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Chapter 3 Rescaling 1111001010001001001000100100100100 1111100101100100100100010100010010 1001010010100100101001010101000101 0000100101001010010001010100101010.

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Presentation on theme: "Chapter 3 Rescaling 1111001010001001001000100100100100 1111100101100100100100010100010010 1001010010100100101001010101000101 0000100101001010010001010100101010."— Presentation transcript:

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2 Chapter 3 Rescaling %

3 What is Rescaling? Conversion of one measurement scale  another Common technique used in quantitative biology %

4 12 rescaling options Nominal [0,0,1,0,0,0] Ordinal [1 st, 2 nd, 3 rd ] Interval [90,180,45] o Ratio [0,1.4,3.2]m Less detail More detail

5 Why bother? Logical rescaling has many applications – Simplify analyses – Even out datasets – Help reveal patterns – Consolidate explanatory variables – Run non-parametric statistics

6 Simplify Gravel Other Ratio  Nominal

7 Even out datasets Fish census Net  # by species 10brown trout 5smelt 1Arctic char Harper cuts Can no longer compare total values 15 6 Salvage long term analysis: Ratio  Nominal

8 Rescaling to a less detailed quantity sometimes makes it easier to see patterns Is the presence/absence of storms associated with number of vagrant birds observed? Help reveal patterns

9 Consolidate explanatory variables Nominal forest/barrens berries present/absent wet/dry Ordinal Habitat rank …

10 Run non-parametric statistics Interval & Ratio  Rank [10.5, 15.6, 19.1, 9.8] ml  [3, 2, 1, 4] Non-parametrics can be useful when parametric test assumptions are violated – Wilcoxon rank-sum test (≈ t-test) But…these test are not a staple these days – Generalized linear models can deal with various error distributions

11 Normalization Common Q ref values: Q max Q mean Q min Q range Q sum Q sd Q and Q ref have the same units Another common technique used in quantitative biology Conversion of quantity to a ratio with no units

12 Scope We can use scope to – compare the capacity of measurement instruments, – compare the information content of graphs, – compare variability of physical systems, or biological systems

13 Physical quantities – larger scope Biological quantities – smaller scope

14 Scope of measurement instruments Defined as the max over min reading

15 Survey scope 1.Defining the sample unit 2.Listing all possible units (the frame), 3.Then survey all possible units (complete census) or sample units at random Salmon survey Unit: 100 km transects Frame: 700 km Scope

16 Survey scope Unit: 100 km transects Frame: sum(rivers) Scope: # possible transects

17 Experiment scope Unit depends on quantity measured or sampling interval – Sampling livers – Census bacteria each day Millions of bacteria recorded each day ?

18 Normalization Relative to a statistic: Q sum Q mean Q range Q sd Q and Q ref have the same units Another common technique used in quantitative biology Conversion of quantity to a ratio with no units

19 Normalization to a sum Taking a percentage – e.g. Mendel’s experiments

20 Normalization to the mean Useful for assessing deviations from the mean – e.g. Number of plant species on the Canary Islands Nplant = [ ] · sp/island mean(Nplant) = n -1 ∑ Nplant mean(Nplant) = 7 -1 · 4061 · species/island = 580 dev(Nplant) = Nplant - mean(Nplant) dev(Nplant) = [ ] · sp/island

21 SST ( o C) SST anomaly ( o C) Date

22 Normalization to the mean Coefficient of Variation Unitless ratio that allows comparisons of two quantities, free of various confounds – e.g. We can use the CV to compare morphological variability in mice and elephants

23 Normalization to a range The range is defined as the largest minus smallest value Ranging uses both the minimum and maximum value to reduce the quantity to the range 0 to 1

24 Normalization to the stdev This is a common form of normalization in statistical treatments of data Returning to example of number of plant species on 7 Canary Islands:

25 Rigid Rescaling Rigid rescaling replaces one unit with another Units disappear because any unit scaled to itself = 1 (no units) – m/m is notation for metre/metre = 1 – kcal/Joules is a number with no units – km 1.2 /m 1.2 has no units: it is the number of crooked m per crooked km

26 1.Write the quantity to be rescaled 2.Apply rigid conversion factors so units cancel 3.Calculate Convert units Generic procedure – Three steps

27 Figure out how much Phelps eats in a day (in lbs) How much do you eat in a day, as a % of body weight? 2000 Kcal/day for women not in training 2200 kcal/day for men not in training


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