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Linear regression,[object Object],Ryan Sain, Ph.D.,[object Object]
Regression Introduced,[object Object],Regression is about prediction,[object Object],Predicting an unknown point based on observations (or measurements),[object Object],Widgets sold based on advertising,[object Object],We can explore known relationships,[object Object],We can explore unknown relationships,[object Object]
The Variables,[object Object],Outcome Variable,[object Object],The thing we are predicting,[object Object],Number of widgets sold,[object Object],Predictor Variable (simple regression),[object Object],The variables that you know about,[object Object],Advertising dollars,[object Object],Predictor Variables (multiple regression),[object Object],We predict values of a dependent variable (outcome) using one or more independent variables (predictors),[object Object]
The model,[object Object],Any prediction follows the basic formula:,[object Object],Outcomei = (model) + errori,[object Object],In regression our model contains several things:,[object Object],Slope of the line (that best fits the data measured) = b1,[object Object],Intercept of the line (at the Y axis) b0 ,[object Object],So our model = Yi = (b0 + b1Xi) + Errori,[object Object],Do you recognize this equation?,[object Object],The model is simply a line,[object Object]
So how do we calculate this line?,[object Object],The Method of Least Squares,[object Object],The line that is the closest to all the data points,[object Object],Residuals = Deviations (distance of actual data points to the line),[object Object],Square these residuals to get rid of negatives,[object Object],Then sum them. ,[object Object]
How well does this line fit?,[object Object],No line is perfect (there are always residuals),[object Object],If our line is a good one it should be better than a basic line (significantly so),[object Object],We compare our line to a basic line:,[object Object],Deviation = SUM (observed – model)2,[object Object],This is basically a ‘mean’ (model),[object Object],The mean is an awful predictor,[object Object],No matter how much you spend on adverts – the sales of your widgets are the same,[object Object]
Fitness continued,[object Object],SSt = total sum of squared differences (using the mean),[object Object],SSr= total residual sum of squares (using our best fit model),[object Object],Represents a degree of inaccuracy,[object Object],SSm (model sum of squares) = SSt – SSr,[object Object],Large = our model is different than the simple model,[object Object],Proportion of improvement:,[object Object],R2 = SSm/ SSt,[object Object],Percentage of variation in the outcome that can be explained by our model,[object Object]
More fitness,[object Object],You can assess this using an F test as well,[object Object],F is simply systematic variance/unsystematic variance,[object Object],In regression that means:,[object Object],Improvement of the model (SSm - systematic) and the difference between the model and the observed data (SSr – unsystematic),[object Object],But we need to look at mean squares,[object Object],Because we need to use the average sums of squares in an F test.,[object Object],So we divide by degrees of freedom,[object Object],For SSm = the number of variables in the model,[object Object],For SSr = number of observations – the number of parameters being estimated (number of beta coefficients or predictors),[object Object],F = MSM / MSR,[object Object]
Individual Predictors,[object Object],The coefficient bis essentially the gradient of the line,[object Object],If the predictor is not valuable then it will predict no change in the outcome as it changes. ,[object Object],This would be b= 0 ,[object Object],This is what the mean does,[object Object],If the predictor is valuable – then it will be significantly different than 0. ,[object Object]
Individual Predictors cont.,[object Object],To test if b is different from 0 we will use a t-test.,[object Object],We are comparing how big the b value is in comparison to the amount of error in that estimate.,[object Object],We will then use the standard error of the bvalue. ,[object Object],t = bobserved – bexpected / SEb,[object Object],Since the expected value is 0 (no change) then we have to simply divide the observed b value by the standard error of b to get the t score.,[object Object],Degress of freedom is calculated using the following: N – p – 1 (p = number of predictors),[object Object]

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