2. Exploratory Data Analysis

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2. Exploratory Data Analysis

• OR An ABC of EDA
• Peter Watson
• http//imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/gra
  phFAQ

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No apriori ideas (model) in EDA!

• For classical analysis, the sequence is
• Problem gt Data gt Model gt Analysis gt
  Conclusions
• For EDA, the sequence is
• Problem gt Data gt Analysis gt Model gt
  Conclusions

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EDA - Exploratory data analysis

• Informal graphical techniques (Tukey, 1977) which
• look at underlying structure
• identify outliers
• check assumptions in later formal analyses
  (Normality, equality of variance)
• Most of the EDA techniques are graphical and
  quite simple
• A picture is worth more than ten thousand words
  Chinese proverb

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Graphical displays

• Histograms
• Boxplots
• Quantile Plots
• Error Bar plots (groups)
• Stem and Leaf Displays
• Scatterplots (especially for checking linearity
  of correlations, residual plots from regressions)
• (see also regression talk)
• Under SPSS EXPLORE

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Symmetry

• Clustered around median
• medianmeanmode
• no skewness
• CIs of mean assume symmetry

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Skew and kurtosis

• Skew
• lt0 upper straggle
• 0 symmetric
• gt0 downward straggle
• Rules of Thumb (Hair et al,1998Simon,2002)
• Negative skew lt-1
• Positive Skew gt 1

• Kurtosis
• lt0 flat (Platikurtic)
• 0 normal peak
• gt0 peaked about mean (Leptokurtic)
• Rules of Thumb (Simon, 2002)
• Positive Kurtosis gt 3
• Negative Kurtosis lt -3

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Peakedness

• Kurtosis measures peakedness
• Dont want too peaked or too uniform
  distributions
• Too peaked -no variation
• Too flat - no one typical value

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Types of Kurtosis (Miles Shevlin, 2001)
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Bimodality

• This is a mixture of two distributions (clear dip
  around the middle multipeaked)
• Histograms are usually good at spotting this
• Suggests modelling the first half and second half
  separately

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Beck Score

• Positive skew (gt1.0)
• Most scores around zero
• Scores above 13 - clinically depressed
• One score of 46!

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Boxplots
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Boxplots

• median line in red box
• Middle half in red box (1.3 sds)
• Outliers circles and stars
• Shape of data

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Outliers in boxplots

• Inner fence - moderately weird. Over 1.5
  boxlengths from upper/lower quartiles Circles in
  SPSS
• (2.67 sds from mean in normal data)
• Outer fence - decidedly weird. Over 3 boxlengths
  from upper/lower quartiles Asterisks in SPSS
• (4.67 sds from mean in normal data)

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Hinges

• Boxplots actually use Hinges to define locations
  of boxes and outliers
• Upper Hinge similar to Upper quartile
• Lower Hinge similar to Lower quartile
• Inter-hinge spread similar to interquartile range

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Boxplot of Beck score

• Positive skew
• Concentration of outliers above median score

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Effect of an outlier

• biases mean (green line)
• inflates variance of mean
• median more robust

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Robustness to outliers

• Number of positive responses (max6)
• 0,0,0,4,4,5,5,5,6,6,6,6,6,6,6
• 95 Bootstrap Confidence Intervals
• Median (4.89,5.11) Observed Median5.00
• Mean (4.11,4.41) Observed Mean4.33

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Consistency of median
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Obtaining 95 CIs for skewed data Bootstrapping
(Efron Tibshirani, 1993)
See also http//www.ruf.rice.edu/lane/stat_sim/s
ampling_dist/index.html
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Example revisited

• 1000 random samples of size equal to original
  sample (N15)
• Results
• Point estimates Mean4.37 Median5
• 95 CIs Mean 3.27, 5.27 Median 4, 6
• Outliers exerting undue influence on the mean
• See http//imaging.mrc-cbu.cam.ac.uk/statswiki/FA
  Q/boot

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The sampling distribution of the variance follows
a chisquare which tends to N(n-1,2(n-1)) as n
increases
shoulds.e.(mean) 5/sqrt(25)1
N25 normal(24,48) observed mean is 23.75,
observed variance6.826.8246.5
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Other approaches to identifying outliers (besides
boxplots)

• Cases with z-scores exceeding 2.5
• (z-score subtracts mean and divides by s.d.)
• Grubbs test (see CBU website for details)

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Quantile Plots

• Raw beck score
• deviates from straight line
• Substantial skew
• Limits choice of statistical tests we can use to
  analyse beck score
• Bump above line positive skew

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Reverse scored Beck

• beck is now negative skewed
• the bump is now under the line

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S shapes

• Symptoms of Kurtosis
• uniformity
• peakedness

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Testing normality more formally

• Kolmogorov-Smirnov test
• Shapiro-Wilks
• Overly sensitive for large samples

Non-Normal
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Symmetric plots

• Rank Beck distances above and below the beck
  score median
• Plots I-th lowest distance above the median
  against I-th lowest distance below the median
• Not many points plotted as so many points below
  the median so doesnt show asymmetry very well
  multiple points with same co-ordinates
• If symmetric points fall on line xy
• distances above median gt distances below median

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Stem and Leaf of Beck Score

• Stem Leaf
• 6 . 0 6.0
• Each leaf4 cases

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Temperature

• What is unusual about this
• distribution?
• Clue spacing.
• Each leaftwo temperatures
• STEM LEAF
• -6 6 -6.6 Degrees C

• Frequency Stem Leaf
• 2.00 -6 . 6
• 4.00 -5 . 00
• 10.00 -4 . 44444
• 6.00 -3 . 338
• 14.00 -2 . 2227777
• 14.00 -1 . 1116666
• 8.00 -0 . 0055
• 12.00 0 . 005555
• 6.00 1 . 616
• 14.00 2 . 7777777
• 16.00 3 . 33333888
• 14.00 4 . 4444444
• 13.00 5 . 555555
• 11.00 7 . 22227
• 7.00 8 . 888
• 6.00 9 . 444

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Scales (c/o RSS News)

• Grain diameters recorded to nearest division 1
  inch apart
• Subsequently told to report in cm 1 inch
  2.5cm approx.
• Raw data (1,0,2,1,1,0,4,1,3,0,1,1) in inches
• (2.50, 0, 5.00, 2.50, 2.50, 0, 10.00, 2.50, 7.50,
  0, 2.50, 2.50) in cm
• The village post office is 1.21 km (2 miles)
  across the valley on the left
• (1 lb) 454 grammes of cheese, (1 pint) 560ml of
  beer
• The human mind likes whole numbers

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Percentage success

• What is wrong with this graph?
• No axis labels or title
• Y axis strangely scaled Cant have percentages lt
  0 or greater than 100
• green markers smaller
• Green and red not distinguishable by colour blind
  person yellow partially hidden by background
• Other caveats
• Joining points can be misleading
• make sure tick marks on scales are not too near
  one another to give false effect.

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Scale invariance or not.

• When is 40 approximately equal to 25?
• When is 73 equal to 111? (asked on University
  Challenge in 2005)
• ANSWERS
• km mph In base 8. Computers think in
  binary (base 2)!
• But
• Februarys temperature was 55F (13C) which is
  three times the average
• So the average equals 55/3 18.3F (13/3
  4.3C)?
• Why is this patently untrue?
• ANSWER
• 18.3F is below freezing (lt32F) but 4.3C is
  above freezing(gt0C).

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One more thing...

• When is Halloween equal to Christmas?
• ANSWER Oct. 31 Dec. 25. I.e. 8x3 1 2x10 5

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Error Bar Charts

• Interactive Bar charts
• Bar length represents 95 Confidence interval for
  the mean
• females have higher depression scores than males

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Bubble Plots (in R) years in education related to
income/prestige
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Multiple scatter plots (R)
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Ladder of Powers (Marsh,1988)

• Powers (double star function in SPSS COMPUTE)
  e.g. 329
• 2 square
• 1 untransformed
• 0.5 square root
• 0 (natural) log
• -0.5 inverse square root
• -1 reciprocal
• -2 inverse square

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Choosing a power

• Trial and Error
• Box-Cox transformation
• SPSS Box-Cox macro available at
• http//stat.tamu.edu/ftp/pub/mspeed/stat653/spss/

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Box-Cox applied to Beck score

• Looks for a power that minimises Beck score
  variance
• Suggests a power of 0.3 (near to log transform
  (power0))
• Regression improve fit of a covariate to predict
  a test score Box-Cox can flag up a non-linear
  relationship
• Can be used to help determine z-scores and means
  but can be misleading for very skewed data e.g.
  when floor and ceiling effects are present

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Predicted test score using a covariate vs actual
test score (raw and square rooted)
More linear relationship taking square root
(right hand side picture)
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Box Cox on residual variance

• test score constant Aitem score residual
• Can use boxcox on residuals of fitting item score
  on test score
• suggests using square root of y
• This is the transform of test score which
  minimizes the residual variance

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Exponential

• Clicks constant AE-B Age
• Another type of non-linear relationship.
  Characterised by ever increasing rates of
  changes as you get older

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Log Beck

• Skew0.60
• Kurtosis-0.06
• Acceptable using rule of thumb

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Quantile plot - log Beck

• Fits closer to a straight line
• Log transform has made the distribution more
  Normal
• Log transform enables the use of more powerful
  statistical tests

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Symmetry of midpoints

• midpoints of percentiles
• average of thresholds marking blue and green
  areas should be equal in symmetric distributions

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Midpoints of beck score

• Beck
• Median6
• 0.5(Sum of Midpoints) - Median
• Quarters 8.3
• Eighths 16.7
• Sixteenths 38

• Log(Beck1)
• Median1.95
• 0.5(Sum of Midpoints) - Median
• Quarters 3.0
• Eighths 14.4
• Sixteenths 26.7
• MORE SYMMETRIC!

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Rank transform

• Downweight outliers
• Useful if power transformations fail
• Useful summary measures
• Medians
• Interquartile ranges (Boxplots)
• Rank sums (Non-parametric tests)

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Using ranks - example

• Compare cost (in ) of two care centres
• Care Centres O R
• Any patient cost saving?

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Centre O stem leaf display

• STEM WIDTH200
• 2 EXTREMES
• POSITIVE SKEW

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Centre R - Stem and Leaf

• (stem width100)
• outliers present
• positive skew
• rank test needed

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RESULTS

• UNRANKED
• t(147) 0.91, p.36
• centre costs the same
• Uses means

• RANKED
• mean Rank
• Study O 65.06
• Study R 85.63
• M-W Z-2.96,p.003
• Centre R costlier
• Uses ranks

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Nonparametric tests

• PROS
• Downweight outliers
• Fewer assumptions
• Useful for skewed distributions

• CONS
• Less powerful
• Lose information
• Limited range of tests

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Equal Group Variances

• Important for t-tests and ANOVAs
• No covariate by group interaction in ANCOVA
  Quades (1967) method is a nonparametric
  equivalent
• May need to transform outcome
• Tests available to identify problems

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Levenes test

• Are group variances equal?
• Gets slope of spread vs location
• Compares slope to 0
• produces F-test

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Proportions

• Variance of a proportion depends on value of
  proportion!
• Arcsine transform resolves this
• In SPSS use function in COMPUTE to do transform
  
• 2 arsin(sqrt(p))

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Funny you should say that...

• There is no truth to the allegation that
  statisticians are mean. They are just your
  standard normal deviates.
• Why don't statisticians like to model new
  clothes?
• Lack of fit.
• Did you hear about the statistician who invented
  a device to measure the weight of trees? It's
  referred to as the ? scale
• ?log
• Old statisticians never die, they just undergo a
  transformation.
• Or in summary.Normal lack of fit try a log
  transformation!
• http//research.microsoft.com/users/lamport/pubs/h
  air.pdf

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And Finally...

• A Statistician is someone who can have their
  head in an oven and their feet in an ice box and
  say that on the whole they are feeling perfectly
  normal
• Check you are using appropriate summary measures

• Further details including references on EDA at
• http//www.itl.nist.gov/div898/handbook/eda/eda.ht
  m
• Thanks to Frank Duckworth RSS News article on
  scales
• Thanks to Chrissy Fletcher for supplying the
  jokes
• Allan Reese (CEFAS, graphical comments)
• Next week (Thursday). 11am
• Ian Nimmo-Smith
• The anatomy of statistical methods models,
  hypotheses, significance and power