z <- a+b*x; # a: intercept, b: slope #plot(x,z,main=paste("sample size=", n1, "(a math. relation)"));abline(a, b, lty=1,col=1); # The true regression line #sd1 <-2y<-z+rnorm(n1)*sd1; # rnorm(M) gives you M random numbers from N(0,1) #plot(x,y, main=paste("sample size=",n1, ",s.d.=", sd1, "(a stat. relation)" ));abline(a, b, lty=1,col=1); # The true regression line #
opar <-par(mfrow =c(1,2), oma =c(0, 0, 2.7, 0)); hist(z,freq=FALSE, main=paste("sample size=",n1, ",s.d.=", 0)) # Histogram of z's #hist(y,freq=FALSE, main=paste("sample size=",n1, ",s.d.=", sd1)) # Do you see any difference in these two histograms? #
You can add options to executable code like this
opar <-par(mfrow =c(1,2), oma =c(0, 0, 2.7, 0)); y<-z+rnorm(n1)*sd1;plot(x,y, main=paste("sample size=",n1, ",s.d.=", sd1));fm <-lm(y~x); # Simple regression; y regress on x # summary(fm); # Check if estimates for intercept and slope good #
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-4.8466 -1.3047 -0.0912 1.5445 5.0028
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.92370 0.19170 15.25 <2e-16 ***
x -1.52351 0.06575 -23.17 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.917 on 98 degrees of freedom
Multiple R-squared: 0.8457, Adjusted R-squared: 0.8441
F-statistic: 537 on 1 and 98 DF, p-value: < 2.2e-16
abline(fm, lty=3,col="red"); # Add the estimated reg. line to the current plot #abline(a, b, lty=1,col=1); # The true regression line #sd2 <-16;t<-z+rnorm(n1)*sd2;plot(x,t,main=paste("sample size=",n1, ",s.d.=", sd2));abline(a, b, lty=1, col=1);fm1 <-lm(t~x);summary(fm1);
Call:
lm(formula = t ~ x)
Residuals:
Min 1Q Median 3Q Max
-34.930 -11.756 0.275 9.198 42.584
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.4956 1.4586 3.768 0.000281 ***
x -1.2288 0.5002 -2.457 0.015788 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 14.59 on 98 degrees of freedom
Multiple R-squared: 0.058, Adjusted R-squared: 0.04839
F-statistic: 6.034 on 1 and 98 DF, p-value: 0.01579
opar <-par(mfrow =c(1,2), oma =c(0, 0, 2.7, 0)); y<-z+rnorm(n1)*sd1;plot(x,y, main=paste("sample size=",n1, ",s.d.=", sd1),ylim=c(-40,40));fm <-lm(y~x); # Simple regression; y regress on x # summary(fm); # Check if estimates for intercept and slope good #
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-4.0602 -1.2436 -0.1034 1.4741 5.3235
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.01993 0.19426 15.55 <2e-16 ***
x -1.43900 0.06662 -21.60 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.943 on 98 degrees of freedom
Multiple R-squared: 0.8264, Adjusted R-squared: 0.8246
F-statistic: 466.5 on 1 and 98 DF, p-value: < 2.2e-16
abline(fm, lty=3,col="red"); # Add the estimated reg. line to the current plot #abline(a, b, lty=1,col=1); # The true regression line #anova(fm); # Obtain the ANOVA table #
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x 1 1760.46 1760.46 466.51 < 2.2e-16 ***
Residuals 98 369.82 3.77
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Call:
lm(formula = t1 ~ x1)
Residuals:
Min 1Q Median 3Q Max
-27.072 -7.881 3.840 10.788 21.049
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.7635 5.4121 1.619 0.144
x1 -0.2665 1.6958 -0.157 0.879
Residual standard error: 17.11 on 8 degrees of freedom
Multiple R-squared: 0.003077, Adjusted R-squared: -0.1215
F-statistic: 0.02469 on 1 and 8 DF, p-value: 0.879
anova(fm3); #the ANOVA table #
Analysis of Variance Table
Response: t1
Df Sum Sq Mean Sq F value Pr(>F)
x1 1 7.23 7.232 0.0247 0.879
Residuals 8 2343.30 292.912
# Can you see the effect of sample size and standard deviation on the fitted line? ## Play with R as much as possible this week, you will need to use R # for your homework shortly. # Always remove all objects before ending your R session # # rm(list=ls(all=TRUE));# q() # To quit R ## Come to my office if you have any question! ^_^ Yes.
