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SIMPLE LINEAR REGRESSION ►Linear regression determines the strength & direction of a relationship between two variables measured at the interval or ratio level. ►Linear regression formalizes the relationship by designating (X) as the independent variable and (Y) as the dependent variable. MORE ON LINEAR REGRESSION.. ►Regression develops an equation that allows us to predict the value of our “outcome” variable Y, on the basis of a specified value of our “predictor” variable X. ►The simple linear regression formula is: Y = a + bX ►Statisticalprediction using linear regression is widely used in “real world” research. Insurance firms use predictor variables to determine auto insurance premiums. Employers administer aptitude & personality tests to predict job performance. Professional schools do the same with undergraduate marks & aptitude tests (e.g., LSAT, MSAT, GRE, GMAT, etc.,). ►If two variables are perfectly correlated, knowing the value of X, allows us to perfectly predict the value of Y. ►As long as 2 variables are significantly correlated, we can use scores on X to predict scores on Y. For example, the strong correlation between social support (X) and mental health (Y) implies that if we know a person’s social support level, we can accurately predict their mental health level (Y). SIMPLE LINEAR REGRESSION EXAMPLE ► Y = a + bx ► Y the sum of: (1) the average duration of employment denoted by the intercept a, and (2) an additional average amount of employment duration due to a counseling session denoted by the slope b. ►The intercept a is the value Y takes when X equals 0. It is the average duration of post-release employment if a young offender attends no counseling sessions at all. ►The slope b (the regression coefficient) is the amount of change in Y (up or down), that is caused by a one unit increase in X. Here it is the average increase in employment duration caused by attending each ► Differencebetween observed & predicted values of Y is the error in the regression. The higher the correlation, the more accurate our predictions of Y using X . ►r²is improvement in our predictions of Y using X….square Pearson’s r = .85, we obtain r² =.72. This is the coefficient of determination and means we improve our predictions of Y by 72% using X. Coefficient of non-determination is 1- r², and it is the percentage of variability in Y not explained by X. Multiple Linear Regression: ►Is a logical and mathematical extension of simple linear regression to situations where we have one interval-level dependent variable, and two or more interval-level independent (predictor variables). Y = a + b1X1 + b2X2 Y … respondent score on the dependent variable. a … the intercept. b1…the regression coefficient for the first predictor (X1). b2…the regression coefficient for the second predictor (X2). X1…respondent score on first predictor variable. X2…respondent score on second predictor variable. Logistic Regression ►Logistic regression predicts the probability of a dependent variable Y (measured as a nominal/ordinal dichotomous outcome) using a predictor variable measured at the interval level. Logistic Regression 2 ►Used in medical research to determine variables that predict whether a tumor is likely to be cancerous or benign, what variables predict probability of a heart attack or stroke, etc., ►In social science, used to predict the odds that convicts will recidivate or not; which variables predict if couples will get divorced or not, etc., ►In linear regression, we predict values of Y using a combination of each predictor variable multiplied by its respective regression coefficient as illustrated in the formula: Y = a + b1X1 z P(Y) = 1/1+ e where z = a + b1X1 P(Y)…probability of Y occurring. e…base of natural logarithms z…sum of simple linear regression coefficients (intercept and slope).
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