# Difference between disease conditions by non-parametric ANOVA

I've the following simplified data as shown.

PATIENT B1  B2  B3  B4
A1  0.19    0.18    0.12    0.85
A2  2.8 0.24    0.06    0.11
A3  0.03    0.24    0.24    0.07
A4  0.65    1.6 0.08    0.31
F1  0.17    0.07    3.86    2.41
F2  0.11    1.74    0.51    0.34
F3  0.11    2.28    0.57    4.06
F4  0.23    0.68    2.51    0.31
S1  0.5 0.19    2.13    0.09
S2  1.21    0.25    2.02    0.2
S3  2.06    0.05    0.16    0.4
S4  0   1.02    0.01    0.37
J1  2.64    0.68    0.1 0.3
J2  2.7 3.89    0.15    0.34
J3  0.09    0.22    0.17    2.74

We have measurements for several biomarkers B1, B2, B3, B4 and for different disease subtypes A, F, S etc. for 4 (3 for J) different patients in each disease group A1-A4, F1-F4, etc.

The data is non-Gaussian so I wish to do a non-parametric Kruskal Wallis test to show if the different disease subtypes are different or not.

I've consolidated the data in the following format (Pastebin Link) throwing away the individual biomarker information and did a Kruskal Wallis test. Is there any way I can use the Biomarker value in the Kruskal Wallis given the biomarker info as shown in this Pastebin link. Can a Friedman Test be done for this case?

Being non-Gaussian doesn't mean that you need go all the way over to non-parametric methods. More importantly, the usual non-parametric methods are typically dead ends once you want to start looking at extra covariates. Unfortunately once you add in another predictor in your case, the small size of the dataset bites even harder.

Looking at your data I see zeros are possible, at least as reported values, and that the distribution is skewed. That suggests something like a gamma distribution as reference distribution and a generalized linear model as framework. These token results (from Stata) are weakly supportive of a contrast between A and S on the one hand and F and J in the other, but your sample sizes are rather small and significance at conventional levels is elusive.

If this is new to you, the magic words are generalized linear models and a good introductory treatment is http://www.amazon.com/Introduction-Generalized-Edition-Chapman-Statistical/dp/1584889500

. glm B i.Subtype, f(gamma)

Iteration 0:   log likelihood = -50.205585
Iteration 1:   log likelihood = -47.356747
Iteration 2:   log likelihood = -47.350664
Iteration 3:   log likelihood = -47.350662

Generalized linear models                          No. of obs      =        60
Optimization     : ML                              Residual df     =        56
Scale parameter =  1.580827
Deviance         =  91.41749283                    (1/df) Deviance =  1.632455
Pearson          =  88.52633799                    (1/df) Pearson  =  1.580827

Variance function: V(u) = u^2                      [Gamma]
Link function    : g(u) = 1/u                      [Reciprocal]

AIC             =  1.711689
Log likelihood   = -47.35066178                    BIC             = -137.8658

------------------------------------------------------------------------------
|                 OIM
B |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Subtype |
2  |  -1.257599    .694457    -1.81   0.070     -2.61871    .1035118
3  |  -1.203282   .7178312    -1.68   0.094    -2.610205    .2036414
4  |  -.5582642   .8008188    -0.70   0.486     -2.12784    1.011312
|
_cons |   2.059202   .6471519     3.18   0.001     .7908076    3.327597
------------------------------------------------------------------------------

. predict Subtype_mean
(option mu assumed; predicted mean B)

. tabdisp Subtype, cell(Subtype_mean) format(%3.2f)

----------------------------
Subtype | Predicted mean B
----------+-----------------
A |             0.49
F |             1.25
J |             1.17
S |             0.67