Assessing normality Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this section some statistical tools will be presented to check whether a given set of data is normally distributed. 1. Previous knowledge of the nature of the distribution Problem: A researcher working with sea stars needs to know if sea star size (length of radii) is normally distributed. What do we know about the size distributions of sea star populations? 1. Has previous work with this species of sea star shown them to be normally distributed? 2. Has previous work with a closely related species of seas star shown them to be normally distributed? 3. Has previous work with seas stars in general shown them to be normally distributed? If you can answer yes to any of the above questions and you do not have a reason to think your population should be different, you could reasonably assume that your population is also normally distributed and stop here. However, if any previous work has shown non-normal distribution of sea stars you had probably better use other techniques. 2. Construct charts For small- or moderate-sized data sets, the stem-and-leaf display and box-andwhisker plot will look symmetric. For large data sets, construct a histogram or polygon and see if the distribution bell-shaped or deviates grossly from a bell-shaped normal distribution. Look for skewness and asymmetry. Look for gaps in the distribution intervals with no observations. However, remember that normality requires more than just symmetry; the fact that the histogram is symmetric does not mean that the data come from a normal distribution. Also, data sampled from normal distribution will sometimes look distinctly different from the parent distribution. So, we need to develop some techniques that allow us to determine if data are significantly different from a normal distribution. 3. Normal Counts method Count the number of observations within 1, 2, and 3 standard deviations of the mean and compare the results with what is expected for a normal distribution in the 68-95- 99.7 rule. According to the rule, 68% of the observations lie within one standard deviation of the mean. 95% of observations within two standard deviations of the mean. 99.7% of observations within three standard deviations of the mean. Example: As part of a demonstration one semester, I collected data on the heights of sample of 25 biostatistics students. These data are presented in the table below. Does the sample shown below have been drawn from normally distributed populations? 59
Table. Heights, in inches, of 25 biostatistics students. 71.0 69.0 70.0 72.5 73.0 70.0 71.5 70.5 72.0 71.0 68.5 69.0 69.0 68.5 74.0 67.0 69.0 71.5 66.0 70.0 68.5 74.0 74.5 74.0 Solution: For normal Counts method, determine the following Heights, in Frequency inches 66 1 67 1 68.5 3 69 4 70 3 70.5 1 71 2 Total = 17 71.5 2 72 1 72.5 1 73 1 74 3 74.5 1 Total 24 x = 70.6; s = 2.3 x ± s is 72.9 to 68.3. 17 out of the 24 observations i.e. 17/24 = 0.70 = 70% fall within x ± s, i.e. between 72.9 and 68.3, which is approximately equal to 68%.There is no reason to doubt that the sample is drawn from a normal population. 4. Compute descriptive summary measures a. The mean, median and mode will have similar values. b. The interquartile range approximately equal to 1.33 s. c. The range approximately equal 6 s. 5. Evaluate normal probability plot If the data come from a normal or approximately normal distribution, the plotted points will fall approximately along a straight line (a 45 degree line). However, if your sample departs from normality, the points on the graph will deviate from that line. If they trail off from a straight-line pack in a curve at the top end, observed values bigger than expected, that s right skewed (see below).if the observed values trail off at the bottom end, that s left skewed. Realize that it is important note that any worthwhile computer statistics package will construct these graphs for you (see below). 60
6. Measure of Skewness and Kurtosis Skewness: The normal distribution is symmetrical. Asymmetrical distributions are sometimes called skewed. Skewness is calculated as follows: 3 n ( xi - x) i= 1 skewness = 3 s ( n -1)( n -2) where x is the mean, s is the standard deviation, and n is the number of data points A perfectly normal distribution will have a skewness statistic of zero. If this statistic departs significantly from 0, then we lose confidence that our sample comes from a normally distributed population. If it is negative, then the distribution is skewed to the left or negatively skewed distribution. If it is positive, then the distribution is skewed right or positively skewed distribution. n Negatively skewed distribution or Skewed to the left Skewness <0 Normal distribution Symmetrical Skewness = 0 Positively skewed distribution or Skewed to the right Skewness > 0 Kurtosis: A bell curve will also depart from normality if the tails fail to fall off at the proper rate. If they decrease too fast, the distribution ends up too peaked. If they don t decrease fast enough, the distribution is too flat in the middle and too fat in the tails. One statistic commonly used to measure kurtosis is typically calculated using the formula, kurtosis 4 2 nn ( + 1) x i x 3( n 1) = ( n 1)( n 2)( n 3) s ( n 2)( n 3) where x is the mean, s is the standard deviation, and n is the number of data points A perfectly normal distribution will also have a kurtosis statistic of zero. If kurtosis is significantly less than zero, then our distribution is flat, it is said to be platykurtic. If kurtosis is significantly greater than 0, the distribution is pointed or peaked, it is called leptokurtic. Platykurtic distribution Low degree of peakedness Kurtosis <0 Normal distribution Mesokurtic distribution Kurtosis = 0 Leptokurtic distribution High degree of peakedness Kurtosis > 0 61
You won t have you calculate it by hand. The calculation itself is sensitive to rounding errors because they are raised to the third and fourth powers. Using SPSS to Evaluate Data for Normality Before the advent of good computers and statistical programs, users could be forgiven for trying to avoid any surplus calculations. Now that both are available and much easier to use, tests for normality should always be carried out as a best practice in statistics. SPSS offers a variety of methods for evaluating normality. Normal probability plot (P-P plot) The P-P plot graphs the expected cumulative probability against the observed cumulative probability. 1. Open the SPSS file containing your data. 2. From the main menu, select Graph and then P-P From the list of available variables, move the variables you wish to analyze to the variable window. If you select multiple variables then SPSS will create separate plots for each. 3. In the box for Test Distribution be sure that the pop-up menu is set for a Normal distribution. In addition, be sure that the Estimate from data box is checked. 4. In the box for Proportion Estimation Formula, select the radio button for the Rankit method. 5. Finally, in the Ranks Assigned to Ties box, select the radio button for High. 6. Click on OK to obtain the plot and complete your analysis 1.00 Normal P-P Plot of Student's heights (inches).75.50 Expected Cum Prob.25 0.00 0.00.25.50.75 1.00 Observed Cum Prob Q-Q plot Repeat the above steps, but in this time select Q-Q (main menu, Graph and then Q-Q) 62
Normal Q-Q Plot of Student's heights (inches) 2.0 1.5 1.0.5 0.0 Expected Normal -.5-1.0-1.5-2.0 64 66 68 70 72 74 76 Observed Value Limitation of visual method One limitation to any visual approach for evaluating normality is that your conclusion is open to some uncertainty. How, for example, can you put a quantitative statement on the confidence of your conclusion? How linear is linear and how much deviation from linearity is acceptable? One approach to obtaining a more quantitative determination of whether a data set is normally distributed is the Kolmogorov-Smirnov test or Shapiro- Wilk s tests. Use data set to tests of normality 1. Open the SPSS file containing your data and from the main menu select Analyze 2. Descriptive Statistics 3. Explore 4. Move your variable from the Variable List window to the Dependent List window. 5. Under Display click Both 6. Click Plots 7. Under Boxplots check Factor levels together 8. Under Descriptive check " Histogram" and stem-and-leaf 9. Check Normality plots with tests 10. Click continue 11. Click OK 12. Evaluate the plot for evidence of normality. 63
Descriptives Student's heights (inches) Mean 95% Confidence Interval for Mean Lower Bound Upper Bound Statistic Std. Error 70.583.4698 69.611 71.555 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis 70.616 70.250 5.297 2.3015 66.0 74.5 8.5 3.375.076.472 -.647.918 Assessment of skewness and kurtosis In fact (like all estimates), we are unlikely to ever see the values of zero in either skewness or kurtosis statistics. The real question is whether the given estimates vary significantly from zero. For this question we need the standard error of skewness, and similarly the standard error of kurtosis. In SPSS, the Explore command provides skewness and kurtosis statistics at once in addition to the standard errors of skewness and kurtosis. The key value we are looking for is whether the value of zero is within the 95% confidence interval. For assessing skewness: 0.076 + 0.472 = 0.548 0.076-0.472 = -0.396 For assessing kurtosis: -0.647 + 0.918 = 0.271-0.647-0.918 = -1.565 Thus the 95% confidence interval for the skewness score ranges from 0.548 to -0.396, and the 95% confidence interval for the kurtosis score ranges from 0.271 to -1.565. In both cases, zero is within our bounds thus we can accept that our statistic is not significantly different from a distribution of zero. Therefore this is normal distribution. Kolmogorov-Smirnov / Shapiro-Wilk s tests for normality The first is the Kolmogorov-Smirnov test for normality, sometimes termed the KS Lilliefors test. The second is the Shapiro-Wilk s test. The advice from SPSS is to use the latter test (Shapiro-Wilk s test) when sample sizes are small (n < 50). The output, using the data from above is presented below. 64
Tests of Normality Student's heights (inches) Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig..129 24.200*.961 24.469 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction The value listed as Asymp. Sig. is probability lies between 0 and 1. In general, a sig. value 0.05 is considered good evidence that the data set is not normally distributed. A value greater than 0.05 implies that there is insufficient evidence to suggest that the data set is not normally distributed. In our example, the significance of 0.469 accordingly, means that our distribution is not significantly different from a normal distribution. Boxplots It is very hard to detect normality using a box plot. But at the very least, look for symmetry and the presence of outliers. Severe skewness and/or outliers are indication of non-normality. Normal distribution: If there are only a few outliers, and the median line evenly divides the box, then data values in a sample that otherwise comes from a normal or near-normal distribution. Skewness: If there are numerous outliers to one side or the other of the box, or the median line does not evenly divide the box, then the population distribution from which the data were sampled may be skewed. Skewness to the right: If the boxplot shows outliers at the upper range of the data (above the box), the median line does not evenly divide the box, and the upper tail of the boxplot is longer than the lower tail, then the population distribution from which the data were sampled may be skewed to the right. Here is a hypothetical example of a boxplot for data sampled from a distribution that is skewed to the right: 65
Skewness to the left: If the boxplot shows outliers at the lower range of the data (below the box), the median line does not evenly divide the box, and the lower tail of the boxplot is longer than the upper tail, then the population distribution from which the data were sampled may be skewed to the left. Here is a hypothetical example of a boxplot for data sampled from a distribution that is skewed to the left. Negatively Skewed Symmetric (Not Skewed) Positively Skewed From the boxplots of student heights in our example, we see that the distribution appear to be reasonably symmetric and approximately normal. 76 74 72 70 68 66 64 N = 24 Student's heights (i 66
Normalizing Transformations Many of the statistical tests (parametric tests) are based on the assumption that the data are normally distributed. However, if we actually plot the data from a study, we rarely see perfectly normal distributions. Most often, the data will be skewed to some degree or show some deviation from mesokurtosis. Two questions immediately arise: A) Can we analyze these data with parametric tests and. if not, B) Is there something we can do to the data to make them more normal? What to do if Not Normal? According to some researchers, sometimes violations of normality are not problematic for running parametric- tests. When a variable is not normally distributed (a distributional requirement for many different analyses), we can create a transformed variable and test it for normality. If the transformed variable is normally distributed, we can substitute it in our analysis. Data transformation Data transformation involves performing a mathematical operation on each of the scores in a set of data, and thereby converting the data into a new set of scores which are then employed to analyze the results of an experiment. To solve for Positive Skew Square roots, logarithmic, and inverse (1/X) transforms "pull in" the right side of the distribution in toward the middle and normalize right (positive) skew. Inverse transforms are stronger than logarithmic, which are stronger than roots. Square root transformation The square-root transformation can be effective in normalizing distributions that have a slightly to moderate positive skew. Data taken from a Poisson distribution are sometimes effectively normalized with a square-root transformation. The square-root transformation is obtained through use of the equation Y = X, where X is the original score (observation) and Y represents the transformed score. Cube roots, fourth roots, etc., will be increasingly effective for data that are increasingly skewed. When you use the square root transformation, be careful; don't have any zeros or negative numbers among your raw data If for example, there are any zero values, add a constant C, where C is some small positive value such as 0.5, and replace each observation by X + 0.5. If there are negative numbers with positive numbers, add a constant to each number to make all values greater than 0. Although this transformation is not used as frequently in medicine as the log transformation, it can be very useful when a log transformation overcorrects. Logarithmic transformation A logarithmic transformation may be useful in normalizing distributions that have more severe positive skew than a square-root transformation. Such distribution is termed lognormal because it can be made normal by log transforming the values. 67
When log transforming data, we can choose to take logs either to base 10 (the 'common' log) or to base e (the 'natural' log, abbreviated ln). The log transformation is similar to the square root transformation in that zeros and negative numbers are taboo. Use the same technique to eliminate them. Some people use the smallest possible value for their variable as a constant, others use an arbitrarily small number, such as 0.001 or, most commonly, 1. The back-transformation of a log is called the antilog; the antilog of the natural log is the exponential, e (e = 2.71828). In medicine, the log transformation is frequently used because many variables have right-skewed distributions. Inverse or reciprocal transformation A reciprocal transformation exerts the most extreme adjustment with regard to normality. It is used to normalize very or absolutely skewed data. Accordingly, the reciprocal transformation is often able to normalize data that the square-root and logarithmic transformations are unable to normalize. The reciprocal transformation is obtained through use of the equation Y = 1/x. If any of the scores are equal to zero, the equation y= l/(x+ 1) should be employed. When inversed, large numbers become small, and small numbers become large. It's possible that you chose a transformation that overcorrected and turned a moderate left skew into a moderate right one. This gains you nothing except heartache. So, if this has happened, go back and try a less "powerful" transformation; perhaps square root rather than log, or log rather than reciprocal. To solve for Negative Skew If skewness is actually negative, "flip" the curve over, so the skew left curves become skewed right, allowing us to use the transformation procedures of positively skewed distributions. Flipping the curve over require reflection of the variable before transforming. Reflection simply involves the following: Before the data are transformed, we can find the maximum value (9), add 1 to it (to avoid too many zeros when we're done), and subtract each raw value from this number. For example, if we started out with the numbers : 1 1 2 4 8 9 then we would subtract each number from 10 (the maximum, 9, plus 1), yielding : 9 9 8 6 2 1 We would then use the transformations for right -skewed data, rather than left-skewed. More transformations Power transformation Y = (X) p : The most common type of transformation useful for biological data is the power transformation, which transforms X to (X) p, where p is power greater than zero. Values of p less than one correct right skew, which is the common situation (using a power of 2/3 is common when attempting to normalize). Values of p greater than 1 correct left skew. The square transformation i.e. p = 2 for example, achieves the reverse of the square-root transformation. If X is skewed to the left (negatively skewed), the distribution of Y = (X) p is often approximately Normal. For right skew, decreasing p decreases right skew. Too great reduction of p will overcorrect and cause left skew. 68
The arcsine (arcsin) transformation: The arcsine of a number is the angle whose sine is that number. The arcsine transformation (also referred to as an angular or inverse sine transformation) is used to normalize data when data are Proportions between 0 and 1 or percentages between 0% and 100%. The arcsine transformation which expresses the value of Y in degrees is obtained through use of the equation Y = arcsin X, where X will be a proportion between 0 and 1. This means that, the arcsine transformation requires two steps. First, obtain the square root of x. Second, by using calculator, find the angle whose sine equal this value. To get the arcsine value for a percentage (e.g. 50%), divide this by 100 ( = 0.5), take the square root (= 0.7071), then press "sin-1" key in your calculator to get the arcsine value (= 45). To get the arcsine value for a proportion (e.g. 0.4), take the square root (= 0.6325), then press "sin-1" to get the arcsine value (= 39.23). In Excel after obtaining the square root X (Fig. 1), find the ASIN function (Fig.2) and then multiply by 57.3 to find the arcsine in degrees. Through use of the equations that compute an arcsine in degrees, is 0 (for a proportion of zero) to 90 (for a proportion of 1). Fig. 1 Fig. 2 After any transformation, you must re-check your data to ensure the transformation improved the distribution of the data (or at least didn t make it any worse!). Sometimes, log or square root transformations can skew data just as severely in the opposite direction. If transformation does not bring data to a normal distribution, the investigators might well choose a nonparametric procedure that does not make any assumptions about the shape of the distribution. SPSS/PC You can use the COMPUTE command to transform the data. Select Transform - Compute - Target Variable (input a new variable name) - Numeric Expression (input transform formula). It's usually a good idea to make up a new variable to hold the transformed data, rather than over-writing the original values; in this way, you can easily undo your mistakes. 69