HERC Economics - 14 -Department of Veterans Affairs

Right-Hand Side Variables

**Transcript of Cyberseminar**

**Department of Veterans Affairs**

**HERC Econometrics with Observational Data**

**Right-Hand Side Variables**

**Ciaran S. Phibbs**

**June 6, 2012**

Ciaran S. Phibbs:    So today’s lecture is focusing on right-hand side variables or the independent variables in a regression model.  And the idea behind this particular lecture is that regression models make a fair number of assumptions about the independent variables and the purpose of the talk is to examine some of the more common problems, some of the methods for addressing them.  The focus is, well, there is some overlap and many of these issues will have been at least touched in the standard econometrics class; the idea of this is to go beyond some of the items that are covered in a standard econometrics class.  And the four things that I’m going to address are heteroskedasticity, clustering of observations, functional form, and the testing for multicollinearity.  Those are all things that you may have heard in a class, but the standard masters level graduate class doesn’t address all of these in the detail that one really needs to proceed.  

And starting with heteroskedasticity, the key regression is that the error terms are independent of the outcome.  And a common pattern of heteroskedasticity is that as X gets bigger, your error gets bigger.  Think of your regression model where you’re trying to predict, you know, take the simple economic theme where your trying to predict consumption behavior and one of your explanatory variables is the person’s income and error in terms of consumption gets a whole lot bigger as the possible errors for the individual observation get bigger as income grows.  That’s the, sort of, classic format.  And your standard class will have taught you that when you have heteroskedasticity error terms, that your parameter estimates are unbiased, they are somewhat inefficient, but you get accurate estimates in the parameter, but your standard errors are not correct.  And the issue is what do you do about it?  

For those of you who are older, you may remember that the old method was you had to try to do some sort of transformation of the variables, but there’s now a – Hal White published a paper in econometrics in the early ‘80s that came up with a relatively simple way to generate robust standard errors that Stata uses and there’s another method it’s called the Hubert-White Correction Method to Correct Standard Errors.  It’s in Stata for virtually every command you can ask them to generate robust standard errors.  In other packages it may be more difficult, but it is possible to get corrected standard errors for essentially this.  

So that the method comes around, one might also consider, you know, what is causing this heteroskedasticity and do you really have the right model?  And we’ll consider this again in terms of the functional form, but to take the example that I talked about with income and consumption, you might ask, well, should I be modeling income in a linear manner?  Maybe I should be using a log of income instead of just income as a right-hand side variable.  And those types of transformations because it’s a nonlinear transformation made also reduced the extent of the heteroskedasticity.  And it just want to make the point that when this is present, that some to these things I’m talking about like multicollinearity and functional form and heteroskedasticity may be interrelated in terms of what you have for your functional form and do you have the right functional form?  And even though, you know, the robust option provides this automatic way of fixing heteroskedasticity, you might want to consider some of these other options in terms of how you specified your model.  The real point here is that in this day and age you should be running robust standard errors for everything you do.  And failure to do that, well, it will be caught in economics journals, it is not always caught in all other journals, but it’s something that you should be doing.  

The second thing I want to talk about, and we’re going to talk about this in a lot more detail, is clustering.  And what clustering assumes is that regressions assume that the error terms are uncorrelated here, these εi’s, the [GEE] individual error terms are not related to each other.  And that is something that in healthcare observations frequently not true.  And we frequently have situations where patients are clustered within groups.  A common example is if you’re running a patient level regression for inpatients, the patients are clustered within hospitals.  And to continue this example, just for simplicity, I’ve got a two-variable model where X1 is a patient level variable, and X2 is some characteristic of the hospitals where those patients are treated.  Now in reality, obviously, you’re probably going to have, at least in terms of X1, you’re going to have a whole vector of patient characteristics, you could have more than one hospital characteristic.  

If I just run this model “as is,” the regression model is going to assume that there are as many hospitals as there are patients.  That’s, you know, essentially an assumption.  And, remember, as you increase your sample size, your standard errors go down.  And, therefore, the standard errors for β2 are going to be too small because there aren’t as many hospitals as there are observations and, therefore, the standard errors are too small.  There is no effect on the parameter estimate.  The parameter estimate is totally unbiased and if you fully correct for clustering, as I’ll address later, there is absolutely no change in β2, all that’s changed is the standard error.   . 

Moving forward, there’s a variety of ways for correcting for this clustering.  And before I go into this, that is a really common problem in healthcare that we look and in example of patients within hospitals, you’re looking for neighborhood effects, and you have neighborhood variables.  And individual variables, you can have patients within physicians, within practices, there’s all kinds of different clustering type structures, and Generalized Estimating Equations will correct the standard errors.  There are formal hierarchical models that can be used and, depending on what you’re looking at, it may with more appropriate to look at a hierarchical structure because it could be that something very different is going on at the patient level versus the hospital level to continue that example of clustering.  

Stata, for virtually every regression command, has a cluster option and it will use a version of this Huber-White correction to correct the stand errors.  It corrects that standard error so that you don’t have this artificially small standard errors.  The Generalized Estimating Equations type solutions versus the Stata Cluster option yield essentially the same answer.  There are small, minor differences in the formulas.  Edward Norton dug into how Stata corrected it versus how [Sue Dan] corrected the standard errors for clustering and it came down to a difference between dividing by N for your sample size versus dividing by N minus 1; so essentially the same answer.  But you can get different answers with hierarchical methods compared to one these methods for correcting for clustering.  They can be very similar, they can be different; it depends on the actual structure of the data.  

Hierarchical modeling, I’m not going to talk about in terms of examples, I’m just going to mention it because that’s really a different topic that really needs a whole lecture to do it.  It’s a method of formally incorporating the hierarchical structures; you can use it for nonlinear models as well.  And I just want to mention that the need for using a formal hierarchical model versus these other methods of correcting the standard of errors, really depends on the structure of the data.  And if you have a structure where something really does matter that you need to use the more formal structuring of the a hierarchical structure versus just correcting for the statistical issues, often the answers are very similar.  One other thing that I will note is that the similarity of results and the robustness also depends on the size of your sample.  With smaller samples, things may be much more sensitive to using an HLM versus another model just because of the smallness of the sample.  And if you have big samples, lots of data will overcome some of the limitations of these other methods when you really have a hierarchical structure.  And then there’s other cases where it’s just a statistical issue and not a hierarchical structure.  

I want to work through an example of clustering and I’m going to apologize that this is not a VA example, it’s actually a newborn intensive care example, and I’m using this because it is a very good example and I’ve worked through all of the issues relating to this.  And when I was looking at the effects of patient volume and NICU level on mortality, and when we looked through this, the point is is that this is a correction, something that I see in the literature all the time when people haven’t corrected for this.  And it’s becoming less common as more people find out about it, but it used to be years ago that no one was making this correction, and it’s easy to fix.  With big samples, the bias is relatively small.  With small samples, the bias can be quite large.  And it’s not just your aggregate sample size, but it is also the number of clusters, and the number of observations within given clusters.  So it’s sensitive to all of that.   

So the example I’m going to use here, as I’m going to actually show you, some of the effect of correcting for this is pretty big.  I had more than 48,000 observations, there were over 200 hospitals in ten years of data, so there were a lot of repeat observations or hospitals, both over time and within year.  And in this example, I had levels of hospitals, so just for the nomenclature, a level 1 is no NICU, working up to high levels of NICU’s.  And then, also, I had variables by the number of infants treated.  You can see here in this first row, the odds ratio, the parameter estimates aren’t biased, but here you can see that the change in the – this is the correct standard error here, and here’s the unadjusted that doesn’t adjust for clustering, and you can see that they’re not that different.  

But if you jump down a couple rows here, you can see that you can get a different answer.  And you see here in this row here that not only did it change from – the unadjusted said it was statistically significant, the adjusted says it’s not statistically significant, but the changes in the standard errors here are at least moderate.  One of the things that I will note in terms of the sensitivity here is that this particular cell was a cell with very few hospitals.  So there weren’t a lot of hospitals, so that made it more sensitive to this correcting for clustering, because there weren’t a lot of hospitals in this group, but they were moderately big, and a couple of them were quite big.  And so it is more sensitive to this correction than it was for others.  Again, this is the structure of the data.  

So are there any questions, Patsy, that I need to address? 

Patsy:  No.  But would you explain the odds of what? 

Ciaran S. Phibbs:  Oh, this is odds of mortality.   

Patsy:  Okay, thanks. 

Ciaran S. Phibbs:  Yeah.  The underlying structure of this model, I’ve given a brief explanation of what it is, what I’m really trying to show is what happens in terms of when you fix this.  And the thing that’s really relevant here is this is a big sample.  If I had been running this on one year of data, so instead of having almost 50,000 observations, I had about 5,000 observations, these changes in the standard errors between the unadjusted and the adjusted standard errors would have been much larger because the sample size would have been much smaller.  So lots of data can actually help you fix the problems.  

All right, in the next case I’m going to move on to functional form.  And this is something that I find in the literature a lot and when I’m reviewing papers that people forgot all of the time.  The regression model βX assumes a linear relationship between X and Y.  Now that linear structure, I mean, some of the nonlinear models like logistic, it’s not a formal linear, but you are still assuming linearity here on this side.  And it is quite possible that you’re going to have relationships that are not linear and if you do not address this, you’re going to end up with a mis-specified model.  And, again, we have lots of things that we use as our regressors where the effect of the independent variable is not linear.  Let’s just think of age and mortality.  Yes, mortality goes up with age, but that’s, you know, once you survive infancy, there’s essentially no effect of age on mortality until you get old.  And then it starts to go up slowly, and then it accelerates, and as you get older, that accelerates.  So that’s a nonlinear function.  And if you just put “age in years” in your model, you’re going to have a misspecified model.  

The same thing will hold if you add – The issue is not just that it’s misspecified model, but how should you check for functional form?  Okay, in general, as a rule, for every variable in your model that is not a binary variable, a yes/no type variable, you need to check for the functional form.  I mean, even if you have integers and you have a small number of integers, say I’m looking at the effect of the number of primary care visits where most of them are zero, one, two, and then you have some there, but that would well have a nonlinear effect and one needs to address how to address this.  And there are formal tests of for model specification, some of which you may have been exposed to in class.  

But some of those tests tend to be fairly weak, and they really don’t show you what you’re looking at and what you need to do.  And there is, actually, a relatively simple way that I’m going to go over to actually look at the effects of functional form.  And that is, what you do is you can use dummy variables to show you what the effect of functional form is.  And what you do is to do that you start by looking carefully at the distribution of each variable and create a set of dummy variables with reasonably small integers and no excluded category.  And then you run the model with no interceptors because you have no excluded category.  You can then graph that out, and that will show you what the relationship is for.  What it looks like, and that can guide you in terms of how to create a variable.  

So I’m going to use this same NICU data set before, because I had to address this, and what I’m going to do is while I was interested in how patient volume and the treatment in NICU affected mortality, and I had overall, was within levels of care.  And so by doing this, when you graph this out, you get an idea of what the function gnat form looks like.  And this can be a good starting point for – this can then guide you in terms of how to fix the problem.  When you graph this out, you might look at it and say, “okay, that’s clearly quadratic, I’ll just do quadratic.  It could be that it has some sort of a complex functional form and, therefore, you have to do other things.  And, okay, I graphed this out and you can see it’s relatively steep, and then it tapers off, and then it tapers off some more.  And what I ended up doing – and then also remember I had different types of hospitals that I was interested in, and you can see here it’s bouncing around, here it’s quite steep and then it goes flat, and then here it’s sort of like this, and you see that the groups don’t overlap because the hospitals here, there weren’t any small hospitals in that group and in the blue series there aren’t any big hospitals.    And so this can be a guidance for how, then, to address this issue.  

For some issues when you look at it you can say, “gee, this is a complicated relationship, and instead of using a continuous variable, or a spline variable, I’m just going to break it into a set of dummy variables.  And one of the things is it can be very difficult to get a continuous function, especially when you move away from linearity to get a continuous function to predict accurately across the entire range of the variables.  And it may be that you have to use some sort of a spline function, which is different functional forms for different ranges, or to use a categorical variable.  And all [inaudible] aside, for many situations, instead of having some very complicated functional form, it may be easier to present a set of categorical variables to less sophisticated medical audiences.  

And I’m going to cycle back; what I ended up doing for this, and I haven’t shown all the categories because there’s even more categories than this, is I broke it down into categories by volume, patient volume and level, so I had two things.  So for each level of care, I had different volume categories and I used the regressions that I had run with sets of dummy variables to guide how I created these categories, I’ve not shown you all of them because it’s a great, big, long list and this just makes it simple, but it’s just some examples.  So what I did was I broke it down; I had two different types of categories that I was worried about and just made a whole set of categorical variables to reflect two underlying patterns.  

Any questions, Patsy?

Patsy:  Yes.  And I’m sorry I didn’t see these when we were talking about clustering.  The first one is, “In Yi = β0 + β1x1 + β2x2 + εi –” got it –

Ciaran S. Phibbs:  Okay?

Patsy:  —that one, “what is the effect of clustering on the standard error of β1? 

Ciaran S. Phibbs:  There is zero effect.  Because in this case, β1 is the patient level variable, β2 is the hospital level variable in this simple example.  And the issue that underlines clustering is the reason there is an effect on the standard errors is because the regressions, statistically thinks there are as many hospitals as patients to follow this example.  And because there are actually that many patients, there’s no effect to the stand error on β1, it’s only when there are fewer of that can characteristic than there are observations, in this case hospitals, that you that you get a problem with the standard error.  

You said that there was one more question? 

Patsy:  Yeah.  The second question is, “Since that accounting for clustering would be more important who you have a small number of big clusters, as opposed to a large number of small clusters, is the intuition correct? 

Ciaran S. Phibbs:  It’s complicated and it varies; the relative effects of sample slides versus the clusters.  So in the example I talked about where the biggest change occurred in the data I had was in a cluster where there were actually, you know, it’s the characteristics of the cluster, so it’s not so much the number of clusters as how many observations are that cluster?  To that was an example where, out of the roughly over 200 hospitals, that one cluster represented three or four hospitals only, and with a moderate number of patients.  And it was because it was a cluster that didn’t have very many cells in it, whereas, come of the other clusters represented large numbers of hospitals.  

And there were other hospitals that had moderate numbers, so the correction was small when there were lots of hospitals, each of whom didn’t have that many patients; the correction was small.  And then there was also the case; there was a cluster of 15 or 20 hospitals, relatively small number of hospitals, but they had lots and lots of observation.  So, again, the lots of observations was overseeing it.  So it’s what’s within that cluster, almost as much as how many different clusters.  And you have different things within the cluster; is it a small number of hospitals, small number of observations?  And they all work together and, in general, more is better of them. 

Patsy:  Okay, this is a follow-up to the first question.   “So if we’re not interested in β1, is it necessary to bother with corrections such as Huber-White?” 

Ciaran S. Phibbs:  So by “not interested” so you’re just putting it in as a control variable.  Well, β1 you don’t have a clustering issue.  So to transform that question—this may answer it—is if you were interested in the patient characteristics and you were just putting in hospital controls as controls, and you weren’t making any interpretation of them, so you weren’t making any inference on the statistical significance, then the fact that you didn’t correct for those standard errors really wouldn’t matter because you weren’t making any inferences of them and that parameter estimates are unbiased.  If you weren’t trying to make any inferences on the hospital level variable represented by β2, and you didn’t correct for clustering because the parameter estimate is unbiased, you’re still correct there.  And as long as it’s just a control variable, it’s technically not correct but because you’re not making the inference on it, it wouldn’t be a big deal.  Whereas, for β1 there is no effect because there is no issue in clustering. 

Patsy:  Okay, she’s saying, “So if we are not interested in β2?”  Right?  So you were just answering that. 

Ciaran S. Phibbs:  If it’s just a control variable you’re not going to make any inference off it, then fact that you didn’t cluster it because you’re not making any inference off it which is what the standard error is implying in terms of significance then you’re okay because the parameter is unbiased. 

Patsy:  Okay.  Okay, great.  So just to emphasize, I still see that there are hands raised, and since we can’t actually call on you, please do use the question box on the right hand side to ask your questions.  I have two more questions right now. 

Ciaran S. Phibbs:  Okay.

Patsy:  “I have a five-point [Lycra] scale of clinical variables, i.e.; symptoms, predicting a continuous one, hours worked, can I leave the clinical variables in the model as is, or should I transform the clinical into binary or some other form?” 

Ciaran S. Phibbs:  For that, who you want in terms of, you know, you have applied point [Lycra] scale, and what you want to do is put them as binary – you know, if you just put them in as one to five, you’re assuming linearity.  And you don’t know, especially because those hours worked may be captured nonlinear there.  What you want to do is create a set of dummy variables, and look to see what the functional form is.  And that will tell you whether or not you can assume linearity or not.  The point is for something like this is don’t assume linearity.  Test it to make sure that that’s a reasonable assumption and if it’s not a reasonable assumption, make appropriate adjustments. 

Patsy:  Okay.  And this next one is, again, about clustering.  “If you include hospital fixed effects in this model, would that obviate the need for clustering at the hospital level?” 

Ciaran S. Phibbs:  Okay, hospital fixed effects; what is a fixed effect?  Just a quick deviation here.  A fixed effect is a dummy variable for each hospital, which he essentially pulls out the hospital.  And to do this, you have to have repeated observations for that hospital and what that does is then the parameter estimates represent things as deviations from the mean and you’re pulled out of the main hospital effected.  And therefore you can’t put “time invariant hospital specific effects” in the model.  So you wouldn’t have them in the model, you’d have an all fixed effects model.  You see people doing this where they estimate a fixed effect and they put in time and variant characteristics, and you actually shouldn’t be doing that.  So if you have a fixed effects format, but a fix effects format is a certain structure that is being imposed on the data and one needs to understand that and that’s a panel data issue.  We talked about that some in other lectures and it’s beyond to scope of what we have a time to really address here.  But the bottom line is if you include fixed effects, you shouldn’t be including time and variable characteristics so you don’t have to correct them.  

Now, if you have a time varying characteristic, so fixed effects are traditionally used on panel data models, which you have repeated observations over time.  So if I’m running a regression with – let’s comeback just to format here – I have patient characteristics, I have hospital characteristics, and I have hospital characteristics, and it’s also a fixed effect so I have an additional hospital dummy here.  And so this hospital characteristic that I’m including a one that varies over time is therefore not captured properly by the fixed effect, which is for time and variance effects.  I still need to correct this variable.  So the fixed effect depends on whether the variable is – if you do fixed effects, you don’t have time and variance models and it’s not an issue.  But if you have a time varying parameter that’s in there, so then you do need to correct the standard errors for that variable.  I hope that answers the question. 

Patsy:  I think so.  So that’s the end of the questions right now. 

Ciaran S. Phibbs:  Okay.  So that’s done; let me get back on track.  And we’re going to talk about multicollinearity.  And, you know, most people are aware of this that if X and Y are strongly correlated, essentially what’s going on is if X and Y move together, the regression will have trouble attributing the variance to one variable versus the other, it can increase the standard errors and it can increase the parameter estimate.  It’s one thing in a much more complicated model where there are lots of other factors.  If you have two variables that are very highly correlated, one thing that you can see is that one variable will take on a big negative value and one variable will take on a big positive value as they, sort of, outlay and, really, the parameter estimates get inflated in offsetting directions as they try to sort out how much is the variance.  So you really can get, if [strings] are strongly correlated, you can actually get parameter estimates that are messed up, in addition to standard errors 

Okay.  Strong, simple correlation, you know you have a problem.  You know, if you have two variables that are correlated .9, and you try to put them both into a regression, you’re going to have problems.  But there can be more subtle problems there are a little harder to detect.  The variance inflation factor, which is the /VIF option in SAS, and there’s an also a VIF in the state of regression diagnostics, measures the inflation of the variances in each parameter due to the collinearities among the regressors.  So it’s the variances and the parameters, your standard errors.  It doesn’t tell you what may be going on in terms of these offsetting effects within the parameter estimates.  Some packages will report something called the tolerance, which is just one over the variance inflation factor.  And, in general, a variance inflation factor greater than ten signifies that you have a collinearity problem.  

General rule of thumb is, you know, when you have correlation between available greater than .5, you’re likely to have a correlation problem.  But can still have collinearity programs.  When the simple correlation between two variables is less than five.  So just looking at the correlation matrix is not enough.  You should at a minimum look at the variance inflation factors and see if you have problems.  So let me just give you an example.  In a study I was doing in nurse staffs, where one of the things, because we thought there might with a nonlinear effect, is we put in the nursing hours for patient day, and nursing hours for patient day squared, and those two were quite highly correlated, not surprisingly.  And the variance inflation factors on those, depending on the subset, ranged between 10 and 40.  And, obviously, that’s going to affect the significance of the results.  

Now, in terms of fixing multicollinearity, more observations is always good because as long as there isn’t perfect correlation, additional observations helps especially if you’ve got some variance in it because I’ll help the regressions sort out what to attribute to what ways.  You can revise the data in ways to try and reduce the correlation of nonlinear transformations.  And in that nurse staffing example that I just referred to, and this comes back to, really, how you’re specifying the functional form, is that, you know, we were concerned that the effects of nurse staffing, there was a nonlinear relationship between the number of nursing hours and patient outcomes, and we were trying to capture that nonlinearity.  But when we were doing a quadratic, we were having collinearity problems, and so we addressed that by using a set of dummies to capture the nonlinearity.  And in the process of doing that, we eliminate the collinearity problem. 

 And so, again, I mentioned earlier the fact that the issues of heteroskedasticity and collinearity can actually be tied to how you’re specifying your functional forms.  

So to give you an example of something going on in terms of it not just affecting the significance, but also affecting the parameter estimates.  This same study of looking at nurse staffing on patient outcomes, I believe these are from, like, the effects on patient length of stay, but I’m not positive.  And one of the things we were looking at was the average RN tenure on the unit.  What’s the average number of years a nurse has been working on the unit?  And we were also interested in the age of the nurses and their correlated very highly, .46.  And if we run our regression where we only put in tenure, we got significant effect that more senior nurses was associated be better patient outcomes.  We got a very small and non-significant effect on age.  When we ran the model with both of them, both were not significant and, as you can see here, there was also huge changes in the parameter estimates relative.  I mean, the age parameter here became much bigger.  This one went to almost a zero after having a negative effect and both were not significant.  Again, it’s the variables correlated with each other and it’s mucking up the parameter estimates.  And so you have to be conscious of this when you have this problem that it can affect not only your significance, but also your parameter estimates when you get collinearity.  Because, essentially, when you have variables that are correlated with each other, the regression has trouble attributing parameter effects to each of them.  

And I mentioned before that simple correlation like I just showed, you have problems, but there can be hidden problems that are more difficult to detect.  And think of multiple regression with lots of different variables in the model.  If you have N variables, you’re really dealing in N-space where N is a moderate integer.  And correlation, and the regression is trying to sort out the information in a bunch of different planes, or dimensions and correlation in any one of those planes can actually effect the parameter estimates and cause collinearity problems.  And there’s an option in SAS and I haven’t found it; it may be in there now, but when I looked a few years ago it wasn’t yet in Stata, there’s something called the Collin option in Proc Reg that looks at how much of the variation for each eigen vector.  And each eigen vector is essentially how much of the variations is explained in that one dimension in N-space is explained by each variable?  So intuitively, you’re looking at the correlation within each plane.  And this is for regular old Proc Reg, and you can say, “How do I do it then if I’m estimating a logistic model?  Well, even with Proc Logistic, or any other thing, the X-matrix, which is where you’re looking for this relationship, is going to be the same, so it is fine to use for doing this type of a diagnostic on your X-matrix.  You can’t use this for interpreting it if you’re running a logistic or some other model, but you can use this to test your X-matrix for collinearity. 

 And I’m going to give you a little example here; in SAS, in the Proc Reg model, you have the model and you put your variables, then you just put “/Collin” and I’m going to give you a very simple model that I ran from my newborn data where birth weight and gestational age among [feature references] is highly correlated.  And for this point I had my programmer rerun a very simple model where I only put in birth weight, gestational age, and a dummy variable for if the infant was black.  And when you get this output, and I’m going to show you some output, you know, what does an output look like, one of the things you get is there’s a condition index and, like a variance inflation greater than 10, if you have a condition index on a given variable, you have a collinearity problem and greater than 100 indicates an extreme problem.  And there’s, you know, if when we look at this here, so what we want to look at is to look across here.  So here’s the eigen value, this is my recreated very abbreviated form of what the output looks like.  Then you’ll see the condition index, and so the first one there’s no problem, the second one there’s no problem, this is borderline, it’s almost 10.  And this one here, you clearly have a collinearity problem.  

And it will also tell you that this fourth eigen vector is loading and a combination of a constant, and gestational age with some correlation with birth weight.  And it tells you which of them.  So you cannot only say that what it says that you have a problem between the constant and this gestational age variable in this dimension.  And so it will tell you what the correlation is in terms of the problem, and in what dimension.  

Again, functional form come back in terms of addressing the collinearity problem.  Okay, I have this multicollinearity problem between birth weight and gestational age, and so what I did is again, went back to the functional form to address it.  And, instead of using continue variables, I used dummy variables for birth weight and, you know, looking at the functional form.  I use smaller intervals for the very, very low birth weights, and then, one an infant’s got above a thousand grams, which is 2.2 pounds, I use smaller ones.  And then I use gestation at two-week intervals and in this particular case, we actually used separate birth weight dummies for singles and males, singles and females, and low weight births because there are different relationships there.  And when we did this, so we spit up we had this relationship and we split this up, the maximum condition index was less than 8 and the variance inflation factors were all down to smaller numbers as well.  So we had broken the collinearity problem by transforming the variables to break the correlations by using setting of dummy variables.  And this is one way, when you have serious correlation, you know, so I really wanted, in this case, I really wanted both birth weight and gestation age in the model, they were collinear.  I couldn’t put them in, so I used sets of dummy variables to break the correlation so I could include the factors that I wanted to include in the model.  

And, again, it’s paying attention to the functional form and you actually have to go back and look at the functional form in detail.  And I will note that when I did this that the model predictions also improved.  So you can see here I just put out a graph of what those three birth weight functions look like in terms of this is the dummy variables to fix it in terms of the mortality odds and you can see it in the variables.  And part of the problem, you can see here at the very low birth weights, this is for kids between 600 and 700 grams, that you can see that the odds for males were much higher than they were for multiples or females, to it was appropriate to break them apart by gender.  And they converge as they get bigger; the mortality risk goes down.  But again, don’t just pay attention – pay attention to the underlying structure of data and look at your data carefully and don’t just look at the variables, your interest here, birth weight, because as this graph shows, there was some other factor that with a binary characteristic that I also needed to sort on to get better prediction.  And it all goes back to very carefully looking at your data, looking at the relationships, and then making adjustments accordingly.  

Included in the slides here, there’s a classic text on regression diagnostics Belsley, who, and Welsch that I put it in – it’s a book – as a reference.  And I’ll note that the next lecture is going to be July 11, Jean Yoon will talk.  And I’ll try and address some other questions. 

Patsy:  Okay.  The first one is, “With a multicollinearity problem, can you address standard errors of predictions as opposed to standard errors of coefficients?” 

Ciaran S. Phibbs:  Those are totally different concepts and if you have a problem, both will be messed up.  Because, remember, if you have a serious collinearity problem, not only is there problems with standard errors of that particular variable because it can affect your parameter estimates, your predictions are also going to be biased.  So if you’ve got collinearity that’s significant, you need to address it, and there’s no simple automatic fix like there is for clustering or heteroskedasticity where you can just correct the standard errors.  When you have collinearity, you’ve got to go back and transform you’re data somehow to address the underlying issue. 

Patsy:  Ciaran, would you go back to the reference for the book on it on the slides? 

Ciaran S. Phibbs:  Okay. 

Patsy:  And just mention the references are in the slide deck as well.  The next question is, “Which approach is preferable for nonlinear continuous variables, dummies or cubic spline?”

Ciaran S. Phibbs:   It depends on the data.  You’ve got to go look at your data and see.  You know, one of the reasons that I say look at the set of dummy variables, you know, break it into a set of dummy variables and run the regression with that set of dummy variables, because that tells you more than any formal test.  That will tell you what the underlying data structure looks like, and then you have to go dig.  And as I mentioned with that birth weight example you may need to look at subsets of the data if there are underlying causal reasons to expect that there might be different relationships for different subsets.  And you’ve just got to carefully work with your data, know your data, and then make appropriate adjustments.  And there’s no shortcut for this; no magic test.  You’ve just got to put the work in, and dig the data, and know your data. 

Patsy:  “In determining your dummy variable, it seems to be subjective, such as certain range of birth weights, and then a smaller range of birth weights.  Isn’t that a problem?” 

Ciaran S. Phibbs:  Yeah.  Because what you’re doing is you’re determining things to fit to the data.  And basically in this case the effect birth weight, mortality is much more sensitive at the very low-range birth weight.  Moving from 500 grams to 600 grams is a huge change in terms of survivor probability, it has a big effect.  And so it’s how is the variable of interest, or your right hand side variable, how is that having an effect on your left hand side variable?  And the fact is because it can be nonlinear, and so it may be much more sensitive to small changes at one part of the distribution than at the other.  And so it is entirely appropriate to use different intervals.  Similarly, if you were modeling age, as in terms of mortality risk, you might want to use smaller intervals as you move up in years, whereas, you know, you could 20 to 50 or 20 to 45, depending on your data, or whatever.  You know, oh there’s no difference in risk for that huge group, just put them in a single group.  And the limit that you have is you have to remember that in terms of cell sizes, you can’t have your cell sizes be too small, and you need to have an adequate number in your reference group.  

So for doing something like this here, in terms of adjusting for age, or adjusting for birth weight, you know, the standard thing is that someone says, “oh, I’m going to look at this and I’ll take a group at the end as my reference group.”  And that may not be appropriate.  It may be appropriate to take a reference group in the middle because that’s sort of where there’s no action.  And remember, you have to have a relatively big group as a reference group.  You cannot have very small groups, that causes all kinds of problems.  And it’s better to have a big group and to have – you know, people tend to say, oh, it’s not at one end and I want to have this.  Get away from that because it may not be appropriate.  Because you may –

Patsy:   So, I –

Ciaran S. Phibbs:  Go ahead. 

Patsy:  I was just going to say so one of the issues for you in making this presentation is that you know this data so intimately. 

Ciaran S. Phibbs:  Yeah.

Patsy:  And you know where the effects are going to occur.  I wonder whether you might take a moment to talk about how to make these decisions about where to create your dummies, or your dummies split up that continuous variable by running some diagnostics on it; the distribution and –

Ciaran S. Phibbs:   Yeah.  So essentially what the dummy variable test is so, I eluded to this, maybe I didn’t make it be clear enough.  So what I did, for the birth weights in terms of making the cells, you know, let’s just take age, something we commonly adjust about, look at your age distribution very carefully.  Print out a full distribution of it so that you know how many you have where.  And then use that information on the distribution of the variable to make up a set of dummy variables that you can control for.  And you will have some understanding of what you’re studying so that it can help guide, you know, where do I need to have more specificity.  And you do that to make your dummy variables.  To make those dummy variables, you have to look at your data to break it up into a number of small groups, and I’m talking, you know, minimum, breaking it up into 15 or 20 cells to run that initial regression to look at what that functional form looks like. 

Patsy:  Okay, thanks. 

Ciaran S. Phibbs:  I’ll just come back here.  What I did here for this double functional form, this is on hospital volume here, is I looked at the distribution of the counts of the number of patients in each cell.  And as you can see here, as I get to very big hospitals the spacing between the cells gets bigger because I didn’t have as many observations with lots of patients.  So I was having to make bigger groups to have an adequate number in those cells.  Whereas up here they’re all right on top of each other because I was making very small groups to run to take a look at this pattern.  So you look at the data to tell you in terms of the distribution of where the patients are, or the sample is within the range of that variable and that guides how you make these dummy variables up.  And then you run the regression, and then you start looking, and you may take it apart in terms of subsets. 

Patsy:  Okay.   Here’s another question. 

Ciaran S. Phibbs:  Yeah?

Patsy:   “It’s analytic and not subjective?  Correct?” 

Ciaran S. Phibbs: What?  It just says “analytic, not subjective?” 

Patsy:  No, no, no, no, no.  I’m commenting on what you just said. 

Ciaran S. Phibbs:  Oh.

Patsy:  It’s not a subjective – this is from a question on whether it was subjective and I’m saying, no, it’s analytic, not subjective –

Ciaran S. Phibbs:  Yeah.

Patsy:  About where you make your classification.

Ciaran S. Phibbs:  Your cuts; yeah.  I mean, there’s got to be some subjective judgment in it.  But what you’re trying to do is let the data tell you where to make these cuts. 

Patsy:  Right.

Ciaran S. Phibbs:  Specify your –

Patsy:  Okay.  I have three more questions.

Ciaran S. Phibbs:  Okay.

Patsy:  “Does clustering affect the residual of the regression, or just the standard errors of the parameter?”

Ciaran S. Phibbs:  Standard errors of the parameter.  It doesn’t affect the residual of the regression because the parameter estimate is unbiased.

Patsy:  Okay.  “If you only care about β1, patient level variable, and β1 is not biased by β2, hospital level variable, then why do reviewers always make such a fuss over cluster effects?”

Ciaran S. Phibbs:  I can’t answer that.  I mean, me personally, I would make an effect if you haven’t clustered, you know, if you’re just putting the hospital variable in as a control and not making any inference on it; who cares?  But that’s me.  Other people may want you to be precise.  And that’s, you know, I can’t—

Patsy:  Okay.  And this is the last question, I believe.  “The robust and cluster option do not work when using survey data and specifying weights.  Primary sampling units and Strata variables; why is this?”

Ciaran S. Phibbs:  There’s a – [Sue Dan] can deal with this; but it’s really complicated.  And let’s leave it at that.

Patsy:  And if you want to call Ciaran directly—

Ciaran S. Phibbs:  I don’t know the precise answer to that one, but, you know, you can’t just simply apply these corrections when you have sampling weights, because you have to adjust the corrections for the sampling weights.  And [Sue Dan] will actually do this and I don’t know, I haven’t dug into or talked to anybody who’s dug into it, but Stata now has the ability to handle sample weight and it wouldn’t surprise me if Stata has those corrections fixed in its [inaudible].  But one would have to look in the Stata manual to see if those options exist or, you know, so if the cluster option or the robust option exists within the Stata commands for survey data, that means that they’ve gone through and made the correction as [Sue Dan] has.

Patsy:  Okey-doke; that’s the last question.

Ciaran S. Phibbs:  All right, turn it back to Heidi, then,

Heidi:  Fantastic, we’re right at the top of the hour there; perfect timing.  Ciaran, I want to thank you so much for presenting today, we really appreciate all of the time and effort that you put into these sessions.  And as we have up on our screen here, the next session in this series is scheduled for July 11 and Jean Yoon will be presenting on research design.  We’ll be sending registration information out to everyone on that shortly.  We hope that you can all join us for that.  Thank you and we will see you next time.

[**End of Recording]**