We start Part II (Quantifying Uncertainty) today Lab 2 - Equations

1. We start Part II (Quantifying Uncertainty) today Lab 2 - Equations Tomorrow - Tue 3-5 or 7-9 PM - SN 4117 Assignment 2 – Data Equations Due Wednesday

2. Chapter 5 Data = Model + Residual Data = Model + Residual Data Equations Data = Model + Residual Data = Model + Residual Data = Model + Residual

3. Data Equations • Central concept in the course • We’ll teach a general approach that will allow you to set up an appropriate analysis of data • You don’t have to worry about whether you have selected the ‘right’ test • We are going to use data equations to compare models to data Data = Model + Residual

4. Flexible approach Data = Model + Residual

5. Symbolic expression  Analyse data • Increase confidence using model triangle • Link symbols with graphical and verbal model Verbal Data Graphical Formal

6. Assessing fit • We’ll use data equations to measure: • How well the model fits the data (goodness of fit) • Error rate • We do not expect a perfect fit • But model and observed values should be close Model Time in sunlight Data Plant height

7. Definition • Three terms: Y = Ŷ + ε Data ModelResidual Observed ExpectedError Fitted • Residuals: ε = Y - Ŷ

8. e.g. Dobzhansky’s Fruit Flies Nothing in Biology Makes Sense Except in the Light of Evolution • Dobzhanskypioneered work on fruit fly evolutionary genetics in the lab and field • One research question he addressed was: • Does genetic variability decrease at higher altitude, due to stronger selection in extreme environments?

9. e.g. Dobzhansky’s Fruit Flies ? Nothing in Biology Makes Sense Except in the Light of Evolution Nothing in Biology Makes Sense Except in the Light of Statistics

10. Model? • Many options • First we’ll check deviance from a single value • Heterozygosity in Drosophila = 40% Data=Model+ Residual H = Ĥ+ε H = 40% + ε

11. Model 1: Deviations from a Single Value Model • With this simple model, we can form 7 data equations Summed residuals is a measure of bias Summed residuals2 is a measure of goodness of fit

13. What if the parameter is unknown? • Use statistical methods to make the "best" estimate • What does "best" mean? • Residuals should sum to zero (unbiased estimate) • Residuals should be as small as possible • The mean meets both criteria • Next model: Deviations from the Mean Data = Model + Residual

14. Model 2: Deviations from the Mean • Form 7 data equations

17. Single value vs. Mean model = 0.3514 ∑ res = 0 ∑ res2 = 0.1171 = 0.4 ∑ res = -0.34 ∑ res2= 0.1336 Mean model: unbiased and better fit But biological criteria have been replaced by statistical criteria

18. e.g. Dobzhansky’s Fruit Flies Nothing in Biology Makes Sense Except in the Light of Evolution What about elevation?...don't you remember my question? • Does genetic variability decrease at higher altitude, due to stronger selection in extreme environments?

19. Model 3: Deviations from a linear trend • What’s all that? ↗ Data = Model + Residual • Where: • is the heterozygositygradient (%/km) • is elevation (km) • is the offset % % %/km km % • Remember ?

20. Estimate slope () and offset ()

21. Model 3: Deviations from a linear trend Parameters estimated using first and last and values With this equation, we can calculate fitted values

22. Model 3: Deviations from a linear trend With this equation, we can calculate fitted values ? ? ? ? ? ? ? ? ? ? ? ? ? ?

24. There’s a better way to estimate slope • “Least squares" estimate of slope () • Estimate offset () from mean values • Line passes through mean coordinates (,) • We know less about Y-intercept

25. Model 4: Deviations from a linear trend (least squares)

28. Model comparison Two unbiased models