Grouping similar subjects based on slopes

Question

I am studying the responses of 500 subjects to temperature increase using a linear regression on each degree of temperature (from 10°C to 28°C). Thus, it was possible for me to compute the intercept and slope for each one of the 500 subjects.

I am interested in identifying three groups of individuals that are "statistically identical":

• Group 1: those who are robust and respond independently of the temperature effect. These are the subjects which have a slope of approximately zero.
• Group 2: Those having negative / downward slopes
• Group 3: Those having positive / upward slopes.

I have computed the slope for each subject and afterwards sorted the data from lowest to highest slope.

Can you please help me to use the most relevant statistical parameters and tools that allow me to identify individuals (statistically homogeneous) of each of these three groups?

I considered using the t-test, clustering techniques, PCA and discrimination but I need your help for the choice and how to do it using SAS or R.

For better illustration, here is part of the data:

Sub. C15    C16     C17     C18     C19     Slope
1    30,55  30,05   29,56   29,07   28,58   -0,49
2    22,22  21,83   21,44   21,05   20,67   -0,39 
3    20,16  19,78   19,39   19,01   18,63   -0,38
4    61,07  60,69   60,31   59,93   59,55   -0,38
5    49,29  48,92   48,55   48,18   47,81   -0,37
6    52,54  52,18   51,81   51,44   51,08   -0,37

238  18,19  18,18   18,18   18,18   18,18   -0,0017
239 -10,23 -10,23  -10,23  -10,24  -10,24   -0,0010
240 -14,44 -14,44  -14,44  -14,44  -14,44   -0,0006
241  19,76  19,75   19,75   19,75   19,75   -0,0006
242  13,55  13,55   13,55   13,55   13,55    0,0010
243  19,93  19,93   19,93   19,93   19,94    0,0012
244  55,69  55,69   55,69   55,69   55,69    0,0016

495 -28,70 -28,43  -28,16  -27,90  -27,63    0,27
496 -9,71  -9,40   -9,10   -8,80   -8,49     0,30
497 -12,29 -11,98  -11,67  -11,35  -11,04    0,31
498 -43,85 -43,48  -43,11  -42,74  -42,37    0,37
499 -29,41 -28,97  -28,52  -28,07  -27,62    0,45
500 -8,54  -7,98   -7,43   -6,87   -6,31     0,56

Columns C15-C19 (for example) are predicted values of the trait of interest at each temperature. These values were the results obtained using a reaction norm model. The slope for each subject is computed here using linear regression and afterwords the data were sorted by slope values.

In this example:

1. Subjects (1 to 6) have the lowest negative slopes and remain of interest in our study according to their decrease in response to increasing temperature.

2. Subjects (238-244) are here as example of subjects having a slope of approximately (zero) and are more interesting in our study because they represent the robust ones and where their performance are not affected by increase of the temperature

3. Subjects (495-500) are those having the highest positive slope. We are interested of them because their performance increase even in harsh conditions and high temperature.

Now our question is:

How can you identify more precisely (based on a statistical tool) the subjects that should correspond to each of the 3 groups?

In a first step, we think we could use a t-test to identify the subjects of each group. Some people advised us to apply clustering or discriminant approaches. But we were not able to figure out how to do it the right way. We have some background in SAS. We asking how to solve our problem and help us with an example of how to solve the problem in SAS.

Answer 1

If you fit a random effects model (SAS proc mixed) with random subject-specific slopes for temperature, then you will get two things that will be useful:

1) A prediction (essentially an estimate) of each subject's slope.

2) An estimate of the variability in individual slopes across the sample.

Then you can define your three groups quite formally, for example Group 1 = subjects with slopes within +/- 0.5 SD of zero, Group 2 = subjects with slopes < -0.5 SD, and Group 3 = subjects with slopes > 0.5 SD.

No matter what approach you use, you'll have to make a scientific decision on what constitutes "close enough" to zero; statistics can't tell you that.

Answer 2

What you want to do is determine which subjects belong to one of three different groups, where you don't have access to true group labels. An analogy would be trying to guess who is male and female when you only have data on subjects' heights. This is unsupervised classification. You need to investigate finite mixture modeling (Wikipedia, SAS documentation, pdf of tutorial) and cluster analysis (Wikipedia, SAS documentation).

You can cluster on only one dimension (see, e.g., How can I group numerical data into naturally forming brackets?) although it is less common. As @RayKoopman notes, your slopes appear to be correlated with your intercepts, so it may be better to fit a single mixed effects model with random intercepts, random slopes, and a correlation between them, then get predicted slopes and intercepts for each subject and cluster those together. With only (possibly) three groups and (up to) two dimensions, finite mixture modeling should be tractable and would be preferable to traditional clustering algorithms (e.g., k-means) if the subpopulations you identify appear reasonably Normal.