Developing "coupling graphs"

[andkov]

@ampiccinin @knighttime
Objective:

[andkov]

See the script ./sandbox/coupling-graphs/coupling-graphs-1.R for the implementation of the graphs.

Graph 1

The graph show 48 random individuals (males) from MAP study. Red indicates process A (physical), black represents process B (cognitive). If a person has less then two data points no lines are drawn.

on more similar scale than original metric

not really sure what this means. Could you please elaborate?

Graph 2

The graph shows a random 48 ids from MAP study.

would it make sense to connect the dots?

here's a version of this graph with connected dots.

You can adjust or further develop graph in the ./sandbox/coupling-graphs/coupling-graphs-1.R script. I have prepared the functions to extract the data. Let me know if you have any questions!

[ampiccinin]

@andkov - wow! That was fast! Thanks!

Graph 1 is exactly what I meant. By more similar metric I meant that both would hover around zero, whereas their original raw metrics might be 0-44 versus 300-600.

Too bad they don't all look like person 198!!

I'm surprised that cognitive is black. I would have expected the physical measurements to be more variable.

For Graph 2, I wondered about a single summary graph that contained all individuals' data. That was why I suggested joining points within person - to have a sense of which points went together. Looking at person 198 again, though, shouldn't all their points fall between -1 and 1?

[andkov]

By more similar metric I meant that both would hover around zero, whereas their original raw metrics might be 0-44 versus 300-600.

ah. of course. this explains why the black (cognitive, in this case boston story immediate) is all over the place, while red (physical, in this case fev) is relatively stable. Here's the descriptives for the observed measures:

> summary(ds_wide$observed_A)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
  0.420   1.720   2.145   2.134   2.560   3.930     749 
> summary(ds_wide$observed_B)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
  0.000   8.000  10.000   9.609  11.000  12.000     395 

Redisualizing helps reducing the span, but it still remain substantially different:

> summary(ds_wide$residual_A)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
-1.0970 -0.0893  0.0033 -0.0001  0.1033  0.6330     749 
> summary(ds_wide$residual_B)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
-7.3840 -0.9522  0.0680 -0.0001  0.9649  3.6590     395 

And of course with other measures, this would be even greater. I turn these residual scores into z-scores, which should help this issue

[ampiccinin]

Thanks @andkov - I'm not sure we want z-scores. Our model is not standardized, and seeing how much a variable jumps around should be informative. When I said "more similar metric" I was indicating that it would be easy to plot with the same axis (both would have scores around 0).

My surprise about the cognitive residuals seeming larger than the physical was based on my memory of the observed raw trajectories. My sense had been that the physical measures jumped around quite a bit.

Are you able to resolve the apparent inconsistencies between Graphs 1 & 2 for person 198? Is it just my misinterpretation that is confusing me? Thanks!!

[ampiccinin]

@andkov ...are you saying that the physical looks like smaller variability because the original range is wider?

[andkov]

a slight diversion.

I have been troubled by such patterns as 153 (see graph 1), so I started looking at the individual level data. Here's the data for this person:

> ds_wide %>% 
+   dplyr::filter(id == 153) %>% 
+   dplyr::select(id,observed_A, observed_B, predicted_A, predicted_B, residual_A, residual_B)
   id observed_A observed_B predicted_A predicted_B residual_A residual_B
1 153         NA          8          NA    9.556000         NA -1.1525397
2 153         NA         10          NA    9.466136         NA  0.3954367
3 153         NA          8          NA    9.370914         NA -1.0154452
4 153         NA         12          NA    9.282460         NA  2.0129001
5 153       2.29          9    2.255416    9.184888  0.2008373 -0.1369478

Even though there is only one observation in A, the model still estimates. I wonder if the peculiar patterns we see in correlations may be due to the fact that many factor scores are coming from cases like these. I've looked at this study (MAP) to get these counts:

count_miss <- ds_wide %>% 
  dplyr::group_by(id) %>% 
  dplyr::mutate(
    n_nonmiss_a = length(observed_A) - sum(is.na(observed_A)),
    n_nonmiss_b = length(observed_B) - sum(is.na(observed_B))
  ) %>% 
  dplyr::ungroup() %>% 
  dplyr::distinct(id,n_nonmiss_a, n_nonmiss_b) 

count_miss %>% 
  dplyr::group_by(n_nonmiss_a) %>% 
  dplyr::count() %>% 
  dplyr::mutate(
    cumulative = cumsum(n)
  )
# A tibble: 6 × 3
  n_nonmiss_a     n cumulative
        <int> <int>      <int>
1           0    26         26
2           1    74        100
3           2    54        154
4           3    53        207
5           4    55        262
6           5    59        321

as you see out of 321 individuals, there were 26 that had no observation on process A and 100 that had up to 0 or 1 observations. For process B (cogntive) the story is similar:

> count_miss %>% 
+   dplyr::group_by(n_nonmiss_b) %>% 
+   dplyr::count() %>% 
+   dplyr::mutate(
+     cumulative = cumsum(n)
+   )
# A tibble: 5 × 3
  n_nonmiss_b     n cumulative
        <int> <int>      <int>
1           1    42         42
2           2    39         81
3           3    34        115
4           4    42        157
5           5   164        321

In fact, there are even cased in which we have 0 observed point on A, only 1 observed point in B and we still get an estimate for the factor score!

> count_miss %>% 
+   dplyr::group_by(n_nonmiss_a, n_nonmiss_b) %>% 
+   dplyr::count() %>% 
+   print(n=100)
Source: local data frame [21 x 3]
Groups: n_nonmiss_a [?]

   n_nonmiss_a n_nonmiss_b     n
         <int>       <int> <int>
1            0           1     9
2            0           2     6
3            0           3     5
4            0           4     3
5            0           5     3
6            1           1    33
7            1           2    19
8            1           3     4
9            1           4     3
10           1           5    15
11           2           2    14
12           2           3    14
13           2           4    10
14           2           5    16
15           3           3    11
16           3           4    16
17           3           5    26
18           4           4     9
19           4           5    46
20           5           4     1
21           5           5    58

Which is, I guess, because of the FIML option? To clarify, i'm using the observation THAT HAS BEEN USED BY MPLUS during the estimation, recovering it from the output.

[ampiccinin]

@andkov - damn. I'm sure there are people who would be fine with this. I am not one of them. Thanks for checking it!!! In theory (at least with MAR), the S-S correlation should be the same with and without these people who have data on only one or the other variable. Might be worth checking, though. What happens if you restrict to only folk who have at least 2 of each variable?

[andkov]

@ampiccinin , I can certainly do that. It will bring down the sample size, but not catastrophically.

[andkov]

Looking at person 198 again, though, shouldn't all their points fall between -1 and 1?

I wonder if it's just hard seeing it on this graph. Here's the data for person 198. There are almost between -1 and 1. Does that resolve your the inconsistency you thought there was in this case?

> ds_wide %>% 
+   dplyr::filter(id == 198) %>% 
+   dplyr::select(id,observed_A, observed_B, predicted_A, predicted_B, residual_A, residual_B)
   id observed_A observed_B predicted_A predicted_B residual_A residual_B
1 198      2.480         12    2.319000    11.09000   0.161000   0.910000
2 198      2.420         12    2.237832    11.01248   0.182168   0.987520
3 198      1.550         11    2.147230    10.92595  -0.597230   0.074050
4 198      2.285         12    2.060188    10.84282   0.224812   1.157180
5 198         NA         11          NA    10.75417         NA   0.245835