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Simple Linear Regression

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Start by exploring the data Construct a scatterplot Does a linear relationship between variables exist? Is the relationship strong? How much variation can be explained by a linear relationship with the independent or explanatory variable?

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Beers and BAC

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Variance “Candy Bar” ExplainedUnexplained The R-sq value: estimates the percentage of variation explained by a linear relationship with the independent or explanatory variable. Unless this estimate is 100% (or very near), it is not sufficient on its own. The amounts of explained and unexplained information due to the model are measured by Sums of Squares

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Decomposition of information into explained and unexplained parts

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Residuals residual A residual is the difference between an observed value of the dependent variable and the value predicted by the regression line. Residual = (observed y) - (predicted y)= y – ŷ They help us assess the fit of a regression line.

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Variance “Candy Bar” ExplainedUnexplained SS explained by model SS Total SS Error Systematic SS + Random SS = Total SS

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Model Assumptions about the residuals (ε) The distribution is NORMAL The mean is ZERO The variance is CONSTANT for all values of x (σ 2 ) Errors associated with any two observations are independent

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Assessing the utility of the model: model variance Variance is variability of the random error (σ 2 ) The higher the variability of the random error, the greater the error of prediction σ 2 is estimated with s 2 (often called the mean square for error, MSE) Variance: s 2 = SSE/degrees of freedom (n-2) Standard error: This is like standard deviation; with standard error, we are looking at deviation from the line Approximately 95% of observed y values will lie within 2s of their respective predicted values

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Assessing the utility of the model: Slope Does y change as x changes? Does x contribute information for the prediction of y? Test this with the t-statistic or p-value (p<.05); these values are included in software output

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Assessing the utility of the model: Correlation Coefficient r Measure of the strength and direction of the linear relationship between x and y Always between -1 and +1 High correlation does not imply causality

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Assessing the utility of the model: Coefficient of Determination (r 2) The R squared value is the % of the variation in y explained by the model. For linear regression, the higher the value, the better the model.

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Using the model for estimation and prediction: Confidence interval for mean response For any specific value of x: A confidence interval for adds to this estimate a margin of error based on the standard error. Confidence intervals widen as the value of x is further from its mean.

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Confidence interval for mean response

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Prediction interval for a future observation Similar to confidence interval for mean response Standard error used in prediction interval includes Variability due to the fact that the least- squares line is not exactly equal to the true regression line Variability of the future response variable y around the subpopulation mean.

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Prediction interval for a future observation

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In the MINITAB regression window, you might want to… Set confidence levels in Options Enter a value for prediction in Options Store Residuals and Fits in Storage Display full table of fits and residuals in Results (select last bullet)

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Beware of Extrapolation Extrapolation is the use of a regression line for prediction far outside the range of values of the independent variable x that you used to obtain the line. Such predictions are not accurate.

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Example from book: p. 138 How can we tell if it is reasonable to fit a linear regression model? Let’s run the analysis and interpret the results

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