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I need to understand how a number of parameters contribute to decomposition in wood.

The parameters in question are (1) temperature, (2) humidity and (3) exposure to sun light. We already know that the all contribute, but we don't onow exactly how they affect the specific decomposition process.

My need to asses to what extent a 1% increase in either of these parameters will increase the decomposition process.

At my disposal, I have a dataset covering some 30 datapoints where the degree of decomposition is recorded together with the three other parameters.

What method would you recommend for this type of problem?

1 Answers 1

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One approach would be to use regression. The ideal, simplest model would be

$$Y_i = \beta_0 + \beta_1 X_{1i}+ \beta_2 X_{2i}+ \beta_3 X_{3i} + e_i,$$

where $e_i \stackrel{indep}{\sim} Norm(0, \sigma),$ for $i=1, 2, \dots, 30.$ Here $Y_i$ are the 30 measurements of deterioration, $X_{1i}$ of temperature, $X_{2i}$ of humidity, and $X_{3i}$ of sunlight exposure.

To check in advance whether this model has a chance of working, you can look at bivariate scatterplots of each the six pairs of the four variables to see whether relationships are roughly linear.

  • If plots of $Y_i$ vs. $X_{ji}$ are nearly linear, that is a positive indication.
  • If any of the plots of $X_{ji}$ vs. $X_{ki},\, j\ne k$ show points almost exactly on a straight line, that shows that temperature, humidity, and sunlight exposure may not all have useful information to contribute.
  • Also, it is possible (but I think in your case unlikely) you'll discover distinctly nonlinear relationships. Then the strictly linear model I proposed above may not be useful. (For example, what if low and high temperature both led to smaller deterioration than medium temperature?)

A multiple regression procedure would provide estimates $\hat \beta_j$ of the $\beta_j,\, j = 0,1,2,3$ and $\hat \sigma$ of $\sigma.$ You could perform standard regression diagnostic procedures to see whether 'errors' $e_i$ are normally distributed and whether their variances are the same. Also there may be indications in the regression output that some 'predictor' variables $X_j$ are much more important than others in predicting deterioration $Y.$ If so, you may get a more useful model by eliminating the least useful variable(s).

If you can get a satisfactory fitted regression model, then you can 'predict' $\hat Y_0$ corresponding to various new values $X_{j0}$ to see how much difference changes in these various 'predictor' variables make in the 'predicted' variable $Y.$ For small changes such as the $\pm 1\%$ you mention, the size of the corresponding $\hat \beta_j$ may be a useful guide.

I have not seen your data and so I can give only the most generic sort of advice. If you try this approach and encounter questions that can't be answered by an elementary statistics book, please leave a Comment, being as specific as possible, and I'll take a look.

  • 0
    Let me also add, to this very complete answer, that you might want to think about interactions of the three variables. Conceivably, it is possible that humidity has an effect only at high temperatures. In terms of the regression model, this amounts to including $X_{1i}X_{2i}$ among the regressors. Other interactions are possible and can be added, although one need to keep in mind that the number of regressors has be to below the number of observations. – 2017-01-25