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

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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
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    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?
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    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
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    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
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    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
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    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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Lec08.Review1 - Biological Statistics II Review 1...

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