1. STATS 330: Lecture 16 Case Study 7/17/2018 330 lecture 16

2. Case study Aim of today’s lecture
To illustrate the modelling process using the evaporation data.
7/17/2018
330 lecture 16
STATS 330 lect 16

3. The Evaporation data Data in data frame evap.df Aims of the analysis:
Understand relationships between explanatory variables and the response
Be able to predict evaporation loss given the other variables
7/17/2018
330 lecture 16
STATS 330 lect 16

4. Case Study: Evaporation data
Recall from Lecture 15: variables are
evap: the amount of moisture evaporating from the soil in the 24 hour period (response)
maxst: maximum soil temperature over the 24 hour period
minst: minimum soil temperature over the 24 hour period
avst: average soil temperature over the 24 hour period
maxat: maximum air temperature over the 24 hour period
minat: minimum air temperature over the 24 hour period
avat: average air temperature over the 24 hour period
maxh: maximum humidity over the 24 hour period
minh: minimum humidity over the 24 hour period
avh: average humidity over the 24 hour period
wind: average wind speed over the 24 hour period.
7/17/2018
330 lecture 16
STATS 330 lect 16

5. Modelling cycle Choose Model Fit model Examine residuals Transform
Bad fit
Good fit
Use model
Plots, theory
7/17/2018
330 lecture 16
STATS 330 lect 16

6. Modelling cycle (2) Our plan of attack: Graphical check
Suitability for regression
Gross outliers
Preliminary fit
Model selection (for prediction)
Transforming if required
Outlier check
Use model for prediction
7/17/2018
330 lecture 16
STATS 330 lect 16

7. Step 1: Plots Preliminary plots
Want to get an initial idea of suitability of data for regression modelling
Check for linear relationships, outliers
Pairs plots, coplots
Data looks OK to proceed, but evap/maxh plot looks curved
7/17/2018
330 lecture 16
STATS 330 lect 16

8. 7/17/2018
330 lecture 16

9. Points to note Avh has very few values
Strong relationships between response and some variables (particularly maxh, avst)
Not much relationship between response and minst, minat, wind
strong relationships between min, av and max
No obvious outliers
7/17/2018
330 lecture 16
STATS 330 lect 16

10. Step 2: preliminary fit Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
avst *
minst
maxst *
avat
minat
maxat
avh
minh
maxh **
wind
Residual standard error: on 35 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 10 and 35 DF, p-value: 2.073e-11
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12. Findings Plots OK, normality dubious
Gam plots indicated no transformations
Point 31 has quite high Cooks distance but removing it doesn’t change regression much
Model is OK.
Could interpret coefficients, but variables highly correlated.
7/17/2018
330 lecture 16

13. Step 3: Model selection Use APR Model selected was
evap ~ maxat + maxh + wind
However, this model does not fit all that well (outliers, non-normality)
Try “best AIC” model
evap ~ avst + maxst + maxat + minh+maxh
Now proceed to step 4
7/17/2018
330 lecture 16
STATS 330 lect 16

14. Step 4: Diagnostic checks
For a quick check, plot the regression object produced by lm
model1.lm<-lm(evap ~ avst + maxst + maxat + minh+maxh, data=evap.df)
plot(model1.lm)
7/17/2018
330 lecture 16
STATS 330 lect 16

15. Outliers ?
Non-normal?
7/17/2018
330 lecture 16
STATS 330 lect 16

16. Conclusions?
No real evidence of non-linearity, but will check further with gams
Normal plot looks curved
Some largish outliers
Points 2, 41 have largish Cooks D
7/17/2018
330 lecture 16
STATS 330 lect 16

17. Checking linearity Check for linearity with gams > library(mgcv)
>plot(gam(evap ~ s(avst) + s(maxst) +
s(maxat) + s(maxh) + s(wind),
data=evap.df))
7/17/2018
330 lecture 16
STATS 330 lect 16

18. Transform avst, maxh ?
7/17/2018
330 lecture 16
STATS 330 lect 16

19. Remedy Gam plots for avst and maxh are curved
Try cubics in these variables
Plots look better
Cubic terms are significant
7/17/2018
330 lecture 16
STATS 330 lect 16

20. 7/17/2018
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21. > summary(model2.lm) Coefficients:
> model2.lm<-lm(evap ~ poly(avst,3) + maxst + maxat + minh+poly(maxh,3), data=evap.df)
> summary(model2.lm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) **
poly(avst, 3) **
poly(avst, 3) *
poly(avst, 3)
maxst e-05 ***
maxat **
minh
poly(maxh, 3) e-05 ***
poly(maxh, 3)
poly(maxh, 3) *
---
Residual standard error: on 36 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 9 and 36 DF, p-value: 4.459e-15
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330 lecture 16

22. New model > influenceplots(model2.lm) Lets now adopt model
lm(evap~poly(avst,3)+maxst+maxat+poly(maxh,3) + wind
Outliers are not too bad but lets check
> influenceplots(model2.lm)
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STATS 330 lect 16

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25. Deletion of points
Points 2, 6, 7, 41 are affecting the fitted values, some coefficients. Removing these one at a time and refitting indicates that the cubics are not very robust, so we revert to the non-polynomial model
The coefficients of the non-polynomial model are fairly stable when we delete these points one at a time, so we decide to retain them.
7/17/2018
330 lecture 16
STATS 330 lect 16

26. Normality?
However, the normal plot for the non-polynomial model is not very straight – WB test has p-value 0.
Normality of polynomial model is better
Try predictions with both
7/17/2018
330 lecture 16
STATS 330 lect 16

27. predict.df = data.frame(avst = mean(evap.df$avst),
maxst = mean(evap.df$maxst),
maxat = mean(evap.df$maxat),
maxh = mean(evap.df$maxh),
minh = mean(evap.df$minh))
rbind(predict(model1.lm, predict.df,interval="p" ),
predict(model2.lm, predict.df,interval="p" ))
fit lwr upr
CV fit:
7/17/2018
330 lecture 16