> On 21 Oct 2015, at 19:57 , Charles C. Berry <[email protected]> wrote:
>
> On Wed, 21 Oct 2015, Ravi Varadhan wrote:
>
>> [snippage]
>
> If half the subjects have a value of 5 seconds and the rest are split between
> 4 and 6, your assertion that rounding induces an error of
> dunif(epsilon,-0.5,0.5) is surely wrong (more positive errors in the 6 second
> group and more negative errors in the 4 second group under any plausible
> model).
Yes, and I think that the suggestion in another post to look at censored
regression is more in the right direction.
In general, I'd expect the bias caused by rounding the response to quite small,
except at very high granularity. I did a few small experiments with the
simplest possible linear model: estimating a mean based on highly rounded data,
> y <- round(rnorm(1e2,pi,.5))
> mean(y)
[1] 3.12
> table(y)
y
2 3 4 5
13 63 23 1
Or, using a bigger sample:
> mean(round(rnorm(1e8,pi,.5)))
[1] 3.139843
in which there is a visible bias, but quite a small one:
> pi - 3.139843
[1] 0.001749654
At lower granularity (sd=1 instead of .5), the bias has almost disappeared.
> mean(round(rnorm(1e8,pi,1)))
[1] 3.141577
If the granularity is increased sufficiently, you _will_ see a sizeable bias
(because almost all observations will be round(pi)==3):
> mean(round(rnorm(1e8,pi,.1)))
[1] 3.00017
A full ML fit (with known sigma=1) is pretty easily done:
> library(stats4)
> mll <- function(mu)-sum(log(pnorm(y+.5,mu, .5)-pnorm(y-.5, mu, .5)))
> mle(mll,start=list(mu=3))
Call:
mle(minuslogl = mll, start = list(mu = 3))
Coefficients:
mu
3.122069
> mean(y)
[1] 3.12
As you see, the difference is only 0.002.
A small simulation (1000 repl.) gave (r[1,]==MLE ; r{2,]==mean)
> summary(r[1,]-r[2,])
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.004155 0.000702 0.001495 0.001671 0.002554 0.006860
so the corrections relative to the crude mean stay within one unit in the 2nd
place. Notice that the corrections are pretty darn close to cancelling out the
bias.
-pd
>
>
> HTH,
>
> Chuck
>
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--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: [email protected] Priv: [email protected]
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