reg_norm_trans.txt

                             jfm 2/24/2012



Regression w/ normal deviates

*********************************************

> x.100 <- c(1:100)

> y.norm <- rnorm(100, 0, 5)

> yy <- abs(x.100 + y.norm)

> plot(x.100, yy)



*********************************************

Now, generate values that violate equal variances assumption

> yz <- abs(x.100 + log(x.100)*y.norm)

> plot(x.100, yz)

> # Variance in Y increases w/ larger Y

> # Try log-transforming Y

> yz.trans <- log(yz + 1)

> plot(x.100, yz.trans)

> # Result: too strong: variance in Y now DECREASES w/ larger Y

> #    also changed distribution from linear to curved



> # Instead, try square root transform of Y

> yz.sqrt <- sqrt(yz + 0.5)

> plot(x.100, yz.sqrt)

> # Result: Variance in Y about equal throughout, but pattern still curved

> # Try transforming X

> x.trans <- sqrt(x.100 + 0.5)

> plot(x.trans, yz.sqrt)

> # Better, although still not perfect



> # Try fitting regression line, then adding line to plot



> reg.trans <- lm(yz.sqrt ~ x.trans)

> abline(reg.trans)

> summary(reg.trans)



Call:

lm(formula = yz.sqrt ~ x.trans)



Residuals:

     Min       1Q   Median       3Q      Max 

-3.70343 -0.95159  0.02777  1.10129  2.83043 



Coefficients:

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

(Intercept) -0.42822    0.40860  -1.048    0.297    

x.trans      1.02547    0.05722  17.923   <2e-16 ***

---

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



Residual standard error: 1.317 on 98 degrees of freedom

Multiple R-squared: 0.7662,     Adjusted R-squared: 0.7639 

F-statistic: 321.2 on 1 and 98 DF,  p-value: < 2.2e-16 



> # Now plot/view residuals

> plot(x.trans, reg.trans$residuals)



> # Add reference line at Y=0 

> min(x.trans)

[1] 1.224745

> max(x.trans)

[1] 10.02497

> abline(1.22, 10.0, 0, 0)