Large Pearson residuals

Question

My aim is to see if there exists a relation between the variables "Category" and "Types", spec. if there is a tendency for a particular category to use a particular type. These are the observed frequencies:

	T1	T2	T3	T4	T5
A	800	1	9	245	2503
B	350	1560	1200	341	790
C	7	290	7	607	1
D	387	1001	1311	932	598
E	16	81	19	7	4
F	493	98	0	39	342
G	32	318	2	1	3
H	777	52	18	127	139
I	3	2	27	9	1
J	128	299	1	4	2
K	10	5	2	4	76
L	139	1	1	29	6

So I conducted a chi-square test. This is the code I used in R:

x <- matrix(c(800, 350, 7, 387, 16, 493, 32, 777, 
    3, 128, 10, 139, 10, 1560, 290, 1001, 81, 98, 
    318, 52, 4, 299, 15, 18, 19, 1200, 7, 1311, 
    19, 0, 2, 18, 27, 10, 2, 8, 245, 341, 607, 
    932, 7, 39, 10, 127, 9, 7, 8, 29, 2503, 790, 
    1, 598, 4, 342, 3, 139, 2, 3, 76, 6), ncol=5)

attr(x, "dimnames") <- list(Category=c("A", "B", 
    "C", "D", "E", "F", "G", "H", "I", "J", "K", 
    "L"), 
    Types = c("T1", "T2", "T3", "T4", "T5"))

x.test <- chisq.test(x, correct = FALSE)

x.test$expected

All the expected frequencies are > 5 and the result of the chi-square test is: X-squared = 13493, df = 44, p-value < 2.2e-16. I have also used Cramer's V to see how strong is the effect independently of the sample size (Cramer's V = 0.4543752).

I have also calculated Pearson and standardized residuals (It's not clear to me what should I use, if any) but the values are very large and I have read that this may indicate large errors, which may imply that the model can be inappropriate for the data.

These are the results of Pearson residuals:

And these are the results for standardized residuals:

Does it make any sense or would it be better to use another measure?

Answer 1

With such a large contingency table with large counts you are better off with visualizing methods, or simply reordering rows/columns of the table so that structure becomes more easily visible. The large chi-squared value, and large residuals makes it apparent that the independence model is untrue.

Correspondence analysis is your help. Some code in R, with your data:

mod.CA <- MASS::corresp(x, nf=2)
plot(mod.CA)   ### A biplot (not shown here)

I will use the row and column scores corresponding to the largest eigenvalue to reorder rows and columns, then the structure will be more visible:

RS <- mod.CA$rscore[, 1]
CS <- mod.CA$cscore[, 1]

X <- x[order(RS), order(CS)]

X
        Types
Category   T5  T1  T4   T3   T2
       A 2503 800 245   19   10
       F  342 493  39    0   98
       H  139 777 127   18   52
       K   76  10   8    2   15
       L    6 139  29    8   18
       B  790 350 341 1200 1560
       J    3 128   7   10  299
       D  598 387 932 1311 1001
       E    4  16   7   19   81
       I    2   3   9   27    4
       C    1   7 607    7  290
       G    3  32  10    2  318

We can also show the table as a mosaicplot:

mosaicplot(X, color=TRUE)