Statistics

This section explains the correlation and explained variance calculations that are used by the Analysis module.

The Analysis module calculates the correlation coefficient for linear models. This shows how well the data are linearly related. The correlation is displayed in the Statistics table.

If you are producing a calibration curve that relates peak area or height to amount or concentration, you aim to achieve a high positive correlation coefficient. A value of +1 indicates a perfect fit of all the data to the straight line. A molecular size curve has a negative slope, so the aim is towards a correlation coefficient of -1.

If you only have two data points for a Linear model, or only one point for a Linear through origin model, the fitted straight line will inevitably pass exactly through the points. By definition, this leads to a correlation of exactly +1, but this does not indicate a good fit, but instead indicates too few data points. In these cases the Statistics table will display a “---” symbol instead of the correlation value.

The correlation is derived as follows:

Explained variance provides a measurement of how much of the variation in the data points (xy pairs) is due to the model. The remaining variation can be attributed to noise, i.e. random errors, or to the fact that an inappropriate model has been selected. This makes it possible to use the explained variance value for model selection, e.g. to decide if a quadratic model fits the data better than a linear model. This would be confirmed by a higher explained variance value.

Note: The explained variance is not calculated for curve models drawn through the origin.

The explained variance is equal to R2 adjusted for degrees of freedom. The illustration below shows the mathematical model:

You can only obtain a value for explained variance if you have sufficient data points on the curve. For instance, if you only have two points for a Linear model, or only three points for a Quadratic model, the fitted curve will pass exactly through the points. By definition, this leads to an undefined value for explained variance. In these cases the Statistics table will show a “---” symbol instead of an explained variance value.