Is there any other method that can be used to model data based on experimental data, when data points are limited(3 or 4 points).I can understand that when we have sufficient (atleast 10 data points) than its relatively easy to get a regression equation equation but what will be a case when we have just 3 to 4 data points?
Dear Imran,
There are two separate issues here: the ability to construct a linear regression model and the validity of the model thus constructed. I would recommend first using a scatter plot to assess the dispersion of your data points, even though there are very few. Then, provided that their dispersion appears to be 'cigar shaped', run a Pearson correlation and look at the significance value. A linear regression solution should only then be quoted if there is evidence of a linear correlation (i.e. a significance value less than 0.05). It is still possible for this to work with small samples.
If you have two data points, there is one and only one straight line pass through those two point, you can find slope (beta) and interception (alpha), but there is no standard error. You don't need statistics, no degree freedom. If use p=the number of parameters, n=the points of observation. Statistics the problem, with n>p, n=3 is the smallest size of n, which the simple linear regression can performed. The aim is to estimate the unknown population parameter alpha and beta from a sample. In practice, your sample is randomly selected, you may get good sample, and you may get bad sample, hence from sample to sample, the estimation of the alpha and beta changes, we have to study the standard deviation to judge if the slope, which shows the effect of X (predictor) on Y, is coming from a chance (not significant) or it exists (describe the association of Y and X). The standard error is proportional of 1/sqrt(n). Hence, if n larger, the standard error will be smaller, the estimation should more accurate.
If you just have 3 or 4 points, you may
1. get biased estimation,
2. you can't make conclusion, more chance to accept H0 beta=0, make type 1 error.
What sample size needed, do power calculation.
Yuanzhang Li has provided a reasonable technical answer of the reasoning here, but the short answer is that a linear regression with only 3 or 4 data points is effectively meaningless.
I would caution against your other generalization that it is "relatively easy" to get a regression equation when you have 10+ data points as well. This is not a trivial assumption, and really depends on the specifics of your research question and how you are collecting the data.
What is the reason for such a small sample?
Is it difficult to find data? Have you temporal data, so for one more observation you need one more year to pass?
Why such a small N?
You cannot estimate a model using only 3-4 points without any prior information.
What you can do is to apply Bayesian analysis here.
This means you can use some formalised prior judgmental info and then add your data to make your estimates more precise.
In case of using the above approach it may be possible to use just 3-4 points to estimate your model. Please see our work on this topic:
https://www.researchgate.net/publication/282136270_A_joint_Bayesian_forecasting_model_of_judgment_and_observed_data
Technical Report A joint Bayesian forecasting model of judgment and observed data
P.S. For linear regression solutions, see http://www.amazon.com/An-Introduction-Bayesian-Inference-Econometrics/dp/0471169374
Bundle of thanks @ Peter Samuels,Andrey Davydenko and Yuanzhang Li others......Well, Andrey , I am already thinking on same lines as you suggested.But the approach looks a bit tricky. I posted my question to confirm i am right to consider about baysian inference.Can you suggest me something about priors?I mean how we decide about priors?Definitly, i will go through what you suggested in the form of an article and a website.
@ Demetris Christopoulos ...When solving real world problems we dont always met with ideal situations.Some time we are limited with resources and that leads us to look the same problems but from a diffrent angle. The study concerned with some material featutres. I can use upto four materials but i am not in position to buy 10 or 20 materials. You can consider as catgorical behaviour.Hope fully it can address what you think as considering more number of observations.
Anyways, once again thanks everybody.
You can obtain informative priors based on either judgment or some relevant data. What type of data are you dealing with?
Thanks Andrey Davydenko ..for quick response----- Well, my study concern to understand the effect of polymer crystallinity on percolation of carbon nanotubes within a semicrystalline polymer. so, i have foure polymers with diffrent crystallinity lets asuum crystallinity is, 0%, 40%, 60% and 70%. I observed a dirrent percolation threshold Xo(minimum cnt insertion for a 3d network) for such polymers. Now you can understand why i have just four points. So, model concerns with polymer crystallinity and percolation threshold.
Its a bit tricky as if i dont have 0 crystallinity sample than i can assum priors as 0 value but as 0% already is one of the sample and thats why i am abit confuse.Can you suggests me?
Dear Imran, I am afraid that your problem has to be solved by a 'budget increase'!
Sorry for this, but it's the only way to make valid inferences about materials.
Good luck!
What is the measurement error for your observations? Do you know what the error can be approximately?
The idea is that if the dispersion of your measurement error is known and the regression function is known as well, then it is realistic to get estimates. If your measurement error is very low, then regression analysis is not needed - just fit the model to the points.
P.S. You will need to transform percentages using logit or arc/sqrt transformation for regression analysis as otherwise there will be difficulties in finding adequate residuals model.
But where from the idea that we are dealing with linear dependence in this case? Having only 3 or 4 points, presumably with small uncertainties in both coordinates, all we can say is that the function is increasing and, perhaps, whether it might be concave or convex. Linear approximation, if it makes sense at all, will give you only a very rough estimates.
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