notes4 - Chapter 1 Timothy Hanson Department of Statistics,...

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Unformatted text preview: Chapter 1 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 24 Functional versus stochastic relations Model: a mathematical approximation of the relationship among real quantities (equation & assumptions about terms). We have seen several models for a single outcome variable from either one or two populations. Now well consider models that relate an outcome to one or more continuous predictors. Functional relationships are perfect. Realizations ( X i , Y i ) solve the relation Y = f ( X ). A statistical relationship is not perfect. There is a trend plus error. Signal plus noise. 2 / 24 Section 1.1: relationships between variables A functional relationship between two variables is deterministic, e.g. Y = cos(2 . 1 x ) + 4 . 7. Although often an approximation to reality (e.g. the solution to a differential equation under simplifying assumptions), the relation itself is perfect. (e.g. page 3) A statistical or stochastic relationship introduces some error in seeing Y , typically a functional relationship plus noise . (e.g. Figures 1.1, 1.2, and 1.3; pp. 45). Statistical relationship: not a perfect line or curve, but a general tendency plus slop. 3 / 24 Whale Selenium Selenium protects marine animals against mercury poisoning. n = 20 Beluga whales were sampled during a traditional Eskimo hunt; tooth Selenium (Se) and liver Se were measured. Would be useful to be able to use tooth Selenium as a proxy for liver Selenium (easier to get). Same idea with biomarkers in biostatistics. data whale; input liver tooth @@; label liver="Liver Se (mcg/g)"; label tooth="Tooth Se (ng/g)"; datalines; 6.23 140.16 6.79 133.32 7.92 135.34 8.02 127.82 9.34 108.67 10.00 146.22 10.57 131.18 11.04 145.51 12.36 163.24 14.53 136.55 15.28 112.63 18.68 245.07 22.08 140.48 27.55 177.93 32.83 160.73 36.04 227.60 37.74 177.69 40.00 174.23 41.23 206.30 45.47 141.31 ; proc sgscatter; plot liver*tooth / reg; * or pbspline or nothing; 4 / 24 Whale Selenium Must decide what is the proper functional form for the trend in this relationship, e.g. linear, curved, piecewise continuous, cosine?...
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notes4 - Chapter 1 Timothy Hanson Department of Statistics,...

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