Linear regression: Adjusted variables plot (AVPLOT)

Linear regression: Adjusted variables plot (AVPLOT)

There is dependent variable Y (weight), explanatory variable Xi(height), Xj(age).

First, regress dependent variable Y on all other explanatory variables except variable Xj. Here regress weight on height.

Call:

lm(formula = wt ~ ht, data = d)

Residuals:

    Min     1Q Median       3Q Max

-2.44736 -0.55708  0.01925 0.49941 2.73594

Coefficients:

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

(Intercept) -8.694768   0.427398 -20.34 <2e-16 ***

ht           0.235050 0.005257   44.71 <2e-16 ***

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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9017 on 183 degrees of freedom

Multiple R-squared:  0.9161,    Adjusted R-squared:  0.9157

F-statistic:  1999 on 1 and 183 DF,  p-value: < 2.2e-16

First, regress dependent variable Xj on all other explanatory variables. Here regress age on height.

Call:

lm(formula = age ~ ht, data = d)

Residuals:

   Min     1Q Median      3Q Max

-18.722  -4.168 -1.222   3.467 20.625

Coefficients:

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

(Intercept) -74.01432    3.11317 -23.77 <2e-16 ***

ht            1.29736 0.03829   33.88 <2e-16 ***

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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.568 on 183 degrees of freedom

Multiple R-squared:  0.8625,    Adjusted R-squared:  0.8618

F-statistic:  1148 on 1 and 183 DF,  p-value: < 2.2e-16

Finally compare the residuals of the model I and the model II. The below is AVPLOT

Call:

lm(formula = residual_WtHt ~ residual_ageHt)

Residuals:

    Min     1Q Median       3Q Max

-2.48498 -0.53548  0.01508 0.51986 2.77917

Coefficients:

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

(Intercept)    2.261e-15 6.624e-02   0.000 1.000

residual_ageHt 5.368e-03  1.014e-02 0.529 0.597

Residual standard error: 0.901 on 183 degrees of freedom

Multiple R-squared:  0.001529,    Adjusted R-squared:  -0.003927

F-statistic: 0.2802 on 1 and 183 DF,  p-value: 0.5972

Also, we could do the process using MLR

Call:

lm(formula = wt ~ age + ht, data = d)

Residuals:

    Min     1Q Median       3Q Max

-2.48498 -0.53548  0.01508 0.51986 2.77917

Coefficients:

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

(Intercept) -8.297442   0.865929 -9.582 <2e-16 ***

age          0.005368 0.010169   0.528 0.598

ht           0.228086 0.014205  16.057 <2e-16 ***

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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9035 on 182 degrees of freedom

Multiple R-squared:  0.9163,    Adjusted R-squared:  0.9154

F-statistic: 995.8 on 2 and 182 DF,  p-value: < 2.2e-16

Both of the slope determined by AVPLOT (model III) and by MLR are 5.368e-03.