It seems to me to be pretty clear what goes on here; there is an over-representation in the (C,Z) and the (W,A) corners,
most easily shown by a mosaic plot; try the following in R:
tab <- rbind( c(27,17, 4),
c(19,31, 8),
c(26,41,13),
c( 9,40,28) )
rownames(tab) <- c("W","X","Y","Z")
colnames(tab) <- c("A","B","C")
addmargins( tab )
chisq.test( tab )
mosaicplot( tab , off=0, col=rainbow(3) )
mosaicplot( t(tab), off=0, col=rainbow(4) )
b.r.
Bendix Carstensen
________________________________
From: A UK-based worldwide e-mail broadcast system mailing list [[log in to unmask]] on behalf of Kim Pearce [[log in to unmask]]
Sent: 19 August 2016 15:58
To: [log in to unmask]
Subject: Post Hoc Contingency table analysis : your views
Hi everyone,
I wonder if anyone has a view on the following?
I am conducting a chi square analysis in SPSS. I'll refer to the following contingency table where, as is usual, we are testing the null hypothesis that our two categorical variables are independent.
More specifically, I am testing that there are no statistical differences among my 3 groups A,B and C i.e. my null hypothesis is that: the proportion in group A with W = proportion in group B with W = proportion in group C with W and that the proportion in group A with X = proportion in group B with X = proportion in group C with X etc.
Group A
Group B
Group C
Total
Row Marginal %
W
27
17
4
48
18.25095
X
19
31
8
58
22.05323
Y
26
41
13
80
30.41825
Z
9
40
28
77
29.27757
Total
81
129
53
263
Say the (omnibus) Chi square test was significant i.e. there was evidence of an association between our two variables. I then go on to break this down to further investigate the nature of the association.
Upon the recommendation of several authors, I generate the 'adjusted’ standardised residual for each cell of the table. A ‘significant’ adjusted standardised residual means that the associated cell(s) significantly contribute to the overall (significant) chi-square statistic.
Say if the ‘Group A with W’ cell had an adjusted standardised residual of +4.2. The adjusted standardised residual is a z score and, as such, can be compared with the normal distribution (after appropriate adjustement for the Type 1 error rate). As there are 12 tests here, I will apply the Bonferroni correction here for an alpha_adj of 0.05/12 = 0.0042 i.e. a 2-tailed critical value of 2.865….so any cell with an adjusted standardised residual greater than 2.865 (in absolute value) can be said to significantly contribute to the overall chi-square statistic.
My question is really a double check on the meaning of a ‘significant’ adjusted standardised residual. If we look at the ‘Group A with W’ cell again which had an adjusted standardised residual of +4.2. We have that its expected frequency is (48*81)/263 = 14.8 and the percentage of Group A with W is (27/81)*100 = 33.3%. Am I correct in thinking that we can interpret the significant adjusted standardised residual either as:
1) The percentage of Group A with W (33.3%) is significantly different from the row marginal percentage (18.25%)
Or
2) The observed frequency of Group A with W (27) is significantly different from the expected frequency of group A with W (14.8).
Similarly, if we looked at the ‘Group C with Z’ which has an expected frequency of 15.5 and an adjusted standardised residual of +4.2, we could either say:
1) The percentage of Group C with Z (52.8%) is significantly different from the row marginal percentage (29.28%)
Or
2) The observed frequency of Group C with Z (28) is significantly different from the expected frequency of group C with Z (15.5).
Do you agree with the above?
An additional question: often authors are not consistent in their decription of the conclusions to their analyses when talking about adjusted standardised/standardised rssiduals…..some say that ‘there is a significant difference’ from the null expectation, others report a direction e.g. ‘the observed frequency is significantly higher than expected’. I’d also appreciate anyone’s view on this too.
Thank you so much in advance for your time.
All the very best,
Kim
Dr Kim Pearce PhD, CStat, Assoc. Fellow HEA
Senior Statistician
Haematological Sciences
Room MG261
Institute of Cellular Medicine
William Leech Building
Medical School
Newcastle University
Framlington Place
Newcastle upon Tyne
NE2 4HH
Tel: (0044) (0)191 208 8142
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