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Slide 2 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 % Slide 3 What is Rescaling? Conversion of one measurement scale another Common technique used in quantitative biology 111100101000100100100010 010010010011111001011001 001001000101000100101001 010010100100101001010101 000101000010010100101001 000101010010101001011110 101101101001010101001010 100100111101011001101001 20 % Slide 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 Slide 5 Why bother? Logical rescaling has many applications Simplify analyses Even out datasets Help reveal patterns Consolidate explanatory variables Run non-parametric statistics Slide 6 Simplify Gravel Other Ratio Nominal Slide 7 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 Slide 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 Slide 9 Consolidate explanatory variables Nominal forest/barrens berries present/absent wet/dry Ordinal Habitat rank Slide 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) Butthese test are not a staple these days Generalized linear models can deal with various error distributions Slide 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 Slide 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 Slide 13 Physical quantities larger scope Biological quantities smaller scope Slide 14 Scope of measurement instruments Defined as the max over min reading Slide 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 1 2 3 456 7 Slide 16 Survey scope Unit: 100 km transects Frame: sum(rivers) Scope: # possible transects Slide 17 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 ? Slide 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 Slide 19 Normalization to a sum Taking a percentage e.g. Mendels experiments 224705 Slide 20 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 Slide 21 SST ( o C) SST anomaly ( o C) Date Slide 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 Slide 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 Slide 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: Slide 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 Slide 26 1.Write the quantity to be rescaled 2.Apply rigid conversion factors so units cancel 3.Calculate Convert units Generic procedure Three steps Slide 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