[R] Linear Regression with slope equals 0

[ONKELINX, Thierry]

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
> 
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