# Positive skewness: what to do when transformations don't help?

I would like to perform General Linear Model with one response variable and two predictor variables (1 numeric, 1 categorical). The response variable is positively skewed and transformations don't seem to bring it closer to normality. I tried sqrt, logarithmic, inverse and Box Cox transformations (performed by SPSS). Is there any other way to transform it? Can Generalized Linear Model work with such skewed data (I'm not very familiar with it)? Here are the data:

identifier CatPredictor NumPredictor Response
1  A                  .9           0
2  A                 2.6           0
3  A                 3.2           0
4  A                 6.6           0
5  A                80.1           0
6  AT               41.4           0
7  T                22.3           0
8  T                29.8           0
9  T                14.5           0
10  A                 9.9          .3
11  A                 5.7          .5
12  A                 6.7          .5
13  AT                9.9          .5
14  T                  19          .5
15  T                  23          .5
16  A                  .3          .8
17  T                  23          .8
18  A                   1           1
19  A                 5.7           1
20  A                 7.4           1
21  T                14.5           1
22  T                14.5           1
23  T                22.3           1
24  A                   7         1.2
25  T                29.8         1.3
26  A                 9.6         1.5
27  AT                9.6         1.6
28  A                  12         1.8
29  A                 4.5           2
30  A                 5.8           2
31  A                 7.6           2
32  A                 7.6           2
33  T                  23           2
34  T                  23         2.5
35  A                 3.2           3
36  A                 5.1           3
37  A                   7           3
38  A                 7.3           3
39  AT                6.6           3
40  T                  23           3
41  A                 5.5         3.4
42  A                  .4         3.4
43  A                 3.6         3.5
44  A                  12         3.5
45  T                22.3         3.8
46  A                 7.6           4
47  T                12.4           4
48  A                  .9         4.2
49  A                   0         4.3
50  A                   1         4.5
51  A                 6.4           5
52  AT                 11           5
53  T                22.3         5.3
54  A                 2.2         5.8
55  T                 4.9           6
56  T                22.3           6
57  A                 4.7         6.5
58  A                 5.2           7
59  A                 2.1         7.2
60  T                14.5         7.5
61  A                  .9         7.7
62  A                 8.3           8
63  T                 4.9         8.7
64  T                22.3         9.3
65  A                 4.5         9.5
66  A                 3.3          10
67  A                 5.1          10
68  A                 9.9        10.5
69  AT               46.3          11
70  A                 1.1        11.6
71  A                  21        12.5
72  A                 3.6          13
73  A                 5.8          14
74  A                  .8        14.5
75  T                22.3        14.7
76  A                  .2          15
77  A                  .4          15
78  A                 3.6          15
79  T                 4.9          15
80  T                  11          16
81  A                 7.9          18
82  A                 9.6          18
83  A                  .1          25


Here are two examples of my distributions:

• Is the bimodality an artifact of the binning for the histogram or a reflection of real behavior? May 31 '16 at 13:38
• (1) The "general linear model" and a "generalized linear model" are two very different things! Which one do you mean? (2) The distribution of the raw response is irrelevant, so why are you trying to transform it? What matters is the distribution of the response conditional on the regressors.
– whuber
May 31 '16 at 13:49
• I re-posted the data. I tried to make it appear in colums but somehow nothing seems to work. I am sorry for the trouble. Jun 2 '16 at 14:53
• Edited the data listing. Jun 2 '16 at 15:23

The distribution you get is good news, not bad. The distribution is close to symmetric on a logarithmic scale. That means that we don't expect the distribution to be problematic to deal with on that logarithmic scale.

Note further that few methods expect the outcome or response variable to have a marginal normal distribution. Regression certainly doesn't. An approximately symmetric distribution like this will be well behaved. That doesn't rule out surprises or complications arising from other variables in your data, but we have no precise information on those variables.

Further, why did you add 1 before taking logarithms? Was it because there are some zeros in your data? Know that generalized linear models with logarithmic link have made that fudge unnecessary. That's too new an idea for some fields to have caught up, as the key work was published as recently as 1972. Generalized linear models with logarithmic link just expect that means are positive, and that doesn't oblige all values to be positive.

Not only do generalized linear models not have problems with skewed responses; dealing with those skewed responses using appropriate links such as the logarithm is arguably one of their main benefits.

NB: General linear models and generalized linear models are not the same family.

EDIT: I plotted the data. It can all go on one graph, but I fall short at offering a model as I have no idea what kind of model makes sense.

I chose a square root scale to pull in the outliers (wilder values) a bit. That's arbitrary, except in so far as it copes cleanly with the zeros, as 0 maps to 0 without fudge.

There is one A standing outside the others at bottom right.

The ATs fall into two groups, perhaps.

The Ts fall into two groups, perhaps.

Perhaps the zero responses are qualitatively different as well as quantitatively. It's tempting to note that two apparent anomalies are for points with zero response. (A small merit of the transformation is that the zeros stand out. That's clearer on a quantile plot than a histogram, so I give a quantile plot too. The quantile plot below shows the distribution of the roots of the Response, but labels it according to the raw Response value. Histograms often obscure fine structure in data.)

Does any of those convey some biological meaning or message? It's likely that any analysis ignoring that fine structure might obscure as much as it clarifies.

To summarize so far: Mild skewness in your data can be handled by a mild transformation. Your bigger problem is identifying what model makes sense for your data.

• I know general LM and generalized LM are not the same thing. I thought, however, that normal distribution of the response variable (+ homosedacity + normal distribution of residuals) is among the assumptions of the general LM. But if I understand this correctly a symmetric distribution is enough? I now ran the analysis with the log-transformed data and the other two assumptions are met - I assume that means I can stick to general LM? However, a new problem emerged as one of the predictor variables has a significant effect but the overall model does not. I'm not sure how to interpret that. Jun 1 '16 at 14:23
• And yes, I added 1 because there are zeros in my data. My stats course didn't include GLM so I'm trying to self learn and hence the big lack of knowledge. Jun 1 '16 at 14:28
• Strictly a marginal normal distribution for the outcome doesn't guarantee a normal distribution for the residuals. Nor is homoscedasticity an assumption for generalized linear models; on the contrary many possible models imply heteroscedasticity. We can't comment easily on your results without any specific details. Can you post your data and indicate exactly what model you fitted? Jun 1 '16 at 14:46
• But homoscedasticity and normal distribution for the residuals are assumptions for the generAL LM? And if these assumptions are met + I have a symetrical (not normal) distribution I can stick to the generAL LM? Not sure how to post the data here. But if this is the case then I know how to proceed. I omitted one category with very few cases from the categorical predictor and now the model and both predictors have a significant effect (using the generAL LM). Jun 2 '16 at 9:43
• You can post the data by copying and pasting a text listing into your original question. You want approval for a possible analysis when we can't even see your data! What is true that some people would transform their data and then apply a general linear model. Much depends on what is acceptable in your field, which you don't specify. Jun 2 '16 at 9:58