Matlab Training Session 12: Statistics II

1. Matlab Training Session 12:Statistics II • Course Website: • http://www.queensu.ca/neurosci/Matlab Training Sessions.htm

2. Course Outline Term 1 • Introduction to Matlab and its Interface • Fundamentals (Operators) • Fundamentals (Flow) • Importing Data • Functions and M-Files • Plotting (2D and 3D) • Plotting (2D and 3D) • Statistical Tools in Matlab Term 2 9. Term 1 review 10. Loading Binary Data 11. Nonlinear Curve Fitting 12. Statistical Tools in Matlab II 13. 14.

3. Week 12 Lecture Outline Statistics II • Basic Matlab Statistics Review • Mean, Median, Variance • Statistics Toolbox • Simple Parametric and Non-parametric statistical tests • Simple Statistical Plotting • Histograms • Box Plots D. Anovas • 1 Way Unrelated Design • Post Hoc vs A Priori Comparisons • N-Way Anovas • Related (Repeated Measures) Design • Unrelated (Between Groups) Design

4. Week 12 Lecture Outline Required Toolboxes: Statistics Toolbox

5. Week 12 Lecture Outline Statistics II Part A: Basic Matlab Statistics Review

6. Part A: Basics • The Matlab installation contains basic statistical tools. • Including, mean, median, standard deviation, error variance, and correlations • More advanced statistics are available from the statistics toolbox and include parametric and non-parametric comparisons, analysis of variance and curve fitting tools

7. Mean and Median Mean: Average or mean value of a distribution Median: Middle value of a sorted distribution M = mean(A), M = median(A) M = mean(A,dim), M = median(A,dim) M = mean(A), M = median(A): Returns the mean or median value of vector A. If A is a multidimensional mean/median returns an array of mean values. Example: A = [ 0 2 5 7 20] B = [1 2 3 3 3 6 4 6 8 4 7 7]; mean(A) = 6.8 mean(B) = 3.0000 4.5000 6.0000 (column-wise mean) mean(B,2) = 2.0000 4.0000 6.0000 6.0000 (row-wise mean)

8. Mean and Median Examples: A = [ 0 2 5 7 20] B = [1 2 3 3 3 6 4 6 8 4 7 7]; Mean: mean(A) = 6.8 mean(B) = 3.0 4.5 6.0 (column-wise mean) mean(B,2) = 2.0 4.0 6.0 6.0 (row-wise mean) Median: median(A) = 5 median(B) = 3.5 4.5 6.5 (column-wise median) median(B,2) = 2.0 3.0 6.0 7.0 (row-wise median)

9. Standard Deviation and Variance • Standard deviation is calculated using the std() function • std(X) : Calcuate the standard deviation of vector x • If x is a matrix, std() will return the standard deviation of each column • Variance (defined as the square of the standard deviation) is calculated using the var() function • var(X) : Calcuate the variance of vector x • If x is a matrix, var() will return the standard deviation of each column

10. Standard Error of the Mean • Often the most appropriate measure of error/variance is the standard error of the mean • Matlab does not contain a standard error function so it is useful to create your own. • The standard error of the mean is defined as the standard deviation divided by the square root of the number of samples

11. Week 12 Lecture Outline Statistics II Part B: Parametric and Non-parametric statistical tests

12. Comparison of Means • A wide variety of mathametical methods exist for determining whether the means of different groups are statistically different • Methods for comparing means can be either parametric (assumes data is normally distributed) or non-parametric (does not assume normal distribution)

13. Parametric Tests - TTEST [H,P] = ttest2(X,Y) Determines whether the means from matrices X and Y are statistically different. H return a 0 or 1 indicating accept or reject nul hypothesis (that the means are the same) P will return the significance level

14. Parametric Tests - TTEST [H,P] = ttest2(X,Y) Determines whether the means from matrices X and Y are statistically different. H return a 0 or 1 indicating accept or reject nul hypothesis (that the means are the same) P will return the significance level

15. Parametric Tests - TTEST Example: For the data from Week 8 exercise 3 [H,P] = ttest2(var1,var2) >> [H,P] = ttest2(var1,var2) H =1 P = 0.00000000000014877 Variable 1 Variable 2

16. Non-Parametric Tests Ranksum • The wilcoxin ranksum test assesses whether the means of two groups are statistically different from each other. • This test is non-parametric and should be used when data is not normally distributed • Matlab implements the wilcoxin ranksum test using the ranksum() function ranksum(X,Y) statistically compares the means of two data distributions X and Y

17. Non-Parametric Tests - RankSum Example: For the data from week 8 exercise 3 [P,H] = ranksum(var1,var2) P = 1.1431e-014 H = 1 Variable 1 Variable 2

18. Week 12 Lecture Outline Statistics II Part C: Simple Statistical Plotting • Histograms • Box Plots

19. Histograms • Histograms are useful for showing the pattern of the whole data set • Allows the shape of the distribution to be easily visualized

20. Histograms • Matlab hist(y,m) command will generate a frequency histogram of vector y distributed among m bins • Also can use hist(y,x) where x is a vector defining the bin centers Example: • >>b=sin(2*pi*t) • >>hist(b,10); >>hist(b,[-1 -0.75 0 0.25 0.5 0.75 1]);

21. Histograms • The histc function is a bit more powerful and allows bin edges to be defined [n, bin] = histc(x, binrange) x = statistical distribution binrange = the range of bins to plot eg: [1:1:10] n = the number of elements in each bin from vector x bin = the bin number each element of x belongs • Use the bar function to plot the histogram

22. Histograms • The histc function is a bit more powerful and allows bin edges to be defined Example: >> test = round(rand(100,1)*10) >> histc(test,[1:1:10]) >> Bar(test)

23. Box Plots • Box plots are useful to graphically display the mean and variance of distributions, as well as the interquartile range and outliers

24. Box Plots • Matlab function boxplot(x) will generate a boxplot of the distribution defined by x Example: % add outlier to test distribution >>test(101) = 16 >>boxplot(test)

25. Box Plots • The box has lines at the lower quartile, median, and upper quartile values. • The whiskers are lines extending from each end of the box to show the extent of the rest of the data. • Outliers are data with values beyond the ends of the whiskers. • If there is no data outside the whisker, a dot is placed at the bottom whisker. +

26. Box Plots • boxplot(X,notch) with notch = 1 produces a notched-box plot. • Notches graph a robust estimate of the uncertainty about the means for box-to-box comparison. The default, notch = 0, produces a rectangular box plot. Example: >>test2 = test * (rand*10) >>boxplot([test test2],1)

27. Week 12 Lecture Outline Statistics II D. Anovas • 1 Way Unrelated Design • Post Hoc vs A Priori Comparisons • N-Way Anovas • Unrelated (Between Groups) Design • Related (Repeated Measures) Design

28. Anovas • ANOVA’s are tests used to make direct comparisons between the amount by which sample means vary and the amount that values in each sample vary around the group means

29. Anovas • ANOVA’s are tests used to make direct comparisons between the amount by which sample means vary and the amount that values in each sample vary around the group means

30. Anovas Terminology Null Hypothesis = Both Means are the same Type I error: Reject Null Hypothesis when it is true. Eg Means are not actually significantly when p < 0.05 Type II error: Accept Null Hypothesis when it is false. Eg means are actually significantly different when p > 0.05

31. Anovas Beta Probability of making type II Error Alpha Probability of making type I Error P < 0.05

32. Anovas Terminology Family Wise Error: The probability of making at least 1 family wise error while making multiple ANOVA comparisons

33. 1 way Anovas The matlab function anova1 calculates a 1 way anova p = anova1(X) performs a balanced 1-way ANOVA comparing the means of the columns of data in the matrix X ** each column must represent an independent sample containing m mutually independent observations. The function returns the p-value for the null hypothesis p = anova1(X,group) group = Each row of group contains the data label for the corresponding column of X

34. 1 way Anovas Assumptions All sample populations are normally distributed All sample populations have equal variance All observations are mutually independent The ANOVA test is known to be robust to modest violations of the first two assumptions.

35. 1 way Anovas • The standard ANOVA table divides the variability of the data in X into two parts: • Variability due to the differences among the column means (variability between groups) • Variability due to the differences between the data in each column and the column mean (variability within groups)

36. 1 way Anovas The ANOVA table has six columns: • Source of the variability • The Sum of Squares (SS) due to each source. • The degrees of freedom (df) associated with each source. • Mean Squares (MS) for each source, which is the ratio SS/df. • F statistic, which is the ratio of the MS's. • The p-value, which is derived from the cdf of F. As F increases, the p-value decreases.

37. 1 way Anovas Example 1 The following example comes from a study of the material strength of structural beams in Hogg (1987). The vector strength measures the deflection of a beam in thousandths of an inch under 3,000 pounds of force. Stronger beams deflect less. The civil engineer performing the study wanted to determine whether the strength of steel beams was equal to the strength of two more expensive alloys.

38. 1 way Anovas Example 1 Steel is coded 'st' in the vector alloy. The other materials are coded 'al1' and 'al2'. S strength = [82 86 79 83 84 85 86 87 74 82 78 75 76 77 79 ... 79 77 78 82 79]; alloy = {'st','st','st','st','st','st','st','st',... 'al1','al1','al1','al1','al1','al1',... 'al2','al2','al2','al2','al2','al2'}; Though alloy is sorted in this example, you do not need to sort the grouping variable.

39. 1 way Anovas Solution: p = anova1(strength,alloy) p = 1.5264e-004 The p-value indicates that the three alloys are significantly different. The box plot confirms this graphically and shows that the steel beams deflect more than the more expensive alloys.

40. 1 way Anovas

41. Post Hoc and A Priori Comparisons If a 1 way anova test indicates a significant difference between at least on mean: Post Hoc Comparisons: The decision to compare means after a significant 1 way anova is caluculated. When all possible comparisons are made after the fact the changes of type 1 error become high. A Priori Comparisons: Comparisons decided upon before the 1 way anova is performed based on the general theory of the study. This minimizes possible type I error.

42. N-way Anovas Unrelated (Between Groups) Design p = anovan(X,group) performs a balanced or unbalanced mult way ANOVA for comparing the means of the observations in vector X with respect to N different factors. • The factors and factor levels of the observations in X are assigned by the cell array group. • Each of the N cells in group contains a list of factor levels identifying the observations in X with respect to one of the N factors. • The list within each cell can be a vector, character array, or cell array of strings, and must have the same number of elements as X.

43. N-way Anovas Related (Repeated Measures) Design NOT IMPLEMENTED IN THE STATISTICS TOOLBOX!!

44. Exercise • Load testdata2.txt from week 8 • Assume the data columns represent independent normally distributed variables • Perform a 1 way ANOVA on the data and interpret the results

45. Getting Help • Help and Documentation • Digital • Accessible Help from the Matlab Start Menu • Updated online help from the Matlab Mathworks website: • http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html • Matlab command prompt function lookup • Built in Demo’s • Websites • Hard Copy • Books, Guides, Reference • The Student Edition of Matlab pub. Mathworks Inc.