# Linear mixed-effects models

It is not straightforward to define a p-Value for RT-DC data (e.g. change in deformation for a treatment vs. its control). This is somewhat counter-intuitive, because one could assume that the large number of events in a single dataset should be enough to compare two datasets. However, Focus changes, chip-to-chip variations, etc. may generate systematic offsets which make a direct comparison (e.g. t-Test) impossible. Linear mixed effect models (LMM) allow to assign a significance to a treatment measurement compared to a control measuerement (fixed effect) while considering the systematic bias in-between the measurement repetitions (random effect).

dclab offers LMM analysis as described in [HMMO18]. The LMM analysis is performed using the lme4 R package.

## Computing p-values with lme4 in dclab

dclab exposes two models from lme4:

**linear mixed-effects models**(“lmer”): This is basically the simplest way of determining whether or not a treatment has an effect.**generalized linear mixed-effects models with a log-link function**(“glmer+loglink”): This model makes use of lme4’s generalized linear effects model (GLMM)`glmer`

function with a log-link function (`family=Gamma(link='log')`

). This is used for data that is log-normally distributed. Log-normal behaviour is quite common, especially in biology. When a physical parameter has a lower limit, and the measured values are close to that limit, the resulting distribution will be skewed, resembling a log-normal distribution. In case of RT-DC this is specially (but not only) true for deformation. Another example is area, which also has a lower limit of zero and may therefore have a skewed distribution. While GLMMs are designed to handle skewed data, it was shown that LMMs already deliver robust results, even for highly skewed data [GH06].

Warning

The decision whether to use LMM or GLMM is not particularly important. Ideally, both LMM and GLMM are consistent. However, never perform both analyses only to then pick the one with the lowest p-value. This is p-hacking! The analysis routine should be defined beforehand. If in doubt, stick to LMM.

An LMM analysis is straight-forward in dclab:

```
import dclab
from dclab import lme4
# Load the data
ds_rep1_ctl = dclab.new_dataset(...) # control measurement, 1st repetition
ds_rep1_trt = dclab.new_dataset(...) # treatment measurement, 1st repetition
ds_rep2_ctl = dclab.new_dataset(...) # control measurement, 2nd repetition
ds_rep2_trt = dclab.new_dataset(...) # treatment measurement, 2nd repetition
# Instantiate Rlme4
rlme4 = lme4.Rlme4(model="lmer", feature="deform")
# Add the datasets
rlme4.add_dataset(ds=ds_rep1_ctl, group="control", repetition=1)
rlme4.add_dataset(ds=ds_rep1_trt, group="treatment", repetition=1)
rlme4.add_dataset(ds=ds_rep2_ctl, group="control", repetition=2)
rlme4.add_dataset(ds=ds_rep2_ctl, group="treatment", repetition=2)
# Perform the analysis
result = rlme4.fit()
print("p-value:", result["anova p-value"])
print("fixed effect:", result["fixed effects treatment"])
print("model converged:", result["model converged"])
```

The `fit()`

function returns the most important results and also exposes
some of the underlying R objects (see `dclab.lme4.wrapr.Rlme4.fit()`

).
An LMM example is also given in the example section.

Note

If a treatment and a control share the same repetition number, it is implied that they are paired. For those measurements, lme4 will perform a paired test. In your experimental design you determine which measurements are paired, before doing any experiments. Pairing can be done e.g. for measurements done on the same day or on the same chip. In cases where you perform the control measurements on one day and the treatment measurements on another day, you could still pair them. Just keep in mind that this could introduce systematic errors, if the measurement conditions (temperature, illumination, etc.) were not identical. Under no circumstances, choose a pairing that yields the lowest p-value (p-hacking).

Alternatively, you can also run an unpaired test by just giving each measurement a different repetition number. For example for 3x control and 3x treatment measurements, you could enumerate the repetition number from 1 to 6.

## Differential feature analysis with reservoir data

The (G)LMM analysis is only applicable if the feature chosen is not pronounced visibly in the reservoir measurements. For instance, if a treatment results in a significant change in deformation already in the reservoir, then the p-value determined for the channel data might be underestimated (too many stars). In this case, the information of the reservoir measurement must be included by means of differential deformation [HMMO18]. The idea of differential deformation is to subtract the reservoir from the channel deformation. Since it is not possible to assign the events in the reservoir to the events in the channel (two different measurements), bootstrapping is employed which generates statistical representations of the two measurements that can then be subtracted from one another. Then, for the actual LMM analysis, only the differential deformation is used.

To perform a differential feature analysis, simply add the reservoir
measurements to the `dclab.lme4.wrapr.Rlme4`

class (they are
recognized as reservoir measurements via their meta data).

```
# Load the data
ds_rep1_ctl = dclab.new_dataset(...) # control measurement, 1st repetition (channel)
ds_rep1_ctl_res = dclab.new_dataset(...) # control measurement, 1st repetition (reservoir)
[...]
# Instantiate Rlme4
rlme4 = lme4.Rlme4(model="lmer", feature="deform")
# Add the datasets
rlme4.add_dataset(ds=ds_rep1_ctl, group="control", repetition=1)
rlme4.add_dataset(ds=ds_rep1_ctl_res, group="control", repetition=1)
[...]
# Perform the analysis
result = rlme4.fit()
assert results["is differential"] # adding "reservoir" data forces differential analysis
```

Keep in mind that the analysis is now performed using the differential
features and not the actual features (`result["is differential"]`

).
For more information, please see `dclab.lme4.wrapr.Rlme4.get_differential_dataset()`

and `dclab.lme4.wrapr.bootstrapped_median_distributions()`

.
A full example, including GLMM and differential deformation, is given in the
example section.