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Research Methods 1998 Graphical design and analysis. Ó Gerry Quinn, Monash University, 1998 Do not modify or distribute without expressed written permission of author. Graphical displays. Exploration assumptions (normality, equal variances) unusual values which analysis? Analysis

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slide1

Research Methods 1998Graphical design and analysis

Ó Gerry Quinn, Monash University, 1998

Do not modify or distribute without expressed written permission of author.

graphical displays
Graphical displays
  • Exploration
    • assumptions (normality, equal variances)
    • unusual values
    • which analysis?
  • Analysis
    • model fitting
  • Presentation/communication of results
space shuttle data1
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
slide5

3

2

Number of incidents

1

0

50

55

60

65

70

75

80

85

Joint temp. oF

O-ring failure vs temperature

Pre 1986

challenger flight
Challenger flight

Jan 28th 1986 - forecast temp 31oF

slide7

3

2

Number of incidents

1

0

50

55

60

65

70

75

80

85

o

Joint temp. F

O-ring failure vs temperature

checking assumptions exploratory data analysis eda
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?
distributions of biological data

Pr(y)

y

Pr(y)

y

Distributions of biological data
  • Bell-shaped symmetrical distribution:
  • normal
  • Skewed asymmetrical distribution:
  • log-normal
  • poisson
common skewed distributions
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
example data set
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
frequency distributions
Frequency distributions

Observations grouped into classes

NORMAL

LOG-NORMAL

Number of observations

Value of variable (class)

Value of variable (class)

number of cellana per quadrat
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

dotplots
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

boxplot

outlier

*

largest value

}

25% of values

hinge

VARIABLE

}

"

median

spread

}

"

hinge

}

"

smallest value

GROUP

Boxplot
slide17

1. IDEAL

2. SKEWED

3. OUTLIERS

4. UNEQUAL VARIANCES

*

*

*

*

*

slide18

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

scatterplots
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
cheviot beach survey 5 n 15

40

30

20

10

0

0

10

20

30

40

50

60

70

Cheviot Beach survey 5 n = 15

Number of Cellana

per quadrat

% cover of Hormosira per quadrat

scatterplot matrix
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
slide22

SPLOM for Cheviot Beach survey 5

CELLANA

- numbers of Cellana

SIPHALL

- numbers of Siphonaria

HORMOS

- % cover of Hormosira

n = 15 quadrats

transformations
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).
log transformation
Log transformation

Lognormal Normal

y = log(y)

Measurement data

power transformation
Power transformation

Poisson Normal

y = Ö(y), i.e. y = y0.5, y = y0.25

Count data

arcsin transformation
Arcsin Ö transformation

Square Normal

y = sin-1(Ö(y))

Proportions and percentages

outliers
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.
assumptions not met
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
category or line plot

Cheviot Beach

Sorrento

30

30

25

25

20

20

Mean number of Cellana per quadrat

15

15

10

10

5

5

0

0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Survey

Category or line plot

Mean number of Cellana per quadrat

Survey

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