# Exploring Various ways to combine transcription profile correlation (r), confidence in that correlation (p value associated with r), and phylogenetic distance (w(d)) between species into a useful PTC index (Phylogenetic Transcription Conservation Index)

## Original solution

$$PTCI = r \cdot (1-p) \cdot w(d)$$

Xiaohui objected that 1-p is not optimal because the 1-p will likely have a distribution skewed toward 1. He suggested loging the variable, among other options (see below).

### Original Results This is a histogram of the three way mean PTCIs using the original solution. Blue=Real data. Grey=Randomizations that abolish the three way orthology relationships. The ‘Null’ distribution [grey] is significantly different from the real data [blue] but has significant structure that is not what one would predict (resemble a normal dist). Notably, there is a large concentration around -0.5 that I can not explain.

### Suggested alterations

1. log the 1-p value
2. convert to Z scores
• not sure exactly what he meant here
• *$$z(r) \cdot (1-p) \cdot w(d)$$
• $$r \cdot z(1-p) \cdot w(d)$$
• $$z(r) \cdot z(1-p) \cdot w(d)$$
• $$z(r \cdot (1-p)) \cdot w(d)$$
3. ‘fiddle’ with it till the null distribution makes sense...

### Distributions of r and 1-p

I will now plot the distributions of the separate variables in the PTCI to determine if they help explain the structure seen above.

In:

# run gfunc_build_n_way_one2one script and capture output in variable 'trap'
# NOTE: for syntax highlighting to work on this blog post, I had to comment out the ipython magic command below:

# %run -i gfunc_build_n_way_one2one.py gfunc/Aa_Ag_Cq_sort_pci.conf

graphHandler,graphBuilder,n_way_ortho_table,ortho_parser = trap

# extract edge data from the graphs
edges = graphHandler.edge_dict.values()

# save original graphHandler in case we alter it and then want to use the original later
from copy import deepcopy
original_graphHandler =  deepcopy(graphHandler)

bestChoose is 'comb' from 'gmpy'.
main() completed.

#### Here I define a function ‘edge_correlation()‘ to calculate and return r and its p

In:

from scipy import stats as sp_stats

def edge_correlation(gFunc_edge):
"""
Returns the pearson r value and corresponding p-value
for a given edge's nodes.
"""
node1,node2 = gFunc_edge.nodes
try:
r_val,p_val = sp_stats.pearsonr(node1.data.expression_vector, node2.data.expression_vector)
if np.isnan(r_val):
pass
else:
return r_val,p_val

except AttributeError as err:
if """'Bunch' object has no attribute""" in err.message:
# if this executes then one of the nodes did not have an expression_vector which means no r is possible
# in this case return None
return None
else:
# if this executes then something ELSE went wrong: thus I will fail.
raise err


In:

# collect all the results using edge_correlation()
edge_r_and_p_values = [edge_correlation(edge) for edge in edges]

# get rid of any results that equal None
edge_r_and_p_values = [x for x in edge_r_and_p_values if x != None]


#### Now we plot a histogram of the r values

In:

r_values = [edge_r_and_p_values[i] for i in range(len(edge_r_and_p_values))]

hist(r_values,bins=50,histtype='stepfilled',cumulative=False, color='b')
xlabel('correlation values')
ylabel('number of edges in each bin')
title('r values for pairwise edge comparisons')


Out:

<matplotlib.text.Text at 0x3861e950> #### And now of the (1 - p_values)

In:

p_values = [edge_r_and_p_values[i] for i in range(len(edge_r_and_p_values))]
one_minus_pvals = [(1-p) for p in p_values]
hist(one_minus_pvals,bins=50,histtype='stepfilled',cumulative=False, color='g')
xlabel('1-p')
ylabel('number of edges in each bin')
title('(1-p) for pairwise edge correlations')


Out:

<matplotlib.text.Text at 0x3ac69b50> #### Now the combined $$r \cdot (1-p)$$

In:

r_by_1minusP = [r_values[i] * (1 - p_values[i]) for i in range(len(r_values))]
hist(r_by_1minusP,bins=50,histtype='stepfilled',cumulative=False, color='c')
xlabel('r * (1-p)')
ylabel('number of edges in each bin')
title('r * (1-p) for pairwise edge correlations')


Out:

<matplotlib.text.Text at 0x40ea4090> In:

hist2d(r_values,p_values,bins=25)
colorbar()
xlabel("r values")
ylabel("p-values")
title('2D histogram of r and p-value pairs')


Out:

<matplotlib.text.Text at 0x3862f1d0> The 2D histogram above shows that as expected the p-values track mostly but not completely with the r values. Generally, a more extreme r value will also come with a “better” p-value, but this is not absolute which warrants keeping the p-value variable in the PTCI in some capacity. Also, this plot illustrates that the most populated bins are those with positive r values close to +1.

#### Now the full PTCI

In:

# define function to calculate the PTCI
from gfunc.maths import weight_d_for_ptci as scale_the_d

def calc_ptci(gFunc_edge):
"""
calculate the PTCI
"""
try:
r_val,p_val = edge_correlation(gFunc_edge)

d_val,d_min,d_max = gFunc_edge.data.divergence

ptci = r_val * (1-p_val) * scale_the_d(d_val,d_min,d_max)

if not np.isnan(ptci):
# If we get a valid ptci store the value in the gFunc_edge object and also return it
gFunc_edge.data.PTCI = ptci
return ptci
else:
# If we get an invalid ptci, store it and return it as None
gFunc_edge.data.PTCI = None
return None
except TypeError as err:
if str(err) == "'NoneType' object is not iterable":
# If we get an invalid ptci, store it and return it as None
gFunc_edge.data.PTCI = None
return None
else:
raise


In:

pairwise_ptci_vals = [calc_ptci(edge) for edge in edges]

# remove any None values
pairwise_ptci_vals = [ptci for ptci in pairwise_ptci_vals if ptci != None]


In:

hist(pairwise_ptci_vals,bins=50,histtype='stepfilled',cumulative=False, color='c')
xlabel('ptci')
ylabel('number of edges in each bin')
title('ptci for pairwise edge correlations')


Out:

<matplotlib.text.Text at 0x476b5110> ## Explore Null distributions for pairwise edge relationships (randomized orthology assignments)

### Generate the Null distributions

In:

# import edge randomizer function
from gfunc.scripts.gfunc_build_n_way_one2one import reset_random_edges

# Scramble the edges many times and collect the pairwise ptci distributions

reps = 50

null_paired_ptci_distributions = []

for rep in range(reps):
# scramble edges for this rep and set new r&p vals
reset_random_edges(graphHandler,graphBuilder,n_way_ortho_table,ortho_parser)
graphHandler.measure_relations()

# calculate null ptci vals
null_edges = graphHandler.edge_dict.values()
null_pairwise_ptci_vals = [calc_ptci(edge) for edge in null_edges]

# remove any None values
null_pairwise_ptci_vals = [ptci for ptci in null_pairwise_ptci_vals if ptci != None]

# collect null ptci distribution
null_paired_ptci_distributions.append(null_pairwise_ptci_vals)


### Graph the Null distributions

In:

# Show what the actual data looks like for comparison
hist(pairwise_ptci_vals,bins=50,histtype='stepfilled',cumulative=False, color='c',alpha=.7, label='Real Data')

# Graph null distributions as grey slightly transparent histograms
null_label = 'Null Data'

for null_dist in null_paired_ptci_distributions:
hist(null_dist,bins=50,histtype='step',cumulative=False, color='k',alpha=.1,label=null_label)
null_label = None

xlabel('ptci')
ylabel('number of edges in each bin')
title('null distributions for pairwise edge correlations (%s reps)' % (reps))
legend()


Out:

<matplotlib.legend.Legend at 0x35fad690> ### Observations and Discussions:

1. The structure of the null distributions in the pairwise cases are what I would have predicted
• most pairs show no correlation
• there is no bias in either the strong negative or strong positive correlation zones (near -1 or +1)
2. [ QUESTION: ] It seems that the PTCI math itself is not the ultimate cause of the strange structure found in the null distributions of the 3-way comparisons at the start of this document.
3. [ QUESTION: ] This seems to negate the need to use z-score conversions on any of the PTCI variables??
• ACTUALLY: this is not totally true, although it will might not fix the shape problem, as the z-score distributions will all have the same “shape” as the un-converted distributions
• BUT: since the z-score distributions are x-shifted so that “0” lies at the mean (or alternately: median if I wanted to do it that way), z-converting the r values may still be a good way to highlight those pairs that are most likely to be positively correlated? (see figure below)
4. [ QUESTION: ] Would this also suggest that the shape of the p-value distribution is not a primary problem either (that taking the log of the p-values would not help the shape of the 3-way null distributions)
5. [ PROPOSITION: ] Perhaps I made a mistake in the way that I calculated the mean 3-way PTCI values for the null distributions?
6. [ PROPOSITION: ] Perhaps I should exclude any mean PTCI info from a 3-way ortholog triplicate if any pairwise PTCI is missing (usualy due to one of the orthologs being undetected). Currently, if one ortholog expression vector is missing (undetected) I simply take the mean of the remaining two pairwise comparisons. Maybe this is introducing strange structural elements in the distribution depending on which species is missing data?
• I thought that the branch length data should correct this but maybe not?

In:

def convert_to_z_scores(data_vector):
vec_mean = mean(data_vector)
vec_median = median(data_vector)
vec_stdev = std(data_vector)

z_vector = (data_vector - vec_median) / (vec_stdev)
return z_vector


In:

z_r_values = convert_to_z_scores(r_values)

subplot(211)

vert_line_color = '0.75'

hist(r_values,bins=50,histtype='stepfilled',cumulative=False, color='b')
axvline(linewidth=4, color=vert_line_color)
xlabel('r values')
ylabel('edges per bin')
title('r values for edges BEFORE z-convertion')

subplot(212)
hist(z_r_values,bins=50,histtype='stepfilled',cumulative=False, color='b')
axvline(linewidth=4, color=vert_line_color)
xlabel('r values')
ylabel('edges per bin')
title('r values for edges AFTER z-convertion')

tight_layout() ### Next Actions:

1. Re try calculating the 3-way null distributions: using new code or code from this notebook rather than gFunc code to see if I simply made a mistake.
2. Replot pairwise distribution using z-converted r values.