Permissible scale transforms

As my earlier article — https://medium.com/swlh/levels-of-measurements-a793c983828 reads, measurements do not carry meaningful information if the level is low; the numbers, even if with high precision, are not accurate. Each level of measurement has a permissible transform, which is the allowed set of mathematical operations that preserve the level of measurement. It is impossible for measurement at lower levels to transform to another level by any set of mathematical operations; the central dogma of statistics, GIGO (Garbage In Garbage Out) is prevalent everywhere. Permissible transforms are hierarchical. If you perform a transform that is only permissible at a lower level, you will automatically drop the level of measurement to that lower level.

For the measurements at the nominal level, any transformation that preserves equivalency and set membership are permissible. Virtually any mathematical operations preserve these basic attributes. For example, in our previous article — https://medium.com/swlh/levels-of-measurements-a793c983828, values in column V can be transformed to another set of values by multiplying with -1, yet Shatakshi and Tejaswini would still take the same time in the race.

Multiplying with zero would also preserve this attribute (this would equalize everyone’s time). Division, Square Root, Cube Root, Exponential, Log etc. also preserve equivalency and set membership attributes. However, note that most of the measurements done at the nominal level are for categorical or qualitative variables, which are non-numerical, therefore such transformations are not possible.

For measurements at the ordinal level, any monotonic increasing functional transform is allowed.

Y = f (X)

For example, addition, subtraction, multiplication with positive numbers, division with positive numbers, cubing are all monotonic increasing functions, and therefore, order-preserving. Squaring is not a permissible transform, as squaring negative numbers would make them positive, disturbing the rank order.

For measurements at the interval level, any linear transform is allowed.

Y= mX + C, where m > 0

The formula above would generate a linear graph. Note that ‘C’ (the y-intercept) need not be positive. It could be a fraction too. Note that variants such as (Y= X+1) or (Y= X-4) are also linear transforms and permissible at the interval level.

For measurements at ratio level, any multiplicative transform is allowed.

Y= mX, where m > 0

The formula above would generate a linear graph that passes through the origin (zero).

For measurements at absolute level (the true measurement), permissible transform is equivalency (which is, rather, not a transform at all)

Y=X

Permissible Statistics

In a practical point of view, levels of measurements are important because certain statistical operations are allowed only at that particular level; if a statistical operation not allowed at that measurement level is performed, conclusions arising therein would not be valid. Therefore, the first step in most of the statistical test is to find out at which measurement level the dataset in question falls into. For example, the ‘dihybrid cross’ dataset of Gregor Mendel is an example of measurements at nominal level (as it is impossible to go one step further and infer that ‘round green’ are better or worse than ‘wrinkled yellow’; no such rank-ordering is possible). Only statistical operators allowed at the nominal level (as explained below) can be performed for this dataset.

• For measurements at the nominal level, mode and Chi-square are two of the allowed statistical operations. Other operators, for example, mean, standard deviations and so on are not permissible at this level. If such operations are performed, conclusions are invalid.
• For measurements at ordinal level, non-parametric rank-based statistics such as median, percentile, Inter Quartile Range etc. are valid. For example, calculating the mean IQ level of a class is arbitrary and meaningless, as IQ scores fall under ordinal level, and mean is not an allowed statistical operation at this level.
• For measurements at the interval level, parametric statistics such as mean, standard deviation, correlation, regression, ANOVA and so on are valid. For example, logarithms of temperatures at the Celsius scale makes no sense, as the Celsius scale comes under interval level, and logarithmic operation is now valid at this level.
• For measurements at ratio level, in addition to all parametric statistics that are allowed at the interval level, logarithms, coefficient of variation, geometric mean, harmonic mean etc. are also allowed.

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