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Stat 6601 project Linear Statistical Models Analysis of Covariance Example

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### Stat 6601 project Linear Statistical ModelsAnalysis of Covariance Example

By

Gadir Marian

Myrna Moreno

Data

- ‘Whiteside’ data, Mr. Derek recorded weekly gas consumption and average external temperature at his house during two ‘heating seasons’ one before and after cavity-wall insulation was installed.
- Variables:

- Insul (levels: before or after insulation)

- Temp (the average outside temperature in

degrees Celsius)

-Gas (The weekly gas consumption in 1000

cubic feet units)

Goal

- Assess the effect of the insulation on gas consumption.

Method

- Linear Model for Analysis of Covariance

Y= +X +

Where:

is a random effect due to treatment.

is a fixed effect due to covariate.

is a random error.

Method(continued)

- Using R:

-A primary model is fitted using a “model

fitting function”

lm (formula, data, weights, subset, na.action)

- A resulting “fitted model object” can be

analysed, interrogated or modified.

Codes

- require(latice)
- xyplot(Gas ~ Temp | Insul, whiteside, panel =
- function(x, y, ...) {
- panel.xyplot(x, y, ...)
- panel.lmline(x, y, ...)
- }, xlab = "Average external temperature (deg. C)",
- ylab = "Gas consumption (1000 cubic feet)", aspect = "xy",
- strip = function(...) strip.default(..., style = 1))
- gasB <- lm(Gas ~ Temp, whiteside, subset = Insul=="Before")
- gasA <- update(gasB, subset = Insul=="After")
- summary(gasB)
- summary(gasA)
- gasBA <- lm(Gas ~ Insul/Temp - 1, whiteside)
- summary(gasBA)
- gasQ <- lm(Gas ~ Insul/(Temp + I(Temp^2)) - 1, whiteside) summary(gasQ)$coef
- gasPR <- lm(Gas ~ Insul + Temp, whiteside)
- anova(gasPR, gasBA)
- options(contrasts = c("contr.treatment", "contr.poly"))
- gasBA1 <- lm(Gas ~ Insul*Temp, whiteside)
- summary(gasBA1)$coef

Results

- The output from fitting regression model:

Residuals:

Min 1Q Median 3Q Max

-0.97802 -0.18011 0.03757 0.20930 0.63803

Coefficients:

Estimate Std. Error t value Pr(>|t|)

InsulBefore 6.85383 0.13596 50.41 <2e-16 ***

InsulAfter 4.72385 0.11810 40.00 <2e-16 ***

InsulBefore:Temp -0.39324 0.02249 -17.49 <2e-16 ***

InsulAfter:Temp -0.27793 0.02292 -12.12 <2e-16 ***

Residual standard error: 0.323 on 52 degrees of freedom

Results(continued)

The output by fitting quadratic regression model:

Estimate Std. Error t value Pr(>|t|)

InsulBefore 6.759215179 0.150786777 44.826312 4.854615e-42

InsulAfter 4.496373920 0.160667904 27.985514 3.302572e-32

InsulBefore:Temp -0.31765873 0.062965170 -5.044991 6.362323e-06

InsulAfter:Temp -0.137901603 0.073058019 -1.887563 6.489554e-02

InsulBefore:I(Temp^2) -0.008472572 0.006624737 -1.278930 2.068259e-01

InsulAfter:I(Temp^2) -0.014979455 0.007447107 -2.011446 4.968398e-02

Results(continued)

The output from the ANOVA

Estimate Std. Error t value Pr(>|t|)

(Intercept) 6.8538277 0.13596397 50.409146 7.997414e-46

InsulAfter -2.1299780 0.18009172 -11.827185 2.315921e-16

Temp -0.3932388 0.02248703 -17.487358 1.976009e-23

InsulAfter:Temp 0.1153039 0.03211212 3.590665 7.306852e-04

Summary

- Whiteside data
- Fitting Linear Regression Model
- Fitting Quadratic Regression Model

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