For companies competing in the same field, how to test if their market share changes over a period of time is siginificant Suppose 3 companies, A, B, and C, compete for market shares in the same field. In year 1, their market shares in percentage are a1, b1, and c1. And in year 2, market shares are a2, b2 and c2. With $\sum(a1, b1, c1) = \sum(a2, b2, c2) = 100$.
How to test if market share distribution in (a2, b2, c2) is the same or different from (a1, b1, c1)? Is it better to use raw sales instead of percent market shares?
 A: Data in the table below might have come from looking
at 100 randomly chosen sales in Year 1 and then another 100 randomly chosen sales in Year 2. 
In each year numbers of these sales by Companies A, B, and C are noted:
Company    A    B    C   Total
Year 1    30   36   34     !00
Year 2    20   29   51     100

It seems that A and B have lost market share to C in Year 2. You can test whether the proportions have changed significantly from from year to year with a chi-squared test in R. 
For my data above, the answer is that
the difference is just barely significant at the 5% level
with P-value 0.0461.
 TBL = rbind(c(30, 36, 34), c(20, 29, 51)) 
 TBL
      [,1] [,2] [,3]
 [1,]   30   36   34
 [2,]   20   29   51
 chisq.test(TBL)

        Pearson's Chi-squared test

data:  TBL
X-squared = 6.1538, df = 2, p-value = 0.0461

In such an analysis, you must use counts of sales, not percentages
or proportions.
For example, if you were to do the same kind of chi-squared test for data with
all counts exactly five times what they were in the table above,
then the proportions would be exactly the same, but the P-value
would be very much smaller (0.0000002082, almost 0):
TBL2 = 5*TBL;  TBL2
     [,1] [,2] [,3]
[1,]  150  180  170
[2,]  100  145  255
chisq.test(TBL2)

        Pearson's Chi-squared test

data:  TBL2
X-squared = 30.769, df = 2, p-value = 2.082e-07

If you do a chi-squared test with all counts as close as possible to half what
they were in the original table, you don't get a significant P-value.
TBL3 = round(TBL/2)
TBL3
     [,1] [,2] [,3]
[1,]   15   18   17
[2,]   10   14   26
chisq.test(TBL3)

        Pearson's Chi-squared test

data:  TBL3
X-squared = 3.3837, df = 2, p-value = 0.1842

