simple linear models straight line is simplest case, but key is that parameters appear linearly in...
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![Page 1: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/1.jpg)
Simple linear models
• Straight line is simplest case, but key is that parameters appear linearly in the model
• Needs estimates of the model parameters (slope and intercept)- usually by least squares
• Makes a number of assumptions, usually checked graphically using residuals
![Page 2: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/2.jpg)
Examples for linear regression
• How is LOI related to moisture?• How should we estimate merchantable volume of wood
from the height of a living tree?• How is pest infestation late in the season affected by
the concentration of insecticide applied early in the season?
![Page 3: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/3.jpg)
Scatterplot of tree volume vs height
![Page 4: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/4.jpg)
Minitab commands
![Page 5: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/5.jpg)
Regression Output
![Page 6: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/6.jpg)
Interpreting the output
• Goodness of fit (R-squared) and ANOVA table p-value?• Confidence intervals and tests for the parameters• Assessing assumptions (outliers and influential
observations• Residual plots
![Page 7: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/7.jpg)
t = distance between estimate and hypothesised value, in units of standard error
t Coef SECoef
vs tcrit
CI Coef tcrit SECoef
Confidence intervals and t-tests
![Page 8: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/8.jpg)
Confidence intervals and t-tests
![Page 9: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/9.jpg)
Confidence intervals and t-tests
![Page 10: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/10.jpg)
Regression output
![Page 11: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/11.jpg)
Outliers
![Page 12: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/12.jpg)
Residual plots
Standardized Residual
Perc
ent
210-1-2
99
90
50
10
1
Fitted Value
Sta
ndard
ized R
esi
dual
5040302010
2
1
0
-1
-2
Standardized Residual
Fre
quency
210-1
8
6
4
2
0
Observation Order
Sta
ndard
ized R
esi
dual
30282624222018161412108642
2
1
0
-1
-2
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for VOLUME
![Page 13: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/13.jpg)
Confidence and prediction intervals
HEIGHT
VOLU
ME
90858075706560
80
60
40
20
0
-20
S 13.3970R-Sq 35.8%R-Sq(adj) 33.6%
Regression95% CI95% PI
volume as a function of heightVOLUME = - 87.12 + 1.543 HEIGHT
![Page 14: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/14.jpg)
Low R-sq
High R-sq
Low p-value: significant High p-value: non-significant
Four possible outcomes
![Page 15: Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope](https://reader035.vdocument.in/reader035/viewer/2022062619/5515c987550346a3758b4a66/html5/thumbnails/15.jpg)
• Not because relationships are linear• Transformations can often help linearise• Good simple starting point – results are well understood• Approximation to a smoothly varying curve
Why linear?