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lm() Function.pptxsfdfsfsfsfsfsfsfsdfsdfsfsfs
- 1. Function lm() inR and its Basic Parameters Jiong Xun
- 2. Learning Objectives - Understandlinear regression - Understand purpose of lm() function - Using lm() to fit regression models - Interpret output of lm() function
- 3. Ever Wondered How… -Maps are able to estimate your travelling time - Surcharge pricing are determined to meet demands for taxi - HDB resale prices are forecasted
- 4. What is LinearRegression? - Interested in the relationship between a dependent variable (y) and one or more independent variables (x) - Models relationships between variables - Simple Linear Regression (1 independent variable) - Multiple Linear Regression (2 or more independent variables) - Finds best-fit line that minimises distance between observed data values and predicted values
- 5. How do weobtain best-fit line?
- 6. Ordinary Least Squares -Find the best-fit line want the line to be as ⇒ close to data points as possible - Minimise vertical distance between each point to line - Residual Sum of Squares (RSS) ⇒ squared sum of residuals for all data points - Squared as we do not want residuals to “cancel off” one another Minimise RSS Minimum total distance between line and points Best-fit line
- 7. Visualised on aSimple Plot Actual Data Points Residuals Regression Line
- 8. How to WeUse R to plot Linear Regression Model?
- 9. Syntax of lm() Functionlm() stands for linear model ⇒
- 10. Example Dataset “trees”Inches ft Cubic ft
- 11. Example Dataset “trees” Yvariable (response) X variable (explanatory) Name of data frame that model is using
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- 13. Example Dataset “trees” Differencebetween observed and predicted values
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- 15. Example Dataset “trees” Usedto predict value of the response variable
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- 17. Example Dataset “trees” Averageamount that estimate varies from actual value
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- 19. Example Dataset “trees” tvalue = Estimate / std. Error
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- 21. Example Dataset “trees” p-valuefor the t-test to determine if coefficient is significant
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- 23. Example Dataset “trees” Standarddeviation of the residuals Number of data points that went into estimation
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- 25. Example Dataset “trees” Givesa measurement of what % of variance in response can be explained by the regression
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- 27. Example Dataset “trees” Indicatesif model as a whole is statistically significant
- 28. Example Dataset “trees” PredictVolume of tree based on Girth and Height of tree?
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Editor's Notes
- #10 Simple linear regression (SLR) (black cherry trees)
- #13 Min ⇒ represents the data point furthest below the regression line 1Q ⇒ 25% of the residuals are less than this number 3Q ⇒ 25% of teh residuals are greater than this number Max ⇒ point that is furthest from the regression line Mean of residuals not shown as it will always be 0 Now, what can we infer just from the residuals alone to determine if this can be a good linear regression model? Median of residuals ideally should be as close to 0 as possible (hard to preface what is close or far as it is relative to your data) Why ⇒ this would imply our model isnt skewed one way or another Another would be that it is symmetrically distributed ⇒ want min and max to have same magnitude, as well as 1Q and 3Q
- #15 Moving on to coefficients, the intercept will always be given, including any x variables that we have provided ⇒ y=mx+c
- #17 Standard error ⇒ responsible for the width of the confidence interval Ideally, you want a lower number relative to the estimate for standard error (e.g. standard error is small compared to est)
- #21 We use 0.05 as a benchmark, it means that it is unlikely that the relationship between the y variable and x variable is due to chance ⇒ statistically significant Stars on the right shows the significant levels, as seen in the significant codes
- #23 Standard deviation of residuals ⇒ average distance of points from the regression line, adjusted with the number of points Degrees of freedom ⇒ number of observations (31) minus number of variables (29)
- #25 Multiple R-squared ⇒ Generally gets better with more x variables Adjusted R-squared ⇒ Penalises for adding useless predictor (x) variables If multi R-squared is much higher than your adjusted, your model might be overfitted
- #27 Want a value way bigger than 1 to be statistically significant, or you can just refer to p-value where if it is <0.05 it would be significant
- #28 Multiple linear regression