The Problem
I have a bunch of data that can be categorized into many small
groups. Each small group has a set of values for an ordered set
of intervals. Having observed that the values for most groups seem
to increase with the order of the interval, I hypothesize that their is
a statisticallysignificant, monotonically increasing trend.
An Analogy
To make this abstract problem more relatable, imagine the following
scenario.
There are many companies (groups) selling products in an industry
that is on the verge of widespread growth. Each company sets a
projection (not a goal) for endofyear sales (value). and adjust
this projection once per quarter (i.e. four times a year) (interval)
after analyzing their todate sales. Having observed that most companies
that follow this scheme tend to increase their goals for nextquarter
sales (based on an initial, beginningoftheyear projection that is too
low), the market analyst (me) wonders if there is a nontrivial,
positive trend across quarters in the year. ^{1}
To summarize how the “variables” in this analogy relate to my abstract
description of the situation,
 groups = companies
 value = sales
 interval = quarteryear
(Actually, there is one more variable in this context, given that I am
interested in relative value—the rank of the value, relative to the
other values.)
And, to make the example concrete, here is what the data (for four
companies) might look like.
company q1 q2 q3 q4
A 3.6 8.5 9.2 9.4
B 0.73 0.04 0.28 0.47
C 6604 4631 6604 7217
D 49 9 400 87
(Bad) Solution: Linear Regression
Just at first thought, running a univariate linear
regression might seem
like a good way of attacking this problem. However, there are a couple
of basic “gotchas” that make ordinary linear regression a notsogreat
idea for this scenario:

There are not many intervals (i.e. independent variables) per group.
(This condition inherently makes any kind of model—not just a linear
regression one—sensitive to the samples. In general, there is a
problem with creating models with small sample
sizes.) 
The values across groups might have very different magnitudes.
(Thus, trying to create a single, general model that groups all of
the data and uses the group labels as a categorical independent
variable would likely lead to unreliable results.) 
The values themselves might be very volatile for a single group.
(This might be interpreted as a violation of the normal
distribution assumption of linear
regression.
Additionally, if the values are truly monotonic, the assumption of
no autocorrelation might also be violated.)
Aside from these caveats, the value for a given interval is not
relevant—rather, its relationship with all other values is, and, more
specifically, its relationships with the previous and subsequent values.
^{2}
(Better) Solution: Spearman’s Rho
Given the nature of the data (which one might say is non linear) and my
intent to quantify ordinality between two variables, it turns out that
Spearman’s
rho,
in theory, provides exactly the measurement that I want—it quantifies
the association between paired samples using the ranks of the variables
(not their values) relative to their samples. Notably, the statistical
significance (i.e. via a pvalue) can be calculated (traditionally,
using a ttest),
which should be handy for my intent on identifying nontriviality.
Nonetheless, even though this metric seems promisable, it will certainly
be sensitive to the small samples of each group (assuming that it is
calculated for each group). Don’t believe me? Check out how the
Spearman’s rho value changes for the numerica columns in the builtin
iris
dataset (which has 150 rows) when it is calculated for just the
first 10 rows.
Another Solution: Custom Heuristic
So, what can be done? Well, even with the hope that the Spearman’s rho
metric provides for quantification and significance inference, I thought
that I would try to create some kind of easily understandable heuristic
that I could explain to someone else without having to delve into
statistical theory. Nonetheless, I would be ignorant to not compare (and
validate) the results of my heurisitc with those of statistical theory
after creating my heurisitc.
Having this plan in mind, I began to think about how I would define my
heuristic, which, in essence, tries to quantify
monotocity. But
what exactly constitutes monoticity? Surprisingly, that’s a more complex
question than it might seem. ^{0}
For my purposes, I don’t necessarily care if the set of values is
strictly increasing or decreasing, but they should be “sufficiently”
increasing or decreasing. For example, while it is clear that the
sequence 1
, 2
, 3
, 4
is strictly monotonic (increasing) and the
sequence 1
, 2
, 4
, 3
is not, I would consider the latter
“sufficiently” monotonic. On the the hand, I would consider something
like 4
, 2
, 3
, 1
because the 1
and 4
are “badly” misplaced in
one another’s appropriate places, which are at the extreme ends of the
sequence. Moreover, if I was intent on identifying increasing monoticity
(as opposed to decreasing monotonicity), I would consider 4
, 3
, 2
,
1
“bad”, even though it is strictly monotonically decreasing. But what
about something like 1
, 4
, 3
, 2
(again, assuming that I am
evaluating increasing monotonicity)? Even though the 2
and 4
are
swapped, I might still consider this sequence “sufficiently” monotonic
because the 1
and 3
are placed correctly and the 2
and 4
are
“not too far apart”. Anyways, it’s easy to see how having some kind of
formal definition/calculation/criteria for monotonicity is handy.
The “Algorithm”
After some thinking, I came up with the following algorithm (if one can
even call it that).
(NOTE: I explicitly list the variable names that I use in the code
that follows to help the reader understand the relationships between
these steps and the implementation.)
 Given an
n
length sequence of arbitrary values, assign each value
an integer value between1
andn
to represent its “actual” rank.
This rank is to be assigned based on relative value in the set of
real numbers.
^{3}
 In the machine learning setting, this set of ranks is the dependent
variable (i.e.y
) to predict.  In the example situation described before, it equates to the rank
that would be assigned to the quarterly interval based on sales
relative to the other quarters.  In the simulation that I run below, this is the variable
y0
.
 Create a matrix of all permutations of actual rank and “assigned”
rank. To be clear, this “assigned” rank is independent of the actual
rank and value. To make things straightforward, these assigned ranks
should be transformed to use only the same set of possible rank
values dictated by the actual ranks (i.e. integers between 1 and
n
).
 In the machine learning setting, this “assigned” rank is the
independent variable (i.e.x
) used as a predictor.  In the example, it is the quarterly interval.
 In the simulation that follows, assigned rank is
x0
, and the
matrix (actually, atibble
) of combinations isdata_permn
.

Calculate the absolute difference between the “actual” and
“assigned” ranks for each value in the sequence. Subtract this
distance from the maximum rank value in the sequence. The resulting
value is what I call the “inverse monotonic distance”
(mntc_distinv
in the following code). 
Repeat the calculation of inverse monotonic distance for all groups
(grp
) of “actual” (y0
) and “assigned” (x0
) ranks. 
Sum up the inverse monotonic distance for each value in the
permutation group and take the average of this sum for each group.
^{4} Rescale this pergroup value to a
0 to 1 basis. ^{0} 
Identify any group (
grp
) corresponding to a sumaveragedrescaled
value (mntc_distinv
) in the upper 50% quantile of all values
(i.e. assigned the value"(0.5,1]"
for themntc_tier2
variable)
as “sufficiently” monotonic. (Themntc_tier2
variable can be
interpreted as my heuristic.)
Notably, even though the result set is split at the 50% threshold (which
is a subjective choice), this does not mean that 50% of all possible
groups are classified as 50%. (According to my method, only 33% are for
n = 4
.)
Implementing the Heuristic
Ok, that is enough discussion. What follows is the implementation.
NOTE: In order to keep focus on how the code implements methodology, I
recommend reviewing the code but not worrying too much about the details
(such as the internal workings of my custom functions). Rather, I’d
recommend inspecting in detail only the parts that are printed out (and
going back later to understand the complexities, if curious).
library("dplyr")
library("ggplot2")
library("tidyr")
# These packages are used, but their functions are called explicitly.
# library("purrr")
# library("broom")
# library("combinat")
The only choice that I need to make to begin is the length of the set of
values (i.e. n
), which should be a “small” integer. I’ll choose 4
,
simply because 3
seems like it is “too small” and because subsequent
visualization(s) becomes “cluttered” (and interpretation becomes less
direct) if a number 5
or greater is chosen. ^{0}
The following code chunk corresponds to steps 1 and 2 in my methodology,
which are basically just setup steps.
create_permns < function(n = 1L) {
n_seq < seq(1L, n, by = 1L)
combs <
combinat::permn(n_seq) %>%
purrr::map( ~ paste(.x, collapse = "")) %>%
unlist() %>%
as.integer()
data_xy <
tibble(grp_x = combs, grp_y = combs) %>%
expand(grp_x, grp_y) %>%
mutate(grp = paste0("x", grp_x, "y", grp_y))
into_seq < seq(1L, n, by = 1L) %>% as.character()
sep_seq < seq(1L, n  1L, by = 1L)
wrangle_data_xy <
function(data = NULL, which = NULL) {
col_grp < rlang::sym(paste0("grp_", which))
col_0 < rlang::sym(paste0(which, "0"))
data %>%
separate(!!col_grp, into_seq, sep = sep_seq, remove = FALSE) %>%
gather(idx, !!col_0, matches("^[09]$")) %>%
mutate_at(vars(idx, !!col_0), funs(as.integer))
}
inner_join(data_xy %>% wrangle_data_xy("x"),
data_xy %>% wrangle_data_xy("y")) %>%
select(idx) %>%
arrange(grp_x, grp_y)
}
data_permns < create_permns(n = n)
data_permns
grp_x
grp_y
grp
x0
y0
1234
1234
x1234y1234
1
1
1234
1234
x1234y1234
2
2
1234
1234
x1234y1234
3
3
1234
1234
x1234y1234
4
4
1234
1243
x1234y1243
1
1
1234
1243
x1234y1243
2
2
1234
1243
x1234y1243
3
4
1234
1243
x1234y1243
4
3
1234
1324
x1234y1324
1
1
1234
1324
x1234y1324
2
3
^1^ # of total rows: 2304
Note(s) about the above code chunk:

Although the “actual” ranks (
y0
) could be any real numbers, I’m
using values between1
andn
because my methodology dictates
that the arbitrary set of values would need to be transformed to
this range anyways. When transformed to this range, the values
should be interpreted as ranks. 
Like the “actual”
y0
ranks (representing the order of the
original, arbitrary values), thex0
ranks could technically be any
real numbers, but they would need to be transformed to the
1
ton
range anyways, so I do that directly. 
The number of combinations of “actual” (
y0
) and “assigned” (x0
)
rank pairs is equal ton! * n!
(i.e. 576). For my implementation,
the data.framedata_permns
actually hasn! * n! * n
(2304) rows
(because it is arranged in a “long” format). 
grp_x
andgrp_y
(and the combination of the two in thegrp
column) identify then
length groups of pairs ofx0
andy0
ranks. These are primarily useful for human interpretability and are
not actually relevant for computations. 
I use a function here (and elsewhere) although it may seem
unnecessary for a single execution because I’ll repeat the
methodology for different values ofn
later.
Now, I implement the initial calculation of “inverse monotonic distance”
(mntc_distinv
).
add_mntc_cols < function(data = NULL) {
data %>%
group_by(grp) %>%
arrange(x0, .by_group = TRUE) %>%
mutate(mntc = ifelse((y0 == cummax(y0))  (y0 == cummin(y0)), 1L, 0L)) %>%
mutate(mntc_distinv = as.integer(x0 * (max(x0)  abs(x0  y0)))) %>%
ungroup()
}
data_mntc < add_mntc_cols(data_permns)
data_mntc
grp_x
grp_y
grp
x0
y0
mntc
mntc_distinv
1234
1234
x1234y1234
1
1
1
4
1234
1234
x1234y1234
2
2
1
8
1234
1234
x1234y1234
3
3
1
12
1234
1234
x1234y1234
4
4
1
16
1234
1243
x1234y1243
1
1
1
4
1234
1243
x1234y1243
2
2
1
8
1234
1243
x1234y1243
3
4
1
9
1234
1243
x1234y1243
4
3
0
12
1234
1324
x1234y1324
1
1
1
4
1234
1324
x1234y1324
2
3
1
6
^1^ # of total rows: 2304
Note(s) about the above code chunk:
 The
mntc
variable is a “running” binary1
or0
to indicate
whether or noty0
is monotonic up through its position in the
sequence. (It does not differentiate between increasing or
decreasing.)  I use the
integer
data type (as opposed to simplynumeric
) where
possible because it is more memory efficient (although memory
efficiency is not a concern with this data).
Next is the calculation of the transformed
(i.e. summedaveragedrescaled) version of the “inverse monotonic
distance” (mntc_distinv
), as well as the split of the mntc_distinv
into upper and lower 50% quantiles (mntc_tier2
).
unitize < function(x = NULL) {
(x  min(x)) / (max(x)  min(x))
}
summarise_mntc < function(data = NULL) {
data %>%
group_by(grp) %>%
summarise_at(vars(mntc_distinv), funs(mean)) %>%
ungroup() %>%
mutate_at(vars(mntc_distinv), funs(unitize)) %>%
mutate(mntc_tier2 = cut(mntc_distinv, 2))
}
summ_mntc < summarise_mntc(data_mntc)
summ_mntc
Now, with the “algorithm” fully implemented, I can begin to evaluate the
results.
Exactly how many values make up each 50% quantile?
mntc_tier2 n n_pct
(0.001,0.5] 384 66.67
(0.5,1] 192 33.33
What does the distribution of all “inverse monotonic distance” values
look like?
The positive identificaiton (in yellow) of combinations along the
lefttoright, lowertoupper diagonal is what I would expect. These are
the values where x0
and y0
are perfectly matched. Conversely, values
along the other diagonal are generally colored in purple, as I would
expect. These combinations consist of sequences of x0
and y0
values
that are “negatively” symmetric (e.g. (1
, 2
, 3,
4
) and (4
,
3
, 2
, 1
)).
Checking the Heuristic
Ok, my heuristic seems valid, but how can I know for sure that it is
reasonable? I mentioned before that Spearman’s rho should serve a good
measure, so I’ll take a look at it now.
add_cortest_cols < function(data = NULL) {
data %>%
group_by(grp) %>%
nest() %>%
mutate(cortest =
purrr::map(data, ~ broom::tidy(cor.test(.$x0, .$y0, method = "spearman")))
) %>%
unnest(cortest, .drop = TRUE) %>%
select(grp, estimate, p.value)
}
summarise_mntc_wcortest < function(data = NULL) {
summ < summarise_mntc(data)
data %>%
add_cortest_cols() %>%
inner_join(summ, by = "grp")
}
summ_mntc_wcortest < summarise_mntc_wcortest(data_mntc)
summ_mntc_wcortest
grp
estimate
p.value
mntc_distinv
mntc_tier2
n_pct
x1234y1234
1.0
0.08
1.00
(0.5,1]
100
x1234y1243
0.8
0.33
0.67
(0.5,1]
100
x1234y1324
0.8
0.33
0.76
(0.5,1]
100
x1234y1342
0.4
0.75
0.38
(0.001,0.5]
100
x1234y1423
0.4
0.75
0.48
(0.001,0.5]
100
x1234y1432
0.2
0.92
0.43
(0.001,0.5]
100
x1234y2134
0.8
0.33
0.86
(0.5,1]
100
x1234y2143
0.6
0.42
0.52
(0.5,1]
100
x1234y2314
0.4
0.75
0.57
(0.5,1]
100
x1234y2341
0.2
0.92
0.14
(0.001,0.5]
100
^1^ # of total rows: 576
What exactly is the distribution of the Pearson’s rho ttest estimates
and pvalues?
abs(estimate) p.value n
0.4 0.75 192
0.8 0.33 144
0.2 0.92 96
0.0 1.00 48
0.6 0.42 48
1.0 0.08 48
Note(s) about the above output:
 By taking the absolute value of the estimate, I am essentially
treating monotonically increasing and decreasing as equal.  There are a relatively “small” number of distinct values.
^{5}
Now, to understand how the Pearson’s rho ttest estimate
s and
p.value
s correspond to my heuristic, I’ll simply overlay the
combinations that are identified as significant to my previous heat map
of rank combinations. Because I’m erring on the side of flexibility in
defining “sufficient” monotonicity, I’ll say that the pairs
corresponding to the bottom two tiers of pvalues (corresponding to
0.0833
and 0.33
) constitute “sufficient” monoticity.
![](viz_summ_mntc_wcortest1.png
It looks like there is a large amount of overlap between my heuristic
classification of “sufficient” monoticity and that identified by a more
statistical approach.
Now I’ll repeat the simulation for other values of n
. (Because
computations start to become intensive with n = 6
, and because the
n = 2
is relatvily trivial, I’ll evaluate values of 3
, 4
, and 5
for n
. ^{6})
ns < tibble(n = 3L:5L)
summ_mntc_byn <
ns %>%
mutate(data = purrr::map(n, ~(create_permns(.x) %>% add_mntc_cols()))) %>%
mutate(summ = purrr::map(data, summarise_mntc_wcortest)) %>%
unnest(summ, .drop = TRUE) %>%
ungroup() %>%
arrange(n)
What is the breakdown of mntc_tier2
values?
n mntc_tier2 nn nn_pct
3 (0.001,0.5] 24 66.7
3 (0.5,1] 12 33.3
4 (0.001,0.5] 384 66.7
4 (0.5,1] 192 33.3
5 (0.001,0.5] 9720 67.5
5 (0.5,1] 4680 32.5
What about the distribution of mntc_distinv
values? And of the
estimate
s and p.value
s?
n
abs(estimate)
p.value
mntc_tier2
nn
nn_pct
3
0.5
1.0
(0.001,0.5]
18
50.0
3
1.0
0.3
(0.001,0.5]
6
16.7
3
0.5
1.0
(0.5,1]
6
16.7
3
1.0
0.3
(0.5,1]
6
16.7
4
0.0
1.0
(0.001,0.5]
48
8.3
4
0.2
0.9
(0.001,0.5]
72
12.5
4
0.4
0.8
(0.001,0.5]
144
25.0
4
0.6
0.4
(0.001,0.5]
24
4.2
4
0.8
0.3
(0.001,0.5]
72
12.5
4
1.0
0.1
(0.001,0.5]
24
4.2
4
0.2
0.9
(0.5,1]
24
4.2
4
0.4
0.8
(0.5,1]
48
8.3
4
0.6
0.4
(0.5,1]
24
4.2
4
0.8
0.3
(0.5,1]
72
12.5
4
1.0
0.1
(0.5,1]
24
4.2
5
0.0
1.0
(0.001,0.5]
600
4.2
5
0.1
1.0
(0.001,0.5]
1800
12.5
5
0.2
0.8
(0.001,0.5]
1440
10.0
5
0.3
0.7
(0.001,0.5]
1680
11.7
5
0.4
0.5
(0.001,0.5]
600
4.2
^1^ # of total rows: 36
The distributions are sparse due to the relatively small number of
unique values for each metric (mntc_distinv
, p.value
, etc.).
^{7} Consequently, it is a bit difficult
to extract much meaningful insight about the relationships among the
metrics. To really understand how the distributions and relationships
scale with larger values of n
, mathematical theory would need to be
applied.
Nonethless, without jumping more into statistical theory, It seems to me
that the identification of rank combinations as significant by my
heuristic classification and Spearman’s rho (assuming that one uses the
traditional pvaluebelowathresshold approach) would become more
dissimilar as the value of n
increases. This is because my
classification simply splits all possible values into two sets for any
value of n
, meaning that the percentage of all possible combinations
is relatively insensitive to the value of n
. ^{8}
On the other hand, the Spearman’s rho pvalues would
become more refined with larger values of n
.
Anyways, I believe that my heuristic serves my purposes well. I only
really intended it to be used for small values of n
. Also, I intended
to create a “relaxed” definition of monotonocity, so having only a very
small percentage of all possible rank combinations meet the criteria
would have actually been undesireable.
Conclusion
In the end, I think I did more work than I really needed to do to answer
my original question about quantifying monotonocity and inferring
significance, but I think, in all, this was a worthwhile exploration.
 The market analyst does not necessarily hypothesize why annual projections tend to be high (i.e. perhaps due to overconfidence)
^{^}  Additionally, prediction is not the concern—rather, quantification of trend is. (While regression certainly can help with trend identification, its capability to create predictions is perhaps its better use.)
^{^}  In reality, the rank values could also be any arbitrary value on the real number scale.
^{^}  A sum of sums (instead of an average of sums) could be used here and the subsequent results would not change.
^{^}  Here, there are only
n + 1
(i.e. 5 uniqueabs(estimate)
s andp.value
s. This result is not generally true. (For example, when choosingn = 5
, there will be more than6
unique values of each metric.)
^{^}  This presents a good opportunity to implement a version of the “nestmutateunnest” idiom that can be very effective for creating many models. The “many models” chapter in the R For Data Science book provides an excellent example of this process.
^{^}  Unfortunately this is due to the nature of the data and the simulation, so nothing can be done about it.
^{^}  Note that the 33% number found for
n = 4
is not generally true, although this percentage does not seem to change drastically with different values ofn
.
^{^}
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