Mean, standard deviation (which is not mean standard error, your formulation « its SD » is somehow ambiguous) and slope may be seen as different summary measures of the trajectory, measuring different aspects of this trajectory. So for all three of them it's quite natural to consider them as predictors. A fourth natural one would be the estimated residual variance after the linear regression.

« SD of the mean », which is standard error, and slope SE describe the uncertainty on the parameters above. They play a quite different role. So usage for prediction is different:

— if you predict a different quantity, Z, than the one used to obtain these summary statistics, then SE on mean or slope are probably best considered as weights;

— if you predict the quantity, Y, that was used to obtain them, then they are instead usefull to obtain prediction *intervals* of the correct width, but not individual prediction values.

At first glance, it seems difficult to understand how SEs would play the same role as mean or slope or SD has predictors... Note that even SD appears only to obtain intervals instead of ponctual predictions, in the second case. For more suited answers, a more detailed description of what you are trying to do would be usefull...

1) There is no problem using the standard deviation of a measured variable in a linear model. Sometimes a treatment will have little effect on the mean but will increase the variability in the response.

2) One could use the standard error in a linear relationship as part of a predictor in some other relationship.

3) A concern in both 1 and 2 is that their utility is also showing that the assumption of homoscedasticity is false. Choose your model accordingly. Discuss as needed.

4) If you use the variance/SD/SE there may be issues in replication. If I run an experiment wherein I collect 20 individuals from a population, I can calculate the variance in some measure from these individuals. I now have one data point. I need multiple collections of 20 individuals to get replicates in the estimated variance.

Egon -

I do not understand your application, but it is apparent that there is one or more definitional problem in the statistical terminology here, which adds to the confusion. 

First, a "predictor" in a regression is an 'independent' variable.  The independent variables are not random variables in the usual case.  Their coefficients can be estimated and standard errors of those coefficients found, as well as confidence intervals. 

A standard deviation for a mean is a standard error. 

When you talk about a mean, you could mean an unconditional mean for the y-data, or the conditional mean of y given x (written y|x) from the regression (the predicted value of y given x), or perhaps you want the unconditional mean of x-data, though I don't know what you'd want to do with that. 

If you have one predictor (which is an independent variable, also called a regressor), in a linear regression, then the coefficient is the slope, and you can obtain a standard error for the slope, and confidence intervals.  For multiple linear regression, the "independent" variables actually can interact, so confidence intervals on each regression coefficient are unstable.  For the random variable y, we can get predicted-y-values and prediction intervals around those. 

All that said, perhaps you might want to restate your question.  You should state what the independent variables are, and describe the dependent variable, y. 

If you still want to talk about means, standard errors, and standard deviations, you best clarify the means, standard errors, and standard deviations of what. 

Best wishes - Jim

Thank you all for your responses. You helped me much to clarify my problem.

"A standard deviation for a mean is a standard error." James makes some very good points, but this one may need more explanation.

In a sample from a population one can calculate the sample mean. One can also calculate how each value in the sample deviates from that mean. We square the difference and then divide by the number of observations (reduced by 1) to get the average difference between the mean and each observation. This is the variance, take the square root to get standard deviation.

Standard error comes in (at least) two forms. It might be the standard deviation divided by the square root of the sample size. Standard error might also appear in some parts of a regression analysis output associated with estimates of model coefficients (as James discusses). In an ANOVA where you ask for a multiple comparison test and the output gives least squared means and standard error, The standard error is in the regression analysis sense and is not just using the standard deviation divided by the square root of the sample size and placing that next to the least squared mean.

Sorry but you have a problem. The SDs, SEs , and variances are measured with error since they are estimates. The requirements for a linear statistical model is that the x variables are fixed. Thus you need to include this additional source of variation in your model in addition to showing that using the variability statistics give good predictors. The models that you want to consider are called errors in variables models. If you Google this term it should give you a start into the literature. Best wishes, David Booth

But all measurements have error. I cannot measure temperature perfectly, and the same for humidity. I cannot measure time perfectly. I cannot measure mass perfectly and if I could I could never get exactly 4 grams of a compound because I would have to split a molecule to do so (and splitting the molecule would change its nature). I cannot measure area exactly. Almost everything has some degree of measurement error. We always hope that it is close enough that it doesn't matter relative to the effect we are trying to measure.

Timothy, of course you are correct about measurements. However, if you look at the way a linear statistical model is built all the error is collected in the random variable error term which clearly is not a function of the Xs since they're on the same side of the equation. As I believe George Box said all models are wrong but some are useful. This has been a long standing problem in statistics that is not fully resolved yet. But much progress has been made since the 1950s when Albert Madansky made one of the first really important contributions to this research. As I said Google errors in variables models and you will start into all of this. The take home message is if there are errors in one or more of the Xs you need to take that variance into account. At present we don't have the complete answer. My students sometimes think statistics is cut and dried. Nothing is further from the truth. Check this stuff out. It is really interesting. Best wishes, David Booth

@ David: my intuitive guess is that as far as error on the X is low enough that, according to its slope, changes in Y due to this error are (much) lower than conditionnal variability of Y knowing X, taking into account errors on X should have very small effect. In other words, if Y/X=x = ax + b + epsilon, as far as |a|*sigma(x)

@Emmanuel,

No offense but your intuition is incorrect. Please look into the literature I cited. David Booth

For the skeptical. Here is a recent review paper from Stanford University on errors in variables models. The authors emphasize economics but as you can see the problem is not confined to economics. This is something that everyone that does regression needs to be aware of. David Booth

@ David: thanks for the paper and the comment. Equation 5 suggests that the criterium should be instead that the measurement error on a given X value is negligible compared to the dispersion of the set of all observed X values, at least for slope estimation. Right? It is probably often the case in the applications in biology/clinics, typically when X values are analytical assays results, for the problems I'm familiar with.

I'm dubious about formula just after 6, because even in the case of no error on X (that is, if I understand correctly, sigma_u = sigma_epsilon), it gives plim â != alpha. I'm not sure what probability limit is exactly in that paper, but that seems contradictory with usual results on classical linear regression, that prove that in these conditions, the estimator is unbiased.

Note also that these models assume X (even without measurement error) is random, but that's not always the case in usual experimental settings.

I've read quite quickly, however. I probably missed something. But clarifications are welcome!

Emmanuel, no it's not the size of an error it is the variance of the distribution of the errors that needs to be accounted for. Anyway how do you tell how small is small?

@ David: error standard deviation was what I meant by « measurement error on a given X » (assumed to be independent of X, which is another debate). Sorry for the unclear expression.

How small is small is always a matter of convention, but according to the formula (5), if sigma( measurement error ) < 10×sigma( X ) the standard deviation ratio should be close enough to 1 for most practical situations. That's at least a typical condition for neglecting a term in several fields (« one order of magnitude lower »), and with the square it would be even two orders of magnitude...

But I definitely agree that this general rule of thumb is not more solid than "n > 30 => Gaussian" or "not less than 5 points for each estimated parameters" rules of thumb in practical, applied statistics and the question should at least be raised. And that it may (badly?) fail in some situations.

Of course, a batch of simulations for check this point would not hurt ;)

A part of the problem is that I have no way to measure the error. Say I am interested in temperature. I do not have any device that will produce a temperature of exactly 25 C, and all the equipment I have to measure temperature will have some error. I have no scenario where I can win.

Oh Timothy! Started my career as a PhD Biophysical chemist. All through college and graduate school we worried about this stuff. The problem was back then nobody really had developed a good approach to dealing with the problem. That's rapidly changing now. From beginning experimental design it is clear that all sources of variation in an experiment are to be accounted for. We are now on the verge of being able to do that certainly we should do that if we want our probability statements to be as accurate as possible. To a large degree if we don't strive for that why should anyone pay attention to our research. The moral is buckle down and keep trying hard to produce the best research we can. If scientists don't do that then science is not worth anything at all. I am NOT ready for that.

Ah David! I fear that boat has already left. The people who have more to say about whether I get a paycheck tomorrow don't care about the statistics. Even something as amazing as the difference between regression and ANOVA seems almost novel.

In some cases I am lucky o get four replicates. They are willing to allow me to run the analysis on my time, but they review the papers and chop out useless and excessive statistical methods. Then their peers get to review the manuscript and do the same. The paper is about biology not statistics.

At this point I would be happy if I could convince them that it is worthwhile to have 10 replicates or (gasp) 20. One poor student got caught with three replicates and had a few failures. Now about 1/4 of the treatments have only one value. My guess is that EIV will not help rescue this project.

Timothy, if that's true why should anyone believe anything you publish? consider climate change. I am aware that some people will never believe but I, as a scientist, have a responsibility to draw the best conclusion that I can about Nature. If not I am worthless as a scientist. I was privileged to know Max Delbruck. He may have made mistakes but he would never accept that kind of thinking. Most serious scientists would not. Best wishes to you. David

So you would believe a published and peer reviewed study that had these problems but would not accept such from me? Why the double standard?

Yes, always draw the best conclusion possible given the available data.

My appologies but ,I am missing the connection between climate change and Max Delbruck.

Timothy, sorry you misunderstood me.i believe that the standard should be the same for all even though we know that some reviews are mistaken. Further that standard should be very high. I learned that from Max and others. Administrators can put us in difficult positions but the goal must be as high as possible.. Hope this is more clear. Best wishes, David