Transformation of specific data

I need help with data transformation. In the picture below the upper left picture shows the histogram of the variable V6. Because it is so right-skewed I tried 3 forms of transformation but none of them seem to make the data more symmetrical. Is there maybe another solution for this? Maybe change of breaks or something else?

The data is:

head of data CanopyCover = V6 91.30 61.50 91.40 92.00 93.20

EDIT: Here is the data:

Id  SqCones Ntrees  DBH TreeHeight  CanopyCover
Abern1  61  32  0.23    20.42   91.30
Abern2  4   4   0.27    15.20   61.50
Abern3  15  34  0.17    15.97   91.40
Abern4  9   22  0.23    22.42   92.00
Abern5  42  22  0.18    19.45   93.20
Abern6  4   21  0.23    23.07   93.50
Abern7  12  19  0.22    21.06   88.50
Abern8  27  15  0.26    18.82   88.00
Abern9  0   12  0.23    19.16   89.80
Abern10 4   9   0.12    6.38    73.30
Abern11 91  5   0.79    25.50   94.80
Abern12 20  12  0.20    12.02   94.20
Abern13 5   15  0.19    9.06    76.80
Abern14 14  42  0.15    8.82    77.20
Abern15 35  74  0.15    17.91   91.30
Abern16 11  23  0.15    15.93   92.20
Abern17 47  67  0.14    13.79   91.80
Abern18 17  33  0.17    14.60   88.60
Abern19 16  12  0.34    13.99   92.40
Abern20 0   7   0.40    16.16   85.20
Abern21 44  14  0.37    20.88   92.90
Abern22 18  23  0.23    15.54   91.50
Abern23 9   13  0.27    16.98   90.70
Abern24 16  7   0.32    19.20   89.00
Abern25 60  11  0.26    20.03   93.50
Abern26 3   7   0.29    15.87   91.90
Abern27 5   10  0.35    20.87   90.70
Abern28 5   11  0.31    21.55   90.40
Abern29 2   3   0.42    20.37   69.90
Abern30 32  11  0.33    18.27   92.60
Abern31 55  15  0.32    24.50   91.40
Abern32 3   11  0.34    19.12   89.20
QEFP33  18  14  0.35    22.98   87.60
QEFP34  0   13  0.27    16.11   54.40
QEFP35  11  7   0.35    22.26   93.10
QEFP36  0   22  0.23    15.55   90.20
QEFP37  6   18  0.33    20.98   93.60
QEFP38  4   18  0.27    19.21   93.10
QEFP39  0   9   0.35    24.12   84.40
QEFP40  48  11  0.37    22.68   86.50
QEFP41  7   16  0.26    21.27   91.10
QEFP42  2   11  0.35    21.70   80.70
QEFP43  3   12  0.35    21.48   83.30
QEFP44  2   8   0.35    21.87   77.30
QEFP45  21  9   0.33    21.65   80.00
QEFP46  22  9   0.32    23.32   88.00
QEFP47  4   12  0.36    22.77   81.10
QEFP48  1   28  0.22    18.53   93.80
QEFP49  3   30  0.19    16.19   84.80
QEFP50  25  30  0.18    19.47   87.20
QEFP51  57  36  0.17    17.08   89.60
QEFP52  12  11  0.26    21.36   87.80

• ummm... maybe there is no solution to this because the data just isn't very normal?! I feel kinda stupid trying to transform this.. ;) – Vera Maria Mar 21 '16 at 21:23
• Why are you trying to make the data more "symmetrical"? How do you intend to use the transformed (or un-transformed) variables? – Vishal Mar 21 '16 at 21:42
• Hey guy from India ;) Ummm.... this actually belongs to an old exam question. I have dataset where I have number of cones (integers) and their abundance is supposed to be explained by environmental variables to which the Canopy Cover variables belong. I'm actually still clueless on how to get an answer (that is which mdoel to use) but I thought I could start with a variable transformation because many models have normality as a prerequesite for usage. So maybe I should recheck the whole thing. Do you have a clue? Thanks :) – Vera Maria Mar 21 '16 at 22:08
• It's cosmetic, but my advice is never to ask a question in terms of names like V6. It's unimportant to us what you call a variable inside a program; we'd rather have information on what it represents. I am not an ecologist, but I can think about percent cover. In fact, it should be important to you, in whatever software to use, never to use a name like V6 but always to use an evocative, informative name. – Nick Cox Mar 21 '16 at 22:21
• I wonder what the histogram of percent non-cover looks like :-). – whuber Mar 21 '16 at 22:34

From your graphs it appears that you have about 50 measurements of percent cover. The values range from about 50% to about 100%. It's possible that some values are recorded as 100%. Presumably values cannot exceed 100% (if not, please tell us otherwise).

Note first of all that they are left-skewed, not right-skewed.

In statistics, the label for skewness is that of the longer tail: the terminology implies that you are looking at a histogram with magnitude axis horizontal. In this case, you are, so no problem there.

So, square root, cube root, logarithm can't possibly help. Those are transformations for right-skewed variables.

There is a more subtle problem too. Over that twofold range, from about 50 to about 100, those transforms are close to linear, as shown by the graphs below. So they will change the units of measurement, but more crucially they won't change the shape of the distribution much at all. That's why -- although they cannot help -- they in fact don't make much difference and why the histograms you show are all more or less the same shape.

It's possible that a logit transformation will help, or a folded power.

Can you post the raw data?

P.S. Normality, or otherwise, is at best marginally relevant for such bounded data. Some analyses might go better if you had a more symmetric distribution, but no more. The bigger deal is what you intend to do with the data.

EDIT on seeing the data:

There are at least two questions bundled together here:

1. What kind of transformation would best symmetrize a variable like canopy cover? Note that in principle such a variable is bounded between 0 and 100%. For that reason alone, many well-known distributions, including the normal, can't fit the data in principle.

2. What kind of scales (including quite possibly the scales on which data arrive) should be used for analysing a response variable in relation to various predictors such as canopy cover?

These questions aren't that closely related. I'll answer them in reverse order.

1. Given that SqCones is a count, I would reach for Poisson regression. Ignoring the other predictors, a Poisson regression on CanopyCover seems reasonable. In principle, this relation could be projected to 100% cover; the graph shows clearly that that really would be an extrapolation.

It's no part of the Poisson regression to assume that any predictor is normally, or even symmetrically, distributed. But if we felt a little squeamish about the skewness and whether it had side effects, we could try transforming to see if it made a difference. As logit is not defined for 100%, I feel hesitant about applying it here, even though all values of cover are below 100%. I tried folded square root, got predictions using that as a predictor and then plotted the predictions on top of the previous predictions:

There is no obvious disadvantage to using the original scale because predictions are close with and without the transformation.

For more about folded powers, see What is the most appropriate way to transform proportions when they are an independent variable? (which gives yet further references).

1. If you were curious about what transformation might make those cover data more symmetrical,

• logit helps a bit, but in principle you should worry about its inapplicability to data that might have been 100% (there are fudges for the latter problem for counted data, but I don't know a good fudge here)

• weaker transformations such as folded cube root or folded root do help, but not much, but at least they are defined for 100%

• to correct left skewness, squares or cubes are available in principle (and perfectly well-defined for 100%), but they don't help much either

• you could try, following @whuber's suggestion, to work on transforming (100% $-$ canopy cover), and my guess is that you could get closer to symmetry, but at the cost of a measure that biologically is the wrong way round (watch out too, for log 0 as a problem in principle).

I've not tried to bring your other variables into the analysis. For completeness, I will mention an unasked question,

1. What kind of scales (including quite possibly the scales on which data arrive) should be used for analysing variables such as canopy cover as response variable in relation to various predictors?

The most important advice I have for this problem is that the precise distribution of a predictor usually doesn't matter much. Normal distributions are not a target: if they were, we could hardly use (0, 1) indicators as predictors, which fail dismally.

• One reason we study transformations of right-skewed variables $X$ so intensively is that they work, essentially without change, on left-skewed variables when applied to $-X$ (or $c-X$ for some constant $c$ which, as in cases like this, has an obvious and natural value). – whuber Mar 22 '16 at 1:48
• Oh Thank you a lot! Taht was really helpful! I had no clue about this whole regression thing but that I should be using a specific regression kinda helps me a lot. I might post your part on our course web thing... Thx again! :) – Vera Maria Mar 23 '16 at 14:44
• Thanks, but I suspect that it makes little sense to analyse #cones without using #trees as the major predictor. So, what I do is not a serious analysis of those data. I would want to read the background and do more work for that. So, everyone should note those reservations. – Nick Cox Mar 23 '16 at 15:03

Oh yess it worked out! I did it liek so: hist(CanopyLogit<-logit(red_squirrel\$V6, percents=max(p, na.rm = TRUE) > 1, 0)) I this actually a good normal distribution? I guess yes!

• As already suggested, labels like V2 V3 don't help us at all! The relevant graphs here are the two on the right, which show that logit(cover) is less left-skewed than cover. It's far from normal, however. Please see my revised answer for much more. – Nick Cox Mar 23 '16 at 12:12

Thanks for that answer! Logit sounds good! (BECAUSE I've hear of that before;) I will try that.

thanks for the information in the left-skewness!

There is a more subtle problem too. Over that twofold range, from about 50 to about 100, those transforms are close to linear, as shown by the graphs below. So they will change the units of measurement, but more crucially they won't change the shape of the distribution much at all.

But what does it mean that the transforms are linear, I mean they were linear before right?

• Thanks for posting the data. I've edited your original question to include the data. Your other points should appear as comments on my answer, not as a new answer. NB, as a small point of English, the term is always skewness, not skewedness (edited above). – Nick Cox Mar 23 '16 at 11:14
• Being "close to linear" is shown by the approximate straightness of the graphs relating transforms to the original. – Nick Cox Mar 23 '16 at 12:13