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Research Methods 1998 Graphical design and analysis

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Research Methods 1998Graphical design and analysis

ÓGerry Quinn, Monash University, 1998

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

- Exploration
- assumptions (normality, equal variances)
- unusual values
- which analysis?

- Analysis
- model fitting

- Presentation/communication of results

- 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

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

Jan 28th 1986 - forecast temp 31oF

3

2

Number of incidents

1

0

50

55

60

65

70

75

80

85

o

Joint temp. F

O-ring failure vs temperature

- 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?

Pr(y)

y

Pr(y)

y

- Bell-shaped symmetrical distribution:
- normal

- Skewed asymmetrical distribution:
- log-normal
- poisson

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

Exploring sample data

- 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

Observations grouped into classes

NORMAL

LOG-NORMAL

Number of observations

Value of variable (class)

Value of variable (class)

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

- 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

outlier

*

largest value

}

25% of values

hinge

VARIABLE

}

"

median

spread

}

"

hinge

}

"

smallest value

GROUP

1. IDEAL

2. SKEWED

3. OUTLIERS

4. UNEQUAL VARIANCES

*

*

*

*

*

Boxplots of Cellana numbers in survey 5

100

80

60

Number of Cellana per quadrat

40

20

0

SFPERRSPCPECBLBCPW

Site

- 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

40

30

20

10

0

0

10

20

30

40

50

60

70

Number of Cellana

per quadrat

% cover of Hormosira per quadrat

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

SPLOM for Cheviot Beach survey 5

CELLANA

- numbers of Cellana

SIPHALL

- numbers of Siphonaria

HORMOS

- % cover of Hormosira

n = 15 quadrats

- 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).

LognormalNormal

y = log(y)

Measurement data

PoissonNormal

y = Ö(y), i.e. y = y0.5, y = y0.25

Count data

SquareNormal

y = sin-1(Ö(y))

Proportions and percentages

- 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.

- 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

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

Mean number of Cellana per quadrat

Survey