R help - Aug 2007 - Linear Regression with slope equals 0

Hi there, am trying to run a linear regression with a slope of 0.

 I have a dataset as follows

 t d
 1 303
 2 302
 3 304
 4 306
 5 307
 6 303

 I would like to test the significance that these points would lie on a
horizontal straight line.

 The standard regression lm(d~t) doesn't seem to allow the slope to be set.

 Any help very welcome.

 ed

On Tue, 14 Aug 2007, E.N.D.Grew at exeter.ac.uk wrote:
>
> Hi there, am trying to run a linear regression with a slope of 0.
>
> I have a dataset as follows
>
> t d
> 1 303
> 2 302
> 3 304
> 4 306
> 5 307
> 6 303
>
> I would like to test the significance that these points would lie on a
> horizontal straight line.
>
> The standard regression lm(d~t) doesn't seem to allow the slope to be
set.
lm(d ~ 1) does, though, to zero.

More generally you can use offset(), e.g. lm(d ~ offset(7*t)) forces a 
slope of 7.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

Hi

Not being a trained statistician regression with slope = 0 seems odd to 
me.

If you do
> fit<-lm(d~t)
> summary(fit)
Call:
lm(formula = d ~ t)

Residuals:
       1        2        3        4        5        6 
 0.04762 -1.43810  0.07619  1.59048  2.10476 -2.38095 

Coefficients:
            Estimate Std. Error t value Pr(>|t|) 
(Intercept) 302.4667     1.7849  169.45 7.28e-09 ***
t             0.4857     0.4583    1.06    0.349 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05
'.' 0.1 ' ' 1

Residual standard error: 1.917 on 4 degrees of freedom
Multiple R-Squared: 0.2192,     Adjusted R-squared: 0.02402 
F-statistic: 1.123 on 1 and 4 DF,  p-value: 0.349 

you will get estimate of your coeficients and AFAIK they are tested 
against Ho that they differ from 0.

So as you see your "t" is not statistically different from 0.

Regards
Petr


r-help-bounces at stat.math.ethz.ch napsal dne 14.08.2007 13:36:37:
> 
>  Hi there, am trying to run a linear regression with a slope of 0.
> 
>  I have a dataset as follows
> 
>  t d
>  1 303
>  2 302
>  3 304
>  4 306
>  5 307
>  6 303
> 
>  I would like to test the significance that these points would lie on a
> horizontal straight line.
> 
>  The standard regression lm(d~t) doesn't seem to allow the slope to be 
set.
> 
>  Any help very welcome.
> 
>  ed
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide 
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

On 14-Aug-07 11:36:37, E.N.D.Grew at exeter.ac.uk wrote:
> 
>  Hi there, am trying to run a linear regression with a slope of 0.
> 
>  I have a dataset as follows
> 
>  t d
>  1 303
>  2 302
>  3 304
>  4 306
>  5 307
>  6 303
> 
>  I would like to test the significance that these points would lie on a
> horizontal straight line.
> 
>  The standard regression lm(d~t) doesn't seem to allow the slope to be
> set.
The model d~1 will fit a constant (the mean), i.e. a regressio with
slope = 0. The model d~t will fit the usual linear regression.
The two con be compared with anova(), as well as getting the details
of the individual fits with summary().

E.g. (with your example):

  d<-c(303,302,304,306,207,303)
  t<-c(1,2,3,4,5,6)
  lm0<-lm(u~1);lm1<-lm(u~t);anova(lm0,lm1)

##Analysis of Variance Table
##Model 1: u ~ 1
##Model 2: u ~ t
##  Res.Df    RSS Df Sum of Sq      F Pr(>F)
##1      5 7785.5                           
##2      4 6641.4  1    1144.1 0.6891 0.4531

summary(lm0)
## Call: lm(formula = u ~ 1)
## Residuals:
##    1     2     3     4     5     6 
## 15.5  14.5  16.5  18.5 -80.5  15.5 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   287.50      16.11   17.85 1.01e-05 ***
##Residual standard error: 39.46 on 5 degrees of freedom

mean(d)
## [1] 287.5

summary(lm1)
## Call: lm(formula = u ~ t)
## Residuals:
##      1       2       3       4       5       6 
## -4.714   2.371  12.457  22.543 -68.371  35.714 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  315.800     37.934   8.325  0.00114 **
## t             -8.086      9.740  -0.830  0.45314   
## Residual standard error: 40.75 on 4 degrees of freedom
## Multiple R-Squared: 0.147,      Adjusted R-squared: -0.0663 
## F-statistic: 0.6891 on 1 and 4 DF,  p-value: 0.4531 

The P-value for the slope in lm1 is the same as the P-value
returned by anova().

If you want to force a particular non-zero slope (e.g. s0)
for comparison with the data, you can use

  lm0 <- lm(d - s0*t ~ 1),

compared with

  lm1<- lm(d- s0*t ~ t)

for instance.

Hoping this helps,
Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 14-Aug-07                                       Time: 13:16:05
------------------------------ XFMail ------------------------------

Dear Ed,

In my opinion you don't need to set the slope to zero. Just test if the
slope in lm(d ~ t) is significant. If it is significant then you have
evidence that the slope is NOT zero. But when it is not significant (and
in your example it is), you can't say that it is zero. But doing some
power calculations allows you to estimate the smallest detectable slope.
The estimated slope in your example is 0.49. Your design has a power of
13% to detect slopes of this size.  The real slope has to be about 1.75
in order to have 80% power. Let's say that it would matter if the slope
is about 2 or larger and it won't matter is it is below 2. The power to
detect a slope of 2 is 89%, hence you would likely get a significant
slope if it was larger then 2. Since the slope was not significant, it's
safe to say that the slope is not larger then 2.
But if a slope of 0.5 would matter, you couldn't make this kind of
assumptions because the power to detect a slope of 0.5 is only 13%. So
the change of rejecting the null hypothesis (slope = 0) when slope = 0.5
is too small.

HTH,

Thierry
> power.trend <- function(repetitions = 5, x = c(0, 1), sd = 1, slope 1,
alpha = 0.05){
+   X <- rep(x, repetitions)
+   ncp <- slope ^ 2 * sum((X - mean(X))^2) / sd ^ 2
+   return(1 - pf(qf(1 - alpha, 1, length(X) - 2), 1, length(X) - 2, ncp
= ncp))
+ }
> 
> df <- data.frame(t = 1:6, d = c(303, 302, 304, 306, 307, 303))
> fit <- lm(d ~ t, data = df)
> summary(fit)
Call:
lm(formula = d ~ t, data = df)

Residuals:
       1        2        3        4        5        6 
 0.04762 -1.43810  0.07619  1.59048  2.10476 -2.38095 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 302.4667     1.7849  169.45 7.28e-09 ***
t             0.4857     0.4583    1.06    0.349    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05
'.' 0.1 ' ' 1

Residual standard error: 1.917 on 4 degrees of freedom
Multiple R-Squared: 0.2192,     Adjusted R-squared: 0.02402 
F-statistic: 1.123 on 1 and 4 DF,  p-value: 0.349 
> power.trend(repetitions = 1, x = df$t, sd = sd(df$d), slope
coef(fit)["t"])
[1] 0.1292668
> power.trend(repetitions = 1, x = df$t, sd = sd(df$d), slope = 1.75)
[1] 0.8021454


------------------------------------------------------------------------
----
ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature
and Forest
Cel biometrie, methodologie en kwaliteitszorg / Section biometrics,
methodology and quality assurance
Gaverstraat 4
9500 Geraardsbergen
Belgium
tel. + 32 54/436 185
Thierry.Onkelinx op inbo.be
www.inbo.be 

Do not put your faith in what statistics say until you have carefully
considered what they do not say.  ~William W. Watt
A statistical analysis, properly conducted, is a delicate dissection of
uncertainties, a surgery of suppositions. ~M.J.Moroney

 
> -----Oorspronkelijk bericht-----
> Van: r-help-bounces op stat.math.ethz.ch 
> [mailto:r-help-bounces op stat.math.ethz.ch] Namens 
> E.N.D.Grew op exeter.ac.uk
> Verzonden: dinsdag 14 augustus 2007 13:37
> Aan: r-help op stat.math.ethz.ch
> Onderwerp: [R] Linear Regression with slope equals 0
> 
> 
>  Hi there, am trying to run a linear regression with a slope of 0.
> 
>  I have a dataset as follows
> 
>  t d
>  1 303
>  2 302
>  3 304
>  4 306
>  5 307
>  6 303
> 
>  I would like to test the significance that these points 
> would lie on a horizontal straight line.
> 
>  The standard regression lm(d~t) doesn't seem to allow the 
> slope to be set.
> 
>  Any help very welcome.
> 
>  ed
> 
> ______________________________________________
> R-help op stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide 
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>