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whether or not an improvement of approximately 2 points is worth the investment
in the training program.
6.5.8 What If the Observations Means Are Different, But We Do
Not See Consistent Movement of Scores?
To see how the results will change if consistent improvement in matched pairs does
not occur, while maintaining the averages, I will shufﬂe the data for training scores.
In other words, the scores in the 36 Women prisoners SC (trained) column will
remain the same, but they will not be associated with the same values in the 36
Women prisoners SC (before training) column. Thus, no change will be made in
values; only the matched pairs will be changed. Table 6.8 shows the new (shufﬂed)
pairs with the same mean scores as before. Table 6.9 shows the new results. Note that
the means remain the same, but the Pearson Correlation value is quite different from
before— –0.15617663 (cell E9). This negative value indicates that as one matched
pair value increases there is generally a very mild decrease in the other value. Now
the newly calculated t-statistic is –0.876116006 (cell E12). Given the critical t-value
of 2.030107915 (cell E16), we cannot reject the null hypothesis that there is no
difference in the means. The results are completely different than before, in spite
of similar averages for the matched pairs. Thus, you can see that the consistent
movement of matched pairs is extremely important to the analysis.
In this section, we progressed through a series of hypothesis tests to determine the
effectiveness of the EB special training program applied to SC prisoners. As you
have seen, the question of the special training’s effectiveness is not a simple one
to answer. Determining statistically the true effect on the mean score improvement
is a complicated task that may require several tests and some personal judgment.
It is often the case that observed data can have numerous associated factors. In
our example, the observations were identiﬁable by state (SC or TX), status of free-
dom (prisoner and non-prisoner), exposure to training (standard or EB special), and
ﬁnally gender, although it was not fully speciﬁed for all observations. It is quite easy
to imagine many more factors associated with our sample observations—e.g. age,
level of education, etc.
In the next section, we will apply Analysis of Variance (ANOVA) to similar prob-
lems. ANOVA will allow us to compare the effects of multiple factors, with each
factor containing several levels of treatment on a variable of interest, for example
a test score. We will return to our call center example and identify 3 factors with
2 levels of treatment each. If gender could also be identiﬁed for each observation,
the results would be 4 factors with 2 treatments for each. ANOVA will split our
data into components, or groups, which can be associated with the various levels of
factors.
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