We cannot reject this. Another options is the Monte Carlo test implemented in chisq.test:

set.seed(18548777)
chisq.test(x, simulate.p.value = TRUE, B = 10000)
#R 
#R Pearson's Chi-squared test with simulated p-value (based on 10000 replicates)
#R 
#R data:  x
#R X-squared = 4.5246, df = NA, p-value = 0.6812

The conclusion is the same. We cannot reject that scores are independent of the of the method. The answer here though ignores that the data is ordinal.