[From Rick Marken
(2009.09.08.2200)]
Bill Powers (2009.09.08.1207 MDT)
I’m not sure what variables you were correlating – could you
explain? What does that word “cumulative” in the figure
mean?
RM: I correlated the week number
against the “controlled items cumulative” (y axis) value for
each baby for just the sequence control graph. That’s what gave me the
.86 correlation. This was for 50 observations, the total for all four
babies.
Then, taking
“controlled items cumulative” at its name, I correlated week
with what I took to be the number solved each week, which would be, for
week n, the y value in week n minus the y number in week n-1. I did this
for only two babies (1 and 2, I believe), resulting in correlations of
-.20 for one and .56 for the other. As I said in my post, I don’t know
what “controlled items cumulative” really means; I don’t think
it can mean " cumulative solved" since some of the curves go
down, giving negative values for the number correct.
BP: But why not do all four? Seems to me the correlation might have been
considerably higher, since Baby 1 was the one outlier here. Or maybe you
picked two of the other three, in which case the correlation is too
high.
I noticed the negative slopes, too, and realized that
“cumulative” can’t mean what it seems to mean. I haven’t read
the actual papers so I don’t know the details. Another possible
interpretation is that there were perhaps 30 different tasks, and the
plots show how many of them were mastered at a given time. The
negative-going changes wouldn’t rule that interpretation out.
RM: I think that, in order
to make any sense of these data graphs we have to know 1) what the kids
were controlling (what were considered events and sequences) and 2) what
is the dependent variable (“controlled items
cumulative”).
BP: I agree. We’ll just have to wait for Frans to get back from wherever
he’s
hiding.
BP earlier: What I see in the figure is that three of the four babies
started increasing the number of sequence variables controlled within
three weeks of each other while one of them didn’t seem to improve much.
At a correlation of 0.86 the odds of something or other are 5 to 1; I
guess that means that predictions of the something on the basis of the
probabilities would be wrong about 1 time in 6, which isn’t too bad if
being wrong isn’t terribly important. The actual data seem to show that
you’d be wrong about 1 baby in 4 here, not too great. I don’t see what is
being predicted, however.
RM: I don’t see anything being predicted here either, any more than I see
anything being predicted in my measure of the relationship between taxes
and growth.
BP: I always look at it that way. Regardless of causation, and which
direction you assume it goes, how well does knowing the independent
variable’s value let you predict what the next observation of the
dependent variable is going to be? If you can’t make any predictions,
what’s the point of the correlations?
RM: What is being done in
both cases is seeing whether there is evidence for a relationship between
two variables; taxes and growth in one case, time (in weeks) and control
ability (measured in a still unknown way) in the other.
BP earlier: I don’t know – depends on what you’re trying to predict.
I’m pleased to see that the population data show the peaks in the correct
order, with zero occurances about halfway between the peaks.
RM: What population data? What peaks? Which graphs are your referring
to?
This is from page 35, the next one after the sequence graph you
used.
![Emacs!]()
The red curves show the distribution of each regression period. As you
can see, there are hardly any points where the red curves intersect (the
last transition shows the biggest overlap, which is a trivial number of
cases). It would be good to see the raw data, of
course.
BHP earlier: Does that really imply that there were no individuals
who developed the levels in the wrong order? That would be very
surprising, though gratifying.
RM: Right, overall, comparing the “event” and
“sequence” graph, all kids seem to develop the ability to
control events before they control sequences.
BP: So: if baby A’s acquisition of level n comes before acquisition of
level n+1, what is the probability that baby B’s order of acquiring the
levels will be the same for any adjacent pair of levels, and for all 8
transitions taken together? Methinks it will be rather close to 1.00.
We’d have to look at the raw data for all the levels and all the
researchers to see how many individuals stuck to the exact timetable. I
suspect that we’re looking at a plot of means rather than individuals.
I’m still looking for a way to discount the seeming perfection of my
subjective guesses about the levels. Frans’ late wife Hetty’s comment was
that the fit of my levels to the data was “uncanny.” That waves
all sorts of red flags! I’d rather find out what the catch is before
someone else does.
BP earlier: As to predicting the exact week at which the new level
would start to appear, I would expect that to vary a lot from one baby to
another;
RM: I would imagine that that’s true. My correlations are not an attempt
to predict the exact week when a control skill develops; I’m just looking
for an increase in control skill with weeks. Since kids develop this
skill at somewhat different times (presumably) the most interesting
correlations are the one’s for each individual subject. But we can’t
really compute these correlations knowledgeably until we know what the
dependent variable was in these experiments. What, that is, is
“controlled items cumulative”?
BP: Yes, without that information it’s hard to know what to correlate
with what.
BP earlier: Is there a way to calculate a probabilty that the
ordering of the steps between levels is off by N steps?
RM: I bet Richard could do that easily; that would be an excellent test.
But right now it looks like we have evidence for N = 2 levels: events and
sequences. At least that’s all I see in the slides. If there is evidence
that all babies develop N=10 levels of control in the same order, then I
think someone’s got a great Science paper.
BP: See the chart above. But it’s still too early to jump to
conclusions.
Best,
Bill P.
···
At 09:58 PM 9/8/2009 -0700, you wrote: