Regression and Models with Multiple Factors. Ch. 17, 18

Transcription

1 Regression and Models with Multiple Factors Ch. 17, 18

2 Mass Scatter Plot Snout-Vent Length

3 Mass Linear Regression Snout-Vent Length

4 Least-squares The method of least squares chooses the line that minimizes the squared deviations along the Y-axis Y = bx + a The slope of the line is the average change in variable Y given a change in variable X The slope is usually of greater interest than the intercept The slope is the covariance between X and Y divided by the variance in X a positive slope implies a positive covariance and a negative slope implies a negative covariance

5 Linear Regression Uses Depict the relationship between two variables in an eye-catching fashion Test the null hypothesis of no association between two variables The test is whether or not the slope is zero Predict the average value of variable Y for a group of individuals with a given value of variable X Note that variation around the line can make it very difficult to predict a value for a given individual with much confidence

6 What are Residuals?

7 What are Residuals? Chimps have larger testes than predicted for their body weight Gorillas have smaller testes than predicted for their body weight Chimps have a positive residual Gorillas have a negative residual

8 What are Residuals In general, the residual is the individual s departure from the value predicted by the model In this case the model is simple the linear regression but residuals also exist for more complex models For a model that fits better, the residuals will be smaller on average Residuals can be of interest in their own right, because they represent values that have been corrected for relationships that might be obscuring a pattern (e.g., the body weighttestes mass relationship)

9 Strong Inference for Observational Studies Noticing a pattern in the data and reporting it represents a post hoc analysis This is not hypothesis testing The results, while potentially important, must be interpreted cautiously What can be done? Based on a post-hoc observational study, construct a new hypothesis for a novel group or system that has not yet been studied For example, given the primate data, a reasonable prediction is that residual testes mass in deer will be associated with mating system Collect a new observational data set from the new group to test the hypothesis

10 Phylogenetic Pseudoreplication

11 Performing Linear Regression in R Run the regression: > reg_newt <- lm(mass ~ svl, data=males) > summary(reg_newt) > confint(reg_newt) #get the CI for slope Add the line to a plot: > plot(mass ~ svl, data=males) > abline(reg_newt) See for code to draw the line s 95% CI on the graph

12 Interpreting the Output Call: lm(formula = mass ~ svl, data = males) Residuals: Min 1Q Median 3Q Max Slope Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) *** svl e-09 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 46 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 46 DF, p-value: 8.712e-09 p-value for slope

13 Mass Least Squares Regression with 95% CI Snout-Vent Length

14 Code for That males <- read.csv("malenewtsampledata.csv") plot(males$mass ~ males$svl, pch=19, xlab=list("snout-vent Length", cex=1.5), ylab=list("mass",cex=1.5)) box(lwd=3) axis(1,lwd=3) axis(2,lwd=3) reg_newt <- lm(mass ~ svl, data=males) abline(reg_newt, lwd=3, col="maroon") summary(reg_newt) #Add confidence limits for the regression line xpt <- seq(min(males$svl),max(males$svl), length.out=100) ypt <- data.frame(predict(reg_newt, newdata=data.frame(svl=xpt), interval="confidence")) lines(ypt$lwr ~ xpt, lwd=2, lty=2) lines(ypt$upr ~ xpt, lwd=2, lty=2)

15 Assumptions of Linear Regression The true relationship must be linear At each value of X, the distribution of Y is normal (i.e., the residuals are normal) The variance in Y is independent of the value of X Note that there are no assumptions about the distribution of X

16 Common Problems Outliers Regression is extremely sensitive to outliers The line will be drawn to outliers, especially along the x-axis Consider performing the regression with and without outliers Non-linearity Best way to notice is by visually inspecting the plot and the line fit Try a transformation to get linearity [often a log transformation] Non-normality of residuals Can be detected from a residual plot Possibly solved with a transformation Unequal variance Usually visible from a scatterplot or from a residual plot

17 residuals(reg_newt) Residual Plot svl plot(residuals(reg_newt) ~ svl, data=males, pch=19)

18 Examples of Problems

19 Outlier Example

20 Multiple Explanatory Variables The reason ANOVA is so widely used is that it provides a framework to simultaneously test the effects of multiple factors ANOVA also makes it possible to detect interactions among the factors ANOVA is a special case of a general linear model

21 General Linear Models GLMs handle categorical factors and continuous factors in a single modeling framework ANOVA is a special case with just categorical explanatory variables Linear regression is a special case with just continuous explanatory variables

22 Two-factor ANOVA Testing: The mean of serum starved versus normal culture The mean of wild-type versus WTC911 The interaction (i.e., wild-type responds differently than WTC911 to the change in culturing conditions [serum starved versus normal culture])

23 ANOVA with Blocking Analysis is similar to two-factor ANOVA (at least it was in 1995) Except: Typically do not test for an interaction term Don t care if the block is significant or not Remember: you don t care about the block; you simply want to reduce the variance introduced by it

24 Analysis of Covariance Used to test for a difference in means, while correcting for a variable that is correlated with the response variable The slopes must not differ in the two groups In other words, the mean comparison is only valid if the interaction term is not significant Also used to compare the slope of two regression lines If the interaction term is significant, then the conclusion is that the slopes are different

25 ANCOVA Example

26 ANCOVA Example

27 Summary Statistical models can be quite complex, with potentially many factors and interaction terms The model is specified by something that looks like an equation: Y = μ + A + B + A*B General linear models allow you to combine categorical and continuous factors into a single model Your sample size will limit the complexity of the model For complex models, you will need to think about procedures to choose the best model (i.e., the model that best explains your observations)