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The Power to See:

A New Graphical Test of Normality

Sivan Aldor-Noiman

Lawrence D. Brown

Robert A. Stine

Department of Statistics

The Wharton School

University of Pennsylvania

PA, USA

Abstract

Many statistical procedures assume the underlying data generating process involves Gaussian errors. Among the well-known procedures are ANOVA, multiple regression, linear discriminant analysis and many more. There are a few popular procedures that are commonly used to test for normality such as the Kolmogorov-Smirnov test and the Shapiro- Wilk test. Excluding the Kolmogorov-Smirnov testing procedure, these methods do not have a graphical representation. As such these testing methods oer very little insight as to how the observed process devi- ates from the normality assumption. In this paper we discuss a simple new graphical procedure which provides condence bands for a nor- mal quantile-quantile plot. These bands dene a test of normality and are much narrower in the tails than those related to the Kolmogorov- Smirnov test. Correspondingly the new procedure has much greater power to detect deviations from normality in the tails. Key words: normality test, condence bands, graphical presentation, power analysis, quantile-quantile plot Research supported in part by NSF Grant DMS-1007657 1

1 Introduction

To motivate our procedure we rst look at two dierent data sets that were pre- viously explored. In both examples the researchers were interested in testing the normality assumption. The rst data series contains monthly log returns of IBM stock from March

1967 to December 2008. This is part of a data set that was examined in

Tsay (2010) and we obtained it from its corresponding website. In Figure 1 we show the time plot for the data and the quantile-quantile plot of the data with both our proposed Tail-Sensitive (TS) 95% condence bands and the Kolmogorov-Smirnov (KS) condence bands. It is apparent that some of the points fall outside the TS condence bands but inside the KS condence bands. It is apparent that ve points in the left tail fall outside the TS condence bands but are well inside the KS condence bands. Therefore, according to the TS condence bands the data do not follow the normal distribution and the log returns for this stock have a heavier left-tail compared to the normal distribution. The KS condence bands do not detect this deviation and as such the researcher may wrongly conclude that the data are normally distributed. The second data series contains measurements from experiments that test the eectiveness of body armors. The data was collected as part of a National Academies report requested by the US Army Unknown (2012). The Army wanted to investigate the dierence between two methods of assessing how deep a bullet penetrates ceramic body armor begin tested for approval for use. In the standard test a cylindrical clay model is layered under the armor vest. A projectile is then red against the vest, causing an indentation in the clay. The 2 -0.3 -0.1 0.0 0.1 0.2 0.3

Log return

Year

19671977198719972007

-3-2-10123 -3 -2 -1 0 1 2 3

KS test

Theoretical Quantiles

Normalized Sample Quantiles

l l l l ll ll l ll l lll ll llllll lll ll lllllllllllllllllllll llllllllllllllllllllllllll lllllllllllllllll llllllll l lllllll ll llll ll l l l l -3-2-10123 -3 -2 -1 0 1 2 3

TS test

Theoretical Quantiles

Normalized Sample Quantiles

l l l l ll ll l ll l l l l ll llllll lll ll lllllllllllllllllllll ll llllllllllllllllllllllll llll lllllllllllllllll llllllll l lllllll ll llll ll l l l lFigure 1: The monthly log returns for IBM stock from March 1967 to December

2008. The upper plot shows the time plot of the data against the time index.

The lower plots shows the 95% Kolmogorov-Smirnov condence bands and the corresponding TS condence bands, respectively. 3 deepest impression in the clay is measured as an indication of the survivability of the soldier using this armor. The traditional method of measuring the depth of this impression involves using a manually controlled digital caliper. A more recently-adopted method is to measure the impression using a computer- controlled laser. The two methods were compared in a calibration experiment involving a series of test rings measured by each method. Figure 2 shows the quantile-quantile plot of measurements from the experiments: the upper plots show the measurements using the digital caliper and the lower plots show the results using the laser based approach. These plots also present the proposed Tail-Sensitive (TS) 95% condence bands and the Kolmogorov-Smirnov (KS) condence bands. Based on the KS bands, observations from both methods are consistent with the normality assumption. However, based on the TS condence bands there is a suspicious outlier on the right tail of the caliper- based measurements. On the laser-based measurements we see two points on the right tail that fall outside the bands and several suspicious data points on the left tail that lie on the boarder of the bands. These points indicate that the data deviate from the normality assumption. Our condence bands procedure indicates that if the Army adopts the laser based method it should not rely on the normality assumption to establish its safety standards. In the next section we list a few common procedures that are used to test the normality assumption. We later compare these testing procedures perfor- mances with our TS procedure. 4 -2-1012 -4 -2 0 2 4

Caliper Measurements with KS test

Theoretical Quantiles

Normalized Sample Quantiles

l l l l lll l l l ll l ll lllll llll ll lllllll llll lllllllllllll ll lllllllllllllllll l lll l ll l l ll l l l -2-1012 -4 -2 0 2 4

Caliper Measurements with TS test

Theoretical Quantiles

Normalized Sample Quantiles

l l l l lll l l l ll l ll lllll llll ll lllllll llll lllllllllllll ll lllllllllllllllll l lll l ll l l ll l l l -2-1012 -4 -2 0 2 4

Laser Measurements with KS test

Theoretical Quantiles

Normalized Sample Quantiles

l l l lll lllll l lll l ll lll l l ll llllll lllll lll llllllllllllll llllllllllllllllllll l lll l l l l ll -2-1012 -4 -2 0 2 4

Laser Measurements with TS test

Theoretical Quantiles

Normalized Sample Quantiles

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