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1 Lesson 15 Linear Regression

2 Lesson 15 Outline Review correlation analysis Dependent and Independent variables Least Squares Regression line Calculating l the slope Calculating the Intercept Residuals and Residual Plots Identifying significant relationship: t-test test of the slope R 2 : coefficient of determination Using the regression line for Prediction of Y from X Relationship between correlation coefficient and linear regression 2

3 Linear Regression and Correlation Both Linear Regression ess and Correlation o Analysis s can be used to explore the linear relationship between two continuous (quantitative) random variables. Correlation analysis is used when the interest is in identifying if a relationship exists and quantifying the strength of the relationship Regression Analysis is used to identify a relationship AND to predict the value of one variable given a value of the other variable(s). 3

4 Review: Correlation Analysis 1. Plot the data using a scatter plot to get a visual idea of the relationship 2. Calculate the correlation coefficient 1. Use Pearson s correlation coefficient if both variables are continuous 2. Use Spearman rank correlation coefficient if both variables are ordinal or one is ordinal and the other continuous. 4

5 Review: Scatter Plots and Association i Plot the 2 variables in a scatter plot (EXCEL) The pattern of the dots in the plot indicates the statistical relationship between the variables (the strength th and the direction) Positive relationship pattern goes from lower left to upper right. Negative relationship pattern goes from upper left to lower right. The more the dots cluster around a straight line with a positive or negative direction the stronger the linear relationship. 5

6 Review: Correlation Coefficient r ( x x )( y y ) [ ( x x ) 2 ][ ( y y) 2 ] The statistic r is called the Correlation Coefficient r estimated the population correlation coefficient: (the Greek letter r ) The correlation coefficient provides a measure of the linear association between two variables r is always between 1 and 1 6

7 Review: Correlation Coefficient i in Excel Use the CORREL function to find the correlation coefficient If data for one variable are in cells A1:A12 and data for other variable are in cells B1:B12, =CORREL(A1:A12,B1:B12) will return the Pearson correlation coefficient. Correlation coefficients i closer to 1 or 1 1i indicate a stronger linear relationship. Correlation coefficients close to 0 indicate a weak linear relationship. However there could be a nonlinear relationship when the correlation coefficient is close to 0. 7

8 Simple Linear Regression Like correlation analysis, Linear regression analysis is a technique that is used to explore the relationship between two continuous random variables that have a linear relationship. Regression analysis allows us to investigate the change in one variable that corresponds to a given change in the other variable. If only ONE variable is used to predict the value of the other variable, the analysis is called simple linear regression. When two or more variables are used to predict the value of the other variable, the analysis is called multiple linear regression (not covered in this course). 8

9 Linear Regression: Background Regression is from a Latin root meaning going back Linear regression as a statistical method was first described by Sir Francis Galton in his paper "Regression Towards Mediocrity in Hereditary Stature published in The Journal of the Anthropological Institute, 1886 Galton described the relationship between mid-parent height (Mid- parent height = the average of the 2 parent s height) and the height of their offspring Taller mid-parent height had children with heights closer to the average height Shorter mid-parent height had children with heights closer to the average height Galton called this phenomenon regression towards mediocrity 9

10 Sir Francis Galton: Regression When mid-parents are taller than mediocrity, their children tend to be shorter than they and When mid-parents are shorter than mediocrity, it their children tend to be taller than they 10

11 Variables in Simple Linear Regression Analysis Dependent or response variable- a variable to be predicted from or explained by the other variable The response variable is typically labeled Y Y is a continuous variable in simple linear regression Independent or explanatory variable the variable used to predict the dependant variable. This variable is typically labeled l X X can also be called the predictive variable or the regressor variable For simple linear regression X is a continuous variable For multiple linear regression X can be continuous or categorical 11

12 Identifying independent and dependent variables. In regression analysis, it s important to correctly identify the dependent d (Y) and independent d (X) variables. The study description should provide you with information about which is the dependent variable and which is the independent variable. If the study description states that the goal is to predict variable 1 from variable 2, 2 then variable 1 is the dependent variable (Y) and variable 2 is the independent variable (X). Typically, if the variables are separated in time, the variable collected first is the independent variable (X) )andthevariable collected later is the dependent variable (Y). In Galton s regression analysis, the mid-parent height was the independent variable and the offspring height was the dependent variable 12

13 Linear Regression Overview Look at a scatter plot of the data Plot Y on the y-axis and X on the x-axis Does the relationship appear to be linear? Estimate the regression line equation Find the slope and intercept of the regression line Check residuals Is the relationship statistically significant? Use a t-test test of the slope to determine significance How well does the estimated regression line equation fit the data? Calculate R 2 - the coefficient of determination Use the estimated regression line equation to predict values of fth the dependent d variable (Y)f for specified values of fth the independent variable (X). 13

14 Simple Linear Regression: An Example Is there a linear relationship between body weight and plasma volume that can be used to predict plasma volume from weight? Plasma volume is the dependent variable Y since we are interested in predicting this from body weight, the independent variable X. Body Plasma Subject Weight(kg) Volume(l)

15 Scatter plot of the Data There is a positive relationship between plasma volume and body weight. With this small number of data points it is difficult to see the linear relationship but there is a general linear trend to the data We want to identify a line that has a good fit to the data. This isn t a deterministic relationship so the points won t fall perfectly on the line. 4 Volume (liter rs) Plasma Body Weight (kg) 15

16 Estimate the Regression Line Equation A few of the many possible lines through the data points are illustrated t in the plot. How do we decide which h line best fits the data? 4 Pla asma Volum me (liters) Body Weight (kg) 16

17 Least Squares Regression Line The linear regression line is the line that gets closest to all of the points. This is called the least squares regression line. The least squares regression line minimizes the sum of the squares of the vertical distance between each observed data point (y i ) and the line minimize n ( y i 1 2 i point on linei) 17

18 Vertical distances between each observed Y (y i ) and the line are in red. The sum of these distances squared is minimized by the least squares regression line 4 Plasma a Volume (L) Body Weight (kg) 18

19 Least Squares Regression Line Equation The equation for a line requires a slope and an intercept In regression analysis, we estimate the population regression line with the least squares regression line calculated l from sample data: the sample regression line The notation for the slope and intercept in the population regression line are Greek letters for the intercept for the slope The notation for the slope and intercept in the sample regression line are Roman letters a for the intercept t b for the slope 19

20 The Population Regression Line 0 is the y -intercept of the line is the slope of the regression line 1 is the error term - the difference between the observed Y and the regression line Y X 20

21 Sample Regression Line 0 and ad 1 are aepopulation o parameters a Sample estimates for the regression parameters are : a is the estimate for b is the estimate for Y a bx is the regression line calculated from sample dt data Y is the predicted value of Y 21

22 Least Squares Regression Line aand and b are estimates of the regression coefficients and The regression coefficients are estimated from the sample data by the least squares method The intercept a is the estimated expected value of Y when X= 0 The slope b is the estimated expected change in Y corresponding to a 1 unit increase in X Y is the expected (or predicted) value of y, the point on the line. It is called the fitted value of y The following slide illustrates the least squares regression ession line 22

23 The Equation of a Regression y y Line Y a bx b a intercept 0 One-unit Change in X slope x 23

24 Interpretation of predicted values of Y The predicted value of y is the expected y-value Since not all observed data points are exactly on the regression line, there is a range of possible y-values (a distribution) for each x-value. In regression analysis the distribution of y-values for each x-value is assumed to be a normal distribution. The predicted values of y represent the mean values of the distributions of y for each specified value of x. The following slide illustrates this for 3 values of X: notice that t the mean of each distribution ib ti is on the regression line equation (the predicted value of y) and that the distribution of y-values are normal distributions. 24

25 Simple Linear Regression Model Illustrated 25

26 Assumptions for Regression Analysis There are several assumptions that should be met for regression analysis: For each value of X, the Y variable is assumed to have a normal distribution the mean of the normal distribution is the predicted value, Y The normal distributions are assumed to have equal variance across the entire range of X values. This assumption is called homogeneity or homoscedasticity. The predicted values of Y fall on the regression line representing the linear relationship between X and Y The Y observations are assumed to be independent The observations are from a random sample 26

27 Interpretation of the Slope of the Regression line The slope b is the expected change in Y corresponding to a 1 unit increase in X b = 0: There is no linear association between Y and X b > 0: There is a Positive linear association between Y and X (as X increases the expected value of Y increases) b < 0: There is a Negative linear association between Y and X (as X increases the expected value of Y decreases) The following slide illustrates a positive, negative and 0 slope. 27

28 Illustration of Negative, Positive slopes y and slope = 0 y b >0 b = 0 b < 0 0 x 28

29 Calculating the Slope of the Regression Line The formula to calculate the slope of the least squares regression line is given below b n ( x x )( y y ) i 1 i i n ( ) x x i i Notice that the numerator is the same as the numerator in the formula for the correlation coefficient. 29

30 b for plasma (Y) and body weight (X) example X Y (X- Xbar) (Y-Ybar) (X-Xbar)(Y-Ybar) (X-Xbar) Mean SUM

31 Slope of regression line From the previous slide the sum of (X-X)(Y-Y) Y) = The sum of (X-X) X) 2 = b = / = Interpretation of the slope: For every one unit increase in X, the expected increase in Y is units (rounded to 4 decimal places) Plasma volume increases liters for every one kg increase in body weight. The slope is positive indicating that as body weight (X) increases, plasma volume (Y) also increases 31

32 Calculating the Intercept of the regression line The intercept a of the regression line is the estimated value of Y when X = 0 a is calculated from the average value of Y, the average value of X and the estimated t slope b by the following formula: a Y bx 32

33 Intercept for Plasma Volume Example X Y b a * The intercept is the estimated expected value of Y when X = 0. Intercepts do not always have realistic interpretations. In this example, plasma volume is predicted to be liters when body weight = 0 kg. which h is not a possibility. 33

34 Regression Line Equation Once the slope and the intercept have been calculated the regression equation can be constructed: t Y a bx Y X This is the equation that will be used to predict plasma volume (l) from body weight (kg). The regression equation calculated from sample data is an estimate of the true population regression equation. 34

35 Regression Line Equation and interpretation i of the slope A 1 unit increase in X for this data = 1 kg so the interpretation of the slope in this regression line equation is: For each 1 kg increase in body weight, the expected increase in plasma volume is.0436 liters. What is the expected plasma volume increase for a 10 kg increase in body weight? For a 10 kilogram increase in body weight, the expected increase in plasma volume = 10* = liters. 35

36 What if the slope of the regression line is negative? If the slope of the regression line is negative we would expect a decrease in Y with each unit increase in X. The slope is a measure of the expected change in Y for each 1-unit increase in X If the slope is positive, the expected change in Y is an increase If the slope is negative, the expected change in Y is a decrease. 36

37 Regression Coefficients in Excel Excel has functions to calculate the slope and the intercept of the least squares regression line: The SLOPE function returns b - the slope =SLOPE(y-range, x-range) The INTERCEPT function returns a -the intercept =INTERCEPT(y-range, x-range) For both of these functions enter the y-range of fd data first and dth then the x-range of fth the data. 37

38 Plasma Volume Example in Excel The Lesson 15 Excel Module works through h the Plasma Volume / body weight regression example: Create a scatterplot of the data work through the calculations of the Slope and Intercept of the regression line Use the Excel Slope and Intercept functions After you ve worked through the calculations once, use the Excel functions to find the slope and intercept for future regression problems 38

39 Residuals The residual is st the ed difference ee cebet between ee the observed (Y) and the expected (Y ) value of Y Residual = Y Y Y is the observed Y for any X Y is the Y-value on the regression line for that t value of X The residual is the component of Y that is not predicted by X The least squares regression line is the line that minimizes the squared residuals 39

40 Residuals for Plasma Volume Example X Y Y' Residual Which point is closest to the regression line? Which point is furthest from the regression line? Calculate Y, the expected value of Y, using the regression line equation. The residual is the difference between Y and Y (74, 3.37) has the smallest residual (70.5, 3.49) has the largest residual 40

41 Regression Line and Residuals Largest residual Plasm ma Volume (L) Body Weight (kg) Smallest residual 41

42 Analysis of Residuals A Residual plot is a plot of the residual values on the Y- axis and the x-values on the X-axis If there is a linear relationship between X and Y, the correlation between X and the residuals should equal 0. The scatterplot will be a random scatter of points with no evident linear pattern. A nonlinear relationship between X and Y will be more evident in the residual plot of the (X, residual) data than in the scatterplot of the original (X, Y) data The Excel Regression analysis tool has an option for selecting the Residual plot. The Residual plot for the plasma volume example is on the following slide. 42

43 Residual Plot for Plasma Volume Body weight data body weight (kg) Residual Plot Re esiduals body weight (kg) No evidence of nonlinearity. The points are equally distributed around the value 0 with no evident positive or negative slope 43

44 (X, Y) Scatterplot for a nonlinear (or curvilinear) relationship When there is a curvilinear relationship between X and Y, the least squares regression line does not represent the relationship 44

45 Residual Plot for Curvilinear Relationship X Residual Plot 6 4 Residuals X This is the residual plot for the relationship on the previous slide. It illustrates that the relationship is not linear. The residual plot points aren t evenly distributed around the value 0. 45

46 Regression analysis for curvilinear relationships Simple linear regression analysis should not be used when X and Y have a curvilinear relationship There are several strategies for dealing with a curvilinear relationship between X and Y One option is to try a logarithmic transformation of the data to see if this improves the linear relationship Another option is to use piecewise regression fit one regression line to the increasing portion of the curve and a second regression line to the decreasing portion of the curve Athid third option is to include X 2 or X 3 in the regression equation (covered in PubH 6415 with multiple regression models). 46

47 Linear Regression Procedure Look at a scatter plot of the data Plot Y on the y-axis and X on the x-axis Add the trend line to the plot Estimate the regression line equation Find the slope and intercept of the regression line Check Residuals Is the relationship between X and Y statistically significant? Use a t-test test t of the slope to determine significance ifi How well does the estimated regression line equation fit the data? Calculate R 2 - the coefficient of determination Use the estimated regression line equation to predict values of the dependent variable (Y) for specified values of the independent variable (X). 47

48 Is the relationship between X and Y significant? ifi If the slope of the regression line = 0, this indicates there is no linear relationship between the variables. If there is no linear relationship the variables are considered to be independent Att t-test test t of the slope estimate t can be done to test t for independence between the X and Y variables Null hypothesis: slope = 0 The null hypothesis states t that t the variables are independent d Alternative hypothesis: slope 0 The alternative hypothesis is that there is a significant relationship between the variables If the t-test test of the slope result is significant (p-value < ), reject the null hypothesis and conclude that there is a statistically significant relationship between the two variables. 48

49 Notation for Population slope and Intercept As in any hypothesis test, the null and alternative hypotheses are stated about the population parameters, not about the estimates. The population parameters for the slope and intercept t of the regression line for the population are the Greek letters 1 and 0 1 is the population parameter for the slope 0 is the population parameter for the intercept The statistic for the t-test test of the slope will use the estimated value of the slope (b) that is calculated from the data. 49

50 t-test test of the Slope 1. State the Hypotheses Null hypothesis: = 0 Alternative hypothesis: 0 2. A t-test test will be used to test the hypothesis 3. Significance level = The degrees of freedom for a t-test test of the slope are n-2 where n=sample size The critical values of the t-test test are found using TINV(0.05, 05 df). For the plasma volume example, n = 8 so the critical values = TINV(0.05, 6) = and

51 t-test test of the slope 5. Calculate the test statistic the slope estimate divided by the standard error of the slope t b 1 SE( b 1 ) The formula for the SE of the slope is complicated so we will use the Excel Data Analysis Tool to do this t- test. The Data Analysis Tool provides the t-statistic and the p-value of the t-test test of the slope 6. State the conclusion. If the test statistic is more extreme than the critical values reject the null hypothesis and conclude that there is a significant relationship between the variables. 51

52 T-test of the Slope in Excel Data Analysis Tool output for the weight / plasma volume example: The t-statistic and p-value for the t-test of the slope are highlighted SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted d R Square Standard Error Observations 8 ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Body weight P-value for t-test test = so reject the null hypothesis and conclude that there is a significant relationship between weight and plasma volume 52

53 Regression Analysis in Excel In Excel Module 15 use the Data Analysis Tool to obtain the Regression Analysis results select Regression under the Data Analysis Tool. Enter the plasma volume data for Y-range and the weight data for X-range Check labels if you highlight the column headers Also check Residuals and Residual Plot Identify the t-statistic t ti ti and the p-value for the t-test test t of the slope. Also identify the slope and the intercept on the output table These are under the Coefficients column 95% confidence intervals for the coefficients are also provided if the Confidence Level box is checked 53

54 T-test of the Intercept The Data Analysis Tool also provides results of a t-test test of the Intercept. The Null hypothesis of this test is that the intercept = 0: = 0 The Alternative ti hypothesis of this test t is that t the intercept 0: 0 Usually there is not much interest in the t-test test of the intercept because testing whether the intercept = 0 does not provide information about the relationship between the two variables. From the Regression Table, you can see that the null hypothesis for the intercept = 0 is not rejected because the p-value = This result does not affect the significant result of the t-test test of the slope. 54

55 Linear Regression Procedure Look at a scatter plot of the data Plot Y on the y-axis and X on the x-axis Add the trend line to the plot Estimate the regression line equation Find the slope and intercept of the regression line Is the relationship statistically significant? Use a t-test test of the slope to determine significance How well does the estimated t regression line equation fit the data? Calculate R 2 - the coefficient of determination Use the estimated regression line equation to predict values of the dependent variable (Y) for specified values of the independent variable (X). 55

56 How well does the regression line equation fit the data? r 2 is st the notation otato for the ecoe coefficient ce to of determination r 2 is equal to the correlation coefficient (r) squared. It can range from 0 to 1. Interpretation of r 2 r 2 is proportion of variation in the dependent d variable (Y) that is explained by the estimated least squares regression equation. Larger values of r 2 indicate a better fit of the regression line to the data which indicates a more useful predictive model. 56

57 Calculating r 2 In Excel, you can use the CORREL function to find the correlation coefficient and square this value to find the coefficient of determination For the plasma / weight data, r = so r 2 = = Or you can find r 2 on the Data Analysis Tool Output: Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 8 Multiple R = the correlation coefficient R square = coefficient of determination (r 2 ) 57

58 Interpretation of r 2 For the plasma volume example r 2 = Interpretation: 57.6% of the variation in plasma volume is explained by the regression line equation with weight as the explanatory variable. Since only 57.6% of the variation in plasma volume is explained by body weight, there are likely other variables that explain some of the variation in plasma volume. Multiple l regression analysis uses more than one explanatory variable to predict the dependent variable This is covered in PubH 6415 If there are other explanatory variables significantly related to plasma volume in a multiple regression model, r 2 will increase 58

59 Linear Regression Procedure Look at a scatter plot of the data we have done this Plot Y on the y-axis and X on the x-axis Does the relationship appear to be linear? Estimate the regression line equation we have done this Find the slope and intercept of the regression line Is the relationship statistically significant? Use a t-test test of the slope to determine significance How well does the estimated t regression line equation fit the data? We have done this Calculate R 2 - the coefficient of determination Use the estimated regression line equation to predict values of the dependent variable (Y) for specified values of the independent variable (X). 59

60 Using the Regression Line equation for Prediction i The regression line equation for the weight and plasma volume data is: Y X For a given value of weight (X), the plasma volume (Y) can be predicted. What is the expected plasma volume for an individual who weighs 60 kg? Insert 60 in the equation in place of X and solve for Y: Y * lite liters 60

61 Predicting plasma volume for weight = 60 kg Plasma a Volume (liters) Body Weight (kg) The predicted plasma volume for weight = 60 kg is the point on the regression line corresponding to x = 60. This point is 2.7 liters. 61

62 Appropriate Applications of the Regression Line Equation Predictions using regression line equations are only valid within the range of x-values in the collected data. For the example data, the range of weight is from kgs. It would not be appropriate to use this regression line equation to predict plasma volume for an individual weighing 100 kg or an individual weighing 25 kg. There may be a different relationship between weight and plasma volume beyond the values of the collected data so the relationship identified by the regression line equation should not be extrapolated much beyond the range of the X values. 62

63 More cautions about application of Regression line predictions Predictions using Regression line equations are only valid for the population represented by the sample data. For Example, if data for a regression analysis are collected for girls age 10-18, predictions using the equation are not necessarily valid for boys, adults or girls younger than 10. You can t assume that the relationship between two variables in one population is the same in other populations. Read the study description carefully to identify the population that was sampled. Regression analysis inferences are valid for this population but not necessarily other populations. 63

64 What if there isn t a significant relationship between the variables? If regression analysis reveals that there is NOT a significant relationship between the two variables (that is if the p-value for the t-test test of the slope > ) )the ) regression equation is not useful for predicting values of the dependent variable from the independent variable. If the t-test test of the slope is NOT significant, end the regression analysis procedure and do not use the regression line equation for prediction. Prediction using the regression line equation is only useful if the null hypothesis of independence between the variables is rejected. 64

65 Relationship between Correlation and Regression The correlation coefficient and the slope of the regression line are related. For a given set of data: They will both have the same sign indicating the direction of the relationship (positive or negative). There is a mathematical ti relationship between the slope and the correlation coefficient: the slope of the regression line is equal to the correlation coefficient times the standard deviation of y divided by the standard deviation of x: b 1 rs y s x 65

66 Hypothesis Test of population correlation coefficient: i We can set up a hypothesis test of independence for the population correlation: Null Hypothesis: no significant linear association between the variables Alternative Hypothesis: 0 significant linear association between the variables The test statistic is a t-statistic with n-2 df After finding the t-statistic,,y you can use EXCEL to find the p-value = TDIST(t, n-2, 2) t r n 1 r

67 T-test of the correlation coefficient i For a given sample data, the t-test test for and the t-test test for the slope, 1, will have the same t-statistic t ti ti and p-value. For the plasma volume data, the t-statistic for the test of the population correlation coefficient = which is the same as the t-statistic t ti ti for the slope of the regression line You can work through the equation in EXCEL to confirm this P-value = TDIST( , 6, 2) = The same conclusion is reached from either hypothesis test: t there is a significant ifi relationship between the two variables The p-value < 0.05 so the null hypothesis of independence e is rejected at significance n level el

68 Linear Regression and Correlation: which to use? Both Linear Regression and Correlation Analysis can be used to explore the linear relationship between two continuous (quantitative) random variables Use Correlation analysis when the interest is primarily in identifying whether a relationship exists. Use the t-test test of the correlation coefficient to determine if the relationship is significant. Use Regression ession Analysis to identify a relationship AND to predict the value of one variable given a value of the other variable. Use the t-test test of the slope to determine if the relationship is significant Regression analysis is most useful when there is an identified interest in predicting one variable from the other(s). If prediction doesn t make sense, use correlation analysis. 68

69 Readings and Assignments Reading Chapter 8 pgs , 194, Complete the Lesson 15 Practice Exercises Lesson 15 Excel Modules Excel Module 15: Plasma Volume works through the example in this Lesson Excel Module 15: BMI works through the example in the text (pages , 206, ) 209) Complete OPTIONAL Homework 11: Use the Data Analysis Tool for the Linear Regression problems 69

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