# Regression Tests of Fit and Probability Plotting Positions

Stephen W. Looney, Thomas R. Gulledge

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

Probability plots are commonly used as a technique for testing distributional assumptions. However, any conclusion about the linearity of such a plot is based strictly on the user's judgment. Regression tests of fit are supposed to make this procedure more objective, but these tests typically are not based on probability plots as they are constructed in practice. This is because the developers of these tests defined probability plots in terms of plotting positions which are not used by practitioners. In this paper, a class of goodness-of-fit test statistics which are calculated directly from probability plots as they are constructed in practice is described. Several realistic plotting positions for the normal distribution are chosen and empirical sampling methods are used to derive the null distribution of each of the corresponding test statistics. These tests are then compared on the basis of 5% power against certain nonnormal alternatives. Results of the comparisons indicate that the test based on the plotting position pi =(i — 0.4)/(j + 0.2) is a reasonable choice for a powerful regression test of fit for normality.

Original language English (US) 115-127 13 Journal of Statistical Computation and Simulation 20 2 https://doi.org/10.1080/00949658408810761 Published - Jan 9 1984

### Fingerprint

Probability Plot
Regression
Statistics
Test Statistic
Normal distribution
Null Distribution
Goodness of Fit Test
Sampling Methods
Sampling
Pi
Linearity
Normality
Gaussian distribution
Testing
Strictly
Alternatives

### Keywords

• Correlation coefficient
• Empirical power comparison
• Filliben test
• Plotting position
• Shapiro-Francia test
• Shapiro-Wilk test

### ASJC Scopus subject areas

• Statistics and Probability
• Modeling and Simulation
• Statistics, Probability and Uncertainty
• Applied Mathematics

### Cite this

Regression Tests of Fit and Probability Plotting Positions. / Looney, Stephen W.; Gulledge, Thomas R.

In: Journal of Statistical Computation and Simulation, Vol. 20, No. 2, 09.01.1984, p. 115-127.

Research output: Contribution to journalArticle

title = "Regression Tests of Fit and Probability Plotting Positions",
abstract = "Probability plots are commonly used as a technique for testing distributional assumptions. However, any conclusion about the linearity of such a plot is based strictly on the user's judgment. Regression tests of fit are supposed to make this procedure more objective, but these tests typically are not based on probability plots as they are constructed in practice. This is because the developers of these tests defined probability plots in terms of plotting positions which are not used by practitioners. In this paper, a class of goodness-of-fit test statistics which are calculated directly from probability plots as they are constructed in practice is described. Several realistic plotting positions for the normal distribution are chosen and empirical sampling methods are used to derive the null distribution of each of the corresponding test statistics. These tests are then compared on the basis of 5{\%} power against certain nonnormal alternatives. Results of the comparisons indicate that the test based on the plotting position pi =(i — 0.4)/(j + 0.2) is a reasonable choice for a powerful regression test of fit for normality.",
keywords = "Correlation coefficient, Empirical power comparison, Filliben test, Plotting position, Shapiro-Francia test, Shapiro-Wilk test",
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AU - Looney, Stephen W.

AU - Gulledge, Thomas R.

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N2 - Probability plots are commonly used as a technique for testing distributional assumptions. However, any conclusion about the linearity of such a plot is based strictly on the user's judgment. Regression tests of fit are supposed to make this procedure more objective, but these tests typically are not based on probability plots as they are constructed in practice. This is because the developers of these tests defined probability plots in terms of plotting positions which are not used by practitioners. In this paper, a class of goodness-of-fit test statistics which are calculated directly from probability plots as they are constructed in practice is described. Several realistic plotting positions for the normal distribution are chosen and empirical sampling methods are used to derive the null distribution of each of the corresponding test statistics. These tests are then compared on the basis of 5% power against certain nonnormal alternatives. Results of the comparisons indicate that the test based on the plotting position pi =(i — 0.4)/(j + 0.2) is a reasonable choice for a powerful regression test of fit for normality.

AB - Probability plots are commonly used as a technique for testing distributional assumptions. However, any conclusion about the linearity of such a plot is based strictly on the user's judgment. Regression tests of fit are supposed to make this procedure more objective, but these tests typically are not based on probability plots as they are constructed in practice. This is because the developers of these tests defined probability plots in terms of plotting positions which are not used by practitioners. In this paper, a class of goodness-of-fit test statistics which are calculated directly from probability plots as they are constructed in practice is described. Several realistic plotting positions for the normal distribution are chosen and empirical sampling methods are used to derive the null distribution of each of the corresponding test statistics. These tests are then compared on the basis of 5% power against certain nonnormal alternatives. Results of the comparisons indicate that the test based on the plotting position pi =(i — 0.4)/(j + 0.2) is a reasonable choice for a powerful regression test of fit for normality.

KW - Correlation coefficient

KW - Empirical power comparison

KW - Filliben test

KW - Plotting position

KW - Shapiro-Francia test

KW - Shapiro-Wilk test

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