Page 202 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

5.1 Inference on One Population 183

Commands 5.4. SPSS, STATISTICA, MATLAB and R commands used to
perform the chi-square goodness of fit test.

SPSS Analyze; Nonparametric Tests; Chi-Square

Statistics; Nonparametrics; Observed
STATISTICA 2
versus expected Χ .
MATLAB [c,df,sig] = chi2test(x)

R chisq.test(x,p)

MATLAB does not have a specific function for the chi-square test. We provide in
the book CD the chi2test function for that purpose.

5.1.4 The Kolmogorov-Smirnov Goodness of Fit Test

The Kolmogorov-Smirnov goodness of fit test is a one-sample test designed to
assess the goodness of fit of a data sample to a hypothesised continuous
distribution, F X (x). The null hypothesis is formalised as:

H 0: Data variable X has a cumulative probability distribution F X (x) ≡ F(x).

Let S n(x) be the observed cumulative distribution of the random sample, x 1,
x 2,…, x n, also called empirical distribution. Assuming the sample data is sorted in
increasing order, the values of S n(x) are obtained by adding the successive
frequencies of occurrence, k i/n, for each distinct x i.
Under the null hypothesis one expects to obtain small deviations of S n(x) from
F(x). The Kolmogorov-Smirnov test uses the largest of such deviations as a
goodness of fit measure:

D n = max | F(x) − S n(x) |, for every distinct x i. 5.10

The sampling distribution of D n is given in the literature. Unless n is very small
the following asymptotic result can be used:

lim P ( D ≤ t ) n =1 − 2 ∑ ∞ ( − ) i −1 e −2 ti 2 2 . 5.11
1
n → ∞ n i =1

The Kolmogorov-Smirnov test rejects the null hypothesis at level α if

D n > d α , n , where d α , n is such that:

P (D > d ) = α . 5.12
H 0 n α , n

Using formula 5.11 the following critical points are obtained:

d n . 01 = . 1 63 / n; d n . 05 = . 1 36 / n; d n . 10 = . 1 22 / n . 5.13
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0
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