Download presentation

Presentation is loading. Please wait.

Published byElvin Conley Modified over 2 years ago

1
Simple Linear Regression

2
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?

3
Beers and BAC

4
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

5
Decomposition of information into explained and unexplained parts

6
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.

7
Variance “Candy Bar” ExplainedUnexplained SS explained by model SS Total SS Error Systematic SS + Random SS = Total SS

8
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

9
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

10
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

11
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

12
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.

13
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.

14
Confidence interval for mean response

15
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.

16
Prediction interval for a future observation

17
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)

18
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.

19
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

Similar presentations

OK

Regression Analysis Regression analysis is a statistical technique that is very useful for exploring the relationships between two or more variables (one.

Regression Analysis Regression analysis is a statistical technique that is very useful for exploring the relationships between two or more variables (one.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Cardiovascular system anatomy and physiology ppt on cells Ppt on osmosis and diffusion Ppt on marketing management Ppt on motivation theories Ppt on digital media piracy control Ppt on condition based maintenance army Urinary bladder anatomy and physiology ppt on cells 3d holographic display ppt online Download ppt on civil disobedience movement in war Ppt on teamviewer 6