Linear vs. Logarithmic Scaling

Graphical representations of data are crucial when performing most analyses . An effective graph will clearly display the relationship between experiment parameters. Graphical scaling and transforms of any parameter (i.e. channel) can be adjusted in order to appropriately interpret a given dataset. The optimal scaling depends on the nature of the data. Linear and logarithmic scaling are two common methods of representing data.

Linear scaling is achieved by plotting events within evenly distributed, equally sized bins. Accordingly, the visual distance between data points is proportional to the numerical distance between the values. Rulers and measuring tapes are other examples of linear scales. Figure 1 is a plot of arbitrary values with a linear relationship. 

Figure 1. Data with a linear relationship

Each graduation along both axes represents a value change of one. This scale is constant for the entire span of the graph. 

Linear scales are most effective for displaying datasets with values spread evenly across a given range. When working with flow cytometry data, linear scaling is commonly used when plotting forward scatter (FSC) and side scatter (SSC). FSC and SSC are relative to cell size and cell granularity, respectively. These measurements are not concentrated in any particular region of the parameters’ scales, so both the parameter’s features are displayed well on a linear plot. Linear plots are less practical when data points consume a larger dynamic range. 

Figure 2. Scatter parameters displayed in linear (left) and log transformed space, within FlowJo.

Logarithmic scales are very powerful when graphing parameters with a wide dynamic range. A log scale is based on exponential (i.e. multiplicative) differences between values. Consecutive graduations along an axis represent equal changes in ratio. Figure 2 is a plot of arbitrary values with a logarithmic relationship.

Figure 3. Data with a logarithmic relationship; log-10 specifically.

The intervals increase successively by a factor of ten.  This allows for observation of percentage changes rather than absolute changes. Consider the scale along the x-axis in Figure 2: a shift of five units between the first and second interval would be significant. A shift of five units between the sixth and seventh intervals would be unnoticeable. The duality of the log scale enables the identification of meaningful shifts around low values as well as around bigger input. 

Log scales are used when plotting fluorescent flow cytometry data. In order to delineate a heterogeneous population of cells, a cell structure of interest is stained with a fluorescent dye. Positive cells (i.e. those that contain a large number of the target structure) emit light, sometimes at an intensity thousands of times higher than negative cells (those with few or no targets). Log scales are suitable for this type of parameter. 

Figure 4. Fluorescence parameters displayed in log (left) and linear transformed space, within FlowJoⓇ.

Log scales do not include values of zero or lower, and may not appropriately illustrate low dynamic ranging details of a dataset. If the dataset being displayed includes negative values, then a linear display may be more appropriate. However FlowJoⓇ does provide transforms which accommodate the best of both worlds - such as “Biexponential”.  

Figure 5. Fluorescence parameters displayed with Biexponential scaling in FlowJo.