Matlab Training Session 12: Statistics II

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

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• 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

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• Week 12 Lecture Outline
• Required Toolboxes
• Statistics Toolbox

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• Week 12 Lecture Outline
• Statistics II
• Part A Basic Matlab Statistics Review

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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

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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)
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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)
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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

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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

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• Week 12 Lecture Outline
• Statistics II
• Part B Parametric and Non-parametric
  statistical tests

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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)

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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

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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

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Parametric Tests - TTEST

• Example
• For the data from Week 8
• exercise 3
• H,P ttest2(var1,var2)
• gtgt H,P ttest2(var1,var2)
• H 1
• P 0.00000000000014877

Variable 1
Variable 2

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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

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

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Histograms

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

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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

gtgtbsin(2pit) gtgthist(b,10) gtgthist(b,-1
-0.75 0 0.25 0.5 0.75 1)
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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 1110
• 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

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Histograms

• The histc function is a bit more powerful and
  allows bin edges to be defined
• Example
• gtgt test round(rand(100,1)10)
• gtgt histc(test,1110)
• gtgt Bar(test)

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Box Plots

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

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Box Plots

• Matlab function boxplot(x) will generate a
  boxplot of the distribution defined by x
• Example
• add outlier to test distribution
• gtgttest(101) 16
• gtgtboxplot(test)

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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.

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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
• gtgttest2 test (rand10)
• gtgtboxplot(test test2,1)

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• 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

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Anovas

• ANOVAs 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

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Anovas

• ANOVAs 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

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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 lt 0.05
• Type II error
• Accept Null Hypothesis when it is false. Eg
  means are actually significantly different when p
  gt 0.05

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Anovas
Beta Probability of making type II Error
Alpha Probability of making type I Error
P lt 0.05
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Anovas

• Terminology
• Family Wise Error
• The probability of making at least 1 family wise
  error while making multiple ANOVA comparisons

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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

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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.
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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)

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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.

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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.
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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','a
l1',... 'al2','al2','al2','al2','al2','al
2' Though alloy is sorted in this example, you
do not need to sort the grouping variable.
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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.
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1 way Anovas
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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.

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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.

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N-way Anovas

• Related (Repeated Measures) Design
• NOT IMPLEMENTED IN THE STATISTICS TOOLBOX!!

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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

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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/tech
  doc/matlab.html
• Matlab command prompt function lookup
• Built in Demos
• Websites
• Hard Copy
• Books, Guides, Reference
• The Student Edition of Matlab pub. Mathworks Inc.