Statistical Analysis for Education and Psychology Researchers

Chi-square 1 15.643 0.000

Phi Coefficient 0.351

Cramer’s V 0.351

Sample Size=127

Figure 6.3: Output for Chi-square analysis

The computed χ^2 value is the same as the value in the worked example. Notice an actual
probability is given rather than p at a pre-specified value (i.e., p≤0.05 or p≤0.01). Clearly
the actual value is statistically significant at the 1 per cent (p≤0.001) level. Each cell in
the contingency table contains a cell frequency count and row and column per cents.

Phi Coefficient and Cramer’s Phi

The χ^2 procedure is sensitive to sample size and is nearly always significant with large
samples. The χ^2 test assesses the statistical significance of an association and not the
strength of the association. Correlational type statistics are therefore required to
determine the strength of any statistically significant association detected by the χ^2
statistic. Two of the most useful measures of association provided in the SAS output are
i) Φ, (Phi Coefficient) and ii) Cramer’s V (sometimes called Cramer’s Phi Coefficient).
Phi should only be used as a measure of the strength of association when both variables
are binary (scored as 0, 1; present, absent; +, −, etc.) and can be used when data is in the
form of a 2 × 2 contingency table. There is a direct relationship between χ^2 and Φ given
by the formulae:

Phi—

6.2

where n is the total sample size and χ^2 is the statistic from the same contingency table.
The value of Phi for the data shown in Figure 6.1 is 0.351 (15.643/127)0.5. Phi has a
lower limit of 0, no strength of association (variables are not related) and an upper limit
of 1, maximum strength of association (variables are perfectly correlated). When a
contingency table has more than four cells Cramer’s V should be used to measure the
strength of association. Similar to Phi the range of this statistic varies between 0 and 1.
Cramer’s V is calculated using the following formulae:

Cramer’s

V—6.3

where χ^2 is the value for the entire contingency table, n is the total sample size and j is the
smaller of the number of rows or number of columns in the contingency table.
The value of Cramer’s V for the data shown in Figure 6.1 is 0.35, (15.643/127
×(2−1))0.5, the same value as Phi. For discussion of the use of measures of association in
conjunction with the Chi-square statistic, see Delucchi (1983).

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