1 Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship between two variables. If the researcher is working with numeric measures and supposes a linear relationship between these two variables, the appropriate measure of association is correlation. Additionally, if a particular set of assumptions is met, we can predict one of the two variables (an outcome) based on the other variable (a predictor ); this is called simple linear regression. Further, a researcher may wish to understand the relationships among more than two variables. This can be done with an extension of simple linear regression, called multiple linear regression. Recall, any statistical hypothesis test is a method for quantifying how much evidence constitutes enough evidence to declare a significant outcome in a research study. The hypothesis being tested by a correlation, and also by simple linear regression, is whether two variables have a significant linear association with each other.
2 Slide 2 Linear Regression: Examples Is higher wine consumption associated with lower rates of hear disease? What is the nature of this relationship? Is the relationship linear? What is the relationship between the number of people living on farms and the passing of time from 1935 to How fast did the number of people living on farms in the US decrease? What is the relationship between plasma volume in the blood and body weight? Do these two measures have a linear relationship? Does estriol level of a mother have a linear relationship with the birth-weight of her baby? Can we predict birth-weight of a baby from a mother s estriol level? Does the age at which a child first begins talking predict a score of mental ability later in childhood? Is there a linear relationship between systolic blood pressure and age? 2 We learned when we have a measure of two continuous variables we can describe this relationship visually with a scatter-plot. In addition, if that relationship appears to be linear, we can measure the strength and direction of the linear association. Finally, if certain assumptions are met, we may be able to predict the value of one measure from another measure. For example, is higher wine consumption associated with lower rates of hear disease? What is the nature of this relationship? Is the relationship linear? What is the relationship between the number of people living on farms and the passing of time from 1935 to In other words, how fast did the number of people living on farms in the US decrease from 1935 to 1990? What is the relationship between plasma volume in the blood and body weight? Do these two measures have a linear relationship? Can we predict plasma volume in the blood from a person s body weight? How well? Does estriol level of a mother have a linear relationship with the birth-weight of her baby? Can we predict birth-weight of a baby from a mother s estriol level? If so, can we anticipate a low birth-weight baby from estriol levels? Does the age at which a child first begins talking predict a score of mental ability later in childhood? Is there a linear relationship between systolic blood pressure and age? In all of these examples, we are investigating the relationship between two quantitative variables. We may begin this investigation with a scatter-plot followed by a correlation analysis. We will now take our investigation further by introducing simple linear regression.
4 Slide 3 Simple Linear Regression Simple Linear Regression(SLR) analysis is used to quantify the linear relationship between two quantitative variables. In this way, it is like correlation, but regression goes farther: It allows us to draw the line that best describes the linear relationship between X and Y. It allows us to predict the value of the outcome Y for a specified value of X. It allows us to quantify how much of a change in the value of Y is seen with a specified change in the value of X. In other studies the goal is to assess the relationships among a set of variables. 3 Simple linear regression analysis is used to quantify the linear relationship between two quantitative variables. In this way, it is like correlation, but regression goes farther: It allows us to draw the line that best describes the linear relationship between X and Y. It allows us to predict the value of the outcome Y for a specified value of X. It allows us to quantify how much of a change in the value of Y is seen with a specified change in the value of X.
5 Slide 4 Variable (X) and Variable (Y) We can describe the relationship or association between two quantitative variables using: Scatterplot Correlation Simple linear regression Usually we identify one variable as the outcome of interest, what we have been mostly thinking of as a disease variable so far. This is often called the response, or dependent, variable. The other variable is the predictor of interest, what we have been mostly thinking of as an exposure variable so far. This is often called the explanatory, or independent, variable. 4 Recall, usually we identify one variable as the outcome of interest, what we have been mostly thinking of as a disease variable so far. This is often called the response, or dependent, variable. The other variable is the predictor of interest, what we have been mostly thinking of as an exposure variable so far. This is often called the explanatory, or independent, variable. When each unit (person) has two measures we usually call one x and one y. If one variable can help predict the value of the other variable we call this variable x. It is also called the predictor, explanatory or independent variables. The other variable, y, is called the outcome, response variable or dependent variable. Sometimes we cannot tell which is the predictor and which is the outcome. Simple linear regression requires we pick one variable as the outcome.
6 Slide 5 Wine Consumption and Heart Disease Is higher wine consumption associated with lower rates of hear disease? What is the nature of this relationship? Is the relationship linear? Moore and McCabe, Introduction to the Practice of Statistics 4 th Edition, W. H. Freeman & Co., New York.. 5 Here is some data on wine consumption and heart disease deaths. Does this data suggest a linear relationship between these two variables?
7 Slide 6 Wine Consumption and Heart Disease 6 The data suggest a negative trend. Can we estimate how much lower heart disease rates are for each extra liter per person per year? How would we draw a line through this data to help us with this estimate? What can we say about the precision of this regression line? How much of the variability in heart disease deaths is explained by the regression line? Do you think these data come from a random sample? What assumptions are we making when using linear regression to make predictions? What confounders must we consider? These are all concepts we will investigate with linear regression.
8 Slide 7 Population Living on Farms What is the relationship between the number of people living on farms and the passing of time from 1935 to How fast did the number of people living on farms in the US decrease? 7 What is the relationship between the number of people living on farms and the passing of time from 1935 to How fast did the number of people living on farms in the US decrease? Does this data suggest a linear relationship between these two variables?
9 Slide 8 Population Living on Farms. How fast did the number of people living on farms in the US decrease? 8 We can see a strong negative trend that appears fairly linear. How might we draw a line through this data? Is there a best way to draw this line?
10 Slide 9 Plasma Volume and Body Weight What is the relationship between plasma volume in the blood and body weight? Do these two measures have a linear relationship? Body Plasma Subject Weight(kg) Volume(l) Consider the association between bodyweight in kilograms and plasma volume in the blood in liters for eight randomly selected people. Do heavier people have more plasma? If so, how much more? Is this relationship linear?
11 Slide 10 Simple Linear Regression Y, plasma volume (liters) Pearson s correlation = X, body weight (kg) 10 When we plot the data we can see a positive relationship between bodyweight and plasma. The data do not fall perfectly in a line. The correlation value when calculated is of We could calculate the value of correlation to help us understand the strength of the linear relationship. We may want to draw a line through this data, thus giving us a mathematical model to estimate plasma volume from weight, but which is the best line? The white line, the green line or the purple line? The technique of least squares regression will help us pick the line of best fit.
12 Slide 11 How Do We Choose the Best Line? The least squares regression line is the line which gets closest to all of the points How do we measure closeness to more than one point? minimize n (y i point_on_line i ) 2 i=1 11 The line of best fit is the regression line is the line that gets `closest' to all the data points. `Closeness' is measured as the vertical distance from the line to the data points. Specifically, the regression line is the one that minimizes the sum of all the squared vertical distances, hence estimation of this line is called least squares and the line is called the least square regression line.
13 Slide 12 Simple Linear Regression 12 Visually, we find the line that minimizes the squares of the vertical distances and the positive measures (points above the line) and the negative measures (points below the line), sum to zero. This could be very difficult to achieve by trial and error. We have some mathematical formulas that help us determine this exact line.
14 Slide 13 Equation of a Line Definition A line is defined by The intercept a (where the line crosses the vertical axis, the value of Y when X = 0), and The slope b (`rise over run,' how much y changes for each 1 unit change in x). y = a + bx 13 Before we move further with linear regression, let s review the equation of a line. That is, how do we represent a line with a mathematical function. A line is defined by the intercept a (where the line crosses the vertical axis, the value of Y when X = 0), and the slope b (`rise over run,' how much y changes for each 1 unit change in x). We write this as y = a + bx.
15 Slide 14 Equation of a Line 14 We can see the line crosses the vertical axis at the value a, when x = 0. We also see that for every one unit increase in x, y will change by the amount b.
16 Slide 15 Equation of a Line: Statistical Notation b b 0 1 = intercept = slope ˆ = b + b x y In statistics, the symbol for the intercept is b knot and the symbol for the slope is b sub one. Then we write the line as : y hat equals b0 + b1x. The reason we use yhat instead of y is to differentiate between the real data value y and our predicted value yhat given a value of x.
17 Slide 16 Equation of a Line: Statistical Notation y ˆ = b + b x y 0 1 b 0 b 1 slope intercept 0 x 16 Using statistical notation, we have the same picture as before. Here the line crosses the vertical axis at the value b knot, when x = 0. We also see that for every one unit increase in x, y-hat will change by the amount b sub 1.
18 Slide 17 Estimating Intercept and Slope b b 0 1 = y b x s = r s y x 1 yˆ = b + b x The least squares line minimizes the sum of squared vertical distances. This translates into: b knot equal ybar slope times xbar. The slope is the correlation times the ratio of the standard deviation of the observed y values divided by the standard deviation of the observed x values. In this way, we see the slope and the correlation are related to one another. The correlation depends on both the slope and the precision. The equations are obtained using mathematics beyond this course. It is enough to understand that these are the equations to help us determine the least squares regression line, y hat = b not plus b sub 1 times x.
19 Slide 18 y y Slope and Correlation b >0 1 b 1 = 0 b 1 < 0 0 x 18 Notice if the slope is positive then the correlation is positive. If the slope is zero then the correlation is zero. If the slope is negative then the correlation is negative.
20 Slide 19 Simple Linear Regression Y, plasma volume (liters) Pearson s correlation = X, body weight (kg) 19 The data points are represented as the dots in our scatter-plot, but the data points don't fall exactly on the line. How do we compute (and write) the least squares line for this data? Once we have the line, for any x value within the range of those values in our dataset, y-hat is the point that will fall exactly on the least squares line, not the data value for y. Thus every x value can be plugged into this equation to calculate a predicted y value which we denote y-hat.
21 Slide 20 Estimating Intercept and Slope sy b1 = r = s x = b = y b x = (66.875) = yˆ = x 20 Using the equations for estimating the slope and intercept for the least squares regression line, we get an intercept of and a slope of We must calculate the slope first because the equation for the intercept requires the use of the estimate of the slope. Generally, we do not do these calculations by hand. We use software to compute these values.
22 Slide 21 Plasma Volume and Weight yˆ = x 21 Using R we plot the least square regression line. This means for every one kilogram increase in body weight there is on average a liter increase in plasma volume. The intercept is the estimated plasma volume for a person who weighs zero kilograms. This estimate does not make biological sense. In this way, the intercept for this model is merely used to help us determine the line, not make a prediction at x = 0. The only meaningful estimates are within the range of our x values. That is weights from about 55 to 75 kilograms.
23 Slide 22 Plasma Volume and Weight Measurement of plasma volume very time consuming Body weight easy to measure: use equation and body weight to estimate plasma volume yˆ = x = (60) = Measuring plasma volume is very time consuming. We may want to estimate the plasma volume of a person outside this study based on the person s weight. For example, what on average would you expect plasma volume to be in liters for a 60 kilogram man? We would put 60 kilograms in for x and then calculate the estimated value to be 2.7 liters. That is, yhat equals * 60. Be very careful only to make estimates within the range of the data that was used to estimate the regression line. Also, be aware that measurement unit is meaningful. We would not want to insert values in pounds when the regression line is based on kilograms.
24 Slide 23 RSQUARE The square of the correlation (r 2= RSQUARE) is the fraction of the variation in the values of y that is explained by the least squares regression of y on x. r 2 variance of predicted values ŷ = variance of observed values of y = SSM SST 23 Recall Pearson s correlation: It measures the strength of the linear relationship between two quantitative variables. There is another measure called the coefficient of determination. It s value is Pearson s correlation squared. For this reason, it is often denoted RSQUARE. When using least squares regression typically the value of the coefficient of determination is used to help understand the amount of total variation that is explained by the regression of y on x. In fact, RSQUARE = SSM/SST. This is the sum of the squares of the model divided by the sum of the squares total. Those values will come from the ANOVA table in the linear regression output from the software. We will discuss the ANOVA table at length in a later lesson.
25 Slide 24 Plasma Volume and Weight This means 57.6% of the variation in plasma volume is explained by the least squares regression line of plasma volume on body weight. r 2 = 2 (0.759) = Recall, the correlation between plasma volume and weight is It we square this value, we have the coefficient of determination. The value is This means 57.6% of the variation in plasma volume is explained by the least squares regression line of plasma volume on body weight. When RSQUARE is close to 1, the regression line (the y-hat values) is representing the original data (the Y values) well. When RSQUARE is close to 0, the regression line is not representing the original data well.
26 Slide 25 Simple Linear Regression: Residuals 25 When we draw the least squares regression line, the line of best fit, the line does not fall directly on all the data points. That is, the y-hat values are different than the actual y values for the data. We call these vertical distances Residuals.
27 Slide 26 Residuals Model ˆ = b + b x y 0 1 ε = i y i yˆ i ε i =difference between observed and predicted value of response for each value of x => Called the residual. 26 y yhat for each piece of data is the residual for that point. This value is often denoted with epsilon sub i. We can calculate the value at any x in our dataset by taking the observed y value minus the predicted value, y-hat from the model. If the residual is positive, it means the data value is above the line. If the residual is negative, the data value is below the line. We will use residuals and residual plots in our next lesson to investigate how well the linear model is fitting the data observed.
28 Slide 27 Estriol and Infant Birth-weight Obstetricians sometimes order tests for estriol levels from 24-hour urine specimens taken from pregnant women who are near term. The level of estriol (mg/24 hours) has been found to be positively related to the birth-weight (grams/100) of the infant. Thus, the test can provide indirect evidence of an abnormally small fetus. [Bernard Rosner, Fundamentals of Biostatistics, page 425] 27 Let s do an another example. Obstetricians sometimes order tests for estriol levels from 24-hour urine specimens taken from pregnant women who are near term, since the level of estriol has been found to be related to the birth-weight of the infant. The test may provide indirect evidence of an abnormally small fetus.
29 Slide 28 Estriol and Infant Birth-weight Pearson' s Correlation, r = Here is the scatter-plot of birth-weight and Estriol for 31 women and babies. We can see that there is a positive relationship between estriol level and birthweight. The relationship is not perfect, but linear regression may still help with predictions. The Pearson s correlation value is Notice that birth-weight is in g/100. We will want to know this unit later for our calculations.
30 Slide 29 Estriol and Infant Birth-weight yˆ = x 29 The values of the slope and intercept can be calculated using software, or by using the equations given in earlier slides. The prediction line shown on the scatter-plot is yhat = x. This means for every one unit increase in estriol level the birth-weight of the infant is on average g/100 higher, about 60 grams.
31 Slide 30 Estriol and Infant Birth-weight Using estriol level to predict infant birth-weight when estriol level is 10mg. yˆ = x = (10) = 27.6 grams/ Suppose we want to estimate the birth-weight of a baby whose mother has an estriol level of 10 mg. Before we begin, we verify 10 mg is in the range of the original data. We can do this by looking at the scatter-plot of the data. We can then put 10 mg in the least squares regression equation for x and calculate an estimated weight of 27.6 g/100. This is 2,760 grams.
32 Slide 31 Estriol and Infant Birth-weight Using estriol level to predict infant birth-weight when estriol level is 30mg. 31 Suppose we want to estimate the birth-weight of a baby whose mother has an estriol level of 30 mg. Before we begin, we verify 30 mg is in the range of the original data. We can do this by looking at the scatter-plot of the data. We see that 30mg is NOT in the range of the x data for our study. We should not use the regression line to estimate infant birth-weight!
33 Slide 32 Estriol and Infant Birth-weight Now let's go in the reverse direction: Low birth-weight may be defined as infant birth-weight less than 2500 grams. For what estriol level is the predicted infant birth-weight equal to 2500 grams? (First convert to the correct units: 2500 grams = 25 grams/100.) 25 = x = 0.608x = x = x 32 Now let's go in the reverse direction: Low birth-weight may be defined as infant birth-weight less than 2500 grams. For what estriol level is the predicted infant birth-weight equal to 2500 grams? First we must convert to the correct units: 2500 grams = 25 grams/100. If you set 25 = x and then solve for x, you will find the estriol level that predicts a low birth-weight baby. The value of x is 5.72 mg.
34 Slide 33 Assumptions L = linear relationship between y and x. I = independence between values of y. (Value of one y does not affect value of another y). N = normality of error around each value of y. E= equality of variance around y for each value of x. 33 Linear regression requires we make some assumptions. Conveniently, these assumptions follow the acronym LINE. These assumptions are: L = = linear relationship between y and x. I = independence between values of y. One value of y does not affect another value of y. N = normality of error around each value of y. E= equality of variance around y for each value of x. Our next lesson will explore techniques to evaluate each of these assumptions.
35 Slide 34 Cautions Predicted values should only be computed for X values that fall within the range of X values in the original data. Just like a correlation, a regression line only summarizes the linear relationship between X and Y. If the relationship is truly non-linear, then using the regression line can be misleading. Seeing a relationship (an association) between X and Y does not imply causation: that changes in X will cause changes in Y. 34 In addition to evaluating linear regression assumptions, we must take caution with the interpretation of our results. Predicted values should only be computed for X values that fall within the range of X values in the original data. Just like a correlation, a regression line only summarizes the linear relationship between X and Y. If the relationship is truly non-linear, then using the regression line can be misleading. Seeing a relationship (an association) between X and Y does not imply causation: that changes in X will cause changes in Y.
36 Slide 35 Cautions In the regression context, a lurking variable is a third variable that may influence the relationship between X and Y. Outliers and skewed data can impact the regression line, just like they can impact the correlation. Either X or Y or both could have outliers or skewness. If including a particular data point changes the regression line compared to when it is not included, the data point is called influential. 35 In the regression context, a lurking variable is a third variable that may influence the relationship between X and Y. Outliers and skewed data can impact the regression line, just like they can impact the correlation. Either X or Y or both could have outliers or skewness. If including a particular data point changes the regression line compared to when it is not included, the data point is called influential. Does that seem like many `cautions'? It is: as we learn methods that are more complicated, there will often be more limits on their use and interpretation.
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