# Body fat percentage data for 102 elite male athletes training at the Australian Institute of Sport. ## Format A data frame with 102 rows and two columns: - **Skinfold**: Skin-fold thicknesses measured using calipers - **Bodyfat**: Percentage of fat content in the body ## Source Data collected by Dr R. Telford who was working for the Australian Institute of Sport (AIS) ```r bodyfat ``` ## Examples ```r summary(bodyfat) plot(bodyfat$Skinfold, bodyfat$Bodyfat, xlab="Skin", ylab="Fat") plot(bodyfat$Skinfold, log(bodyfat$Bodyfat), xlab="Skin", ylab="log Fat") plot(log(bodyfat$Skinfold), log(bodyfat$Bodyfat), xlab="log Skin", ylab="log Fat") old.par <- par(no.readonly = TRUE) # par(mfrow=c(2,2)) # draws four plots in a graph plot(bodyfat$Skinfold, bodyfat$Bodyfat, xlab="Skin", ylab="Fat") plot(bodyfat$Skinfold, log(bodyfat$Bodyfat), xlab="Skin", ylab="log Fat") plot(log(bodyfat$Skinfold), log(bodyfat$Bodyfat), xlab="log Skin", ylab="log Fat") plot(1:5, 1:5, axes=FALSE, xlab="", ylab="", type="n") text(2, 2, "Log both X and Y") text(2, 1, "To have the best plot") # Keep the transformed variables in the data set bodyfat$logskin <- log(bodyfat$Skinfold) bodyfat$logbfat <- log(bodyfat$Bodyfat) bodyfat$logskin <- log(bodyfat$Skinfold) par(old.par) # Create a grouped variable bodyfat$cutskin <- cut(log(bodyfat$Skinfold), breaks=6) boxplot(data=bodyfat, Bodyfat~cutskin, col=2:7) head(bodyfat) require(ggplot2) p2 <- ggplot(data=bodyfat, aes(x=cutskin, y=logbfat)) + geom_boxplot(col=2:7) + stat_summary(fun=mean, geom="line", aes(group=1), col="blue", linewidth=1) + labs(x="Skinfold", y="Percentage of log bodyfat", title="Boxplot of log-bodyfat percentage vs grouped log-skinfold") plot(p2) n <- nrow(bodyfat) x <- bodyfat$logskin y <- bodyfat$logbfat xbar <- mean(x) ybar <- mean(y) sx2 <- var(x) sy2 <- var(y) sxy <- cov(x, y) r <- cor(x, y) print(list(n=n, xbar=xbar, ybar=ybar, sx2=sx2, sy2=sy2, sxy=sxy, r=r)) hatbeta1 <- r * sqrt(sy2/sx2) # calculates estimate of the slope hatbeta0 <- ybar - hatbeta1 * xbar # calculates estimate of the intercept rs <- y - hatbeta0 - hatbeta1 * x # calculates residuals s2 <- sum(rs^2)/(n-2) # calculates estimate of sigma2 s2 bfat.lm <- lm(logbfat ~ logskin, data=bodyfat) ### Check the diagnostics plot(bfat.lm$fit, bfat.lm$res, xlab="Fitted values", ylab = "Residuals") abline(h=0) ### Should be a random scatter qqnorm(bfat.lm$res, col=2) qqline(bfat.lm$res, col="blue") # All Points should be on the straight line summary(bfat.lm) anova(bfat.lm) plot(bodyfat$logskin, bodyfat$logbfat, xlab="log Skin", ylab="log Fat") abline(bfat.lm, col=7) title("Scatter plot with the fitted Linear Regression line") # 95% CI for beta(1) # 0.88225 + c(-1, 1) * qt(0.975, df=100) * 0.02479 # round(0.88225 + c(-1, 1) * qt(0.975, df=100) * 0.02479, 2) # To test H0: beta1 = 1. tstat <- (0.88225 -1)/0.02479 pval <- 2 * (1- pt(abs(tstat), df=100)) x <- seq(from=-5, to=5, length=500) y <- dt(x, df=100) plot(x, y, xlab="", ylab="", type="l") title("T-density with df=100") abline(v=abs(tstat)) abline(h=0) x1 <- seq(from=abs(tstat), to=10, length=100) y1 <- rep(0, length=100) x2 <- x1 y2 <- dt(x1, df=100) segments(x1, y1, x2, y2) abline(h=0) # Predict at a new value of Skinfold=70 # Create a new data set called new newx <- data.frame(logskin=log(70)) a <- predict(bfat.lm, newdata=newx, se.fit=TRUE) # Confidence interval for the mean of log bodyfat at skinfold=70 a <- predict(bfat.lm, newdata=newx, interval="confidence") # a # fit lwr upr # [1,] 2.498339 2.474198 2.52248 # Prediction interval for a future log bodyfat at skinfold=70 a <- predict(bfat.lm, newdata=newx, interval="prediction") a # fit lwr upr # [1,] 2.498339 2.333783 2.662895 #prediction intervals for the mean pred.bfat.clim <- predict(bfat.lm, data=bodyfat, interval="confidence") #prediction intervals for future observation pred.bfat.plim <- suppressWarnings(predict(bfat.lm, data=bodyfat, interval="prediction")) plot(bodyfat$logskin, bodyfat$logbfat, xlab="log Skin", ylab="log Fat") abline(bfat.lm, col=5) lines(log(bodyfat$Skinfold), pred.bfat.clim[,2], lty=2, col=2) lines(log(bodyfat$Skinfold), pred.bfat.clim[,3], lty=2, col=2) lines(log(bodyfat$Skinfold), pred.bfat.plim[,2], lty=4, col=3) lines(log(bodyfat$Skinfold), pred.bfat.plim[,3], lty=4, col=3) title("Scatter plot with the fitted line and prediction intervals") symb <- c("Fitted line", "95% CI for mean", "95% CI for observation") ## legend(locator(1), legend = symb, lty = c(1, 2, 4), col = c(5, 2, 3)) # Shows where we predicted earlier abline(v=log(70)) summary(bfat.lm) anova(bfat.lm) ```

bodyfat dataset