Designing randomized trials to understand treatment effect heterogeneity

November 17, 2020

Biostatistics Seminar - November 17, 2020

Elizabeth Tipton, PhD
Associate Professor, Department of Statistics, Northwestern University

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00:00So welcome everyone,
00:01it is my great pleasure to introduce
00:04our seminar speaker today,
00:06Doctor Elizabeth Tipton.
00:07She's an associate professor
00:09statistics the Co director of the
00:12statistics or evidence based policy
00:14and practice Center and a faculty
00:16fellow in the Institute for Policy
00:18Research at Northwestern University.
00:20Unducted sentence research focuses on
00:23the design and analysis of randomized
00:25experiments with a focus on issues for
00:28external validity and generalizability,
00:30as well as meta analysis with the
00:33focus on dependent effect sizes.
00:35Um, today she's going to share with us
00:37how to design randomized experiments
00:39to better understand treatment effects.
00:41Head virginity welcome best.
00:43The floor is yours.
00:44Thank you.
00:45Thank you. I'm very excited to be here today.
00:48I really wish I hear I wasn't talking
00:50about my office slash closet and was
00:52actually with you guys in person and
00:55this is my first time doing slides where
00:58I'm on the slide so it's a little.
01:00It's a little strange.
01:01I don't know what is the protocol
01:04for questions. How do you guys?
01:06How do you usually set this up?
01:08Do people what's the norm?
01:10Do you guys usually up jump in
01:12with questions or save them for
01:13the end? So I think as you prefer,
01:16we can do either way. OK, I'm
01:18just I won't be very good at
01:20checking the chat, so if there's
01:22a question if somebody can just
01:23speak up that would be will do that.
01:26I'll do that on the chat. OK, thank you.
01:28OK so I just want to set out background
01:31for what I'm talking about today,
01:33which is I'm talking about randomized
01:34trials an I realized that in a
01:36Biostatistics Department, you guys.
01:39The idea that randomized trials are
01:41common is probably almost absurdly basic
01:43for the world that you operate in,
01:45but I do a lot of my statistical
01:47work in the areas of education and
01:50psychology and kind of in the field
01:52experiments world and those areas.
01:55Randomized trials have only
01:56become common really,
01:57I'd say the last 20 years.
02:00So almost 20 years ago,
02:03the Institute for Education Sciences
02:05was founded in the Department of
02:08Education in the US government,
02:10and that has funded almost 500
02:12what are called efficacy and
02:15effectiveness trials and education.
02:17Previous to that there were
02:19very few of these.
02:21There's also an increasing number of
02:24nudge experiments in social psychology
02:27experiments that are occurring in the world.
02:31I know that there's a lot of rain in mice
02:34trials occurring in developing countries,
02:38so this is late in parallel,
02:40maybe 2.
02:42In public health,
02:43they're being randomized trials there,
02:45so I'm just sort of pointing out
02:46that these are becoming increasingly
02:48common for policy decisions,
02:50not just individual decisions.
02:53But the trials as there as they are
02:56designed currently are not necessarily
02:58ideal ideal in the sense that they are
03:00not as big as we would like them to be.
03:03In order to be able to really explore
03:06the data well, there often in,
03:08you know,
03:09sort of somewhat small samples
03:11of clusters in the in the kind of
03:13education world that I work in it.
03:16They're very often just simple to
03:18arm designs 5050 treatment control.
03:19I much less common to see things
03:22like step wedge or smart designs,
03:24so those are trickling in, I think.
03:28And the goal of these the is often
03:30to get into some things like clearing
03:32House of some places so that policy
03:35making decision makers can use the
03:38information from the trials to
03:40make decisions.
03:41But the problem which is the focus
03:43of my talk is that there very
03:45often been taking place in samples
03:48that are purely of convenience,
03:50which makes thinking about generalizability
03:52and heterogeneity rather difficult.
03:58If the treatment of X very if treatment
04:00effects vary across individuals or
04:01they vary across clusters in some way,
04:04then it's pretty straightforward to
04:05see as a group of statisticians here
04:07that the average treatment effect
04:09you would get in the population.
04:11Is probably not exactly the same thing as
04:14the average treatment effect in the sample,
04:16and that these could be quite different
04:18if treatment effects vary a lot aniff
04:21depending upon how the sample is selected.
04:23So there has been an increasing
04:25amount of work in this area.
04:27There's a couple of papers I think that
04:30are particularly helpful if there's
04:32a paper in education where they're
04:34looking at bias from non random treats.
04:37Non random treatment assignment or
04:39they show that the bias of external
04:41validity is on the same order as
04:44internal validity bias and so to do
04:46so they hear they sort of leverage
04:49and natural experiment with a
04:51randomized trial to look at this.
04:52And that's worked by Bell,
04:54Olson, Oregon, Stewart.
04:55There's also work showing that.
04:58In education and the kinds of schools
05:00and school districts that take part in
05:03randomized trials are different than
05:05the populations of various populations.
05:08At something like the Institute of
05:10Education Sciences might be interested in,
05:13so I have a paper out with Jessica Spy,
05:16Brooke Ann are students looking
05:18at 37 randomized trials and the
05:20samples of schools taking part in
05:22those studies and comparing them
05:24to various populations of schools.
05:27In the US.
05:28There's also work hidden behind me.
05:31By Liz Stewart.
05:33San colleagues looking at school districts
05:36and a couple of other papers as well,
05:40and these find fairly consistent things.
05:43For example,
05:44that large school districts are
05:47overrepresented in research.
05:49Relative to the size of districts in the US.
05:52There's been also a lot of work
05:55in this area of generalizability
05:57and post hoc corrections.
05:59I started into this work looking at
06:02using post stratification as a way
06:05of estimating a population average
06:07treatment effect from a sample.
06:09There's also been work using
06:11inverse probability weighting,
06:12maximum entropy weighting
06:14bounding approaches.
06:15There have been some approaches
06:17that focus on little,
06:18so I'm thinking like.
06:22Here's the paper and Stuart San Green.
06:24I think that does that,
06:25so there's been like a kind of
06:27a flurry of method development.
06:29I think here in this area of
06:31thinking you know,
06:32how do I actually estimate this?
06:34If I have population data of different forms
06:36and I have sample data of different forms,
06:39how can I actually estimate a
06:41population average treatment effect?
06:43But when I first started doing this work,
06:46I realized in a series of examples
06:47that I was working on that the
06:50effectiveness of these methods is
06:51often severely limited in practice
06:53because of undercoverage and
06:55what I mean is that it you can't.
06:57If it turns out that your population
07:00has there's a part of the population
07:02that's just not represented in the trial,
07:04there's really not much statistical
07:06magic you can do.
07:07You can make some assumptions,
07:09but you can't really re wait
07:11something that doesn't exist, and the.
07:14It's it's really a reflection of lack
07:16of positive ITI in the study. Yes,
07:18exactly thanks. Yeah exactly.
07:19And yeah, I'm just using survey
07:21sampling language for the same thing.
07:23That's right, yeah, and so.
07:25And that's the lack of positive ITI
07:27often arises because people aren't
07:29thinking about what the population is
07:31in advanced and so it's very tricky
07:33for them after the fact to generalize,
07:35because it turns out that maybe
07:36this what I as analyst him now
07:38trying to think of as the population
07:40isn't exactly the population,
07:42but it's very hard for people to
07:44articulate what the population is,
07:46and so spent a lot of time just trying to.
07:49You're out what the population actually is,
07:51and if that's population is meaningful,
07:53as if that's a population that even matters.
07:55So I realized I pivoted a bit.
07:57I realized that you could do
07:58a lot of this with statistics,
08:00but you were going to be limited
08:02if you didn't design better trials.
08:08And so that's allowed me to think.
08:10Well, why don't we just start?
08:13The beginning and do this
08:14do a better job this so why?
08:17What have we started at the
08:18beginning of our studies by asking
08:20what the target population of the
08:22intervention was thinking about
08:24inclusion and exclusion criteria,
08:25I think it helped.
08:27This probably matters even more with
08:29like comorbidities and you know.
08:31I like rolling out people.
08:32You're doing a study on depression,
08:34but you rule out people with anxiety,
08:36and that's like a big problem
08:38for the interpretation since they
08:39were highly related to each other.
08:41This is true in education as well.
08:43So like,
08:44what are the inclusion exclusion
08:45criteria for your trial and how might
08:48that affect where you can generalize?
08:50And then also thinking about
08:51background characteristics and
08:52contextual variables that might
08:54moderate the intervention's effect
08:55and the tricky part here is,
08:56there's a little bit of a circularity
08:59which I'm going to keep coming
09:00back to in what I'm talking about,
09:02which is in order to know these,
09:05you know,
09:05we don't know what these are in advance,
09:08and we don't have a lot of
09:10knowledge generated to date about
09:11what these variables are,
09:13because studies have not been designed to
09:14estimate or test hypothesis about moderation,
09:17and so instead we have to sort of think
09:19through what we think might matter.
09:21Using,
09:22you know,
09:22not a great source of knowledge here,
09:25but the idea is that you sort of
09:27take all of this information and
09:29then you use this to create to
09:31use sampling methods to actually
09:33design recruitment procedures
09:35like using stratified sampling.
09:36Figuring out if you should
09:38you know within Strata,
09:39using balanced sampling or random
09:41sampling and thinking about sort of
09:44ways in which you can increase the
09:46coverage so you can have positive
09:48ITI for the whole target population,
09:50so that when you do you know.
09:52When you do need to make adjustments
09:54at the end of your trial using
09:56these statistical methods,
09:57they are in a realm in which
09:59they can perform well.
10:03This this sort of lad me to thinking about
10:05tools for general for generalization,
10:07and so I just want to highlight this
10:10because I think this is a good strategy
10:13for methods people to think about.
10:16So I I thought will nobody is going to do
10:19what I'm telling them to do if I don't
10:21build a tool because the kind of people.
10:24Clan randomized trials,
10:25at least in my domain,
10:27don't often have statisticians ready at
10:28the ready to work with them on things,
10:31and they are often writing grant
10:33proposals before they've got funding,
10:34and so it's very possible that they're not
10:37going to think about generalization or or
10:39have the training or tools to do it so.
10:42I got a grant from the Spencer
10:44Foundation and then I've had follow up
10:46money from the Institute of Education
10:49Sciences to build this tool called
10:51the Generalize are that uses some
10:53basic design principles standalone.
10:54It's got.
10:55It's very focused on the user
10:58experience and it in the background
11:01has the Common Core of data which is.
11:04An annual census of the public
11:06schools in the US so that the data
11:08is already been cleaned and set up.
11:11We're adding in right now the iPads data,
11:13which is higher Ed data in the US and
11:16so the idea is somebody could go in
11:18and walk through inclusion exclusion
11:20criteria identified moderate orsan.
11:22It would build you stratified
11:24recruitment plan in less than an hour.
11:26You could leave with a list of all
11:29the schools and start being able
11:31to recruit with the.
11:35Great, I've had this going since 2015
11:37and it was very slow going for awhile.
11:39This is sort of. I just realized this
11:42year that I could actually extract
11:44a lot of user data and So what you
11:47can see here is actually it was slow
11:49going and I had some early adopters.
11:51These are people that would be star users,
11:54so many of them are planning
11:56randomized trials,
11:56but there was actually a very big
11:59jump that occur this summer and
12:01that's based on this jump.
12:02I actually started digging through
12:04things and realized that.
12:05Institute of Education Sciences that actually
12:07enacted requirements for generalizability.
12:09In their request for proposals,
12:11and so you can see that what
12:13I already always speculated,
12:15which is that funders really drive change.
12:17So once funders said you need to
12:20pay attention to generalizability,
12:22people actually started paying attention
12:24to generalizability in their proposals.
12:27OK, so this I just wanted to give
12:28you all of this background as a
12:30way of explaining sort of my like
12:32where I'm coming from in the in
12:34the in heterogeneity and how I'm
12:36thinking about this.
12:37Um?
12:37So everything I've talked about
12:39is sort of averaging over hedge
12:42and 80 when we talk about analyze.
12:44Estimate an average treatment
12:46effect for a population,
12:47assuming that there's variation of
12:49effects and we're averaging over those.
12:52But to average over those requires
12:54that we know something about how
12:56treatment affects very and very often,
12:58and I would say this is the
13:00in general we don't,
13:02and the reason that we don't have a
13:04great handle on this is because sample
13:06size and sample sizes in randomized trials.
13:09I've been very focused on
13:11the after treatment effect.
13:12Moderators have only become more of a focus,
13:15at least in education.
13:17More recently,
13:17and I think that's true in
13:20psychology and related areas as well.
13:23And they are often more like
13:24exploratory analysis at the end,
13:26so you end up with these problems
13:28where moderador effects don't get
13:29replicated and they don't get
13:31replicated because there was.
13:32You know,
13:33who knows how many statistical tests
13:35conducted in order to find those moderators.
13:38So they're not very stable,
13:39and we don't really necessarily
13:41understand or their underpowered
13:42deeply underpowered like you just
13:44have a very homogeneous sample.
13:45And so how are you going to find
13:47a treatment effect variation if
13:49there's not much variation in your
13:51sample to start with,
13:52so they're often an afterthought,
13:54but I what I noticed overtime is
13:56that as generalizability has become
13:57something people are paying attention to,
13:59people are also starting to pay
14:01attention to the idea that you
14:03could predict treatment effects,
14:04or that you could identify subgroup
14:06effects and that this might
14:08be very useful information.
14:10Which led me to start thinking about
14:12how you would design trials for this.
14:15So what I'm going to,
14:16what I'm leading up to is talking
14:18about designing trials to think
14:20about heterogeneity.
14:20So I'm just going to start with
14:22like a little
14:23bit of a background here.
14:25So we're going to assume that you've got.
14:27I'm assuming we've got units
14:28which are usually here.
14:29Let's say students insights
14:30which might be schools,
14:32and I'm doing a randomized trial,
14:33and I've got these potential outcomes.
14:36And so we've got both an
14:39average and intercept in these,
14:41and we've also got some sort of
14:43fixed variation that we can explain.
14:46And then we have this other parts that
14:48are not affected by the treatment.
14:51We've got some site level and individual
14:54residuals and some idiosyncratic errors.
14:56But what we're interested in
14:58really is in these these moderate
15:00yrs of treatment effects,
15:02and so you could say that Delta
15:040 is the difference in averages.
15:07I'm assuming these are centered variables,
15:09so this is nicely the difference in
15:12averages and that the vector Delta
15:14is the difference between these
15:16effects of the treatment and then
15:18under treatment and under control.
15:26A lot of so as you have to
15:27think about interpretability
15:28here of what I mean by Delta.
15:31By this by these deltas and
15:33how to standardize because we
15:34wanted to talk about these,
15:36they need to have a mean of 0,
15:38but also in order to talk
15:40about treatment effects.
15:41Sort of done in general for
15:43developing things like power,
15:44we often standardize them so
15:45often we have effect sizes for
15:47the average treatment effect,
15:48their standardized in relation
15:49to the variation in the in the
15:52sample and the population.
15:53And so here I'm going to sort of
15:55say we what we need to do is we
15:58need to standardize the covariates
15:59and we need to standardize the
16:02covariates in relation to the
16:03population standard deviation.
16:04This might not seem like this
16:06is like a radical statement,
16:08but if you look into the power analysis
16:10literature on how to conduct power
16:12analysis for moderate are tests,
16:14they are typically standardizing in
16:15relation to the sample standard deviation,
16:17and in doing so,
16:18it makes it impossible to see
16:20how your sample actually how you
16:22choose your sample might matter.
16:24Isibaya standardizing by this
16:25fixed value by the population,
16:27you've identified a population,
16:28and now we're standardizing by
16:29that population standard deviation.
16:31That will make the role that the
16:34sample plays here much more clear.
16:37OK,
16:37so the fact that we randomized
16:39to treatment and control allows
16:41us to estimate these dealt these
16:43Spectre Delta using some generalized
16:46least squares of some sort,
16:48and I'm being a little big here
16:49because I'm trying to encapsulate
16:51cluster randomized randomized
16:53block individual randomizer,
16:54all like versions of this.
16:56OK, so I can do so, I can separate.
17:00These are at additive or rather subtractive.
17:02I guess the treatment and
17:04the control their step.
17:06You can separate them.
17:07And and through this I can
17:10think about statistical power,
17:12and for each of these moderador effects.
17:16And so,
17:16one way you can do that is through the
17:18minimum detectable effect size difference.
17:20I don't know how common this
17:22is used in the sort of.
17:23Biostats world,
17:24but it's a pretty common metric that's
17:27used in cluster randomized trials in.
17:30The world I work in,
17:32and so it's nice because it's
17:34sort of easily interpretable,
17:36so this is the smallest affect size
17:39that you could for a for a given
17:42Alpha level which is affecting this.
17:44Msub knew this is.
17:47That's like the critical value.
17:49This is sort of the smallest
17:51true effect that you could detect
17:53with the power that you with like
17:5580% power for example.
17:58And so this is like a general form for this,
18:01and So what I'm showing is that its function,
18:04can I like move my hands?
18:06I don't know.
18:07I'm just going to involve a
18:09lot of never mind,
18:10so it's a function of the variation
18:13in the population in that covariate.
18:15It's also a function of S,
18:17which is you could think of as
18:19the sort of covariance matrix
18:20of the X is in the sample,
18:23so those are different.
18:24And then it's a function of N,
18:26which is the sample size per cluster.
18:29I'm assuming it's constant here.
18:30J is the number of clusters and P
18:33is the proportion in treatment. So
18:36that what is Sigma XK squared? Is?
18:38The population SD of effect modifier or
18:40the population variance effect modifier?
18:42Is the population variance
18:44of the effect moderate or modifier?
18:46But then your square rooting it so
18:48it's going to be gradual scale.
18:53OK, so just to give you a couple
18:55of special cases where you can
18:57sort of parse out some things.
18:59So there's been previous work.
19:01I meant to include a citation here by
19:04Jessica Spy, Brooke and colleagues.
19:05That's looking at power
19:06for moderate are tests.
19:08And so here's 2 cases we have.
19:10Site lab, site level,
19:12moderate yrs and individual
19:14level moderate yrs and.
19:15I'm I'm taking basically what they've got,
19:17but re tweaking part of it.
19:21Because I'm factoring out this Sigma
19:24squared and noting that you can
19:26actually pull out this thing called
19:28RXK at the front and the RXK is
19:30this ratio of the standard deviation
19:32of the covariate in the sample
19:34compared to the standard deviation
19:36of the covariate in the population.
19:38And So what you can see here is by
19:40doing that you by rewriting it.
19:43This way you can see that our XK is
19:45having just as much of an effect on
19:48statistical power as things like the
19:50square root of N or the square root of P.
19:54These other parameters that most power
19:56analysis has spent has focused on,
19:57and that's true.
19:58You know, in any of these designs.
20:00Love seeing it in any of these designs.
20:03RX shows up. OK,
20:06so if RX is something that matters for power,
20:10a question will be well.
20:12What are people doing in
20:14practice right now right?
20:16So maybe maybe people are choosing
20:19fairly heterogeneous samples,
20:20and So what I've got here is 19.
20:23This is 19 randomized trials
20:26in education that we extracted
20:28information from and we've got.
20:31So these are box plots of
20:33values across each of these 19,
20:35and for each of them I've calculated for
20:38holding like the US population of school.
20:40So this is like the US population
20:43of let's say elementary schools.
20:45I'm looking at the ratio of this
20:47moderate are in the sample in these
20:49studies to the ratio of that to that
20:52standard deviation in the population OK,
20:55and then I'm looking at boxplots of
20:57this and what you can see like do this,
21:00don't?
21:01OK, what you can see here is that
21:03the bar at the bottom.
21:05Can you see my cursor?
21:08Can't tell if you guys can see my curse.
21:10No,
21:11you can't see my cursor.
21:12OK,
21:12so the bar at the bottom there's an R
21:14X = sqrt 1/2 and then there's a line
21:17across the top that's like a dashed one.
21:19That's the R X = sqrt 2.
21:22OK,
21:22and so you can see that most
21:24studies are actually below there,
21:26less heterogeneous than the population there.
21:28Below this line for one,
21:30and they're actually far less heterogeneous
21:32than the than the population there are.
21:35Actually.
21:35If you look at these median values,
21:38many of them are closed 2.5,
21:40so they are about 1/4 of the variation
21:43as we're seeing in the population.
21:45So this gives you a sense that if.
21:49That there's, uh,
21:50an opportunity to improve, right?
21:52Like I could increase power not just by
21:55increasing my sample size or increasing.
21:58My sample size in schools or
21:59my sample size of the number of
22:01schools which are pretty expensive,
22:03but I could also increase my power by
22:05changing the kinds of samples that I select.
22:10And so that's where these numbers came from.
22:12They should have gone to
22:14slightly different order.
22:15So the main point is that design
22:17sensitivity, the way we think,
22:19whether that statistical power or
22:21standard errors or whatever framework
22:22that there this is proportional in
22:25some way to this RX value that we can
22:27improve our design sensitivity by
22:29choosing a more heterogeneous sample.
22:32And so funny, I must have like put
22:34this in here twice on accident.
22:36So this is the same thing but
22:38with a line through it.
22:40OK so if once you have that insight
22:42that heterogeneity matters,
22:43that it's actually something that
22:45we can include in our power analysis
22:47and that is something that is not
22:49actually happening in practice.
22:50Then we can start thinking about how
22:52we might plan studies differently.
22:54OK, so if.
22:56So how can we improve statistical power?
22:58Well,
22:58a lot of the literature as I was saying,
23:01is focused on improving power by
23:03increasing sample size or instead.
23:04But what I'm arguing here is that you
23:07could increase instead this ratio.
23:08You could increase the variation
23:10in your sample choosing more
23:11heterogeneous sample annual have
23:12more statistical power for test
23:14of heterogeneity of moderators,
23:15and So what would you do with this?
23:18It would mean you know purposefully
23:19choosing sites that were more extreme,
23:21it might end,
23:22and that's easy enough to do in one variable.
23:25And I'm going to talk a little bit about
23:28how to do that with multiple variables.
23:31So with a simple,
23:32let's just say we had one single continuous.
23:35Moderate are like this is a normal
23:37distance normally distributed.
23:38This theory would tell us that we
23:41should choose half of our sample.
23:42We would choose half of our sample
23:45from the upper from the upper an
23:47lower tails and choosing them from
23:49the upper and lower tails were
23:51actually getting an RX of sqrt 2.
23:53This is actually a rather large,
23:55so this is going to create a much
23:57more homogeneous heterogeneous sample,
23:59thus increasing our statistical power
24:01because it's more heterogeneous than the.
24:03In the population.
24:07Similarly, if we had two
24:09correlated normal variables,
24:10when we this is, you know,
24:12we could imagine getting the corners of this.
24:14These are all principles, by the way,
24:17straight up from experimental design.
24:18If you think if you think about it,
24:21there are principles from like 2.
24:23You know two factor studies or
24:25multi factor studies where you're
24:26manipulating and instead I'm just saying
24:29instead of manipulating these factors
24:31were now measuring these factors.
24:32Someplace you could choose them
24:34to be extreme design points.
24:36It gets a little harder once
24:37things become correlated,
24:38so when they become correlated,
24:40I don't have as much sample available
24:42to me because there's just fewer
24:44population units in those corners,
24:46and so it's going to become
24:47increasingly hard as I add variables,
24:49it might become harder and harder in
24:52order to figure out what these units
24:54are that I could be sampling from.
25:02So I started thinking about
25:04how you would do this,
25:06and I realized that there is actually
25:08a literature on this in in the world
25:11of sort of industrial experiments
25:12and industrial experiments,
25:14and in psychology people again are
25:16thinking about multi factor studies.
25:18So they're thinking about things you could
25:21better in the experimenters control.
25:25But we could instead bout sampling in
25:27the same as the same kind of thing.
25:30Except that we don't have control
25:32over manipulating them.
25:33We can find these units and as as
25:35an alternative approach,
25:36so one of the things we want to
25:38do is we want to make sure that
25:40we observe the full range of
25:43covariate values in the population,
25:45so it requires us to actually think,
25:47you know,
25:47explore the population data and
25:49make sure that we can understand
25:51what that range of values is.
25:53We might need to think carefully about
25:55moderators that are highly correlated.
25:57It can be very hard to D alias these effects,
26:00so if you have two highly correlated
26:02moderators. I think about that.
26:04I have two highly correlated
26:06moderators like this.
26:06If I want to estimate and
26:08understand moderators of X,
26:10if I want to explore X&Z and
26:11these are highly correlated,
26:13I'm going to really need to make sure
26:15I have those off diagonals that are
26:17kind of more rare in order to help me
26:20separate these effects an understand
26:22the unique contribution of each.
26:23The other is that if we might
26:25have many potential moderators
26:27that we're interested in,
26:28and so we're going to have to
26:30anticipate this in advance and think
26:32carefully about sort of compromises,
26:34we might need to make here.
26:36But also think very carefully,
26:37like we're not going to be able to expand
26:40this study to have a much bigger sample.
26:42So a lot of what I'm trying to
26:44operate under the constraint here is,
26:46let's not change the sample size
26:47if we don't change the sample size,
26:49but we instead change the height
26:51types of units in our study,
26:52how much better can we do?
26:55OK,
26:56so this leads to a principle found
26:58in response surface models called D
27:00optimality and so AD optimal design.
27:03This is work from the 40s and 60 Forties,
27:0750s and 60s.
27:08A lot of work here by Walt Kiefer,
27:11and a lot of people in
27:13industrial experiments.
27:14The idea is that you can instead focus
27:16on the generalized variance an you want
27:19to minimize the generalized variance,
27:22which is the determinant.
27:24So D is for determinant.
27:26And so the design that meets
27:27this criteria is one that also
27:29conveniently minimizes the maximum
27:31variance of any predicted outcome
27:33based upon these covariates.
27:34So this is great if what you're
27:36headed for is trying to make predict
27:39individual treatment effects or
27:40site specific treatment effects.
27:43The nice thing about a method
27:44that's been around for a while
27:46is that there's been algorithms
27:48developed for doing this.
27:49Better out Federov win algorithm
27:51is widely used and variations
27:53of it and that these are package that
27:55there are like statistics package
27:57already available that do this.
27:59So in our there's something called
28:01the ALG design package that is set
28:04up to actually work through this.
28:06So designs that we know are optimal.
28:08In other contexts.
28:09You know like our designs like Latin squares,
28:12designs etc all become special cases of this.
28:15So this is a much more general framework
28:19that doesn't require as many assumptions.
28:22OK, so once you start down this
28:24path you realize too that there are
28:27some tradeoffs here, so we have.
28:29You can easily imagine that the design that
28:32is optimal for an average treatment effect,
28:35which might be a representative sample.
28:37That sort of like a miniature of the
28:39population on covariates is likely not
28:42optimal for some of these standardized
28:44effect size differences where we might
28:46need to oversample in order to estimate,
28:49estimate, estimate these,
28:50and so there's another.
28:52Benefit of this approach,
28:53which is that you can focus on
28:56augmentation approach and what that
28:58means is you can actually say using
29:00these algorithms better billable 30
29:02sites or already for I've already got
29:0530 design run so the language of this
29:08is these sites become designed runs.
29:11And I need to select 10 more.
29:15Meaning population units, what?
29:1710 units can I augment it with that will
29:20improve that will make this as D optimal
29:23as possible given these constraints,
29:26and so instead so we're thinking
29:29of population units as possible
29:31design runs and sample as design
29:34runs that we've chosen to use.
29:36OK, so I'm just going to go through
29:40an example to talk about this.
29:45Don't have a ton more slides
29:47I should say so success.
29:49OK, so here's an example.
29:51The success for all evaluation was
29:53an elementary school reading program
29:55evaluated between 2001 and 2003.
29:57The reason I like to use this example.
30:00Is that it's old enough that strangely,
30:02they actually published in their
30:04paper a list of schools they
30:06actually named the schools in their
30:08study and characteristics of them.
30:09I have other data on other studies
30:11where people have shared with me
30:13the names of the schools involved,
30:15but it's all like I have to keep it
30:17secret for the for IR be reason,
30:20so that the fact that this is
30:22available makes it easier to use.
30:23So what I did is I went back
30:25and looked at the Common Core of
30:28data I identified based upon the
30:30study that they were.
30:31In the way that they
30:32talked about their study,
30:34that Title One elementary schools
30:35in the US at that time might be a
30:38reasonable population to think that
30:40they were trying to sample for.
30:42Title one schools have at least 40%
30:45students on free or reduced lunch and
30:46meet a few other characteristics and
30:48then they identified in the paper 5
30:51variables that they thought were possible.
30:53Moderators that would be really
30:55important to include here,
30:56so they talked about total school
30:58enrollment being a factor,
30:59racial and ethnic composition
31:01of the students.
31:02So I'm using that here as the
31:04proportion of students that are black
31:06and the proportion that are Hispanic
31:08and SES meaning and a professor but
31:10proportion at free and reduced lunch.
31:12And they also talk about Urbanicity
31:14because they tried to make sure they
31:16had some urban schools in rural
31:18schools and some other schools.
31:19So I should say in previous work of
31:21mine I've used this as an example
31:23and then this study actually ends
31:25up being a fairly representative
31:27sample of the population,
31:28which is interesting.
31:29Is it because they had no real way
31:31of they weren't doing it totally in
31:33a way that allowed them to compare
31:35this or to choose this in a way,
31:37but they did a lot of work to try
31:40to be representative,
31:41and this is much more representative
31:43sample then.
31:44I take the modal study is in this domain.
31:47OK,
31:47So what I did for for this example
31:50as I'm comparing for you the actual
31:53sample that they selected,
31:55so it's always these five moderators.
31:57The actual sample selected a
31:59representative sample selected.
32:00If I instead I use something like
32:03stratified random sampling.
32:04The optimal sample based upon these
32:06five covariates using this ALG
32:08design package and then various
32:10augmentation allocations.
32:11And So what I would do here as I'd say.
32:16So if I if I took 41, you know.
32:19So if I used 36 sites that were
32:21selected with random sampling with
32:23stratified random sampling and
32:25then I reserved five of them that
32:27were selected using D optimality
32:28and then I would change, you know,
32:30the number of those.
32:32So you could see this sort of effect.
32:34You know that augmentation would
32:36have and then for each of these I
32:38calculated a few different statistics.
32:40So you can see how this works.
32:42So one of them is D.
32:44This measure of the optimality.
32:46And I'm going to show you relative
32:48measures because it's a little easier
32:51to see with relative measures.
32:52I'm also including B,
32:54which is generalizability
32:55index that I developed.
32:57It ranges from zero to one and one means
32:59that the sample isn't exact miniature
33:02of the population on these covariates.
33:040 means they like are completely
33:07orthogonal to each other and.
33:10Chip in its the index is highly related
33:14to measures of undercoverage and how and
33:18the performance of reweighting methods.
33:21And then the mean are meaning the ratio
33:24between the the ratio between the
33:27standard deviation in the sample and
33:30population across these five covariates.
33:34OK, so this is what we get out of this,
33:36and so I just want to talk through this and
33:38I'm happy to answer questions if there's.
33:40I know there's a lot going on here.
33:43Really wish I could figure out how to do a.
33:47Pointer.
33:49I don't think I can point out that way.
33:52OK, so OK.
33:52So what I have going on here is the number
33:55of sites randomly selected is left to right?
33:58So on the left is the
33:59is the D optimal sample,
34:01meaning the whole all 41 sites
34:03were actually selected using a
34:05D optimal algorithm on the.
34:07Right is the ideal for the
34:09average treatment effect.
34:10We've used random sampling to stratified
34:13random sampling and just like the
34:15the sample an in the bar right right
34:17there that like right up there.
34:19This Gray vertical bar is the actual
34:22study values for each of these.
34:24OK so you can see the actual study
34:26and then what I've got are three
34:29different lines going on here.
34:31So one line that's sloping down in
34:33solid is the relative D optimality value,
34:36so this is.
34:37You know the highest value is if
34:40it was a D optimal allocation.
34:42This is a ratio,
34:44and then I've got the B index,
34:46which is the generalizability index.
34:48Is the other solid line going up,
34:50and so, not surprisingly,
34:52that's increasing as we get
34:53to stratified sampling,
34:54so these are going in opposition
34:57to each other.
34:58Is what I'm saying and then this
35:00relative average standard deviation.
35:01Is this dotted bar line?
35:03So what so the main message of
35:05this is that these are going in
35:08opposite directions right that?
35:09The the sample that is optimal for the
35:12average treatment effect is on the right.
35:14The sample that is optimal for
35:16moderate are effects is on the left,
35:18and so there's there's tradeoffs
35:20involved in these that what's best
35:23for one is not best for the other.
35:25But there's other lessons in here,
35:27wow, so the the B index is,
35:29which is a measure of similarity
35:31between the sample and population,
35:32is actually not that bad
35:34for the optimal sample.
35:35So these these the sample is
35:36different from the population.
35:38You'd have to do some re waiting,
35:40but it wouldn't be a tremendous
35:41amount of re waiting to be able to
35:44estimate the average treatment effect.
35:45And so one lesson that you
35:47could think of it from.
35:48This is if you actually if we designed
35:50randomized trials to test moderators,
35:52we'd actually be in a pretty
35:54good space to test moderators.
35:55And to estimate the average treatment effect,
35:57it wouldn't be that far off.
35:58It wouldn't be.
36:00It wouldn't be terrible,
36:01and that makes sense because we're
36:03covering so much of the population
36:06by getting her across a bunch of
36:09moderators that we can do so that
36:11we can re wait when in a domain in
36:13which there's no act extrapolations,
36:15we have positive ITI we can re wait next.
36:18Another sort of I think finding here is
36:21if we look over at the right hand side.
36:24If we do, you know the trade off is.
36:27If I do select for the average
36:30treatment effect.
36:31I do get a tremendously,
36:32you know I can select for the average
36:34treatment effect and do pretty
36:36well for the average human effect,
36:37but not do so well for that.
36:39For the moderators,
36:40and so what's ideal for average
36:42is definitely not deal for
36:43the moderate are tests.
36:44So and then the third thing would
36:46be if you look at the actual study.
36:48As I was saying,
36:49they actually did a pretty good job
36:51in terms of representativeness.
36:52You can see that that top dot,
36:54but if you look at the bottom
36:56at the other two dots you can
36:59see they didn't do so well for.
37:01Being able to test these these moderators.
37:07OK, so in case that was not intuitive
37:09another way you could look at this
37:11is to actually just look at what
37:14these samples these these features
37:16of these samples would look like.
37:18So in the top the top row here
37:20are population distributions.
37:22Of these five covariates that
37:24were sort of identified,
37:25and then at the bottom row is
37:27actually the study that they had.
37:29So what their actual sample looked like.
37:33And then the middle is what AD
37:35optimal sample would look like.
37:36And then I've overlaid on here.
37:38These are values,
37:39so giving you a sense if R is greater is 1.
37:43It means the sample is like the same
37:46standard deviation as in the population.
37:48If R is greater than one,
37:50it means I've got more heterogeneity
37:52in my sample than in my population,
37:54which improves my ability to
37:55estimate moderate are effects.
37:57And So what you see are a few things.
38:00One is in that the optimal sample is.
38:02It pushes things towards the extremes,
38:05right?
38:05It's pushing them towards the
38:07extremes to get endpoints which we
38:09know from basic experimental design,
38:11improved abilities.
38:12The other nice thing though,
38:14is a concern always when you're
38:16doing experimental design like this
38:18is that you're going to get your
38:20highly focused on like a linearity
38:22assumption that you're going to your.
38:25Your ideal sample would have a
38:27strong linearity assumption to it,
38:29but because you have multiple variables an
38:31because not all design runs are possible.
38:34In the population,
38:35you end up with these middle points
38:37as well so you don't end up with
38:39only things on both extremes.
38:41You end up with some middle points
38:43which allow you to be able to estimate
38:47nonlinear relationships as well.
38:48Me and a Third Point with me.
38:50You can see that you would just end
38:52up with a lot more variation and so
38:54not surprisingly, total students,
38:56which, again schools studies,
38:57tend to over represent very large
38:58schools and large school districts.
39:00You can see this is a place where
39:02there would be really a real
39:04opportunity for a change that in
39:05the sample this was less than one
39:07an in the in the optimal sample
39:09it would be greater than three.
39:11But you can see this for most of
39:14these variables that you could.
39:16You could potentially improve your
39:17power and ability to estimate things
39:19related to demographics as well.
39:21And in my paper I actually show that
39:23because many of these are proportions,
39:26you can actually also think about
39:27student level moderate yrs because
39:29proportions conveniently like the
39:31variation in proportions at the
39:33individual level as a function of
39:34the proportion at the aggregate.
39:36And so you can actually kind of workout
39:39a way to select your samples so that you can.
39:42Estimate individual affects,
39:44not just cluster aggregates
39:47for those variables.
39:50OK, and so then the final point.
39:52I just want to make is that the
39:54other thing that this shows is that
39:57there's real benefit to augmentation
39:59so. Maybe? You know,
40:00maybe I'm not going to be able to
40:03convince people to go switch to selecting
40:05their samples based upon extremes.
40:07But maybe you can convince people
40:10that they could preserve 5 or 10.
40:12You know 10% or 25% of their
40:15sample for D optimality.
40:16So you choose.
40:17In this case it would be like choose
40:1930 of your sites using stratified
40:22sampling to represent the population,
40:24and then look for like an additional
40:27class tenor 11 sites that might be
40:29more extreme that allow you to make
40:32sure that you can estimate these.
40:34These moderate are effects
40:36that you're interested in.
40:37And you can see that doing so key
40:39file with these little lines you can
40:41see that doing so doesn't have a huge
40:43effect on the average treatment effect,
40:44but it does greatly improve
40:46your ability to test moderators.
40:50OK, so just to wrap up my take home
40:53points today, I suppose would be that
40:56the design of randomized trials has big
40:59implications for ability to generalize.
41:02And that I think we, I think what I've
41:04seen over time is that people who are
41:07starting to pay attention to that,
41:08and they're starting to think
41:10about how populations you know.
41:11What are the populations I would
41:13add as a side benefit of this is
41:15I've I've watched as people in
41:17asking people to scientists to think
41:19about what the population is.
41:20It actually sometimes make some change
41:22with the intervention is because you kind
41:24of have to realize like is this is this.
41:27If this is the population,
41:28is this the right intervention?
41:31The second sort of point I would say,
41:34is that if we want to sort of estimate
41:36and test hypothesis and moderators
41:38that we would be wise to actually
41:41plan to do so and to think about how
41:44to have better design sensitivity
41:45and statistical statistical power for
41:47doing so instead of waiting until
41:49the end and then the last point is
41:51just that this augmentation approach
41:53indicates that we don't have to
41:55be perfect at this like that,
41:57we could just, you know,
41:58use do this for part of our sample.
42:01And we would be better off and then
42:04I guess I would say maybe my general
42:06philosophy in all of this design is that.
42:08What I'm trying to do is to get people
42:10to think differently and plan differently,
42:13and by doing so,
42:14even if you don't succeed 100%,
42:15you're better off than you would
42:17have been before,
42:18and you're now able to be in the
42:20realm in which you have positive
42:22ITI and heterogeneity,
42:23and you're able to actually
42:25use statistical methods.
42:26To get better estimators at the end.
42:31Thank you, this is all my contact
42:33information and this is the paper
42:36that this talk is really about.
42:38I'm happy to answer questions.
42:41Thanks so much. Best,
42:43I think that's really nice talk and
42:45thank you for being so inspiring.
42:47And maybe let's open to questions
42:491st to see if we have any
42:52questions from the audience.
42:55If so, please speak up or, you know,
42:58send a chat. Either one is OK.
43:07And if not, I can go first.
43:09'cause I do have a couple of questions.
43:13So, so first of all, I think you
43:15know there is a constant tension.
43:17Of course, like you know when we work with
43:20really large trials in the healthcare system,
43:23I think there is a tension between how do we
43:26better represent the population of interest?
43:29Because we want to get effectiveness
43:31information 'cause we're
43:32spending millions of dollars.
43:33But also I think there is a concern
43:36on you know how to really better
43:38engage these large clusters,
43:40large healthcare systems or large clinics,
43:42etc. And so I think.
43:44People end up getting convenience
43:46samples because that's reality.
43:47Even though I do believe that there's
43:49so much more to improve because
43:51they're spending so much money right.
43:53And then in the end,
43:55you know they may be answering a
43:57different question if they have a
43:59very highly selected sample and then
44:01people also worry about you know,
44:03like you know there are some
44:05disparities in their sample selection,
44:06so that you're basically not covering
44:08you know people with maybe more
44:10vulnerable conditions etc in your study,
44:12but you wish to answer questions.
44:14What is population?
44:15So I feel like all of this very,
44:18very relevant, at least to my work.
44:21And so I really appreciate you know this
44:24aspect of how to design styles better.
44:271 one of the questions I have
44:29is that generally,
44:30you know we may not really know priority
44:33what the effect modifiers are in planning.
44:36The trial that we may have
44:38not enough knowledge amount.
44:40So how does that generally come into
44:43the discussion in the design stage?
44:45Is it the tradition that in educational
44:48studies we have a lot of prime knowledge
44:51on what these effect modifiers are
44:53or? No, so I think this is actually
44:55one of the hardest parts, right?
44:57Like I just laid out.
44:58Sort of, if we knew what the
45:00why zeros and Y ones were,
45:01this is what we would.
45:02You know this is that would be optimal,
45:04but I could be wrong on
45:06what those are, right?
45:09And I don't know.
45:10I mean, I think so.
45:12There's sort of what I call the
45:14usual suspects in education,
45:15which are like race class and gender,
45:18which are really more of concerns
45:19about disparity or about closing
45:21achievement gaps in various ways.
45:23And so those in depth and
45:24urbanicity I would add seems to be
45:27something that people often like.
45:28What add into that as characteristics.
45:30Those are the ones that
45:32people most often use.
45:33But the and those are
45:35available in population data,
45:36which is the other thing
45:38that your limit your.
45:39A real limiter is what is available
45:41in the population, sure.
45:43What I gather is more likely to be a moderate
45:46are or something like baseline achievement,
45:49right?
45:49So if my outcome is achievement then I would,
45:52I would think that what the
45:55achievement is baseline in any
45:56of these places would matter.
45:58That's harder to get an education.
46:00I mean that information from places,
46:02so there's been some work trying to
46:05equate tests across across states.
46:06I guess that they do.
46:08Sometimes
46:08they use gain scores just to subtract off
46:11that baseline achievement, right? They
46:13do. Yeah, exactly,
46:14but the problem is that like if you
46:16wanted to use state tests or something,
46:19there are different tests in every state,
46:21and so there's all of these
46:23equating issues that go in with it.
46:25My guess is that implementation is another
46:27one that people often come up with is
46:29like something with implementation.
46:31Now this is tricky because implementation
46:33is coming after assignment and
46:34so it's really like a mediator.
46:36But if you think about often,
46:38if you think implementation
46:39may be part of what is leading
46:41to treatment effect variation,
46:42then you can kind of think well what.
46:45Affects implementation and so
46:46people can sometimes think a little
46:48more carefully about what affects
46:50implementation like Oh well,
46:51it's probably.
46:52You know, it's probably easier to implement
46:54this in schools that are like this.
46:57Then schools that are not like that.
47:01You might try to find various
47:03measures of this for the
47:04implementation that sounds more like a.
47:08It's sort of a version of multiple
47:10treatments, and it's a violation of
47:11the suitable condition, probably.
47:12Yeah, yeah, exactly yeah.
47:13So I mean so it it gets.
47:15It gets tenuous. Yeah, I don't.
47:17I don't have this is, you know,
47:19this is like I when I first started
47:21doing this work I was like,
47:23well assuming moderate yrs and assume a
47:25population moving on as a statistician.
47:26But actually those are the two
47:28hardest things when working with
47:29people in planning these trials
47:31is thinking about what they are.
47:32I'll give you an example though.
47:34Uh, like a positive case which was.
47:37I was part of designing something called
47:39the National Study of learning mindsets,
47:41which is we randomly sampled
47:43100 high schools in the US,
47:45and then we randomly and then the students.
47:48There were.
47:49Ninth graders were in the study
47:51and so 9th graders were randomly
47:53assigned to either using a computer
47:55based intervention to a growth
47:57mindset intervention or something
47:59that was not growth mindset that was
48:02just sort of control condition and.
48:05And in doing that we had the social
48:07psychologist I was working with had
48:09a lot of questions like we had a
48:11lot of hard questions about these
48:13moderators and they had a lot of
48:15theories about what they might be like.
48:17So we oversampled like we.
48:19Looked at for example,
48:22proportion of students that are
48:25minorities in the school and then.
48:28And when we started we wanted to
48:30stratify on that as well as school
48:32at a measure of sort of school
48:34achievement as well,
48:35and so we needed to be able to
48:38cross these in a way in order to.
48:41In order to D alias these trends and so
48:43that they could estimate one without,
48:46you know,
48:46without estimating with separated
48:48from the other. So a lot of it.
48:50So I mean,
48:51in some places people are much more better,
48:53much better theoretically.
48:54Thinking about this,
48:55I think some fields are better at
48:57thinking about these mechanisms than
48:59other fields are, but yeah, it's really hard.
49:02It's really hard.
49:02So my my other other than my
49:04like standard set, you know,
49:06race, class and gender is,
49:08I often ask people to to think about.
49:11Watch what variables might just be
49:13related to other things, right?
49:14That if you could.
49:15If you can think of it as like I
49:17ultimately want to test moderators
49:19that I don't really know exactly
49:21what they are,
49:22but I need to get variation in them,
49:24and that means probably by getting
49:26variation in something else.
49:27I'm going to get variation in those as well,
49:29so.
49:31The size of your site, you know,
49:33I think,
49:34is one place where you know an education.
49:37You can see that everybody's in
49:39very large sites.
49:40And So what if we increase the variation?
49:43The variation of district size
49:44and school size?
49:45It seems like has to increase
49:47variation of some other things as
49:49well. Agreed, agreed.
49:50Yeah, I think another aspect why I so
49:53appreciate like the aspect of effect
49:55modifiers is that it really is a way to
49:58move forward with information from Co.
50:00Buryats and then when we talk
50:02on 80 in a randomized study,
50:05we often ignore covariates and
50:06then just hold that the unadjusted
50:09analysis provides unbiased estimates,
50:11even though that may come with
50:13a larger variation.
50:14So by really talking about effect modifiers,
50:17we somehow incurve those information,
50:19but perhaps even in the estimation
50:21of the average affect,
50:22which can increase precision.
50:24So yeah.
50:27Yeah.
50:30Yeah I haven't questioned,
50:32so I actually have two questions,
50:35so seems you're you're interested
50:37in both individual level
50:39an cluster level moderators right? When
50:41you have cluster level moderators,
50:44how does that work with
50:46the augmentation design?
50:47'cause you mentioned that
50:49in the orientation design,
50:51you might want to pick like.
50:5410 or 30% of the sides.
50:56An kind of like choose
50:58them samples from those.
50:59But how do you choose those 3%?
51:02You choose those third percent with
51:04respect to the cluster level modelers.
51:06You could do it with respect to either.
51:09You can do it with respect there
51:12because it depends the way you
51:14enter them into the model so.
51:18You work out so you can work out that if
51:20if I'm interested in the individual level.
51:23Moderate are that that what I need
51:25to do is I need the I actually
51:27need to include as a covariate the
51:30interaction between like X an 1 -- X.
51:32That's what I included here
51:34as the covariates.
51:35I'm 'cause I want to increase the
51:37variation within sites right?
51:38And so you could do it either way,
51:41because what it's doing,
51:42what the augmentation approach does?
51:44Is it assess is how much variation
51:46you have in those 30 sites already.
51:49And then it looks for possible design runs,
51:52meaning other samples.
51:54Other places that would greatly improve that.
51:58And it just it doesn't algorithmically,
51:59which is nice.
52:00The that I would say I should add an
52:02extra benefit of this is concerned
52:03with all of this sample recruitment
52:05is that there's non response.
52:07You're never going to get,
52:08you know it's not like I can just say like.
52:11Here's your.
52:11Here's your like 40 sites go ask
52:13them and they're going to say yes,
52:15but with the augmentation approach if
52:16somebody says no you can like throw
52:18that out and then go look for it
52:20like what's the next best alternative
52:21so you can keep kind of iterating.
52:24So,
52:25so in our current application,
52:26I think the attributes are all cluster level.
52:30Information right summary statistics
52:31yeah yeah, well that's what
52:33I have right here.
52:34That's in the in the slides but
52:36I didn't include in here though
52:38is like you could it's but it's
52:40in the paper is you could also do
52:43this with individual level only.
52:44For proportions mean because just because
52:46the proportion works out that you can get.
52:48You can think about this with
52:50the same statistics you would
52:52get at the cluster level.
52:53You can't get the variation you can with
52:55a normal like a continuous variable.
52:57I can't get the.
52:59I don't have the standard deviation.
53:01Insights I can't do that,
53:03right, right?
53:05Also, the other question
53:06is about so it seems
53:08like all these designs are
53:09under the assumption that you're
53:11interested in all the moderators,
53:13like equally like meaning that you're
53:15not like you don't have like primary
53:17moderators that you're interested in
53:19estimating the moderate effect on and
53:20then you have a couple of them that.
53:23I mean, if you you
53:25can. So I mean, what's great?
53:27I mean, I think about this like this area
53:29is that it's been so richly developed
53:32in this other sort of design runway
53:34is that you can actually add weights.
53:36So you can say like I'm more like or
53:39more interested in this variable than
53:41that variable, and it will focus.
53:43You know it will focus on one
53:45variable over the other.
53:47Because you Can you imagine like
53:49that ask like that D matrix.
53:50The determinant of S.
53:52You could just add weights into that.
53:54So if you add weights into that
53:56then you can start looking at the
53:58determinant of that weighted version.
54:00Right, so you would add weights
54:02in that matrix and optimize that.
54:03Yeah exactly,
54:04if you add weight so that some
54:06of the Kobe rates are getting
54:07more weight than others.
54:10So I guess just maybe more precisely,
54:13I think the D optimality criteria.
54:16Shouldn't that be the X
54:18transpose V universe in general?
54:20Just because you're working with
54:22clustered randomized studies so
54:23that the outcome correlation is
54:25somehow included in that variance?
54:27Is that what the algorithm is
54:29trying to get in general for?
54:32Yeah yeah.
54:34Inverse, yeah, it's the X prime X inverse,
54:37which is the covariance. Yeah,
54:38but but really not so it you don't.
54:41You don't need to have the
54:42variance matrix of the outcome.
54:46Exactly, you don't need to have the outcome,
54:48it's all about the inputs, right?
54:50But that's, which is why you
54:51can do it in advance, right?
54:53So it's all about the Android just nicely.
54:56You can leverage population
54:57data that you have totally.
54:58And again, I assume in all of this
55:01that like there's measurement error
55:02and that you know you can just
55:04sort of assume that like you're
55:06not going to get it exactly right,
55:08but my baseline comparison is always
55:10what are we doing now versus what could
55:12we be doing an like frankly anything.
55:15Any you know it looks to me like we
55:17have fairly homogeneous samples and
55:19that any effort we can make to increase
55:22that heterogeneity is an improvement.
55:28So, well, I think we're about the hour,
55:31but let's see if we have any
55:33final questions from the audience.
55:40Alrighty, if not, I think you know I'm,
55:43I'm sure if you have any questions
55:45that petition will we have to
55:47answer them offline by email?
55:48So thanks so much. Again, bath.
55:50It's really nice to have you and thanks to
55:53everybody for attending or see all of you.
55:55Hopefully after the break so have
55:58a great holiday. See you later.
56:02Totally not master connect
56:04alright, thanks again.
56:05Talk to you later. Bye take care.