# Linear model or generalised linear model for quantifying effect size of model terms when using a log transformed response

Length – Weight Relationships (LWR) are widely used in fisheries studies; weight and length are recorded, log transformed (e or 10), then the linear regression equation of log weight ~ log length derived. This equation can be used in future studies to estimate weight when only length is known.

I've an abundant fish species for which I wish to test how Length-Weight Relationship (LWR) may be altered due to site (3 level factor), season (2 level factor) and year (2 year factor).

I use a simple linear regression analysis (in R) to test log weight as the dependent variable against the main effects and interaction effects of log length, site, season and year:

log weight ~ log length * site * season * year

I derive a minimum adequate model using analysis of deviance between models with and without each term included to identify non-significant terms for elimination (starting with higher order interactions).

This analysis has 2 aims:

1) to determine if interactive and then main effects of length, site, season and year significantly impact the LWR

2) to quantify the model variance accounted for by each significant model term, and thus the relative importance of each variable (quantified as the model term’ Sum of Squares expressed as a proportion of the SS of the null model)

Note, 4- and 3-way interactions are difficult to interpret - this is not my objective; my objective is simply to establish variables' significance and quantify effect size.

My model residuals look to be normally distributed and have equal homogeneity of variances.

I am aware there is discussion in the literature regarding potential caveats of using log transformed response variables which may affect:

• quantifying the model variance partitioned to the independent variables and interactions terms

• transformed data may not estimate the mean accurately, as the mean of log-transformed responses is not the same as the logarithm of the mean response

Some advices instead suggest using a non-transformed response variable within a generalised linear model with a log-link function. (Linear model with log-transformed response vs. generalized linear model with log link)

However, as fish LWR use log transformed data by standard, so it would seem more suitable to test the effects of site, season and year when using log transformed length and weight data.

I therefore wondered if my analysis above was 1) appropriate and robust (i.e. not wrong), or 2) could be bettered?

• Tthe von Bertalanffy growth equation is not normally log transformed and is considered a more difficult regression due to the requirement for non-linear regression. Most certainly a non-transformed response variable makes interpretation of the fitting results much more straightforward at the same time that it becomes more difficult for professional colleagues to replicate (as they use the simpler log transformed linear regression). So the standard is due to earlier historical lack of tools such as R that make non-linear regression easy to perform, however this is not a modern restriction. – James Phillips Feb 12 '18 at 14:25
• Thanks - I agree with these perspectives. Using a non-transformed response in a non-linear regression (i.e. von Bertalanffy growth equation) is more complex to partition the model variance in weight accounted for by each independent variable (length, site, season, year) and their interactions. Therefore, if possible, I would keep my analysis as above. However, is my analysis above valid? i.e. is it undermined by the discussion in the literature regarding potential caveats of working with a transformed response variable? – user2890989 Feb 12 '18 at 16:52
• As I understand it, the main problem with log transform is interpretation of the fitted parameter values and fit statistics rather than a question of validity. For example, if the reason you have the current mathematical model is for explanation of relationships among the variables then the transform makes this interpretation more difficult than if the variables are not transformed. If on the other hand you wish to predict or interpolate new values for untested conditions within the range of the fitted data, a data transformed model can be quite useful. – James Phillips Feb 12 '18 at 18:11
• That's helpful. The main aim of the analysis is to determine significance of each model term, and then quantify the model variance explained by each model term. This is so I can say that model term 1 explains 90% of model variance while model term 2 explains 1%. Hence, model term 1 is far more important than model term 2 in determining the value of the dependent variable. As you suggest my current analysis seems valid per se, then as I understand it, as I am primarily interested in partitioning variance not interpretation of parameter values, so my current analysis is probably an OK approach. – user2890989 Feb 13 '18 at 11:43
• It might be worth tryng a test regression where the weight is not log transformed, but the length is log transformed. This would still be a linear regression and it is possible that some insight could be gained towards your analytical goal. – James Phillips Feb 13 '18 at 14:05