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Chapter 3 Rescaling 1111001010001001001000100100100100 1111100101100100100100010100010010 1001010010100100101001010101000101 0000100101001010010001010100101010 0101111010110110100101010100101010 0100111101011001101001010001010100 0000010111110100100001011010001010 0101000100000101101111111100101000 1001001000100100100100111110010110 0100100100010100010010100101001010 0100101001010101000101000010010100 1010010001010100101010010111101011 0110100101010100101010010011110101 1001101001010001010100000001011111 0100100001011010001010010100010000 0101101111111100101000100100100010 0100100100111110010110010010010001 0100010010100101001010010010100101 0101000101000010010100101001000101 0100101010010111101011011010010101 0100101010010011110101100110100101 0001010100000001011111010010000101 1010001010010100010000010110111111 1100101000100100100010010010010011 20 %

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What is Rescaling? Conversion of one measurement scale another Common technique used in quantitative biology 111100101000100100100010 010010010011111001011001 001001000101000100101001 010010100100101001010101 000101000010010100101001 000101010010101001011110 101101101001010101001010 100100111101011001101001 20 %

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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

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

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Simplify Gravel Other Ratio Nominal

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Even out datasets Fish census 2005 20062007 20082009 201020112012 Net # by species 10brown trout 5smelt 1Arctic char Harper cuts Can no longer compare total values 15 6 Salvage long term analysis: Ratio Nominal

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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

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

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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

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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

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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

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Physical quantities – larger scope Biological quantities – smaller scope

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Scope of measurement instruments Defined as the max over min reading

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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 1 2 3 456 7

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

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Experiment scope Unit depends on quantity measured or sampling interval – Sampling livers – Census bacteria each day 10 887665443 123456789 Millions of bacteria recorded each day ?

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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

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Normalization to a sum Taking a percentage – e.g. Mendel’s experiments 224705

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Normalization to the mean Useful for assessing deviations from the mean – e.g. Number of plant species on the Canary Islands Nplant = [ 366 348 763 1079 539 575 391 ] · sp/island mean(Nplant) = n -1 ∑ Nplant mean(Nplant) = 7 -1 · 4061 · species/island = 580 dev(Nplant) = Nplant - mean(Nplant) dev(Nplant) = [ -214 -232 +182 +498 -41 -5 -189 ] · sp/island

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

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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

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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

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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:

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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

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

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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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