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Generalizing to a Population:
MODELS: IMPORTANT DETAILS continued

Log Transformation for Better Fits
In log transformation you use natural logs of the values of the variable in your analyses, rather than the original raw values. Log transformation works for data where you can see that the residuals get bigger for bigger values of the dependent variable. Such trends in the residuals occur often, because the error or change in the value of an outcome variable is often a percent of the value rather than an absolute value. For the same percent error, a bigger value of the variable means a bigger absolute error, so residuals are bigger too. Taking logs "pulls in" the residuals for the bigger values. Here's how.

A percent error in a variable is actually a multiplicative factor. For example, an error of 5% means the error is typically 5/100 times the value of the variable. When you take logs, the multiplicative factor becomes an additive factor, because that's how logs work: log(Y*error) = log(Y) + log(error). The percent error therefore becomes the same additive error, regardless of the value of Y. So your analyses work, because your non-uniform residuals become uniform. This feature of log transformation is useful for analysis of most types of athletic performance and many other measurements on humans.

Percent Effects from Log-Transformed Variables
If the percent error in a variable is similar from subject to subject, it's likely that treatment effects or differences between groups expressed as percents are also similar from subject to subject. It therefore makes sense to express a change or difference as a percent rather than as a raw number. For example, it's better to report the effect of a drug treatment on high-jump performance as 4% rather than 8 cm, because the drug affects every athlete by 4%, but only those athletes who jump 2 m will experience a change of 8 cm. In such situations, the analysis of the log-transformed variable provides the most accurate estimate of the percent change or difference. Make sure you use natural logs, not base-10 logs, then analyze the log-transformed variable in the usual way.

Suppose you end up with a difference of 0.037 (you'll often get small numbers like this).
 Explanation of 100(ediff  -  1) and 100diff If Z = log(Y) and Z' = log(Y'), then diff = Z' - Z = log(Y') - log(Y) = log(Y'/Y). But Y'/Y = 1+(Y'-Y)/Y = 1+(percent change in Y)/100. Therefore ediff = Y'/Y = 1+(percent change in Y)/100. Therefore percent change in Y = 100(ediff  -  1). For small diff, ediff  = 1 + diff, so percent change in Y is approximately 100diff.
Now multiply it by 100, and hey presto, the difference in your mean is 3.7%. Actually, multiplying by 100 is an approximation, and it's near enough only for differences <0.05 (5%). The exact percent difference is given by 100(ediff  -  1), where e is exponential e and diff is the difference provided by the analysis of the log-transformed variable (see explanation box). This formula simplifies to 100diff only for diff <0.05.

I find it easier to interpret the diffs (differences or changes) in a log-transformed variable if I use 100x the log of the variable as the log transformation. That way the diffs are already approximately percents. For example, instead of getting a change of 0.037, you will get 3.7, which means approximately 3.7%. To convert this diff to an exact percent, the formula is 100(ediff/100 - 1), obviously! A diff of 3.7 is really 100(e3.7/100 - 1) = 3.8%.

It's easy to get confused when the percent change is large. For example, a change of 90% means that the final value is (1 + 90/100) or 1.90 times the initial value. A change of 100% therefore means that the final value is (1 + 100/100) or 2.0 times the initial value. A 200% increase means that the value has increased by a factor of 3, and so on. A negative percent change can also be confusing. (In a previous version of this paragraph, my interpretation of large negative changes was wrong!) A change of -43% means that the final value is (1 - 43/100) or 0.57 times the initial value. An 80% fall means that the final value is only 0.20 times the initial value, and so on.

When variables need log transformation to make them normal, how do you represent their means and standard deviations? I think a hybrid approach is best. Convert the mean of the log-transformed variable back to raw units using the back-transformation Y = emean (if your transformation was Z = logY) or Y = emean/100 (if you used Z = 100logY). Keep the standard deviation as a percent variation or coefficient of variation (CV). Calculate this CV in the same way as for differences or changes in the variable: if SD is the standard deviation of the log-transformed variable, the approximate CV is simply 100SD, and the exact CV is 100(eSD - 1). If you used 100log for your transformation, the approximate CV is simply the SD itself, and the exact CV is 100(eSD/100 - 1).

You can convert the CV into a raw standard deviation, but it's messy and I don't recommend it. Back-transforming the SD as eSD is incorrect. Instead, you have to show the upper and lower values of the mean ± standard deviation as emean + SD and emean - SD. With a bit of algebra, you can show that emean + SD is equal to the back-transformed mean times 1 + CV, and emean - SD is the back-transformed mean times 1/(1 + CV). Hence a CV of, say, 23% represents a typical variation in the mean of ×1.23 through ×1/1.23. As I explain on the page about calculating reliability as a CV, it's OK to write ±CV, provide you realize what it really means.

CAUTION. With log and other non-linear transformations, the back-transformed mean of the transformed variable will never be the same as the mean of the original raw variable. Log transformation yields the so-called geometric mean of the variable, which isn't easily interpreted. Rank transformation yields the median, or the middle value, which at least means something you can understand. The square-root and arcsine-root transformations for counts and proportions yield goodness-knows-what. Usually it's the effects you are interested in, not the mean values for groups, so you don't need to worry. But if the means are important, for example if you want the true mean counts of injuries to come out of your analysis, you will have to use a cutting-edge modeling approach that does not require transformation, such as binomial regression.

If you're graphing means and standard deviations of a variable that needed log transformation, use a log scale on the axis. Here's how. Plot the values you get from the log-transformed data without back-transformation, but delete the tick marks and put new ticks corresponding to values of the original raw variable that you would normally show on a plot. (You will struggle to understand what I am getting at here. Persevere. And if you use Excel to do your graphs, paste the graph into Powerpoint and do the editing there.) The error bar or bars go onto the plot without and fiddling. In fact, you can put the error bar anywhere on the axis.

More Examples of Log Transformation
Let's get back to the issue of goodness-of-fit with log transformations. In a previous example with weights and heights (see the figure at right), it's clear that people's weights get more variable for heavier people--quite reasonable when you think about it--so taking logs of the weight would be a good thing to try. When you fit a straight line, log transformation of the independent variable may also remove the "dip" in the residuals that we saw with this example on the previous page. So taking logs of the heights and the weights in the above example would make the model much fitter!

Many relationships that have a curve in them respond well to log-log transformation. To get technical, all models of the form Y = aXn convert to simple linear models when you take logs: logy = loga + nlogX. The relationship between weight (Y) and height (X) is a particularly good example. The value of the parameter n is given by the slope of the log-log plot, and it is about 1.7, or nearly 2, which is why we normalize body weights by dividing by the height squared to get the so-called body mass index. It would be better to divide by height to the power of 1.7, but that's another story.

Now check out the figure at right, from our example of the effect of sex on height. Do you think there's a need for log transformation here? You bet! Just look at the differences in the standard deviations on the bar graph: the males have a bigger standard deviation and a bigger mean, so log transformation is indicated (provided the means and standard deviations are pretty-much in proportion). Analysis of log-transformed height will give the difference between the females and males as a percent. You can also analyze these data without transformation by using the t test with unequal variances. What you will get then is the absolute difference in height between the average female and the average male. There's nothing wrong with that, if it's what you want.

Another case for some sort of transformation is where the standard deviation is about the same size as, or even bigger than, the mean. This sort of thing sometimes happens when variables have very skewed distributions. Example: the level of deliberate physical activity in adults, where you have most people hovering around zero hours per week, and the rest doing up to 10 hours a week or even more. So the mean ± SD might be 0.7 ± 1.8 hours per week. It doesn't mean that some people are doing negative hours per week! For such awful data we could use rank transformation: see the next page.

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Last updated 16 Jan 03