Dear Kunjan, didn't you, by any chance, wrote "...low P value as well as low R squared" when you meant "...high P...." and vice-versa ".. the resultant model has high R squared value" when it should be "low R"? I can't figure out how these two values can get higher (or lower) simultaneously as a result of a model fitting procedure (either simple linear to more complex with interactions).

Regards Luis

Hi Kunjan Patel ,

Most likely, this indicates that the model with interaction terms is better to explain your data than the model with no interaction terms. In general, large contribution of interaction terms to the fit of the model means that the effects of explanatory variables are not independent but interact to each other. You may think this as one variable potentiating the effect of the other (synergism) or one variable cancelling out the effect of other (antagonism).

If there are reasonable reasons to include the interaction terms a priori (theoretical reasons), you can go straight to the model with the interaction terms. If there are no clear reasons supporting interaction effects a priori, you can fit both models, calculate their AIC (Akaike Information Criterium, indicates the parsimony of the model) and perform a likelihood test to contrast both models. A significant likelihood test indicates that the model with the smaller AIC is the most parsimonious to explain your data.

I hope that helps.

Hello Eduardo R. Cunha , that exactly is the case. The interaction has larger contribution to the fit of the model.

Thanks alot for the answer.

I still have one doubt though. If the model with higher interaction between the terms result in high R squared, does it mean that it is better for the prediction compare to the model with no interaction terms and low R squared?

Dear Luis F. Gouveia ,

Yes, more interactions between the terms result in higher R squared as well as higher P value for each term. Since the number of terms have been increased because of the more interactions and the significance of single term is divided into the interaction of the same term with other terms. This is my understanding.

For example,

Model without interactions is,

Y= C+3X+2Y+4Z.

here,

X, Y and Z has 0.000 P value but R squared is also 72%.

But for the model with interactions,

Y=2+0.5X+0.2Y+0.7Z+2XY+0.9XZ+0.6YZ+3X^2+4Y^2

here,

X, Y and z has p value more than 0.02 but R squared is 98.4%.

Dear Kunjan Patel , I appreciate you answer/clarification. In fact, whenever you increase the number of model parameters (increasing the order of the terms and/or including interactions) the R^2 always increases (or remains the same) as you increase the explained variation and the total variation is not changed. That is why in the example you gave the R^2 increased from 0.72 to 0.984 (the ratio "explained variation"/"total variation" increased from 72% to 98.4%). The calculation of the R^2 does not take into consideration the number of parameters included in the model. However, and opposite to the R2, the p-values depend on the degrees of freedom associated to the model as well as the ones left to the "error" (unexplained SS).

The reason why you get a higher p-value in the second case (2nd degree model + interactions) is because you have a larger number of df for the model (8 parameters in the second model against 3 in the first) and a smaller number for the error component (df(total)-df(model)) and not because the model itself is better or worst than the simpler model.

In this particular case, the increase in the explained variation (SS reg or SS model) and consequent decrease in the unexplained variation (SS error or SS residual) is not enough to compensate the "large" increase in the number of model parameters (from 3 to 8). That is why it is advisable (and common) to use the R2adjusted or AIC (as Eduardo R. Cunha mentioned), to name a few, when comparing models with different number of parameters.

Both the above metrics take into consideration the number of parameters in the model and allow you to make "fair" comparisons. As such, my question regarding the simultaneously increase of p and R2 remains to be clarified in face of the complexity of the model chosen to describe your set of observations. Without additional information (number of total observations, SS model, SS total, replicates, etc) it is usually not possible to state if the additional parameters make any statistical sense - they may make practical and/or theoretical sense.

It is not clear if the models you show in the example are the ones you had in mind when posting the question. If they are, it is not be clear if the fitting improvement is due to the interaction of variables or to the curvilinear nature of response surface (the squared terms).

Hope this contributes to clarify the point I raised.

Regards, Luis

Dear Luis F. Gouveia ,

Thanks alot for further information. Its been much clearer now.

Sorry that i did not form my question correctly. when i said "increase in P value and Rsquared, i meant increase in P value of predictor A and increase in Rsquared of the model.

And yes, the values i showed before were just for indication.

the original result is as follows.

Case 1 :

Model without interactions

Rsquared = 74.91% ; Rsquared (Adj) = 73.06%

P values

A = 0.028

B = 0.089

C = 0.000

D = 0.000

Case 2:

Model with interactions (through 4th order)

Rsquared = 94.24% ; Rsquared (Adj) = 92.25%

P values

A = 0.810

B = 0.607

C = 0.547

D = 0.162

A*B = 0.839

A*C = 0.464

B*C = 0.523

C*D = 0.000

A*B*C = 0.753

Dear Kunjan Patel , you have a 3-level design and no single squared factor/variable is present in the "final" model? Are you willing to share the data (without revealing any private/sensitive details on the variables) - just the combination of the factors and the result (I guess you should have 81 experiments = 3^4) so we may give some additional insight? Regards, Luis

Dear Luis F. Gouveia ,

sorry for the dealy and thanks alot for the interest. I am sharing the actual data attached to this message.

The input parameters are Length, Height, Thickness, Fiber% and material.

The output parameters are Weldline length and warpage.

Dera Kunjan, I'll be back ASAP - I've been/still am a "bit" busy lately. I noticed, however, that you have a mixed design, (one categorical variable with 2 levels plus 4 continuous variables, 2 with 2 levels and the other 2 with 3 levels, as well as 2 dependent variables). Regards, Luis

Dear Luis, its totally fine. whenever you get a time to look at it. and yes...you are right, its a mixed design.