Results II (Figures) Numbers & Statistics Forestry 545 March 4 2014

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Results II (Figures) Numbers & Statistics

Forestry 545March 4 2014

Dr Sue WattsFaculty of ForestryUniversity of British ColumbiaVancouver, BC Canada [email protected]

mailto:[email protected]

General manuscript format

Title Authors Abstract Introduction Materials & Methods Results Discussion References

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Illustrations

=

Tables & Figures

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Figures• Photographs• Drawings• Gazintas• Algorithms• Maps• Line graphs• Bar graphs• Pie charts• Pictographs

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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

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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

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Year

0

10

20

30

40

50

60

70

90

100

80

1900 19101905 1915 1920

Local index

Year

0

10

20

30

40

50

60

70

90

100

80

1900 19101905 1915 1920

Local index

Avoid chart junk

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Katz 2008

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

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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)

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Gustavii 2002

Improved caption & graph

Gestational duration did not differ between the treated women and control

Gustavii 2002

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Figures• Photograph – used for documentary

illustration

• Drawing• Gazinta• Algorithm• Map• Line graph• Bar graph• Pie chart• Pictograph

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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

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13animals.nationalgeographic.com

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mnn.com

15amazingdata.com

Figures• Photograph

• Drawing• Gazinta• Algorithm• Map – all used as explanatory artwork

• Line graph• Bar graph• Pie chart• Pictograph

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Drawing

Can show perspective and detail (insides, layers) not possible with a photograph

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Drawing allows control of detail

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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

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“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.

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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

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Flow chart algorithmAbout 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…

PERFORM A HEARTWORM ANTIGEN TEST

WE NEED ANTIGEN TESTING

Is test positive?

Examine blood with a Knotts or Filter testnoHas dog been on a monthly

heartworm preventative?

yes

Suspect lapse in protection

yesIs test negative?

yes

Is there any history or clinical evidence to suggest

heartworm infection?

no

Dog is free from heartworm infection. May begin

preventative regimen

Are microfilariae present?

Is there any history or clinical evidence to suggest

heartworm infection?

no

yes

noyes

What kind?

Dog has a heartworm infection. Evaluate extent of disease. Determine

treatment protocol. Regard antigen test as false negative.

D. immitis

Suspect error in testing procedure.

Repeat antigen test

Begin further diagnostic

procedures.

no yes

Retest in 3-6 months or contact test manufacturer for consultation

Negative or uncertain results - retest

Is infection confirmed?no

yesPositive retest

D. reconditum

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Map

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Figures• Photograph• Drawing• Gazinta• Algorithm• Map

• Line graph• Bar graph• Histogram• Pie chart• Pictograph – all used to promote

understanding of numerical results24

Line graph

Graphs are a good choice when you think that a relationship is more important to the reader than the actual numbers

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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

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Table or figure?

Time(hour)

midnight2:004:006:008:00

10:00noon2:004:006:008:00

10:00

Normal(mg/dl*)

100.393.688.2

100.5138.6102.4

93.8132.3103.8

93.6127.8109.2

Diabetic(mg/dl)

175.8165.7159.4

72.1271.0224.6161.8242.7219.4152.6227.1221.3

Blood glucose levels

* decaliters/milligram

Blood glucose levels for normal individual and diabetic

Hour

12:00 6:00 am 12:00 6:00 pm 12:00

BloodGlucoseLevel(mg/dl)

300

250

200

150

100

50

0

BreakfastLunch Dinner

Normal

Diabetic

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Gustavii 2002

Line graph

0

2000

4000

6000

8000

10000

1988 1989 1990 1991 1992

Num

ber

of

confirm

ed c

ases

USA

YearChanges in rabies disease incidence over time.

Canada

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-20

0

20

40

60

80

100

0 30 60 90 120 150

Right eye

Left eye

Minutes

Pup

il d

iam

ete

r (%

cha

nge

)

-20

0

20

40

60

80

100

Minutes

Pup

il d

iam

ete

r (%

cha

nge

) Right eye

Left eye

0 30 60 90 120 150

Tyramine Tyramine

Line graph labeling

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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

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visible pattern

0

2

4

6

8

10

12

14

16

2

y

0 4 6 8 10 12 14 16x

no visible pattern

0

2

4

6

8

10

12

14

16

2

y

0 4 6 8 10 12 14 16 x

Scatter graphs

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Katz 2006

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

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Consumption of pure alcohol (litres)

Bar graph

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Gustavii 2002

Comparative bar graph

This effective bar graph relates insect type to turning choices.

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Keep bar graph simpleDo not use 3-D on 2-D data

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Gustavii 2002

Use 3-D only if necessary

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Jamie Myers

Histogram

• An estimate of the probability distribution of a continuous variable

• Used to present continuous variables in a forceful way

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Comparative histogram

Probability of dying in a coronary care unit after admission with initial working diagnosis of acute myocardial infarction.

0

0.1

0.2

0.3

0.4

<45

Pro

ba

bili

lty

Can replace legend with symbols

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Gustavii 2002

Comparative histogram

Time (min)

pH

0

1

2

3

4

5

6

0 0.5 10 20 30 40 60 80

MD K lowNA highNA

HighNaK H W C

Maximum three groups per category 39

Gustavii 2002

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

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Pie graph

Dandelion (50%)

Apple (25%)

Violet (20%)

Rose (5%)

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.

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Pictograph

Bar graphs made of pictures

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Pictograph

Number of Flowering Plant Species in West Suffolk County

1985 1990 1995 2000

110

75

65

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In this effective pictograph, the length of the flower stems corresponds to the number of plant species.

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Numbers and Statistics

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Numbers and Statistics

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Using statistics

Using statistics properly is a skill

Never be afraid to ask for advice

Dr Tony KozakWednesdays 8:30 – 11:00 amFSC 2027 by appointment [email protected]

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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

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thingsbiological.wordpress.com

Common descriptive statistics

• Size

• Range

• Middle– Mean– Mode– Median

• Spread– Standard deviation– Central 50%

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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 Distance between smallest and largest data valueslargest data values

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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

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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

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SD

Standard deviation

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Central 50%

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Referring to mean and standard deviation

Use mean (SD) = 44% (3)

mean of 44% (SD 3)

NotSD = 44 3%

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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

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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

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Inferential statistics

• Statistical judgments are made by working on the formula in the mathematical universe

• Inferences are covered in your Discussion

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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

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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

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Normal distribution

SD

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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

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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

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Skewed distribution(grades in Forestry 545)

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When you have a non-normal distribution you cannot use mean and standard deviation to describe the distribution – you must use median and rangeConsider 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

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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

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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

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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?

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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

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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

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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

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Statistical reportingAlways 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

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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

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• Report statistics parenthetically with individual elements of a test separated by commas

…were significant (2=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)

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• 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

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• 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

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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

BetterThe differences in the egg cannibalism rates of the two species (Table 1) were not significant (Fisher Exact Probability Test, p=0.05)

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Statistical significance & scientific importance

Scientific research yields 2 kinds of significanceScientific

Statistical

Scientific importance is often ignored as it involves some subjectivity

Statistical significance is easy to convey but may lack scientific vigour

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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

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Statistically significant results should always be accompanied by a discussion of the scientific importance of the findings

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Drug lowered blood pressure by a mean of8 mm Hg from 100 – 92 mm HgStatistically significant (p<0.05)

Better way to present this is with 95% confidence interval (CI)

Here, CI was 2 – 14 mm HgScientifically important to decrease blood pressure by as much as 14 mm Hg, reduction of 2 mm Hg would not be important

Example from Gustavii 200282


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

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Potentially problematic statistical terms (CSE 2006)

Random sample implies true randomizationOften confused with “sampling without known bias”Confidence interval or limit better to use interval as limit implies 2 discrete and unchanging valuesStandard deviation better to note as SD rather than S. Does not need sign

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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

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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

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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

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Rounding numbers (CSE 2006)

To retain 3 significant digitsIf 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

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Numbers and units

Ranges and units – can use single unit after second number

23 to 47 km or 23 km to 47 kmNot 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

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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

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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

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Writing numbers

Example following this rule:

Three thousand eight hundred and seventy-six seedlings were measured at 8-12 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?

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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.

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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

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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

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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%)

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Using percentages

The original data should always be includedOrder 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

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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%)….

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Assignments

• Assignment #2 “Abstract” due today

• Assignment #3 “Introduction” due in 2 week’s time – March 18

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Documents

Results II (Figures) Numbers & Statistics Forestry 545 March 4 2014