Multiple Regression Analysis – How to Find the Range of Independent Parameters for Valid Regression Equations

correlationexploratory-data-analysismachine learningmultiple regressionmultivariate analysis

I have generated a multiple-regression equation/model from the data.

Parameter A in the data varies from $0.1$ – $10$.

Parameter B in the data varies from $100$ – $1800$.

Parameter C in the data varies from $0.1$ – $0.5$.

The model is,

$y = c + \log A + \frac{1000}{B} + \log C$

Is this regression equation valid outside the range of the above parameters?

Is there any method to find out in what range of A, B and C the above regression model is applicable/valid?

Best Answer

Mathematically, the equation would be valid when $A>0, C>0$ and $B\neq 0$, you can substitute and find some $y$ for any given $A,B,C$ satisfying these conditions.

Estimating outside the ranges is extrapolating and since you do not have data to validate those ranges, it'd be nearly impossible to know if your model does a good job there; except the domain know-how you can apply on the found target values.

Related Solutions

Solved – Multiple regression analysis with spatial data as independent variable

It's a little unclear what your objectives are, so other methods might be preferable depending on those. Polynomial regression may suit continuous variables, but wouldn't make sense for categorical ones. You can add higher-order terms for the continuous variables alongside categorical predictors though.

Nominal predictors can be added to a multiple regression model using dummy codes. In your case, you could enter lithology as two dummy variables: using clastic rocks as the reference group (for example; not necessarily the one you want to choose), you could create one binary variable indicating whether a case involves carbonate rocks (1 if so, 0 if not), and another equivalent one for metamorphic/igneous rocks. The same process of dummy coding can work for any number of levels, and won't be too much harder to interpret for vegetation. Here's an example dataset: $$\begin{array}{c|cccccc}\rm Case&\rm Carbonate&\rm Metamorphic/Igneous&\rm Farm&\rm Grass&\rm Forest&...\\\hline\small\rm Clastic\ bare&0&0&0&0&0&...\\\small\rm Carbonate\ bare&1&0&0&0&0&...\\\small\rm Igneous\ bare&0&1&0&0&0&...\\\small\rm Clastic\ farm&0&0&1&0&0&...\\\small\rm Carbonate\ farm&1&0&1&0&0&...\\\small\rm Igneous\ forest&0&1&0&0&1&...\\...&...&...&...&...&...&... \end{array}$$See how that works? (I can elaborate if not.) Just enter these binary predictors like any other. The corresponding $\beta$s represent the differences between given groups and the reference group.
E.g., $\beta_{\rm Carbonate}$ represents the difference between carbonate and clastic in the above example.

Maximum Likelihood – How to Perform a Two-Dimensional Grid-Search for the MLE in R?

Here is an algorithm for an exhaustive search (i.e. searching the full grid rather than searching until you have a local max. You need to choose a step size for the $\pi$ loop and for the $\mu$ loop (it looks like you chose .01 in your code, so I will use that here).

mustep<-.01
pistep<-.01

mu_seq<-seq(min(x), max(x),mustep)
pi_seq<-seq(0,1,pistep)

ll_save<-matrix(, nrow = length(pi_seq), ncol = length(mu_seq))

     for (i in pi_seq){   
        for (j in mu_seq){

        loglike(i,j,x)

        }
     }

indices<-which(ll_save == max(ll_save), arr.ind=TRUE)

pi<-pi_seq[indices[1]]
mu<- mu_seq[indices[2]]

Best Answer

Related Solutions

Solved – Multiple regression analysis with spatial data as independent variable

Maximum Likelihood – How to Perform a Two-Dimensional Grid-Search for the MLE in R?

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