#log-log

[unknownbirds]

[learnerworld]

Multiplicative Regression Model

The general linear model assumes various predictors affects the response variable additively.

For example, in the model 

Ŷ   =   β0  +  β1 X1  +  β2 X2                                                          ...(1)

the predictors  X1 and X2 are assumed to contribute additively to the response variable Y.

The coefficient β1 can be interpreted as slope of the line or constant of proportionality. It represent the absolute change in Y for one unit absolute change in predictor X1 keeping other predictor x2 fixed. Similarly, if X2 increases by one unit, X1 kept constant, Y is expected to increase by β2 units.

And if both X1 and X2 increase by one unit, then Y is expected to change by (β1 + β2) units. In other words, the total expected change in Y is determined by adding the effects of the separate changes in X1 and X2.

However, in some cases, predictors contribute multiplicatively to the response variable. Then, the model can be expressed as: 

Ŷ   =   β0 *( X1 β1 )*( X2 β2)                                                         ...(2)

Unlike additive models, here the expected percentage change in the response variable Y is proportional to percentage change in the predictor x1 (and similarly for X2).

And if X1 and X2 both change, then the expected total percentage change in Y should be the sum of the percentage changes that would have resulted separately.

The multiplicative model can be converted into linear model by using logarithm transformation.

Applying log transformation to equation (2), we get

Log(Ŷ) = Log(β0) + β1*Log(X1) + β2*Log(x2) 

The resulting model (log-log) model is linear, so can be analyzed by using usual techniques. One need to be careful while interpreting the results of log-log model.