Stat 6601 project Linear Statistical Models Analysis of Covariance Example

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Stat 6601 project Linear Statistical
Models Analysis of Covariance Example

• By
• Gadir Marian
• Myrna Moreno

2
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)

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Goal

• Assess the effect of the insulation on gas
  consumption.

4
Plotting the data
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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.

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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.

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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 lt- lm(Gas Temp, whiteside, subset
  Insul"Before")
• gasA lt- update(gasB, subset Insul"After")
• summary(gasB)
• summary(gasA)
• gasBA lt- lm(Gas Insul/Temp - 1, whiteside)
• summary(gasBA)
• gasQ lt- lm(Gas Insul/(Temp I(Temp2)) - 1,
  whiteside) summary(gasQ)coef
• gasPR lt- lm(Gas Insul Temp, whiteside)
• anova(gasPR, gasBA)
• options(contrasts c("contr.treatment",
  "contr.poly"))
• gasBA1 lt- lm(Gas InsulTemp, whiteside)

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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(gtt)
• InsulBefore 6.85383 0.13596
  50.41 lt2e-16
• InsulAfter 4.72385 0.11810
  40.00 lt2e-16
• InsulBeforeTemp -0.39324 0.02249 -17.49
  lt2e-16
• InsulAfterTemp -0.27793 0.02292 -12.12
  lt2e-16
• Residual standard error 0.323 on 52 degrees of
  freedom

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Results(continued)

• The output by fitting quadratic regression
  model
• 
  Estimate Std. Error
  t value Pr(gtt)
• InsulBefore 6.759215179
  0.150786777 44.826312 4.854615e-42
• InsulAfter 4.496373920
  0.160667904 27.985514 3.302572e-32
• InsulBeforeTemp -0.31765873 0.062965170
  -5.044991 6.362323e-06
• InsulAfterTemp -0.137901603 0.073058019
  -1.887563 6.489554e-02
• InsulBeforeI(Temp2) -0.008472572 0.006624737
  -1.278930 2.068259e-01
• InsulAfterI(Temp2) -0.014979455 0.007447107
  -2.011446 4.968398e-02

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Results(continued)

• The output from the ANOVA
• Estimate Std.
  Error t value Pr(gtt)
• (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
• InsulAfterTemp 0.1153039 0.03211212
  3.590665 7.306852e-04

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Summary

• Whiteside data
• Fitting Linear Regression Model
• Fitting Quadratic Regression Model