Some important Statistical Tests

All tests for Normality assumption

Why to test the normality of data:

• Various statistical methods used for data analysis make assumptions about normality, including correlation, regression, t-tests, and analysis of variance. Central limit theorem states that when sample size has 100 or more observations, violation of the normality is not a major issue. Although for meaningful conclusions, assumption of the normality should be followed irrespective of the sample size. If a continuous data follow normal distribution, then we present this data in mean value. Further, this mean value is used to compare between/among the groups to calculate the significance level (P value). If our data are not normally distributed, resultant mean is not a representative value of our data. A wrong selection of the representative value of a data set and further calculated significance level using this representative value might give wrong interpretation. That is why, first we test the normality of the data, then we decide whether mean is applicable as representative value of the data or not. If applicable, then means are compared using parametric test otherwise medians are used to compare the groups, using nonparametric methods.

Methods used for test of normality of data

• An assessment of the normality of data is a prerequisite for many statistical tests because normal data is an underlying assumption in parametric testing. There are two main methods of assessing normality: Graphical and numerical (including statistical tests). Statistical tests have the advantage of making an objective judgment of normality but have the disadvantage of sometimes not being sensitive enough at low sample sizes or overly sensitive to large sample sizes. Graphical interpretation has the advantage of allowing good judgment to assess normality in situations when numerical tests might be over or under sensitive. Although normality assessment using graphical methods need a great deal of the experience to avoid the wrong interpretations. If we do not have a good experience, it is the best to rely on the numerical methods. There are various methods available to test the normality of the continuous data, out of them, most popular methods are Shapiro–Wilk test, Kolmogorov–Smirnov test, skewness, kurtosis, histogram, box plot, P–P Plot, Q–Q Plot, and mean with SD. The two well-known tests of normality, namely, the Kolmogorov–Smirnov test and the Shapiro–Wilk test are most widely used methods to test the normality of the data. Normality tests can be conducted in the statistical software “SPSS” (analyze → descriptive statistics → explore → plots → normality plots with tests).

• The Shapiro–Wilk test is more appropriate method for small sample sizes (<50 samples) although it can also be handling on larger sample size while Kolmogorov–Smirnov test is used for n ≥50. For both of the above tests, null hypothesis states that data are taken from normal distributed population. When P > 0.05, null hypothesis accepted and data are called as normally distributed. Skewness is a measure of symmetry, or more precisely, the lack of symmetry of the normal distribution. Kurtosis is a measure of the peakedness of a distribution. The original kurtosis value is sometimes called kurtosis (proper). Most of the statistical packages such as SPSS provide “excess” kurtosis (also called kurtosis [excess]) obtained by subtracting 3 from the kurtosis (proper). A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. If mean, median, and mode of a distribution coincide, then it is called a symmetric distribution, that is, skewness = 0, kurtosis (excess) = 0. A distribution is called approximate normal if skewness or kurtosis (excess) of the data are between − 1 and + 1. Although this is a less reliable method in the small-to-moderate sample size (i.e., n <300) because it can not adjust the standard error (as the sample size increases, the standard error decreases). To overcome this problem, a z-test is applied for normality test using skewness and kurtosis. A Z score could be obtained by dividing the skewness values or excess kurtosis value by their standard errors. For small sample size (n <50), z value ± 1.96 are sufficient to establish normality of the data. However, medium-sized samples (50≤ n <300), at absolute z-value ± 3.29, conclude the distribution of the sample is normal. For sample size >300, normality of the data is depend on the histograms and the absolute values of skewness and kurtosis. Either an absolute skewness value ≤2 or an absolute kurtosis (excess) ≤4 may be used as reference values for determining considerable normality. A histogram is an estimate of the probability distribution of a continuous variable. If the graph is approximately bell-shaped and symmetric about the mean, we can assume normally distributed data. In statistics, a Q–Q plot is a scatterplot created by plotting two sets of quantiles (observed and expected) against one another. For normally distributed data, observed data are approximate to the expected data, that is, they are statistically equal . A P–P plot (probability–probability plot or percent–percent plot) is a graphical technique for assessing how closely two data sets (observed and expected) agree. It forms an approximate straight line when data are normally distributed. Departures from this straight line indicate departures from normality . Box plot is another way to assess the normality of the data. It shows the median as a horizontal line inside the box and the IQR (range between the first and third quartile) 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 IQR from either end of the box (i.e., Q1 − 1.5* IQR and Q3 + 1.5* IQR). Scores >1.5 times and 3 times the IQR are out of the box plot and are considered as outliers and extreme outliers, respectively. A box plot that is symmetric with the median line at approximately the center of the box and with symmetric whiskers indicate that the data may have come from a normal distribution. In case many outliers are present in our data set, either outliers are need to remove or data should treat as nonnormally distributed. Another method of normality of the data is relative value of the SD with respect to mean. If SD is less than half mean (i.e., CV <50%), data are considered normal. This is the quick method to test the normality. However this method should only be used when our sample size is at least 50.

SKEWNESS:

In statistics, skewness is a measure of the asymmetry of the probability distribution of a random variable about its mean. In other words, skewness tells you the amount and direction of skew (departure from horizontal symmetry). The skewness value can be positive or negative, or even undefined. If skewness is 0, the data are perfectly symmetrical, although it is quite unlikely for real-world data. As a general rule of thumb:

• If skewness is less than -1 or greater than 1, the distribution is highly skewed.

• If skewness is between -1 and -0.5 or between 0.5 and 1, the distribution is moderately skewed.

• If skewness is between -0.5 and 0.5, the distribution is approximately symmetric.

Q-Q Plot(How does it work?)

• We plot the theoretical quantiles or basically known as the standard normal variate (a normal distribution with mean=0 and standard deviation=1)on the x-axis and the ordered values for the random variable which we want to find whether it is Gaussian distributed or not, on the y-axis. Which gives a very beautiful and a smooth straight line like structure from each point plotted on the graph.

• Now we have to focus on the ends of the straight line. If the points at the ends of the curve formed from the points are not falling on a straight line but indeed are scattered significantly from the positions then we cannot conclude a relationship between the x and y axes which clearly signifies that our ordered values which we wanted to calculate are not Normally distributed.

Durbin-Watson Test