PubHlth 640 Intermediate Biostatistics Unit 2 - Regression and Correlation. Simple Linear Regression Software: Stata v 10.1

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1 PubHlth 640 Intermediate Biostatistics Unit 2 - Regression and Correlation Simple Linear Regression Software: Stata v 10.1 Emergency Calls to the New York Auto Club Source: Chatterjee, S; Handcock MS and Simonoff JS A Casebook for a First Course in Statistics and Data Analysis. New York, John Wiley, 1995, pp Setting: Calls to the New York Auto Club are possibly related to the weather, with more calls occurring during bad weather. This example illustrates descriptive analyses and simple linear regression to explore this hypothesis in a data set containing information on calendar day, weather, and numbers of calls. Data File: ERS.dta - This is a stata data set. Variable Name Label Coding/ DAY Date using informat MMDDYY6. Example: is January 16, 1993 CALLS Calls answered FHIGH Forecasted high temperature FLOW Forecasted low temperature HIGH High temperature LOW Low temperature RAIN Rain Forecast 0 = NO 1 = RAIN SNOW Snow Forecast 0 = NO 1 = SNOW WEEKDAY Type of Day 0 = NO 1 = Weekday YEAR 0 = = 1994 SUNDAY 0 = NO 1 = SUNDAY SUBZERO 0 = NO 1 = SUBZERO \stata_howto\simple linear regression ny auto club.doc Page 1 of 10

2 Key - Green: comments (note that a comment begins with an asterisk) Black: Stata command syntax. Note You do not type the leading period. Blue: Output I have also inserted some remarks. *. * Simple Linear Regression Using Stata v * toggle off the screen by screen pausing of output. set more off. * Use FILE > OPEN to read in the stata data set ers.dta. use "/Users/carolbigelow/Desktop/ers.dta". * Use the command CODEBOOK followed by a comma and the option COMPACT. * to see a compact description of the data. codebook,compact Variable Obs Unique Mean Min Max Label day calls fhigh flow high low rain snow weekday year sunday subzero * 1. Create dictionary of variable values for readability. label define rainf 0 "0=no" 1 "1=rain". label define snowf 0 "0=no" 1 "1=snow". label define weekdayf 0 "0=no" 1 "1=weekday". label define yearf 0 "0=1993" 1 "1=1994". label define sundayf 0 "0=no" 1 "1=Sunday". label define subzerof 0 "0=no" 1 "1=subzero". * 2. Associate the discrete variables with their dictionary of value codes. label values rain rainf. label values snow snowf. label values weekday weekdayf. label values year yearf. label values sunday sundayf. label values subzero subzerof \stata_howto\simple linear regression ny auto club.doc Page 2 of 10

3 . *2. Use command LIST to produce a listing of the data.. list day calls fhigh flow high low rain snow weekday year sunday subzero day calls fhigh flow high low rain snow weekday year sunday subzero =no 0=no 0=no 0=1993 0=no 0=no =no 0=no 0=no 0=1993 1=Sunday 0=no =no 0=no 0=no 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =rain 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 0=no 0=1993 0=no 0=no =rain 0=no 0=no 0=1993 1=Sunday 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =rain 1=snow 1=weekday 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 1=weekday 0=1993 0=no 0=no =no 0=no 0=no 1=1994 1=Sunday 1=subzero =rain 1=snow 0=no 1=1994 0=no 0=no =rain 0=no 1=weekday 1=1994 0=no 0=no =no 0=no 1=weekday 1=1994 0=no 1=subzero =rain 1=snow 1=weekday 1=1994 0=no 1=subzero =no 0=no 1=weekday 1=1994 0=no 1=subzero =no 0=no 0=no 1=1994 0=no 0=no =no 0=no 0=no 1=1994 1=Sunday 0=no =no 0=no 1=weekday 1=1994 0=no 0=no =no 0=no 1=weekday 1=1994 0=no 0=no =rain 1=snow 1=weekday 1=1994 0=no 0=no =no 0=no 1=weekday 1=1994 0=no 1=subzero =rain 1=snow 1=weekday 1=1994 0=no 0=no =rain 1=snow 0=no 1=1994 0=no 0=no \stata_howto\simple linear regression ny auto club.doc Page 3 of 10

4 . *3. Look at your data first! Plot of Y=calls versus X=low. * The following command of SET SCHEME is optional. I downloaded this particular scheme previously.. set scheme lean1. graph twoway (scatter calls day, symbol(d)), title("calls to NY Auto Club "). * At the top bar of the GRAPH window click on the SAVE ICON to save your graph!. * Suggestion: From the drop down menu, choose.png extension. It s nice for cut and paste. * I saved my graph as nyauto_graph01.png" Source: nyauto_graph01.png The scatterplot suggests, as we might expect, that lower temperatures are associated with more calls to the NY Auto club. \stata_howto\simple linear regression ny auto club.doc Page 4 of 10

5 . *4. Descriptives on the outcome variable Y=calls. * Use command SUMMARIZE followed by comma and then followed by option DETAIL. summarize calls, detail calls Percentiles Smallest 1% % % Obs 28 25% Sum of Wgt % 3062 Mean Largest Std. Dev % % Variance % Skewness % Kurtosis *4. continued - Assess assumption of normality both graphically and with hypothesis test. * There are multiple graphs you might consider.. * Here I do a histogram with the y-axis defined as frequency and with an overlay normal. histogram calls, frequency normal title("histogram of Y=CALLS with overlay NORMAL") (bin=5, start=1674, width=1454.6). * save graph as nyauto_graph02.png Source: nyauto_graph02.png The graph shows what we suspected nonnormality of Y=CALLS. \stata_howto\simple linear regression ny auto club.doc Page 5 of 10

6 .* There are also a variety of tests of normality. One is the Shapiro Wilk Test..* See Unit 2 lecture notes page 54.* Null : Distribution of calls is normal. Under Null, test statistic W is close to 1.* Evidence of NON normality is reflected in W < 1 and small p-value. swilk calls Shapiro-Wilk W test for normal data Variable Obs W V z Prob>z calls The null hypothesis of normality of Y=CALLS is rejected. Take care, sometimes the cure is worse than the problem. For now, we ll continue along anyway; this will give us a chance to see some interesting diagnostics!. * 6. Least Squares estimation and analysis of variance table.. regress calls low Source SS df MS Number of obs = F( 1, 26) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = calls Coef. Std. Err. t P> t [95% Conf. Interval] low _cons alls ˆ = 7, *[low] The fitted line is c R 2 =.51 indicates that 51% of the variability in calls is explained. The overall F test significance level <.0001 suggests that the straight line fit performs better in explaining variability in calls than does Y = average # calls From this output, the analysis of variance is the following (next page) \stata_howto\simple linear regression ny auto club.doc Page 6 of 10

7 Source Df Sum of Squares Mean Square Model 1 n Regression ( Yˆ ) 2 i Y = 100,233,719 SS(model)/1 i= 1 = 100,233,719 Residual (n-2) = 26 n 2 Error ( Y ˆ i Yi i= 1 3,673, Total, corrected (n-1) = 27 n 2 Y Y ) = 95,513,596.2 SS(residual)/(n-2) = ( i ) = 195,747,315 i= 1. *7. Overlay of straight line fit on the scatter plot. graph twoway (scatter calls low, symbol(d)) (lfit calls low), title("calls to NY Auto Club ") subtitle("overlay Straight Line Fit"). * save graph as nyauto_graph03.png Source: nyauto_graph03.png The overlay of the straight line fit is reasonable but substantial variability is seen, too. There is a lot we still don t know, including but not limited to the following --- Case influence, omitted variables, variance heterogeneity, incorrect functional form, etc. \stata_howto\simple linear regression ny auto club.doc Page 7 of 10

8 . *8. Residuals Analysis - Assessment of Normality of Residuals. * Stata requires that you use post-estimation commands to obtain residuals. * We consider a few here... * Use command PREDICT varname, RESIDUALS to save residuals to a variable you name. predict e, residuals. * Use command PNORM to plot residuals e versus percentiles of normal. * Reasonableness is suggested by points falling along the line. pnorm e, title("normality of Residuals of Y=calls v X=low"). * save graph as nyauto_graph04.png Source: nyauto_graph04.png Not bad, actually. \stata_howto\simple linear regression ny auto club.doc Page 8 of 10

9 . *9. Residuals analysis - Detection of Outliers Using Cooks Distance. * See Unit 2 lecture notes page 60. * Use command PREDICT varname, COOKSD to save residuals to a variable you name. predict cook, cooksd. * Preliminary to plot of cook s distance, we need to create an ID variable. * This is because the data set ers.dta does not have an ID variable. Most do.. * Use command GENERATE varname=_n to save the system variable _n to a variable you name.. generate id=_n. * Plot Cook s distance values on Y-axis versus id on the X-axis. Look for extreme values. graph twoway (scatter cook id, symbol(d)), title("cook's Distance Values") subtitle("simple Linear Regression of Y=calls on X=low"). * save graph as nyauto_graph05.png Source: nyauto_graph05.png For straight line regression, the suggestion is to regard Cook s Distance values > 1 as significant.. Here, there are no unusually large Cook Distance values. Not shown but useful, too, are examinations of leverage and jackknife residuals. \stata_howto\simple linear regression ny auto club.doc Page 9 of 10

10 . *10. Assessing Assumptions of Linearity, Heteroscedasticity, Independence using Jacknife Residuals. * See Unit 2 notes page 59. * In Stata, jackknife residuals are referred to as studentized residuals. * Use command PREDICT varname, XB to save predicted outcomes to a variable you name. predict predicted, xb. * Use command PREDICT varname, RSTUDENT to save jackknife residuals to a variable you name. predict jack, rstudent. graph twoway (scatter jack predicted, symbol(d)), title("jacknife Residuals versus Predicted"). * save graph as nyauto_graph06.png Source: nyauto_graph06.png Recall A jackknife residual for an individual is a modification of the solution for a studentized residual in which the mean square error is replaced by the mean square error obtained after deleting that individual from the analysis. This plot in SAS is nice for its inclusion of some useful summaries the fitted line, the R 2 Departures of this plot from a parallel band about the horizontal line at zero are significant. The plot here is a bit noisy but not too bad considering the small sample size. \stata_howto\simple linear regression ny auto club.doc Page 10 of 10