Checking assumptions - exploratory data analysis (EDA)

1
Research Methods 1998Graphical design and
analysis
Ó Gerry Quinn, Monash University, 1998 Do not
modify or distribute without expressed written
permission of author.
2
Graphical displays

• Exploration
• assumptions (normality, equal variances)
• unusual values
• which analysis?
• Analysis
• model fitting
• Presentation/communication of results

3
Space shuttle data
4
Space shuttle data

• NASA meeting Jan 27th 1986
• day before launch of shuttle Challenger
• Concern about low air temperatures at launch
• Affect O-rings that seal joints of rocket motors
• Previous data studied

5
O-ring failure vs temperature Pre 1986
6
Challenger flight
Jan 28th 1986 - forecast temp 31oF
7
O-ring failure vs temperature
8
Checking assumptions - exploratory data analysis
(EDA)

• Shape of sample (and therefore population)
• is distribution normal (symmetrical) or skewed?
• Spread of sample
• are variances similar in different groups?
• Are outliers present
• observations very different from the rest of the
  sample?

9
Distributions of biological data

• Bell-shaped symmetrical distribution
• normal

• Skewed asymmetrical distribution
• log-normal
• poisson

10
Common skewed distributions

• Log-normal distribution
• m proportional to s
• measurement data, e.g. length, weight etc.
• Poisson distribution
• m s2
• count data, e.g. numbers of individuals

11
Exploring sample data
12
Example data set

• Quinn Keough (in press)
• Surveys of 8 rocky shores along Point Nepean
  coast
• 10 sampling times (1988 - 1993)
• 15 quadrats (0.25m2) at each site
• Numbers of all gastropod species and cover of
  macroalgae recorded from each quadrat

13
Frequency distributions
Observations grouped into classes
NORMAL
LOG-NORMAL
Number of observations
Value of variable (class)
Value of variable (class)
14
Number of Cellana per quadrat
30
Survey 5, all shores combined Total no. quadrats
120
20
Frequency
10
0
0
20
40
60
80
100
Number of Cellana per quadrat
15
Dotplots

• Each observation represented by a dot
• Number of Cellana per quadrat, Cheviot Beach
  survey 5
• No. quadrats 15

0
10
20
30
40
Number of Cellana per quadrat
16
Boxplot
17
(No Transcript)
18
Boxplots of Cellana numbers in survey 5
100
80
60
Number of Cellana per quadrat
40
20
0
S FPE RR SP CPE CB LB CPW
Site
19
Scatterplots

• Plotting bivariate data
• Value of two variables recorded for each
  observation
• Each variable plotted on one axis (x or y)
• Symbols represent each observation
• Assess relationship between two variables

20
Cheviot Beach survey 5 n 15
Number of Cellana per quadrat
cover of Hormosira per quadrat
21
Scatterplot matrix

• Abbreviated to SPLOM
• Extension of scatterplot
• For plotting relationships between 3 or more
  variables on one plot
• Bivariate plots in multiple panels on SPLOM

22
SPLOM for Cheviot Beach survey 5
CELLANA - numbers of Cellana SIPHALL - numbers
of Siphonaria HORMOS - cover of Hormosira n
15 quadrats
23
Transformations

• Improve normality.
• Remove relationship between mean and variance.
• Make variances more similar in different
  populations.
• Reduce influence of outliers.
• Make relationships between variables more linear
  (regression analysis).

24
Log transformation
Lognormal Normal y log(y) Measurement data
25
Power transformation
Poisson Normal y Ö(y), i.e. y y0.5, y
y0.25 Count data
26
Arcsin Ö transformation
Square Normal y sin-1(Ö(y)) Proportions and
percentages
27
Outliers

• Observations very different from rest of sample -
  identified in boxplots.
• Check if mistakes (e.g. typos, broken measuring
  device) - if so, omit.
• Extreme values in skewed distribution -
  transform.
• Alternatively, do analysis twice - outliers in
  and outliers excluded. Worry if influential.

28
Assumptions not met?

• Check and deal with outliers
• Transformation
• might fix non-normality and unequal variances
• Nonparametric rank test
• does not assume normality
• does assume similar variances
• Mann-Whitney-Wilcoxon
• only suitable for simple analyses

29
Category or line plot
Mean number of Cellana per quadrat
Survey