Statistics 333 Chapter 10 Exercise 28 Spring 2003 The questions of interest in Exercise 28 from Chapter 10 are to explore the effects of El Nin o temperature and rain in West Africa on the number of tropical storms hurricanes and a storm index for the Atlantic Basin For the lecture I will examine the effects of these variables on the number of tropical storms Your homework assignment asks you to do a similar analysis on each of the response variables There is no single correct analysis of this data For example my analysis will differ substantially from that in the solution manual for instructors This document shows the R commands but none of the output or graphs to carry out a rather thorough analysis of the data from this exercise El Nin o temperature is categorized as cold neutral and warm and West African seasons are classified as wet or dry The data set in the textbook creates an artificial numerical coding of El Nin o temperature as 1 0 and 1 and also has the West African variable coded as 0 or 1 Our first task will be to create a new data set that has these variables as proper categorical variables ex1028 read table sleuth ex1028 csv header T sep attach ex1028 africa rep A ncol ex1028 africa west africa 0 dry africa west africa 1 wet x data frame year el nino africa storms hurricanes index storm index rm africa detach attach x Next we should make some plots of the response storm versus the explanatory variables year el nino and africa I will make two scatterplots of storms vesus year one showing elnino with different symbols and one showing africa with different symbols par mfrow c 1 2 levelsElNino levels as factor el nino levelsAfrica levels as factor africa plot year storms type n for i in 1 length levelsElNino set el nino levelsElNino i points year set storms set pch i legend 1950 19 5 levelsElNino pch 1 length levelsElNino plot year storms type n for i in 1 length levelsAfrica set africa levelsAfrica i points year set storms set pch i legend 1950 19 5 levelsAfrica pch 1 length levelsAfrica par mfrow c 1 1 It is clear from the plots that there is a relationship of storms with both el nino and africa There is no obvious time trend We can also consider similar plots for the log transformed variable par mfrow c 1 2 levelsElNino levels as factor el nino levelsAfrica levels as factor africa plot year log storms type n for i in 1 length levelsElNino set el nino levelsElNino i points year set log storms set pch i legend 1950 19 5 levelsElNino pch 1 length levelsElNino Bret Larget March 17 2003 Statistics 333 Chapter 10 Exercise 28 Spring 2003 plot year log storms type n for i in 1 length levelsAfrica set africa levelsAfrica i points year set log storms set pch i legend 1950 19 5 levelsAfrica pch 1 length levelsAfrica par mfrow c 1 1 We can consider a linear model to predict storms based on all three variables fit1 lm storms year el nino africa summary fit1 plot fitted fit1 residuals fit1 abline h 0 lty 2 lines lowess fitted fit1 residuals fit1 The summary indicates that there is at most marginal evidence of a time effect but that both el nino and africa have significant effects The residual plot does not indicate any great need for a transformation although we could try a log transformation for the fun of it I have added a local regression fit to the residual plots to make spotting nonlinear trends easier fit2 lm log storms year el nino africa summary fit2 plot fitted fit2 residuals fit2 abline h 0 lty 2 To compare the two fits we could examine normal probability plots of the residuals par mfrow c 1 2 qqnorm residuals fit1 qqnorm residuals fit2 Next let me try a fit without year but with an interaction between elnino and africa for both the untransformed and transformed variable fit3 lm storms el nino africa summary fit3 plot fitted fit3 residuals fit3 abline h 0 lty 2 lines lowess fitted fit3 residuals fit3 fit4 lm log storms el nino africa summary fit4 plot fitted fit4 residuals fit4 abline h 0 lty 2 lines lowess fitted fit4 residuals fit4 In both cases the interaction term is not significant So lets make fits without the interaction terms fit5 lm storms el nino africa summary fit5 plot fitted fit5 residuals fit5 abline h 0 lty 2 lines lowess fitted fit5 residuals fit5 fit6 lm log storms el nino africa summary fit6 plot fitted fit6 residuals fit6 abline h 0 lty 2 lines lowess fitted fit6 residuals fit6 Bret Larget March 17 2003 Statistics 333 Chapter 10 Exercise 28 Spring 2003 Now there is only marginal evidence that the variable africa is significant when storms is transformed or not Finally I will fit a model with log storms as the response using el nino as the sole explanatory variable fit7 lm log storms el nino summary fit7 plot fitted fit7 residuals fit7 abline h 0 lty 2 lines lowess fitted fit7 residuals fit7 Again only one of the levels of el nino is significant There is little evidence for treating cold and neutral values separately Here is an eighth fit with a single indicator variable that el nino is warm warm as factor el nino warm notWarm as factor el nino warm fit8 lm log storms warm summary fit8 plot fitted fit8 residuals fit8 abline h 0 lty 2 Finally here is a ninth fit with the log transformed variable and notWarm as the only explanatory variable to make find 95 confidence intervals easier fit9 lm log storms notWarm summary fit9 Here are some calculations useful for the summary below fit8 coef summary fit8 coefficients fit9 coef summary fit9 coefficients tcrit qt 0 975 46 est8 round exp fit8 coef 1 1 est9 round exp fit9 coef 1 1 lo8 round exp fit8 coef 1 1 tcrit hi8 round exp fit8 coef 1 1 tcrit lo9 round exp fit9 coef 1 1 tcrit hi9 round exp fit9 coef 1 1 tcrit fit8 coef 1 fit8 coef 1 fit9 coef 1 fit9 coef 1 2 2 2 2 After all of this fitting and plotting and analysis I would conclude that the simplest model with an indicator only for warm El Nin o temperatures is sufficient Such a simple model may not be sufficient for the other response variables you are considering My solution to the problem would be this Exercise 10 28 part a I considered several models for estimating the number of tropical storms in the Atlantic Basin based on El Nin o temperature whether or not West Africa was rainy or dry and time The variables were el nino temperature levels cool neutral and warm africa levels rainy and dry and year Models for the logarithm of the number of storms were more nearly linear as judged by residual plots and had residuals with more similar variability
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