Working with normality

scale() : to convert a normal distribution to a standardised normal distribution with mean 0, and sd 1.

For PCA, Ivanek et al. has another solution which may or may not be the same. (chk that post)

 Log transformation is a useful technique, but results need to be back transformed for reporting. while means can eb back-transformed directly, CIs cannot.

Back transformation

Even though you’ve done a statistical test on a transformed variable, such as the log of fish abundance, it is not a good idea to report your means, standard errors, etc. in transformed units. A graph that showed that the mean of the log of fish per 75 meters of stream was 1.044 would not be very informative for someone who can’t do fractional exponents in their head. Instead, you should back-transform your results. This involves doing the opposite of the mathematical function you used in the data transformation. For the log transformation, you would back-transform by raising 10 to the power of your number. For example, the log transformed data above has a mean of 1.044 and a 95 percent confidence interval of 0.344 log-transformed fish. The back-transformed mean would be 101.044=11.1 fish. The upper confidence limit would be 10(1.044+0.344)=24.4 fish, and the lower confidence limit would be 10(1.044-0.344)=5.0 fish. Note that the confidence limits are no longer symmetrical; the upper limit is 13.3 fish above the mean, while the lower limit is 6.1 fish below the mean. Also note that you can’t just back-transform the confidence interval and add or subtract that from the back-transformed mean; you can’t take 100.344 and add or subtract that to 11.1.

Transformations possible:

There are many kinds, but the log and the sqrt are the most common. Best way to try out these transformations and see how it affects normality and homoscedasticity.
 Log transformation: most variables in biology have log-normal transformations. This is because usually the dependent var is a product of many independent vars. meaning, height of a tree is a product of sunlight, water, nitrogen and insects, which mathematically results in a log-normal distribution. For count data, add 0.5 to all numbers. otherwise, sometimes, ‘c’ is added to all  numbers, such that ‘c’+min(X) = 1 or slightly greater than zero.

Square root transformation:
It is usually used for counts of something like number os colonies in a pteri dish, number of beech per sq.m., number of infected colonies in an apiary

Arcsince transformation:
usually used for ratios and proportions (ranges of 0 to 1) . Back tranformation is by squaring the sine of the value

inverse hyperbolic sine transformation: This is an alternative to box-cox transformation


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