VIEWS: 7 PAGES: 12 POSTED ON: 9/30/2012
Simple Linear Regression In many scientific investigations, one is interested to find how something is related with something else. For example the distance traveled and the time spent driving; one’s age and height. Generally, there are two types of relationships between a pair of variable: deterministic relationship and probabilistic relationship. Deterministic relationship s s0 vt distance S: distance travel S0: initial distance v slope v: speed S0 t: traveled intercept time Probabilistic Relationship In many occasions we are facing a different situation. One variable is related to another variable as in the following. height age Here we can not definitely predict one’s height from his age as we did in s s0 vt Linear Regression Statistically, the way to characterize the relationship between two variables as we shown before is to use a linear model as in the following: y a bx Here, x is called independent variable Error: y is called dependent variable is the error term a is intercept y b is slope b a x Least Square Lines Given some pairs of data for independent and dependent variables, we may draw many lines through the scattered points y x The least square line is a line passing through the points that minimize the vertical distance between the points and the line. In other words, the least square line minimizes the error term . Least Square Method For notational convenience, the line that fits through the points is often written as y a bx ˆ The linear model we wrote before is y a bx If we use the value on the line, ŷ , to estimate y, the difference is (y- ŷ) For points above the line, the difference is positive, while the difference is negative for points below the line. y y a bx ˆ ŷ (y- ŷ) Error Sum of Squares For some points, the values of (y- ŷ) are positive (points above the line) and for some other points, the values of (y- ŷ) are negative (points below the line). If we add all these up, the positive and negative values can get cancelled. Therefore, we take a square for all these difference and sum them up. Such a sum is called the Error Sum of Squares (SSE) n SSE ( y y ) 2 ˆ i 1 The constant a and b is estimated so that the error sum of squares is minimized, therefore the name least square. Estimating Regression Coefficients If we solve the regression coefficients a and b from by minimizing SSE, the following are the solutions. n ( x x )( y i i y) b i 1 n ( xi x ) 2 i 1 a y bx Where xi is the ith independent variable value yi is dependdent variable value corresponding to xi x_bar and y_bar are the mean value of x and y. Interpretation of a and b The constant b is the slope, which gives the change in y (dependent variable) due to a change of one unit in x (independent variable). If b> 0, x and y are positively correlated, meaning y increases as x increases, vice versus. If b<0, x and y are negatively correlated. y y a b<0 b>0 a x x Correlation Coefficient Although now we have a regression line to describe the relationship between the dependent variable and the independent variable, it is not enough to characterize the relationship between x and y. We may see the situation in the following graphs. (1) (2) y y x x Obviously the relationship between x and y in (1) is stronger than that in (2) even though the line in (2) is the best fit line. The statistic that characterizes the strength of the relationship is correlation coefficient or R2 How R2 is Calculated? y ˆ y y y y ( y y) ( y y ) ˆ ˆ If we use y_bar to represent y, the error is (y-y_bar). If we use ŷ to represents y, the error is (y- ŷ ). Therefore the error is reduced to (y- ŷ ). Thus (ŷ- y_bar ) is the improvement over using y_bar. This is true for all points in the graph. To account how much total improvement we get, we take a sum of all improvements, (ŷ -y_bar). Again we face the same situation as we did while calculating variance. We take the square of the difference and sum the squared difference for all points R Square Regression Sum of Squares n y SSR ( yi y ) ˆ 2 ˆ y i 1 Total Sum of Squares y n SST ( yi y ) 2 i 1 SSR R2 SST R square indicates the percent variance in y explained by the regression. We already calculated SSE (Error Sum of Squares) while estimating a and b. In fact, the following relationship holds true: SST=SSR+SSE An Simple Linear Regression Example The followings are some survey data showing how much a family spend on food in relation to household income (x=income in thousand $, y=is percent of income left after spending on food) x y x-x_bar y-y_bar (x-x_bar)(y-y_bar) (x-x_bar)^2 y_hat (y-y_bar)^2 (y_hat-y_bar)^2 (y-y_hat)^2 6.5 81 1.185714 1.571429 1.863265306 1.40591837 73.254325 2.46938776 38.12130132 59.99548121 4 96 -1.31429 16.57143 -21.77959184 1.72734694 86.2722 274.612245 46.83527158 94.63009284 2.5 93 -2.81429 13.57143 -38.19387755 7.92020408 94.082925 184.183673 214.7501205 1.172726556 7.2 68 1.885714 -11.4286 -21.55102041 3.55591837 69.60932 130.612245 96.41767056 2.589910862 8.1 63 2.785714 -16.4286 -45.76530612 7.76020408 64.922885 269.897959 210.4148973 3.697486723 3.4 84 -1.91429 4.571429 -8.751020408 3.6644898 89.39649 20.8979592 99.35942913 29.12210432 5.5 71 0.185714 -8.42857 -1.565306122 0.0344898 78.461475 71.0408163 0.935272739 55.67360918 sum 37.2 556 -135.7428571 26.0685714 953.714286 706.8339631 246.8814117 mean 5.31429 79.4286 slope -5.2071 intercept 107.101 SST 953.714 SSR 706.834 SSE 246.881 SST+SSR 953.715 R-square 0.74114