comparing regression coefficients between different models to see if they are similar

#1
Hi all;

This is a very elementary problem; but unfortunately, my strengths do not like in statistics:
I am trying to describe plant growth over time using accumulated temperature (dd) as my independent variable and the plants 'growth stage' (gs) as my dependent variable.
The shape of the data is sigmoid and I have three different sets of data for different geographic areas.
I decided to use a third order polynomial to describe the data (which it does well in all three cases). Example below:

GS = - 0.969 + 0.02909 DD + 0.000062 DD**2 - 0.000000 DD**3

Now; what I really want to know, is if the relationship described in one place is similar to another place: i.e. can I describe plant growth in a geographic area using a polynomial model that was derived in a different location.
I thought the best way to go about doing this was by using the standard errors associated with the coefficients of one model and checking to see if the other two models coefficients fell within these error limits? Am I going about this in the wrong way? Minitab doesn't seem to provide the standard errors when using a 'fitted line plot' to fit the polynomial.
Many thanks for any help..
 

noetsi

Fortran must die
#2
Re: comparing regression coefficients between different models to see if they are sim

I am not sure what you want to know. If you want to compare estimated parameters from different samples, I have never seen that done.
 
#3
Re: comparing regression coefficients between different models to see if they are sim

Hi,
Thanks for the reply. What I ultimately want to find out is if the three polynomial equations are different from one another statistically. If they are not different then I describe plant growth in one geographic area using a polynomial model that was derived in a different location.
For example; below are two polynomials from two different locations describing plant growth stages (GS) in relation to accumulated temperature (DD). How can I find out if they are ststistically different from one another?i.e. is their output very different when supplied with the same DD?
I thought I would use the standard errors associated with the coefficients to do this....but perhaps that is not correct?

Many thanks

GS = - 0.969 + 0.02909 *DD + 0.000062 * DD**2 - 0.0000001 * DD**3
GS2 = 0.847 + 0.03232* DD + 0.000060 * DD**2 - 0.0000002 * DD**3
 

noetsi

Fortran must die
#4
Re: comparing regression coefficients between different models to see if they are sim

If by different you mean they predict the model differently (one adds predictive value over another) than probably the best of many statistics is the AIC or BIC. The model that has the lower value in either will be the best predictor. I have never seen a formal statistical test for if the difference between a AIC/BIC between two models is signficant. If two sets of parameters nest inside each other (that is one set of parameters is entirely included in the other) there is a formal test if one adds predictive value - such as a chi square difference test.

If you mean do the same set of parameters (the same model) predict differently in different samples -I don't know a formal test. The R squared value, for example, is one way you can compare this.
 
#5
Re: comparing regression coefficients between different models to see if they are sim

Thanks for that.
I don't think I have explained myself properly.
While I appreciate what you are suggesting regarding the AIC/BICs, all of the models have the same number of parameters, which reduces the utility of such an approach does it not?

What I actually want to do; is estimate polynomial parameters based on the relationship between temperature (DD) and plant growth stages (GS) in New York (for example); then, when I have my parameterised polynomial 'model';I could input temperature data (DD) from Colorado (for example) and see if the model which was trained in a different location (New York) can give me the plant growth stages in Colorado (I would have the growth stage data to check this).

I have a parameterised polynomial for three such locations, and I want to pick one of them that can robustly produce approximate growth stages for all three locations. But, of course, I must be able to back up this decision statistically.
Does that make sense?

Many thanks
 

noetsi

Fortran must die
#6
Re: comparing regression coefficients between different models to see if they are sim

I think what you are asking is, given the same model, with data from several states, which state data allows me to predict all three states the best (that is which parameters and SE would be best to use to predict each state individually rather than as a whole if you had to chose just one set)? If that is what you are asking then I would think a goodness of fit test (like Hosmer-Lemeshow for logistic regression - I think linear regression has a deviance test to do the same) would be one approach.

It tells you how well your predictions fit the observed data. But I have not seen it used this way, and if there were discrepancies (say one set of parameters worked better in state x and another in state y) it would not tell you which was best overall in any formal way. Nor is there any statistical test if the predictions were signficantly better.
 
#7
Re: comparing regression coefficients between different models to see if they are sim

That's almost exactly what I'm asking except:
taking each of the three models above individually; I want to use the temperature data from each state to produce the GS predictions.
That would give me three sets of GS predictions from each model (9 in total). Then I want to be able to quantify how good or bad each of the models are at predicting GS at the three locations; in order to pick the one which describes the observed GS in all three locations the best (i.e. minimises the error).
I will have a look at the test you suggest. If it provides a quantifiable answer for how well the model fits the observed, then I can compare the quantities between all three models for all three states and see which one maximises the fit?
Thanks
 
#8
Re: comparing regression coefficients between different models to see if they are sim

That's almost exactly what I'm asking except:
taking each of the three models above individually; I want to use the temperature data from each state to produce the GS predictions.
That would give me three sets of GS predictions from each model (9 in total). Then I want to be able to quantify how good or bad each of the models are at predicting GS at the three locations; in order to pick the one which describes the observed GS in all three locations the best (i.e. minimises the error).
I will have a look at the test you suggest. If it provides a quantifiable answer for how well the model fits the observed, then I can compare the quantities between all three models for all three states and see which one maximises the fit?
Thanks
 

noetsi

Fortran must die
#9
Re: comparing regression coefficients between different models to see if they are sim

You will be able to tell which one produces estimated data that is closest to observed data if that is what you mean.

Another possibility is to look at the error terms used in forecasting (although I assume they could be used generally). I am new to these, but as I read this the other day I wondered if you might be able to use it do do what you want. I don't know enough about them to comment yet, but in case you are interested...

http://people.duke.edu/~rnau/compare.htm they all get at how your predictions fit the actual data.