Microsoft Office Tutorials and References
In Depth Information
exp val ij =
expected frequency of observations in the ith row and jth column of
the contingency table, when independence of the variables is assumed. 1
2 statistic is calculated, then it can be compared to a benchmark value
Once the
χ
2
of
χ
that sets a limit, or threshold, for rejecting the null hypothesis. The value
α
2 statistic can achieve before we reject the null hypothesis.
These values can be found in most statistics books. To select a particular
2
of
χ
is the limit the
χ
α
2
χ
,the
α
α
(the level of signiﬁcance of the test) must be set by the investigator. It is closely
related to the p-value —the probability of obtaining a particular statistic value or
more extreme by chance, when the null hypothesis is true. Investigators often set
α
2 statistic (or greater)
when the null is true. So, in essence, our decision maker only wants a 5% chance of
erroneously rejecting the null hypothesis. That is relatively conservative, but a more
conservative (less chance of erroneously rejecting the null hypothesis) stance would
be to set
to 0.05; that is, there is a 5% chance of obtaining this
χ
α
to 1%, or even less.
Thus, if our
2
2
χ
is greater than or equal to
χ
, then we reject the null.
α
Alternatively, if the p-value is less than
we reject the null. These tests are
equivalent. In summary, the rules for rejection are either:
α
2 >
2
Reject the null hypothesis when
χ
= χ
α
or
Reject the null hypothesis when p-value <
= α
(Note that these rules are equivalent)
Exhibit 6.1 shows a worksheet that performs the test of independence using
the chi-square procedure. The exhibit also shows the typical calculation for con-
tingency table expected values. Of course, in order to perform the analysis, both
tables are needed to calculate the
2 statistic since both the observed frequency and
the expected are used in the calculation. Using the Excel CHITEST (actual range,
expected range) cell function permits Excel to calculate the data’s
χ
2 and then
return a p-value (see cell F17 in Exhibit 6.1). You can also see from Exhibit 6.1 that
the actual range is C4:F5 and does not include the marginal totals. The expected
range is C12:F13 and the marginal totals are also omitted. The internally calculated
χ
χ
2 value takes into consideration the number of variables for the data, 2 in our case,
and the possible levels within each variable—2 for risk preference and 4 for mutual
fund types. These variables are derived from the range data information (rows and
columns) provided in the actual and expected tables.
From the spreadsheet analysis in Exhibit 6.1 we can see that the calculated
2
χ
value in F18 is 106.8 (a relatively large value), and if we assume
α
to be 0.05, then
1 Calculated by multiplying the row total and the column total and dividing by total number of
observations—e.g. in Exhibit 1 expected value for conservative/growth cell is 120
90.
Note that 120 is the marginal total Income/Growth and 450 is the marginal total for Conservative.
450/600
=
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