# [R] Rsquared in summary(lm)

Thomas Lumley tlumley at u.washington.edu
Fri May 10 17:21:54 CEST 2002

```On Fri, 10 May 2002, Wataru Shito wrote:

> 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.

Can you give an example of a data set where this doesn't agree with lm()?

The code below doesn't work because the show method refers to an
apparently non-existent t.values object, but looking directly at the slots
of the value of ols1 seems to give the same results as lm

> x<-1:20
> y<-x+rnorm(20)
> summary(lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
Min       1Q   Median       3Q      Max
-1.84289 -0.81606  0.04298  0.96198  1.57927

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.53721    0.53134   1.011    0.325
x            0.99449    0.04436  22.421 1.33e-14 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 1.144 on 18 degrees of freedom
Multiple R-Squared: 0.9654,     Adjusted R-squared: 0.9635
F-statistic: 502.7 on 1 and 18 DF,  p-value: 1.328e-14

> ols1(y,x)->a
> a at coefficients
\$intercept
 0.5372087

\$coefficient
 0.994491

> a at standard.errors
\$intercept
 0.5313436

\$coeficient
 0.04435571

> a at r.square
 0.9654307

It would be surprising if linear regression didn't work in R in general ,
but there may be some system-specific issues

-thomas

> 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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Thomas Lumley			Asst. Professor, Biostatistics
tlumley at u.washington.edu	University of Washington, Seattle
^^^^^^^^^^^^^^^^^^^^^^^^