[R] Rsquared in summary(lm)

[apjaworski@mmm.com]

I am not sure where the problem might be and I missed the original message
by Wouter, but here is a simple example I just ran (Win2000, 1.5.0) which
shows no apparent problems.

> x <- c(-3, -1.5, 2, 5, 7, 8)
> y <- c(2, 4, 5, 6, 6, 9)
> sxx <- sum((x-mean(x))^2)
> sxy <-  sum((x-mean(x))*(y-mean(y)))
> bb <- sxy/sxx
> bb
[1] 0.4761517
> aa <- mean(y) - bb*mean(x)
> aa
[1] 3.944558
> ee <- y - aa - bb*x
> sst <- sum((y-mean(y))^2)
> sse <- sum(ee^2)
> ssr <- sst-sse
> ssr/sst
[1] 0.8477822
> summary(lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
      1       2       3       4       5       6
-0.5161  0.7697  0.1031 -0.3253 -1.2776  1.2462

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   3.9446     0.5098   7.737  0.00150 **
x             0.4762     0.1009   4.720  0.00917 **
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 1.02 on 4 degrees of freedom
Multiple R-Squared: 0.8478,     Adjusted R-squared: 0.8097
F-statistic: 22.28 on 1 and 4 DF,  p-value: 0.009172

Cheers,

Andy

__________________________________
Andy Jaworski
Engineering Systems Technology Center
3M Center, 518-1-01
St. Paul, MN 55144-1000
-----
E-mail: apjaworski at mmm.com
Tel:  (651) 733-6092
Fax:  (651) 736-3122


                                                                                                                                               
                    Wataru Shito                                                                                                               
                    <shito at seinan-g      To:     wouter buytaert <wouter.buytaert at yucom.be>                                                    
                    u.ac.jp>             cc:     r-help at stat.math.ethz.ch                                                                      
                                           (bcc: Andrzej P. Jaworski/US-Corporate/3M/US)                                                       
                    05/09/2002           Subject:     Re: [R] Rsquared in summary(lm)                                                          
                    20:24                                                                                                                      
                                                                                                                                               
                                                                                                                                               





Hi, Wouter

Actually, I have a similar problem too with a simple regression.
In my case, not only the R-square but also the estimates of intercept
and coefficient by lm() seem different from the calculation with the
well known formula for a simple regression.

What I used is the following code.  (I have just started to use R last
week so don't blame my inmature code, please!)
Simply,
  > ols1( y, x )
will give you the result of the simple regression.

Wouter, could you try the following code on your data and see whether
that's what you expect or not?

I will appreciate if anyone can give me some advice why this
differenct happens.

Thankk you.

Wataru Shito


-----------------------------------------

# Single Explanatory Variable Least Square Regression
#
library(methods)
# create ols class
setClass("ols", representation
         ( coefficients="list", standard.errors="list",
          r.square="numeric" ))
setMethod("show", "ols",
          function(object)
          {
            # create row names for data.frame
            rownames <- c("(Intercept)", "X")
            # create data.frame
            z <- data.frame( row.names=rownames,
                            Estimate=object at coefficients,
Std.Error=object at standard.errors,
                            t.value=t.values )
            cat("\n")
            print(z)
            cat( "\nR-Square:", object at r.square, "\n\n" )
          }
          )

ols1 <- function( y, x ){
  size <- length(x) # number of ovservations
  xbar <- mean(x)
  ybar <- mean(y)
  Sxx <- sum( (x-xbar)^2 )
  b <- sum( (x-xbar)*(y-ybar) )/Sxx # coefficient
  a <- ybar - b*xbar # interception
  e <- y - a - b*x   # residuals
  # SSE (error sum of squares)
  SSE <- sum( e^2 )
  # SST (total sum of squares)
  SST <- sum( (y-ybar)^2 )
  # SSR (regression sum of squares)
  SSR <- b^2 * Sxx
  # Coefficient of determination
  r2 <- SSR / SST

  # unbiased estimator of sigma^2
  s.square <- sum(e^2)/(size - 2)
  # standard error for b
  std.error.b <- sqrt( s.square/Sxx )
  # standard error for intercept
  std.error.a <- sqrt( s.square*(1/size + xbar^2/Sxx) )
  standard.errors <- list( intercept=std.error.a, coeficient=std.error.b )
  coefficients <- list( intercept=a, coefficient=b )
  new("ols", coefficients=coefficients, standard.errors=standard.errors,
r.square=r2 )
}
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