1. Linear Regression Exploring relationships between two metric variables

2. Correlation The correlation coefficient measures the strength of a relationship between two variables The relationship involves our ability to estimate or predict a variable based on knowledge of another variable

3. Linear Regression The process of fitting a straight line to a pair of variables. The equation is of the form: y = a + bx x is the independent or explanatory variable y is the dependent or response variable

4. Linear Coefficients Given x and y, linear regression estimates values for a and b The coefficient a, the intercept, gives the value of y when x=0 The coefficient b, the slope, gives the amount that y increases (or decreases) for each increase in x

6. x=1:5 y <- 2.5*x + 1 plot(y~x, xlim=c(0, 5), ylim=c(0, 14), yaxp=c(0, 14, 14), las=1, pch=16) abline(lm(y~x)) points(0, 1, pch=8) points(mean(x), mean(y), cex=3) segments(c(1, 2), c(3.5, 3.5), c(2, 2), c(3.5, 6)) text(c(1.5, 2.25), c(3, 4.75), c("1", "2.5")) text(mean(x), mean(y), "x = mean(x), y = mean(y)", pos=4, offset=1) text(0, 1, "y-intercept = 1", pos=4) text(1.5, 5, "slope = 2.5/1 = 2.5", pos=2) text(2, 12, "y = 1 + 2.5x", cex=1.5)

7. Least Squares Many lines could fit the data depending on how we define the “best fit” Least squares regression minimizes the squared deviations between the y-values and the line

8. lm() Function lm() performs least squares linear regression in R Formula used to indicate Dependent/Response from Independent/Explanatory Tilde(~) separates them D~I or R~E Rcmdr Statistics | Fit model | Linear regression

9. > RegModel.1 <- lm(LMS~People, data=Kalahari) > summary(RegModel.1) Call: lm(formula = LMS ~ People, data = Kalahari) Residuals: Min 1Q Median 3Q Max -86.400 -24.657 -2.561 24.902 86.100 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -64.425 44.924 -1.434 0.175161 People 12.868 2.591 4.966 0.000258 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 47.88 on 13 degrees of freedom Multiple R-squared: 0.6548,Adjusted R-squared: 0.6282 F-statistic: 24.66 on 1 and 13 DF, p-value: 0.0002582

11. plot(LMS~People, data=Kalahari, pch=16, las=1) RegModel.1 <- lm(LMS~People, data=Kalahari) abline(Line) segments(Kalahari$People, Kalahari$LMS, Kalahari$People, RegModel.1$fitted, lty=2) text(12, 250, paste("y = ", as.character(round(RegModel.1$coefficients[[1]], 2)), " + ", as.character(round(RegModel.1$coefficients[[2]], 2)), "x", sep=""), cex=1.25, pos=4)

12. Errors Linear regression assumes all errors are in the measurement of y There are also errors in the estimation of a (intercept) and b (slope) Significance for a and b is based on the t distribution

13. Errors 2 The errors in the intercept and slope can be combined to develop a confidence interval for the regression line We can also compute a prediction interval which is the confidence we have in a single prediction

14. predict() predict() uses the results of a linear regression to predict the values of the dependent/response variable It can also produce confidence and prediction intervals: –predict(RegModel.1, data.frame(People = c(10, 20, 30)), interval="prediction")

16. RegModel.1 <- lm(LMS~People, data=Kalahari) plot(LMS~People, data=Kalahari, pch=16, las=1) xp<-seq(10,25,.1) yp<-predict(RegModel.1,data.frame(People=xp),int="c") matlines(xp,yp, lty=c(1,2,2),col="black") yp<-predict(RegModel.1,data.frame(People=xp),int="p") matlines(xp,yp, lty=c(1,3,3),col="black") legend("topleft", c("Confidence interval (95%)", "Prediction interval (95%)"), lty=c(2, 3))

17. Diagnostics Models | Graphs | Basic diagnostic plots –Look for trend in residuals –Look for change in residual variance –Look for deviation from normally distributed residuals –Look for influential data points

19. Diagnostics 2 influence(RegModel.1) returns –Hat (leverage) coefficients –Coefficient changes (leave one out) –Sigma, residual changes (leave one out) –wt.res, weighted residuals

20. Other Approaches rlm() fits a robust line that is less influenced by outliers sma() in package smatr fits standardized major axis (aka reduced major axis) regression and major axis regression – used in allometry