Statistical analysis

A potential source of confusion in working out what statistics to use in analysing data is whether your data allows for parametric or non–parametric statistics. Non– parametric statistical procedures are less powerful because they use less information in their calculation. For example, a parametric correlation uses information about the mean and deviation from the mean while a non–parametric correlation will use only the ordinal position of pairs of scores (Altman and Bland, 2009).

The basic distinctions for parametric versus non–parametric are:

x If the measurement scale is nominal or ordinal then non–parametric statistics

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x If the measurement represents interval or ratio scales (the case of this study)

parametric statistics must be applied (Altman and Bland, 2009).

In statistical analysis, all parametric tests assume certain characteristics about the data. Violation of these assumptions can change the conclusion of the research and interpretation of the results. For example, the assumption that interval–scale variables

are approximately normally distributed33 and have equal variances (Levene´s test34)

are required in order to use one–way analysis of variance (one–way ANOVA35) for the

identification of significant differences between means.

The one–way ANOVA test must meet three main assumptions: (1) The dependent variable is normally distributed in each group that is being compared in the one–way ANOVA; (2) The population variances in each group are equal; (3) Independence of observations (survey design).

There are two methods for assessing normality, visual inspection of frequency distributions (using one or more of the following: histogram, box plot, Q–Q plot, stem–

and–leaf plot, P–P plot) and normality tests (Skewness36, Kurtosis37, Shapiro–Wilk’s 'W'

test, Kolmogorov–Smirnov38 K–S test).

A histogram, the frequency distribution that plots the observed values against their frequency, provides both a visual judgment about whether the distribution is bell

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The distribution of interval-scale data is bell-shaped, symmetrical about the mean (McCrum-Gardner E., 2008). 34

The Levene test (Levene 1960) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variance. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Levene test can be used to verify that assumption. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm). Accessed 18/06/2016.

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ANOVA is a statistical method used to test differences between two or more means (McCrum-Gardner E., 2008). 36

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it

looks the same to the left and right of the center point. Available at

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm). Accessed 18/06/2016. 37

Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. Available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm. Accessed 18/06/2016.

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The Kolmogorov–Smirnov test (KS–test) tries to determine if two datasets differ significantly. The KS–test has the advantage of making no assumption about the distribution of data. It will enable you to view the data graphically which can help you understand how the data is distributed. Available at http://www.physics.csbsju.edu/stats/KS–test.html). Accessed 18/06/2016.

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shaped, symmetrical about the mean and insights about gaps in the data and outliers outlying values (Ghasemi and Zahediasl, 2012).

The Q–Q plot, or quantile–quantile plot, is a graphical tool to help us assess if a set of data plausibly came from some theoretical distribution such as a Normal or Exponential distribution. The definition of the Q–Q plot may be extended to any continuous density. The Q–Q plot will be close to a straight line if the assumed density is correct, then the data is normally distributed. Moreover, the Q–Q plots are easier to interpret in case of large sample sizes (Ghasemi and Zahediasl, 2012).

The box plot shows the median as a horizontal line inside the box and the interquartile range (range between the 25th to 75th percentiles) as the length of the box. The whiskers (line extending from the top and bottom of the box) represent the minimum and maximum values when they are within 1.5 times the interquartile range from either end of the box (Barton and Peat, 2014). Scores greater than 1.5 times the interquartile range are out of the box plot and are considered as outliers, and those greater than 3 times the interquartile range are extreme outliers. A box plot that is symmetric when the median line is at approximately the Center of the box, and when the symmetric whiskers are slightly longer than the subsections of the Center box, suggests that the data may have come from a normal distribution (Ghasemi and Zahediasl, 2012).

For small sample sizes, normality tests have little power to reject the null hypothesis and therefore small samples most often pass normality tests (Oztuna et al., 2006). For large sample sizes, significant results would be derived even in the case of a small deviation from normality (Oztuna et al., 2006), although this small deviation will not affect the results of a parametric test (Ghasemi and Zahediasl, 2012). Because of sample size differences (Brazil, n = 2,775; New Zealand, n = 658), the Brazil data may be more sensitive to significance tests than New Zealand, but this does not compromise the results. Graphical methods are typically not very useful when the sample size is small. However, in studies with relatively large sample sizes graphical

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methods may be more appropriate for checking normality when tests are violated (Ghasemi and Zahediasl, 2012).

Lack of symmetry (Skewness) and pointiness (Kurtosis) are two main ways in which a distribution can deviate from normal. The values for these parameters should be zero in a fully normal distribution. However, it is more difficult to determine how extreme either the Skewness or the Kurtosis values must be before they indicate a problem with the assumption of normality (Ghasemi and Zahediasl, 2012).

The Kolmogorov–Smirnov test (K–S test) is an empirical distribution function (EDF39

) in which the theoretical cumulative distribution function of the test distribution is contrasted with the EDF of the data (Oztuna et al., 2006). A limitation of the K–S test is its high sensitivity to extreme values; the Lilliefors correction renders this test less conservative (Barton and Peat, 2014). It has been reported that the K–S test has low power and it should not be seriously considered for testing normality (Thode H.J., 2002). In this study, when the K–S test was violated, a histogram was used to verify the normality of data distribution.

The Shapiro–Wilk test (W test) is based on the correlation between the data and the corresponding normal scores (Barton and Peat, 2014) and provides better power than the K–S test even after the Lilliefors correction (Steinskog, 2007). Power is the most frequent measure of the value of a test for normality – the ability to detect whether a sample comes from a non–normal distribution (Thode, 2002).

The one–way ANOVA is considered a robust test against the normality assumption. This means that it tolerates violations to its normality assumption rather well. The one–way ANOVA can tolerate data that is non–normal (skewed or kurtotic distributions) with only a small effect on the Type I error rate (Garson, 2012).

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An EDF plot is a graph that you can use to evaluate the fit of a distribution to your data, estimate percentiles, and compare different sample distributions. Available at http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and- graphs/graphs/graphs-of-distributions/empirical-cdf-plots/empirical-cdf-plot/. Accessed 18/06/2016.

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