This article offers a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function (EDF). Five of the leading statistics are examined—those often labelled D, W 2, V, U 2, A 2—and three important situations: where the hypothesized distribution F(x) is completely specified and where F(x) represents the normal or exponential distribution with one or more parameters to be estimated from the data. EDF statistics are easily calculated, and the tests require only one line of significance points for each situation. They are also shown to be competitive in terms of power.

https://www.jstor.org/stable/pdfplus/2286009.pdf

EDF Statistics for Goodness of Fit and Some Comparisons

Summary Using the Lagrange multiplier procedure or score test on the Pearson family of distributions we obtain tests for normality of observations and regression disturbances. The tests suggested have optimum asymptotic power properties and good finite sample performance. Due to their simplicity they should prove to be useful tools in statistical analysis.

A test for normality of observations and regression residuals

Researchers at the National Cancer Institute developed a new cognitively based food frequency questionnaire (FFQ), the Diet History Questionnaire (DHQ). The Eating at America's Table Study sought to validate and compare the DHQ with the Block and Willett FFQs. Of 1,640 men and women recruited to participate from a nationally representative sample in 1997, 1,301 completed four telephone 24-hour recalls, one in each season. Participants were randomized to receive either a DHQ and Block FFQ or a DHQ and Willett FFQ. With a standard measurement error model, correlations for energy between estimated truth and the DHQ, Block FFQ, and Willett FFQ, respectively, were 0.48, 0.45, and 0.18 for women and 0.49, 0.45, and 0.21 for men. For 26 nutrients, correlations and attenuation coefficients were somewhat higher for the DHQ versus the Block FFQ, and both were better than the Willett FFQ in models unadjusted for energy. Energy adjustment increased correlations and attenuation coefficients for the Willett FFQ dramatically and for the DHQ and Block FFQ instruments modestly. The DHQ performed best overall. These data show that the DHQ and the Block FFQ are better at estimating absolute intakes than is the Willett FFQ but that, after energy adjustment, all three are more comparable for purposes of assessing diet-disease risk.

Comparative Validation of the Block, Willett, and National Cancer Institute Food Frequency Questionnaires The Eating at America's Table Study

SUMMARY Shapiro and Wilk's (1965) W statistic arguably provides the best omnibus test of normality, but is currently limited to sample sizes between 3 and 50. Wis extended up to n = 2000 and an approximate normalizing transformation suitable for computer implementation is given. A novel application of Win transforming data to normality is suggested, using the three-parameter lognormal as an example. RESEARCH into tests of non-normality was given new impetus with the introduction of the so- called analysis of variance test by Shapiro and Wilk (1965). The test statistic Wwas constructed by considering the regression of ordered sample values on corresponding expected normal order statistics, which for a sample from a normally distributed population is linear; Wwas obtained as an F-ratio from generalized least-squares analysis to judge the adequacy of the linear fit. Percentage points of the null distribution of Wwere tabulated for p = 0 01, 002, 005, 0 1, 0 5, 0-9, 0 95, 0 98 and 0 99, for sample sizes n = 3 (1) 50. A normalizing transformation for Wusing a Johnson SB approximation in the region n = 7(1)50 was later proposed (Shapiro and Wilk, 1968), though tables were still required for n = 4(1) 6. Extensive empirical comparisons of Wwith other tests of non-normality using computer- generated pseudo-random numbers indicated that W had good power properties against a wide range of alternative distributions, and was therefore truly an omnibus test (Shapiro et al., 1968). Subsequently, other statistics of the Wtype, namely Y(D'Agostino, 1971), W' (Shapiro and Francia, 1972) and r (Filliben, 1975), were developed and shown to have power properties broadly comparable with those of W Another approach was to combine ,lb and b2, the

https://www.jstor.org/stable/pdfplus/2347973.pdf

An Extension of Shapiro and Wilk's W Test for Normality to Large Samples

This paper introdLlces the normal probability plot correlation coefficient as a test statistic in complete samples for the composite hypothesis of normality. The proposed test statistic is conceptnally simple, is compntationally convenient, and is readily extendible to testing non-normal distributional hypotheses. An empirical power strldy shows that the normal probability plot correlation coefficient, compares favorably with 7 other normal test statistics. Percent points are tabulated for n = 3(l)50(5)100.

https://www.jstor.org/stable/pdfplus/1268008.pdf

The Probability Plot Correlation Coefficient Test for Normality

This article presents a modification of the Shapiro-Wilk W statistic for testing normality which can be used with large samples. Shapiro and Wilk gave coefficients and percentage points for sample sizes up to 50. These coefficients required obtaining an approximation to the covariance matrix of the normal order statistics. The proposed test uses coefficients which depend only on the expected values of the normal order statistics which are generally available. Results of an empirical sampling study to compare the sensitivity of the test statistic to the W test statistic are briefly discussed.

S. S. Shapiro

Papers

An Approximate Analysis of Variance Test for Normality