Hello,
I've a question regarding how researchers come up with equations in their work?
For instance,
y=ax + b or y=x^a +b
whether a weight has to be multiplied or raised at a power of another variable?
I hope someone can understand what my problem is :/
Huma -
It sounds like you are talking about continuous data. The first thing to do would be to graph a scatterplot with x on the horizontal axis, and y on the vertical axis. Scatterplots can be very useful. You should also consider what makes sense for your subject matter. For example, for establishment surveys for Official Statistics, there is often a linear relationship through the origin (zero intercept, so no intercept term), with one regressor and . heteroscedasticity. When x is zero, y is expected to be zero. A graph that is slightly off of that may just be impacted by noise, so we may leave out the intercept term, even if the graph looks like there might be a very small one.
But suppose that there is an intercept term needed, or there is nonlinearity. Scatterplots can help you determine that. There may be a theoretical curve or several competing ones for your subject matter. You can see what the scatterplot supports.
You should also do residual plots. They can help you to see what curves might "fit" well, with what coefficients.
You should not just use a p-value alone - without a type II error analysis - from some "test" as many do, as a lone p-value is not meaningful.
So, plot the (x,y) data, do residual plots, compare to subject matter theory, and you should have a good idea of what your data are telling you. You could also look at the estimated variances of the prediction errors for predicted y-values. The square root of that in SAS PROC REG, for example, is STDI. You can look at standard errors of regression coefficients relative to the coefficient values. But r-square values are rather volatile and not so useful as you might hope.
You might look at a few transformations - logs are popular - but results may be easily misinterpreted, so be cautious.
If your relationship is complicated, you could look at splines or lowess.
But i suggest that you just start with a scatterplot and see what it tells you. You might consider any other graphics options you have, as well.
Cheers -Jim
PS -
If you over-complicate your model, so that you "overfit" your model to the data, then true variance could be high, though residuals for that dataset would appear to be low - because they are 'estimated' residuals. If you underfit your model, you could have 'omitted variable bias,' or bias due to different curvature, but these would show up in your estimated residuals, and such residual plots could be interesting. (Again ... various scatterplots can be very enlightening.)
You would do well to develop your model with one set of data (a "training set" of data is the term used in "statistical learning"), and then validate your model using other applicable data. You might want to research the term "cross-validation."
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