Hello Soheil,

Confidence intervals are used to estimate parameter values. If your interest is in individual values, a better approach is to construct tolerance intervals. See the text on statistical quality control by Douglas C Montgomery.

Hi Soheil,

Raid is right. Confidence intervals are used to give the imprecision of estimations.

You want to identify the genes above or below the prediction limits or above or below the tolerance limits.

Here is a simple (and approximative) way to proceed

1) Perform a regression

in R :

result_reg= lm(Y~yourmodel) # if you use a linear model

2) read the standard deviation of the residuals of your model

or compute it using the following command

sd=sqrt(sum(result_reg$residuals^2)/result_reg$df.residual)

3)compute the standardized residuals and identify which of these have an abolute value  larger than 2

The limit 2 comes from the Gaussian distribution (actually the Student distribution)

select=which(abs(result_reg$residuals/ss)>2)

Now, the variable named select contains the row number of the genes outside the 95% prediction interval

This should work if

a) the variance of the residuals does not change with your covariates (the model is homoscedastic)

b) the distribution of your residuals is Gaussian.

Thank you very much. I want to select genes lower than Q1 and upper than Q3 according to this plot?

Is this method useful?

OK,

The new step 2 relies on computing the residuals quartiles

u=quantile(result_reg$residuals)

u[2] and u[4] are respectively Q1 and Q3

3) select the residuals u[4]

sel1=which(result_reg$residualsu[4])

This works only if homoscedasticity holds ie the points dispersion about the curve does not change with x

Hi Didier and Soheil.

You can use a process control approach and then work directly with the data (X), or you can use an approach by which you idenify "outliers", which are based on the residuals (what cannot be modeled).

The advantage of using tolerance limits is that such a simplified density estimation approach gives you also information on the distribution of the genes characteristic that you are studying. There are parametric tolerance limits (based on the Normal distribution), and there are nonparametric tolerance limits (see William Conover's test on Nonparametric Statistics).

Parametric tolerance limits are narrower (more precise) than nonparametric tolerance limits (based on order statistics), but they have distributional assumptions, such as having a Gaussian distribution.

The simplest nonparametric tolerance limits are (minimum, maximium) of a random sample.

A side-product here is to get range=max-min, which gives you information on the variability.  

Hi Didier and Raid,

Thanks a lot for your response. I did your commands but not observed proper results.

I have some data like this:

                name                                 cv            mean.express

1 ENSG00000000003         0.46087231        9.166223

2 ENSG00000000419         0.10033507       14.379947

3 ENSG00000000457         0.03337102       14.395454

4 ENSG00000000460         0.05965897       12.566220

5 ENSG00000000460         0.03570741       13.088759

I draw the scatter plot as showed in last question.

I want to extract point (name "genes") lower and upper Q1 and Q3.

I did your command and also filter my data separately one time for "cv" and one time for "mean.express"  but not get  proper results?

do you have any solution in this way?

How can I calculate quantile for this plot?

Go over this link for a  parametric tolerance interval:

http://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm

This link is for the nonparametric tolerance interval. In your case, you don't want to use min and mx as then there are no values sticking out for possible further investigations. 

http://www.itl.nist.gov/div898/handbook/prc/section2/prc264.htm

Interestingly, the attached article is by people at your university, Soheil!

This is also very useful material, Soheil.

http://people.stat.sfu.ca/~cschwarz/Stat-650/Notes/PDF/ChapterPercentiles.pdf

See page 10 for the table:

http://salserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/pdf-files/emat08.pdf

 Thanks a lot Raid for your response.