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maiyishan merged 1 commit into from
Mar 17, 2023
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Delta-delta docs amendment #136

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30 changes: 20 additions & 10 deletions dabest/_classes.py
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Original file line number Diff line number Diff line change
Expand Up @@ -857,23 +857,31 @@ def _all_plot_groups(self):

class DeltaDelta(object):
"""
A class to compute and store the delta-delta statistics. In a 2-by-2 arrangement where two independent variables, A and B, each have two categorical values, two primary deltas are first calculated with one independent variable and a delta-delta effect size is calculated as a difference between the two primary deltas.
A class to compute and store the delta-delta statistics for experiments with a 2-by-2 arrangement where two independent variables, A and B, each have two categorical values, 1 and 2. The data is divided into two pairs of two groups, and a primary delta is first calculated as the mean difference between each of the pairs:

.. math::

\\hat{\\theta}_{B1} = \\overline{X}_{A2, B1} - \\overline{X}_{A1, B1}
\\Delta_{1} = \\overline{X}_{A_{2}, B_{1}} - \\overline{X}_{A_{1}, B_{1}}

\\hat{\\theta}_{B2} = \\overline{X}_{A2, B2} - \\overline{X}_{A1, B2}
\\Delta_{2} = \\overline{X}_{A_{2}, B_{2}} - \\overline{X}_{A_{1}, B_{2}}

where :math:`\overline{X}_{A_{i}, B_{j}}` is the mean of the sample with A = i and B = j, :math:`\\Delta` is the mean difference between two samples.

A delta-delta value is then calculated as the mean difference between the two primary deltas:

.. math::

\\hat{\\theta}_{\\theta} = \\hat{\\theta}_{B2} - \\hat{\\theta}_{B1}
\\Delta_{\\Delta} = \\Delta_{B_{2}} - \\Delta_{B_{1}}

and:

and the standard deviation of the delta-delta value is calculated from a pooled variance of the 4 samples:

.. math::

s_{\\theta} = \\frac{(n_{A2, B1}-1)s_{A2, B1}^2+(n_{A1, B1}-1)s_{A1, B1}^2+(n_{A2, B2}-1)s_{A2, B2}^2+(n_{A1, B2}-1)s_{A1, B2}^2}{(n_{A2, B1} - 1) + (n_{A1, B1} - 1) + (n_{A2, B2} - 1) + (n_{A1, B2} - 1)}
s_{\\Delta_{\\Delta}} = \\sqrt{\\frac{(n_{A_{2}, B_{1}}-1)s_{A_{2}, B_{1}}^2+(n_{A_{1}, B_{1}}-1)s_{A_{1}, B_{1}}^2+(n_{A_{2}, B_{2}}-1)s_{A_{2}, B_{2}}^2+(n_{A_{1}, B_{2}}-1)s_{A_{1}, B_{2}}^2}{(n_{A_{2}, B_{1}} - 1) + (n_{A_{1}, B_{1}} - 1) + (n_{A_{2}, B_{2}} - 1) + (n_{A_{1}, B_{2}} - 1)}}

where :math:`s` is the standard deviation and :math:`n` is the sample size.

Example
-------
Expand All @@ -887,16 +895,16 @@ class DeltaDelta(object):
>>> y = norm.rvs(loc=3, scale=0.4, size=N*4)
>>> y[N:2*N] = y[N:2*N]+1
>>> y[2*N:3*N] = y[2*N:3*N]-0.5
>>> # Add drug column
>>> # Add a `Treatment` column
>>> t1 = np.repeat('Placebo', N*2).tolist()
>>> t2 = np.repeat('Drug', N*2).tolist()
>>> treatment = t1 + t2
>>> # Add a `rep` column as the first variable for the 2 replicates of experiments done
>>> # Add a `Rep` column as the first variable for the 2 replicates of experiments done
>>> rep = []
>>> for i in range(N*2):
>>> rep.append('Rep1')
>>> rep.append('Rep2')
>>> # Add a `genotype` column as the second variable
>>> # Add a `Genotype` column as the second variable
>>> wt = np.repeat('W', N).tolist()
>>> mt = np.repeat('M', N).tolist()
>>> wt2 = np.repeat('W', N).tolist()
Expand All @@ -909,10 +917,12 @@ class DeltaDelta(object):
>>> df_delta2 = pd.DataFrame({'ID' : id_col,
>>> 'Rep' : rep,
>>> 'Genotype' : genotype,
>>> 'Drug': treatment,
>>> 'Treatment': treatment,
>>> 'Y' : y
>>> })

>>> unpaired_delta2 = dabest.load(data = df_delta2, x = ["Genotype", "Genotype"], y = "Y", delta2 = True, experiment = "Treatment")
>>> unpaired_delta2.mean_diff.plot()




Expand Down
57 changes: 31 additions & 26 deletions docs/source/deltadelta.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -35,7 +35,7 @@ Effectively, we have 4 groups of subjects for comparison.
<thead>
<tr style="text-align: right;">
<th></th>
<th>Wildtype</th>
<th>Wild type</th>
<th>Mutant</th>
</tr>
</thead>
Expand All @@ -60,7 +60,7 @@ Effectively, we have 4 groups of subjects for comparison.
</div>


There are 2 ``Treatment`` conditions, ``Placebo`` (control group) and ``Drug`` (test group). There are 2 ``Genotype`` s: ``W`` (wildtype population) and ``M`` (mutant population). In addition, each experiment was done twice (``Rep1`` and ``Rep2``). We shall do a few analyses to visualise these differences in a simulated dataset.
There are 2 ``Treatment`` conditions, ``Placebo`` (control group) and ``Drug`` (test group). There are 2 ``Genotype``\s: ``W`` (wild type population) and ``M`` (mutant population). In addition, each experiment was done twice (``Rep1`` and ``Rep2``). We shall do a few analyses to visualise these differences in a simulated dataset.

Simulate a dataset
------------------
Expand All @@ -83,18 +83,18 @@ Simulate a dataset
y[N:2*N] = y[N:2*N]+1
y[2*N:3*N] = y[2*N:3*N]-0.5

# Add drug column
# Add a `Treatment` column
t1 = np.repeat('Placebo', N*2).tolist()
t2 = np.repeat('Drug', N*2).tolist()
treatment = t1 + t2

# Add a `rep` column as the first variable for the 2 replicates of experiments done
# Add a `Rep` column as the first variable for the 2 replicates of experiments done
rep = []
for i in range(N*2):
rep.append('Rep1')
rep.append('Rep2')

# Add a `genotype` column as the second variable
# Add a `Genotype` column as the second variable
wt = np.repeat('W', N).tolist()
mt = np.repeat('M', N).tolist()
wt2 = np.repeat('W', N).tolist()
Expand All @@ -112,7 +112,7 @@ Simulate a dataset
df_delta2 = pd.DataFrame({'ID' : id_col,
'Rep' : rep,
'Genotype' : genotype,
'Drug': treatment,
'Treatment': treatment,
'Y' : y
})

Expand Down Expand Up @@ -206,8 +206,7 @@ for slopegraphs. We use the ``experiment`` input to specify grouping of the data
.. code-block:: python3
:linenos:

unpaired_delta2 = dabest.load(data = df_delta2, x = ["Genotype", "Genotype"], y = "Y", delta2 = True,
experiment = "Drug")
unpaired_delta2 = dabest.load(data = df_delta2, x = ["Genotype", "Genotype"], y = "Y", delta2 = True, experiment = "Treatment")

The above function creates the following object:

Expand Down Expand Up @@ -279,26 +278,31 @@ administered, the mutant phenotype is around 1.23 [95%CI 0.948, 1.52]. This diff
and ``Drug`` group are plotted at the right bottom with a separate y-axis from other bootstrap plots.
This effect size, at about -0.903 [95%CI -1.26, -0.535], is the net effect size of the drug treatment. That is to say that treatment with drug A reduced disease phenotype by 0.903.

Mean difference between mutants and wild types given the placebo treatment is:

.. math::

\hat{\theta}_{P} = \overline{X}_{P, M} - \overline{X}_{P, W}
\Delta_{1} = \overline{X}_{P, M} - \overline{X}_{P, W}

Mean difference between mutants and wild types given the drug treatment is:

\hat{\theta}_{D} = \overline{X}_{D, M} - \overline{X}_{D, W}

.. math::

\Delta_{2} = \overline{X}_{D, M} - \overline{X}_{D, W}

\hat{\theta}_{\theta} = \hat{\theta}_{D} - \hat{\theta}_{P}
The net effect of the drug on mutants is:

and:

.. math::

s_{\theta} = \frac{(n_{P, M}-1)s_{P, M}^2+(n_{P, W}-1)s_{P, W}^2+(n_{D, M}-1)s_{D, M}^2+(n_{D, M}-1)s_{D, M}^2}{(n_{P, M} - 1) + (n_{P, W} - 1) + (n_{D, M} - 1) + (n_{D, M} - 1)}

\Delta_{\Delta} = \Delta_{2} - \Delta_{1}


where :math:`\overline{X}` is the sample mean, :math:`\Delta` is the mean difference.


where :math:`\overline{X}` is the sample mean, :math:`\hat{\theta}` is the mean difference, :math:`s` is the variance and :math:`n` is the sample size.
Specifying Grouping for Comparisons
-----------------------------------


In the example above, we used the convention of "test - control' but you can manipulate the orders of experiment groups as well as the horizontal axis variable by setting ``experiment_label`` and ``x1_level``.
Expand Down Expand Up @@ -334,28 +338,29 @@ We produce the following plot:

.. image:: _images/tutorial_108_0.png

We see that the drug had a non-specific effect of -0.321 [95%CI -0.498, -0.131] on wildtype subjects even when they were not sick, and it had a bigger effect of -1.22 [95%CI -1.52, -0.906] in mutant subjects. In this visualisation, we can see the delta-delta value of -0.903 [95%CI -1.21, -0.587] as the net effect of the drug accounting for non-specific actions in healthy individuals.
We see that the drug had a non-specific effect of -0.321 [95%CI -0.498, -0.131] on wild type subjects even when they were not sick, and it had a bigger effect of -1.22 [95%CI -1.52, -0.906] in mutant subjects. In this visualisation, we can see the delta-delta value of -0.903 [95%CI -1.21, -0.587] as the net effect of the drug accounting for non-specific actions in healthy individuals.

.. math::

\hat{\theta}_{W} = \overline{X}_{D, W} - \overline{X}_{P, W}

\hat{\theta}_{W} = \overline{X}_{D, M} - \overline{X}_{P, M}
Mean difference between drug and placebo treatments in wild type subjects is:

.. math::

\hat{\theta}_{\theta} = \hat{\theta}_{M} - \hat{\theta}_{W}
and:
\Delta_{1} = \overline{X}_{D, W} - \overline{X}_{P, W}

Mean difference between drug and placebo treatments in mutant subjects is:

.. math::

s_{\theta} = \frac{(n_{D, W}-1)s_{D, W}^2+(n_{P, W}-1)s_{P, W}^2+(n_{D, M}-1)s_{D, M}^2+(n_{P, M}-1)s_{P, M}^2}{(n_{D, W} - 1) + (n_{P, W} - 1) + (n_{D, M} - 1) + (n_{P, M} - 1)}
\Delta_{2} = \overline{X}_{D, M} - \overline{X}_{P, M}


The net effect of the drug on mutants is:

where :math:`\overline{X}` is the sample mean, :math:`\hat{\theta}` is the mean difference, :math:`s` is the variance and :math:`n` is the sample size.
.. math::

\Delta_{\Delta} = \Delta_{2} - \Delta_{1}

where :math:`\overline{X}` is the sample mean, :math:`\Delta` is the mean difference.


Connection to ANOVA
Expand Down