Statistical Procedures for Fuzzy Data in Medical Research - Fuzziness and Medicine - page 281

19.3.3 One-Way ANOVA

Gil et al. [5] and González-Rodríguez et al. [6] developed a multiple-sample test of
equality of fuzzy means. This is one-way ANOVA for fuzzy data; it is performed
to statistically examine the effect of a factor with at least two levels. Suppose that
there are four hospitals, call them hospitals 1, 2, 3, and 4, and that we want to
know whether they have the same mean degree of happiness. Figure 19.5 illustrates
the type of data that one may obtain for this study; hypothetical fuzzy observations
obtained from hospitals 1, 2, 3, and 4 are graphed with solid gray lines, dashed lines,
dotted lines, and solid black lines, respectively.

1

0.8

0.6

0.4

0.2

0

-100 -80 -60 -40 -20 0 20 40 60 80 100

Happiness

Fig. 19.5 Hypothetical fuzzy responses obtained from hospitals 1 (solid gray lines), 2 (dashed
lines), 3 (dotted lines), and 4 (solid black lines)

The statistical model considered for one-way ANOVA can be expressed as fol-
lows. Suppose that the factor has n levels. Let X ij denote a fuzzy random variable
(also described as a random fuzzy set) that represents the j th observation under the
i th level of the factor. Denote the overall mean and the mean of the i th level by μ

and α i , respectively. Then for each i and j , the model is expressed as

X ij = μ + α i + ε ij .

where ε ij represents the random component of the measurement. The null hypothe-

sisisthat

α 1 = α 2 = ··· = α n . The alternative hypothesis is that there exist α i 1 and

α i 2 α i 1 = α i 2 .

We can perform one-way ANOVA to determine whether to reject the null hypoth-
esis with a predetermined significance level. For mathematical details, see Gil et al.
[5] and González-Rodríguez et al. [6].

such that