Simple RegressionPowerPoint PresentationSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Simple Regression •correlation vs. prediction research•prediction and relationship strength•interpreting regression formulas–quantitative vs. binary predictor variables–raw score vs. standardized formulas•selecting the correct regression model•regression as linear transformation (how it works!)•“equally predictive” vs. “equivalent predictors”•process of a prediction studyCorrelation Studies vs. Prediction StudiesCorrelation research (95%) • purpose is to identify the direction and strength of linear relationship between two quantitative variables• usually theoretical hypothesis-testing interestsPrediction research (5%)• purpose is to take advantage of linear relationships between quantitative variables to create (linear) models to predict values of hard-to-obtain variables from values of available variables• use the predicted values to make decisions about people (admissions, treatment availability, etc.)However, to fully understand important things about the correlation models requires a good understanding of the regression model upon which prediction is based...Linear regression for prediction... • linear regression “assumes” there is a linear relationship between the variables involved• “if two variables aren’t linearly related, then you can’t use one as the basis for a linear prediction of the other”• “a significant correlation is the minimum requirement to perform a linear regression”• sometimes even a small correlation can lead to useful prediction (if it is not a Type I error)• must have a “meaningful” criterion in order to obtain a useful prediction formulaLet’s take a look at the relationship between the strength of the linear relationship and the accuracy of linear prediction.• for a given value of X• draw up to the regression line• draw over the predicted value of YPredictor XCriterionXY’When the linear relationship is very strong, there is a narrow range of Y values for any X value, and so the Y’ “guess” will be closeNotice that everybody with the same X score will have the same predicted Y score.There won’t be much error, though, because there isn’t much variability among the Y scores for any given X score.YPredictorCriterionXY’However, when the linear relationship is very weak, there is a wide range of Y values for any X value, and so the Y’ “guess” will be less accurate, on the average.There is still some utility to the linear regression, because larger values of X still “tend to” go with larger values of Y.So the linear regression might supply useful information, even if it isn’t very precise -- depending upon what is “useful”? Notice that everybody with the same X score will have the same predicted Y score.Now there will be more error, because there is a lot of variability among the Y scores for any given X score.PredictorCriterionXY’When there is no linear relationship, everybody has the same predicted Y score – the mean of Y.This is known as “univariate prediction” – when we don’t have a working predictor, our best guess for each individual is that they will have the mean.Some key ideas we have seen are:• everyone with a given “X” value will have the same predicted “Y” value• if there is no (statistically significant & reliable) linear relationship, then there is no basis for linear prediction (bivariate prediction)• the stronger the linear relationship, the more accurate will be the linear prediction (on the average)XXPredictors, predicted criterion, criterion and residualsHere are two formulas that contain “all you need to know”y’ = bx + a residual = y - y’x the predictor -- variable related to criterion that you will use to make an estimate of criterion value for each participanty’ the predicted criterion value -- “best guess” of each participant’s y value, based on their x value --that part of the criterion that is related to (predicted from) the predictory the criterion -- variable you want to use to make decisions, but “can’t get” for each participant (time, cost, ethics)residual difference between criterion and predicted criterion values -- the part of the criterion not related to the predictor -- the stronger the correlation the smaller the residual (on average)Simple regression y’ = bx + a raw score formb -- raw score regression slope or coefficient a -- regression constant or y-intercept For a quantitative predictor a = the expected value of y if x = 0b = the expected direction and amount of change in y for a 1-unit change in xFor a binary x with 0-1 coding a = the mean of y for the group coded “0”b = the direction and amount of difference in the mean of y between the group coded “0” and the group coded “1”Let’s practice -- quantitative predictor ... #1 depression’ = (2.5 * stress) + 23apply the formula -- patient has stress score of 10 dep’ =interpret “b” -- for each 1-unit increase in stress, depression is expected to by interpret “a” -- if a person has a stress score of “0”, their expected depression score is #2 job errors = ( -6 * interview score) + 95apply the formula -- applicant has interview score of 10, expected number of job errors is interpret “b” -- for each 1-unit increase in intscore, errors are expected to by interpret “a” -- if a person has a interview score of “0”, their expected number of job errors is 48increase 2.52335decrease 695Let’s practice -- binary predictor ... #1 depression’=(7.5 * tx group) +15.0 code: Tx=1 Cx=0interpret “b” -- the Tx group has mean than Cxinterpret “a” -- mean of the Cx group (code=0) is so … mean of Tx group is #2 job errors = ( -2.0 * job) + 8 code: mgr=1 sales=0the mean # job errors of the sales group is the mean difference # job errors between the groups is the mean # of job errors of the mgr group is 7.5 more1522.58-26standard score regression Zy’ = Zx for a quantitative predictor  tells the expected Z-score change in the criterion for a 1-Z-unit change in that predictor, for a binary predictor,  tells size/direction of group mean difference on criterion variable in Z-unitsWhy