Data Analysis Using R ASC & OIR

Transcription

1 Data Analysis Using R ASC & OIR

2 Overview } What is Statistics and the process of study design } Correlation } Simple Linear Regression } Multiple Linear Regression 2

3 What is Statistics? Statistics is a science that uses mathematical theory and probability to develop rigorous methods for collection, classification, analysis, and interpretation of numerical facts or data. Statistical inference: drawing a conclusion about a population or a general phenomenon on the basis of a limited sample. 3 -

4 Study Design Study subject - one member of a set of entities being studied. Goal: describe population characteristics. 4

5 5 Correlation

6 Quantity of interest: Correlation Measures the relative strength of the linear relationship between two variables } Ranges between 1 and 1 } Unit-less } The closer to 1, the stronger the negative linear relationship } The closer to 1, the stronger the positive linear relationship } The closer to 0, the weaker any positive linear relationship 6

7 Scatter Plots of Data with Various Correlation Coefficients r = -1 r = -.6 r = 0 r = 1 r =.3 7 nslide from: Statistics for Managers Using Microsoft Excel 4th Edition, 2004 Prentice-Hall r = 0

8 Linear Correlation Linear relationships Curvilinear relationships r =.1 r =.6 r = -.8 r = +=.3 8 nslide from: Statistics for Managers Using Microsoft Excel 4th Edition, 2004 Prentice-Hall

9 Linear Correlation Strong relationships Weak relationships 9 nslide from: Statistics for Managers Using Microsoft Excel 4th Edition, 2004 Prentice-Hall

10 Linear Correlation No relationship 10 nslide from: Statistics for Managers Using Microsoft Excel 4th Edition, 2004 Prentice-Hall

11 11 Simple Linear Regression

12 Introduction } Regression analysis is a technique of modeling relationship between an outcome (or dependent) variable and a set of explanatory (or independent) variables } For correlation calculations, the two variables are treated as equals. In regression, one variable is considered independent (=predictor) variable () and the other the dependent (=outcome) variable. } Used to } Predict outcome for a given set of values of independent variables and asses prediction precision; } Determine the effect (if any) of each of the independent variables on the outcome, controlling for all other variables. 12

13 What is Linear? } Equation for a line: =m+b m = slope: A slope of 2 means that every 1-unit change in yields a 2-unit change in. B = intercept: =B if =0 m B 13

14 Example: Data on Professor Evaluations load(url(" variable score rank ethnicity gender language age cls_perc_eval cls_did_eval cls_students cls_level cls_profs cls_credits bty_f1lower bty_f1upper bty_f2upper bty_m1lower bty_m1upper bty_m2upper bty_avg pic_outfit pic_color description average professor evaluation score: (1) very unsatisfactory - (5) excellent. rank of professor: teaching, tenure track, tenured. ethnicity of professor: not minority, minority. gender of professor: female, male. language of school where professor received education: english or non-english. age of professor. percent of students in class who completed evaluation. number of students in class who completed evaluation. total number of students in class. class level: lower, upper. number of professors teaching sections in course in sample: single, multiple. number of credits of class: one credit (lab, PE, etc.), multi credit. beauty rating of professor from lower level female: (1) lowest - (10) highest. beauty rating of professor from upper level female: (1) lowest - (10) highest. beauty rating of professor from second upper level female: (1) lowest - (10) highest. beauty rating of professor from lower level male: (1) lowest - (10) highest. beauty rating of professor from upper level male: (1) lowest - (10) highest. beauty rating of professor from second upper level male: (1) lowest - (10) highest. average beauty rating of professor. outfit of professor in picture: not formal, formal. color of professor s picture: color, black & white. 14

15 Regression line: Estimates average Evaluation Score by Beauty Score score=0.067(bty_avg)+3.88+error m = slope = B = intercept =

16 Linear Regression in R > LM_model <- lm(score ~ bty_avg, data = evals) Ø summary(lm_model) Call: lm(formula = score ~ bty_avg, data = evals) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** bty_avg e-05 *** Residual standard error: on 461 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 461 DF, p-value: 5.083e-05 Regression formula Ignore this part Estimates of regression parameters along with their estimation errors Output that describes model quality 16

17 Regression Model Assessment Residual standard error: on 461 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 461 DF, p-value: 5.083e-05 } Residual standard error measures the spread of points around the line. } Adjusted R-squared, roughly, measures proportion of variability in outcomes explained by =bty_eval: } is usually between 0 and 1, } values closer to 1 are better, } for most applications it is difficult to obtain adj-r 2 above

18 Estimating Response at a Particular Value of score=0.067(bty_avg)+3.88+error 18

19 Estimation Using Simple Linear Regression score=0.067(bty_avg)+3.88+error Interpolation: What is the estimated average evaluation score for a professor with beauty average score of 5? Average(score)=0.067(5)+3.88=4.21 Extrapolation: What is the estimated average evaluation score for a professor with beauty average score of 1? Average(score)=0.067(1)+3.88=

20 Estimation Error for Future Response at a Particular Value of = 0 Estimated evaluation score for a professor with beauty average score of 5 is What is the error of this prediction? > predict(lm_model, newdata=data.frame(bty_avg=5), + interval = "prediction") fit lwr upr Evaluation score is estimated to be 4.21, with a 95% prediction interval (3.16, 5.27*). Adjust the upper limit: (3.16, 5.00). 20

21 21 Multiple Linear Regression

22 Multiple Linear Regression More than one predictor! = a + b 1 * + b 2 *W + b 3 *Z + Each regression coefficient is the amount of change in the outcome variable that would be expected per one-unit change of the predictor, if all other variables in the model were held constant. Why adjust for covariates? Control for confounders Test for interactions between predictors (effect modification) Improve predictions 22

23 Multiple Linear Regression > m_bty_gen <- lm(score ~ bty_avg + gender, data = evals) > summary(m_bty_gen) Call: lm(formula = score ~ bty_avg + gender, data = evals) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** bty_avg e-06 *** gendermale *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 460 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 2 and 460 DF, p-value: 8.177e-07 23

24 Multiple Regression Model Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** bty_avg e-06 *** gendermale *** score= (bty_avg)+0.172(Male)+error } Intercept is significantly different from zero. } First slope is significantly different from 0. For male and female professors, a 1-unit increase in beauty score is associated with increase in the average evaluation score. } Second slope is significantly different from 0. For a male and a female professors with the same beauty scores, male professors average evaluation score is expected to be higher by points. 24

25 Recommended Reading } Practical Regression and Anova Using R, by Julian Faraway, available for free: } Data Analysis Using Regression and Multilevel/Hierarchical Models, Gelman and Hill, with emphasis on R coding. } The Statistical Sleuth: A Course in Methods of Data Analysis, by Ramsey and Schafer, 2 nd or 3 rd edition, with R commands that reproduce the textbook displays available here 25

26 Questions or comments? me at 26