Lec08.Review1

# Lec08.Review1 - Biological Statistics II Review 1 What are...

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Biological Statistics II Review 1 What have we learned thus far? What tools do we now have available?

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What are we attempting to do? Find a relationship between a predictor  and a response variable? Characterize the relationship  parsimoniously Be able to predict mean and standard  error of response given predictor Check validity of assumptions
Example Relationships Intellectual ability vs. blood lead content Weight vs. length Board feet vs. diameter breast height Size vs. age Mortality vs. maximum size Human years vs. dog years

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Example Analyses Are these two variables related to one  another? How well can we predict one from the other? Hypothesis test:  Does growth rate differ between groups? Which model acts as a better predictor? Does our model (theory of how the universe  works) fit what we observe?
Topics Covered Simple Linear Regression Parameter Estimation and Prediction Assumptions and Diagnostics Matrix Notation Regression Through the Origin Bootstrap Applications

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An Example Predicting weight from height We are looking for a means of  characterizing how weight increases with  height We may want a simple way of predicting  weight from a simple measure of height We may be preparing to contrast this  characterization with others
Height 160 170 180 60 70 80 90 Weight Human Weight vs. Height

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Height 160 170 180 60 70 80 90 Weight - = + = + + = 737 . 0 1 . 57 ˆ 1 0 β ε Y i i i X Y ε β Human Weight vs. Height
Estimating Intercept and Slope H.lm  =  lm(H.weight ~ H.height) coef(H.lm)  (Intercept)  H.height     -57.10367 0.7371859 X  =  cbind(rep(1, length(H.height)),  H.height) solve(t(X) %*% X) %*% t(X) %*% H.weight                 [,1]           -57.1036722 H.height   0.7371859 Y X X X β ' ) ' ( ˆ 1 - =

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Linear Regression Package summary(H.lm) Call: lm(formula = H.weight ~ H.height) Coefficients:                   Value Std. Error   t value  (Intercept) -57.1036722  18.329628 -3.115375    H.height   0.7371859   0.106293  6.935416 (Dispersion Parameter for Gaussian family taken  to be 22.48341 ) Null Deviance: 1328.769 on 12 degrees of freedom Residual Deviance: 247.3175 on 11 degrees of  freedom
Measures of Linear Association Coefficient of Determination Coefficient of Correlation 2 2 1 R r SSTO SSE SSTO SSR R ± = - = =

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anova(sparrow.lm) Analysis of Variance Table Response: wing.length Terms added sequentially (first to last)           Df Sum of Sq  Mean Sq  F Value         Pr(F)   age.days  1  19.13221 19.13221 401.0875 5.267053e-010 Residuals 11   0.52471  0.04770                        > attach(sparrow) > SSTO = sum((wing.length-mean(wing.length))^2) > SSTO [1] 19.65692 > SSR = sum((predict(sparrow.lm)-mean(wing.length))^2) > SSR [1] 19.13221
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## This note was uploaded on 04/17/2010 for the course STSCI 3200 taught by Professor Sullivan during the Spring '10 term at Cornell.

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Lec08.Review1 - Biological Statistics II Review 1 What are...

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