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## Linear Regression

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**Linear Regression**William P. Wattles, Ph.D. Psychology 302**Correlation**• Teen birth rate correlated with our composite religiosity variable with r = 0.73; 95% CI (0.56,0.84); n = 49; p < 0.0005. Thus teen birth rate is very highly correlated with religiosity at the state level, with more religious states having a higher rate of teen birth. A scatter plot of teen birth rate as a function of religiosity is presented in Figure 1.http://www.reproductive-health-journal.com/content/6/1/14**“Victor, when will you stop trying to remember and start**trying to think?” --Helen Boyden**You can use linear regression to answer the following**questions about the pattern of data points and the significance of a linear equation: • 1. Is a pattern evident in a set of data points? • 2. Does the equation of a straight line describe this pattern? • 3. Are the predictions made from this equation significant?**Using Regression to predict college performance and college**satisfaction.**Dependent and Independent Variables**• Dependent Variable-or Criterion Variable The variable whose variation we want to explain. • Independent Variable-or Predictor Variable A variable that is related to or predicts variation in the dependent variable.**Examples**• SAT score, college GPA • Alcohol consumed, score on a driving test • type of car, Qualifying speed • level of education, Income • Number of boats registered, deaths of manatees**Correlation**• The relationship between two variables X and Y. • In general, are changes in X associated with Changes in Y? • If so we say that X and Y covary. • We can observe correlation by looking at a scatter plot.**Correlation example**• Is number of beers consumed associated with blood alcohol level?**Correlation**• Correlation coefficient tells us the strength and direction of the relationship between two variables.**Prediction**• If two variables are related then knowing a value for one should allow us to predict the value of the other.**Regression**• Allows us to predict one variable based on the value of another.**Regression**• Using knowledge of the relationship between X and Y to predict Y given X. • X the independent variable (predictor) used to explain changes in Y • Y the dependent variable (criterion)**Linear regression**• Regression line-a straight line through the scatter plot that best describes the relationship. • Regression line-predicts the value of Y for a given value of X.**Regression Line**• A straight line that describes how a dependentvariable changes as the independentvariable changes.**Least squares regression.**• A method of determining the regression line that minimizes the errors (residuals)**Least squares regression**• residual is the error or the amount that the observed observation deviates from the regression line. • goal to find a solution that minimizes the squared residuals • Least squares (the smallest possible sum of the squared residuals)**Least squares regression.**• a is the intercept the value of y when X=0 • b is the slope the rate of change in Y when X increases by 1**Regression formula**• a=Ybar-bXbar • b=sum of deviation products/sum of Xdev squared**The Regression Equation**• x-the independent variable, the predictor • y-the dependent variable, what we want to predict • a-the intercept • b-the slope**Population**Sample βBeta Slope α Alpha Intercept b Slope a Intercept**Relationship**• The scatterplot suggests a relationship between crying and IQ. • Can use knowledge of crying to predict IQ**Babies who cry easily may be more easily stimulated and have**higher IQ’s**Steps to Analyze Regression Data**• Plot and interpret • Numerical summary • Mathematical model**Plot and Interpret**• Plot independent variable on the X axis • Plot dependent variable on the Y axis. • Examine form, direction and strength of relationship**Correlation coefficient tells direction and strength of**relationship. r = +.455 Numerical Summary**r squared**• r2 percent of variance in Y explained by X. • =21%**Use model to predict IQ based on knowledge of crying**Least Squares regression line. Y predict=a + bx a(the intercept) =91.27 b the slope = 1.493 Mathematical Model**The slope and intercept are statistics because they are**calculated on the sample. We are really interested in estimating the population parameters Sample Statistics PopulationParameter Sample Statistic**Residuals**• Residuals-The difference between the observed value of the dependent variable and and value predicted by the regression line.**Coefficient of determination**• R2 the square of the correlation coefficient. • The amount of the variation in Y that can be explained by changes in X**Regression and correlation**• correlation tells us about the relationship • regression allows us to predict Y if we know X**Serotonin**• 5-HT levels predict mood in healthy males. • SSRI, Zoloft, Prozac**Privitera page 531**• Do levels of serotonin predict positive mood in subjects?