Download Forestry 545

Results II (Figures)
Numbers & Statistics
Forestry 545
March 4 2014
Dr Sue Watts
Faculty of Forestry
University of British
Columbia
Vancouver, BC
Canada
[email protected]
1
General manuscript format








Title
Authors
Abstract
Introduction
Materials & Methods
Results
Discussion
References
2
Illustrations
=
Tables & Figures
3
Figures
•
•
•
•
•
•
•
•
•
Photographs
Drawings
Gazintas
Algorithms
Maps
Line graphs
Bar graphs
Pie charts
Pictographs
4
Figures
• As with tables, figures should be
independent and indispensable
• Good visual material will spark reader
interest
• Interested readers will look to the text for
answers
5
Figures
• Need to be attractive but not glitzy
• Watch out for size and scale (reduction
may accentuate some flaws)
• After reduction to publication size capital
letters should be about 2 mm high
• X and Y axis lines should be no wider than
lettering
6
Avoid chart junk
Local index
L
o
c
a
l
i
n
d
e
x
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
1900
Katz 2008
1905
1910
Year
1915
1920
1900
1905
1910
Year
1915
1920
7
Figure captions
• Reader looks at figures then legends
• Title should explain meaning without need
to read manuscript
• Does not need to be a complete sentence
• Like table title, usually in two parts
– Descriptive title
– Essential details
8
Figure captions
• Captions for figures
go below figure
• In a manuscript,
figure captions are
placed on a
separate sheet
• How could you
improve this caption
and graph?
Cumulative weeks to delivery of the women
in group A (n =78) and group B (n = 78)
9
Gustavii 2002
Improved caption & graph
Gestational duration did not differ
between the treated women and control
10
Gustavii 2002
Figures
• Photograph – used for documentary
illustration
•
•
•
•
•
•
•
•
Drawing
Gazinta
Algorithm
Map
Line graph
Bar graph
Pie chart
Pictograph
11
Photograph
• Value to article can range from Ø to more
valuable than any text!
• If you need a photo, pick a journal that
produces high quality reproduction
• Crop or mark with arrows to highlight
important detail
12
13
animals.nationalgeographic.com
14
mnn.com
15
amazingdata.com
Figures
• Photograph
•
•
•
•
Drawing
Gazinta
Algorithm
Map – all used as explanatory artwork
•
•
•
•
Line graph
Bar graph
Pie chart
Pictograph
16
Drawing
Can show perspective and detail (insides,
layers) not possible with a photograph
17
Drawing allows control of detail
18
Jamie Myers
Gazinta
Visuals that show hierarchy, organization or
interaction
• Tree gazintas show sub-assemblies of the
same relative importance
• Block diagrams are interaction gazintas
19
“Gazinta” (organization tree)
ELECTRON MICROSCOPE LABORATORY
TRANSMISSION EM
SCANNING EM
IMAGE PROCESSING
TECHNICAL
PERSONNEL
SAMPLE
SECTIONING
SAMPLE
STAINING
A Typical drawing tree gazinta describes a relatively stable situation.
20
Mathews and Mathews 2008
Algorithm
• Flowcharts & taxonomic keys
• Algorithms are illustrations of a means of
making a decision by considering only
those factors relevant to that decision
• Algorithms are usually easier to follow
than the written text equivalent
21
Flow chart algorithm
About to receive a
heartworm preventative
for the first time…
On a monthly macrolide
heartworm
preventative...
Resuming a daily DEC
preventative for the
coming HW season…
History and heartworm
status unknown…
WE NEED ANTIGEN TESTING
PERFORM A HEARTWORM ANTIGEN TEST
Is test positive?
yes
Has dog been on a monthly
heartworm preventative?
yes
no
Examine blood with a Knotts or Filter test
yes
no
yes
D. reconditum
D. immitis
Is there any history or
clinical evidence to suggest
heartworm infection?
no
no
What kind?
Is there any history or
clinical evidence to suggest
heartworm infection?
no
yes
Retest in 3-6
months or contact
test manufacturer
for consultation
yes
Are microfilariae present?
Suspect lapse in
protection
Suspect error in
testing procedure.
Repeat antigen
test
Negative or uncertain
results - retest
Is test negative?
Dog is free from heartworm
infection. May begin
preventative regimen
Dog has a heartworm infection.
Evaluate extent of disease. Determine
treatment protocol. Regard antigen
test as false negative.
Begin further
diagnostic
procedures.
yes
no
Is infection confirmed?
22
Mathews and Mathews 2008
Map
23
Figures
•
•
•
•
•
Photograph
Drawing
Gazinta
Algorithm
Map
•
•
•
•
•
Line graph
Bar graph
Histogram
Pie chart
Pictograph – all used to promote
understanding of numerical results
24
Line graph
Graphs are a good choice when you think
that a relationship is more important to the
reader than the actual numbers
25
Line graph
• Line graphs, scatter graphs, bar graphs,
histograms, pies and pictographs are used
to promote understanding of numerical
results
• Tables present results
• Graphs promote understanding of results
and suggest interpretation of their
meaning
26
Table or figure?
Blood glucose levels
300
Breakfast
Lunch
Time
(hour)
Normal
(mg/dl*)
Diabetic
(mg/dl)
midnight
2:00
4:00
6:00
8:00
10:00
noon
2:00
4:00
6:00
8:00
10:00
100.3
93.6
88.2
100.5
138.6
102.4
93.8
132.3
103.8
93.6
127.8
109.2
175.8
165.7
159.4
72.1
271.0
224.6
161.8
242.7
219.4
152.6
227.1
221.3
* decaliters/milligram
Dinner
250
Diabetic
Blood
200
Glucose
Level
150
(mg/dl)
100
Normal
50
0
12:00
6:00 am
12:00
6:00 pm
12:00
Hour
Blood glucose levels for normal
individual and diabetic
27
Gustavii 2002
Line graph
Number of confirmed cases
10000
8000
6000
4000
USA
2000
Canada
0
1988
1989
1990
1991
1992
Year
Changes in rabies disease incidence over time.
28
Mathews and Mathews 2008
Line graph labeling
80
100
Tyramine
Right eye
Pupil diameter (% change)
Pupil diameter (% change)
100
60
40
Left eye
20
0
-20
80
Right eye
Tyramine
60
40
Left eye
20
0
-20
0
30
60
90
Minutes
120
150
0
30
60
90
Minutes
120
150
29
Gustavii 2002
Line graph symbols
• Use standard symbols on line graphs
(order below is suggested)
• In some cases there can be symbolic
use of symbols, i.e. filled circle for
treatment and unfilled circle for the
control
Symbols for Line Graphs
30
Scatter graphs
y
y
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
2
4
6
8
10
12
visible pattern
Katz 2006
14
16
0
x
0
2
4
6
8
10
12
no visible pattern
14
16
x
31
Bar graph
• Used to present discrete (unrelated)
variables in a forceful way
• Downside is that they present a relatively
small amount of information in quite a
large space
32
Bar graph
Consumption of pure alcohol (litres)
Gustavii 2002
33
Comparative bar graph
This effective bar graph relates insect type to turning choices.
34
Mathews and Mathews 2008
Keep bar graph simple
Do not use 3-D on 2-D data
35
Gustavii 2002
Use 3-D only if necessary
36
Jamie Myers
Histogram
• An estimate of the probability distribution
of a continuous variable
• Used to present continuous variables in a
forceful way
37
Comparative histogram
Can replace legend with
symbols
Probabililty
0.4
0.3
0.2
0.1
0
<45
Probability of dying in a coronary care unit after admission with
initial working diagnosis of acute myocardial infarction.
38
Gustavii 2002
Comparative histogram
6
MD
K
lowNA
highNA
5
HighNaK
H
W
C
pH
4
3
2
1
0
0
0.5
10
20
30
40
60
80
Time (min)
Maximum three groups per category
Gustavii 2002
39
Pie graph
• Good for getting attention
• Show relationship of a number of parts to
the whole
• Arrange segments in size order with
largest at 12 o’clock
• Downside is that you cannot compare
areas
40
Pie graph
Rose
(5%)
Violet
(20%)
Dandelion
(50%)
Apple
(25%)
Typical Honeybee Pollen Load Composition (n = 1,034 pellets)
This effective divided-circle graph shows which flowers contribute to a typical honeybee pollen
load. To help readers compare the proportions, percentages are included.
Mathews and Mathews 2008
41
Pictograph
Bar graphs made of pictures
42
110
Pictograph
75
65
55
1985
1990
1995
2000
Number of Flowering Plant Species in West Suffolk County
In this effective pictograph, the length of the flower stems corresponds to the number of plant
species.
Mathews and Mathews 2008
43
Numbers and Statistics
44
Numbers and Statistics
45
46
Using statistics
Using statistics properly is a skill
Never be afraid to ask for advice
Dr Tony Kozak
Wednesdays 8:30 – 11:00 am
FSC 2027 by appointment
[email protected]
47
Descriptive statistics
Usually want to reduce the volume of your
data to a few characteristic numbers
These characteristic numbers are
descriptive statistics
Certain descriptive statistics are
particularly helpful in your Results section
48
49
thingsbiological.wordpress.com
Common descriptive statistics
• Size
• Range
• Middle
– Mean
– Mode
– Median
• Spread
– Standard deviation
– Central 50%
50
Size and range
• Size – this is the total number of data
points referred to as N
• Real world data is referred to as the
sample and the output of the
mathematical formula is called the
population
• Range – Distance between smallest and
largest data values
51
Middle
• Mean – Average data value
• Mode – Data value that occurs most often
• Median – Value such that half the data
values are less than this and half are
greater
52
Spread
• Standard deviation – Deviation of each
data point from the mean
• Large standard deviation means data
points are more spread out
• Central 50% – Boundaries in which the
middle half of the data points lie when all
placed in order
53
Standard deviation
SD
54
Central 50%
55
Referring to mean
and standard deviation
Use
mean (SD) = 44% (3)
mean of 44% (SD 3)
Not
SD = 44  3%
56
Standard error or standard
deviation?
• Standard error (SE) is not a measure of
variability
• Standard error is the standard deviation of
a statistic and as such is a measure of
precision for an estimate
• However, SE is often used descriptively
and must be properly identified to avoid
confusion
57
Inferential statistics
• Pure mathematics exists in an abstract
universe, parallel to the real world
• Inferential statistics is done in the
mathematical universe and infers the
identity of the mathematical formula from
the real world sample
58
Inferential statistics
• Statistical judgments are made by working
on the formula in the mathematical
universe
• Inferences are covered in your Discussion
59
Normal distribution
• A curve with a smooth bell shape
• Mean, median and mode have same value
• The exact shape of any normal distribution
can be defined with just 2 numbers
– Its mean and
– Its standard deviation
60
Normal distribution
• In the real world no data set makes a
perfect curve with infinite smoothness
• Nevertheless, we frequently call real world
data sets “normally distributed”
• Many large sets of real world data CAN be
well approximated with a normal
distribution (baby birth weights). Normal
distributions are frequently used in
statistical analyses
61
Normal distribution
SD
62
Normal distribution
• Examine your data set carefully
• Look at its shape and do not make any
assumptions based on a normal
distribution if you are not sure
• Check with a statistician to be certain
63
Non-normal distribution
Many sets of real world data are not
normally distributed
– Consider the assignment grades in a
graduate level communications course where
data points are concentrated asymmetrically
in the upper percent numbers
– Consider the histogram of the number of
people dying at each age where asymmetry is
in the upper ages
64
Skewed distribution
(grades in Forestry 545)
65
Non-normal distribution
When you have a non-normal distribution you
cannot use mean and standard deviation to
describe the distribution – you must use median
and range
Consider the “hand-to-floor stretch” of pregnant
women (Gustavii 2002)
– reported as mean of 12 cm (SD 14)
(Does this suggest some poked their fingers through the
floor?)
– should have used median and percentile range
66
Non-normal distribution
Rule of thumb
If SD is greater than half the mean, the data
are unlikely to be normally distributed
Most results in biomedical science are
asymmetrically distributed
67
Hypothesis testing
• In hypothesis testing need to specify
probability of a type I error or significance
level (α)  Usually use α = 0.05
• Results from hypothesis testing should
include
– Test statistic
– Degrees of freedom
– P value
68
Choosing a significance test
Do not begin with a test in mind
Answer yes/no questions about what you
want to assign confidence levels to
Is my data normally distributed?
Is my data random?
Does my data match someone else’s?
Does my data from exp A differ from data set
of exp B?
69
Choosing a significance test
Now pick a significance test that will directly
answer your questions using the data in the
form that you have generated
Do not be afraid to ask for advice
70
Probability values
• P value is the probability of obtaining a
value of test statistic as large as that
observed by chance alone
• Do not confuse this P value with the
significance level of the test (α)
• Simply stating that a P value was greater
or less than a significance level reduces
interpretation to a yes or no
71
Probability values
• Yes/no answers do not indicate the
chances of getting a more extreme result
• A P value of 0.04 and 0.06 could be
interpreted similarly
• Reporting an actual P value allows the
reader to evaluate the actual probability
72
Statistical reporting
Always report
• Name of test
• If data conformed to assumptions of test
• Absolute differences between groups
• 95% confidence interval for each
difference
• Practical relevance of each difference
73
Statistical reporting
Always report
• Name of statistical software package that
you have used – commercially available
packages have usually been well
validated, may not be case for custom
packages
74
Statistical reporting
• Report statistics parenthetically with
individual elements of a test separated by
commas
2
c
…were significant ( =18.2, df=2, P<0.001)
• Use zero to left of decimal when reporting
P values and correlation coefficients
...means differed by 17.8 g (p=0.23)
75
Statistical reporting
• Do not use more than 3 decimal places
when reporting P values
• Use exact values rather than inequalities
• Smallest P value that needs to be reported
is p<0.001
76
Statistical reporting
• Statistical methods do not need elaborate
presentation – a simple statement of the
chosen test and the probability level is
usually all that is needed
• Reference a text that details the procedure
if you feel that this is necessary
77
Statistical reporting
(Mathews et al 2000)
To determine whether the two species differed in their egg
cannibalism rate (Table 1), we used the Fisher Exact
Probability Test, with =(A+B)!(C+D)!(A+C)!(B+D)!/N!A!B!C!D!,
to obtain a p=0.05, which was not significant
Better
The differences in the egg cannibalism rates of the two
species (Table 1) were not significant (Fisher Exact
Probability Test, p=0.05)
78
Statistical significance
& scientific importance
Scientific research yields 2 kinds of significance
Scientific
Statistical
Scientific importance is often ignored as it
involves some subjectivity
Statistical significance is easy to convey but
may lack scientific vigour
79
Statistical significance
& scientific importance
A test result may be statistically significant but
the difference between the means tested may
be so small that it is scientifically irrelevant
Also, the power of a test increases with sample
size and large samples may reveal differences
that small ones would not
80
Statistical significance
& scientific importance
Statistically significant results should always
be accompanied by a discussion of the
scientific importance of the findings
81
Statistical significance
& scientific importance
Drug lowered blood pressure by a mean of
8 mm Hg from 100 – 92 mm Hg
Statistically significant (p<0.05)
Better way to present this is with 95% confidence
interval (CI)
Here, CI was 2 – 14 mm Hg
Scientifically important to decrease blood pressure by as
much as 14 mm Hg, reduction of 2 mm Hg would not be
important
Example from Gustavii 2002
82
Statistical significance
& scientific importance
In this example could have said
Blood pressure was lowered by a mean of 8 mm Hg
from 100-92 mm Hg (95% CI=2-14 mm Hg; p=0.02)
P values estimate statistical significance
CI values also estimate scientific importance
When CI is used readers can judge for themselves
83
Potentially problematic
statistical terms (CSE 2006)
Random sample implies true randomization
Often confused with “sampling without known
bias”
Confidence interval or limit better to use
interval as limit implies 2 discrete and
unchanging values
Standard deviation better to note as SD rather
than S. Does not need  sign
84
Potentially problematic
statistical terms (CSE 2006)
Standard error of the mean (SE) has little
practical value on its own
Use SD (or interpercentile range) not SE to
indicate variability in a set of data
Use CI rather than SE as a measure of
precision for an estimate
85
Significant digits
(CSE 2006)
• Calculated values (means, standard
deviations) should be to no more than one
significant digit beyond the accuracy of the
data
• Only when sample sizes are large (>100)
should percentages be expressed to one
decimal place
86
Rounding numbers
(CSE 2006)
To retain 3 significant digits
If 4th digit is less than 5, leave 3rd unchanged
4.282 becomes 4.28
If 4th digit is greater than 5, increase 3rd by 1
4.286 becomes 4.29
87
Rounding numbers
(CSE 2006)
To retain 3 significant digits
If 4th digit is 5 and 5th is zero, leave 3rd digit
unchanged when third digit is even
4.285 becomes 4.28
When 3rd digit is odd, increase it by 1
4.275 becomes 4.28
If 4th digit is 5 and 5th is not zero, increase 3rd by 1
4.2851 becomes 4.29
88
Numbers and units
Ranges and units – can use single unit after
second number
23 to 47 km or 23 km to 47 km
Not so with percentages
10% to 15% not 10 to 15% (but 10-15% is
acceptable)
Close up numbers and non-alphanumeric symbols
3 mm 44% $98
89
Scientific notation (CSE 2006)
Express very large numbers to the power of
10 (scientific notation)
2.6 x 104 ……. not 26 000
4.23 x 108……not 423 000 000
7.41 x 10-6 ……not 0.000 007 41
90
Writing numbers
Some rules
Most style manuals now suggest writing out
all numbers (not just those <10)
New rule: In 1 of the 19 forest stands…
Still need to spell out numbers at beginning
of sentence
91
Writing numbers
Example following this rule:
Three thousand eight hundred and
seventy-six seedlings were measured at 812 weeks following fertilizer treatment.
One hundred and sixty-six (4.3%) were
found to have increased height growth.
Correct, but do you find this difficult to
grasp?
92
Writing numbers
Better to re-write so that numbers fall
somewhere in the middle
Height measurements of 3 876 seedlings
at 8-12 weeks following fertilizer treatment
showed that 166 (4.3%) had increased
growth.
93
Writing numbers
Numbers side by side:
The spiders with dorsal stripes had an
average of 257, 112 red and 145 other
colours
Need to separate:
The spiders had an average dorsal stripe
count of 257, of which 112 were red and
145 were other colours
94
Writing numbers
• American and British practice is to indicate
thousands with commas
• However, to avoid confusion with decimal
marker, many style manuals recommend
the use of a space to mark off thousands
12 345 (not 12,345)
Follow your journal style
95
Using percentages
• If the total number is less than 25, do not use
percentages
• If the total number is between 25 and 100,
percentages should be expressed without
decimals (7%, not 7.1%)
• If the total number is between 100 and 100 000,
one decimal place may be added (7.1%, not
7.13%)
• Only if the total number exceeds 100 000 may
two decimals be added (7.13%)
96
Using percentages
The original data should always be included
Order of presentation is important
Height growth occurred in 209 (7.5%) of the
2,801 trees
Do not write
Height growth occurred in 7.5% (209) of the
2,801 trees
97
Using percentages
Do not use prose descriptions for
numerical data without the actual numbers
When 51 researchers were asked to quantify “often”,
the range was between 28 and 92 percent (average
59%)
Better to say
Most of the trees (82%)….
98
Assignments
• Assignment #2 “Abstract” due today
• Assignment #3 “Introduction” due in 2
week’s time – March 18
99