Odds ratios are the bane of many data analysts. Interpreting them can be like learning a whole new language. This teleseminar will go over an example to show how to interpret the odds ratios in binary logistic regression. You will learn:

• how probability and odds both measure the same thing on different scales
• the meaning of odds
• how to interpret an odds ratio for continuous and categorical predictors in logistic regression

Date: April 4, 2012

Time: 1pm Eastern Time (12pm Central, 11am Mountain, 10am Pacific)

Where: Anywhere you have a fast internet connection

Length of Program: An Hour

Cost: Always FREE

Space is limited.

Last month I did a webinar on Poisson and negative binomial models for count data. With a few hundred participants, we ran out of time to get through all the questions, so I’m answering some of them here on the blog.

This set of questions are all related to when it’s appropriate to treat count data as continuous and run the more familiar and simpler linear model.

Q: Do you have any guidelines or rules of thumb as far as how many discrete values an outcome variable can take on before it makes more sense to just treat it as continuous?

The issue usually isn’t a matter of how many values there are.  I see what you mean in that a discrete scale that goes from 0 to 8 for example feels more discrete because there are only nine possible values, compared to a discrete scale that goes from 0 to 200.  201 values just feels more continuous.

But that’s not really the issue in most count models. The issue with count variables is that they bounded at zero. This wreaks havoc on the assumptions of a linear model, which require continuous data.

If none of your data are near zero, it would be less of an issue.  Treating that count variable as continuous would give you predicted values that are non-integers, but perhaps that’s not a big issue in your particular data set.

Q: How high does the count scale have to be before you can consider it continuous?

I suspect you’re getting at the same issue as in the last question. It’s certainly true that when you get into very large numbers, many of the issues with count variables aren’t issues anymore.

For example, most incomes are not measured using decimals, just whole numbers. You could consider them a count of the number of dollars. Likewise, demographic variables like the number of children vaccinated in a state over the course of the year are truly counts, but the smallest values are likely to be in the hundreds of thousands, or even millions.

As long as there are no data along the bound of zero, and you don’t mind predicted values that include decimals, there’s no problem treating it as continuous.

Q: For count data distributing not skewed, but in a symmetric/or even normal shape, are poisson and NB still the best choice?

Sometimes.  The Poisson distribution is only skewed when the mean is very small. When the mean gets up to only 10, the distribution will become symmetric and bell shaped.

Depending on the effects of the predictors, and actual range of the data, i.e. whether there are actual 0 pounds or not, you may get identical results from running a linear model compare to a Poisson or negative binomial model.

If you do run a linear model, it will be possible to get predictive values below zero, and you need to consider whether that’s problematic in your situation. If the point of your model is prediction, it may be more of an issue.

Q: If count data can be normalized by log transformation, will you recommend using poisson or linear regression?

It’s never wrong to run a Poisson model, so what you’re asking is if the increased accuracy is worth the trouble of running the more complicated model.  There are certainly cases where running a linear model simplifies things a lot and still gives you the same results.  (You just won’t know you have the same results unless you run both).

When the mean count is very small, and zero is the most common value in the data set, it will be impossible to normalize using a log transformation.  It just won’t work. The mode will always be at the lowest value.  In that situation, you have no choice.

However, if the mean count is a little bit larger, zero may not be the most common value. When the mode is not the lower bound, it will be possible to use a log transformation to normalize the data. In fact, not too many years ago, when Poisson and negative binomial models were not readily available in software, textbooks did suggest this approach.  You may still have some on your shelf that do so.

It’s not necessarily a bad approach.  You may very well get the exact same results. If so, and if, for example, you are writing a report for an audience with out the statistical sophistication to understand the Poisson model, it may be a better choice.

However, it’s not exactly the same thing. A Poisson model uses a log link function, which applies the log to the mean–not each individual data point. So it’s harder to back transform coefficients when you’ve got a log transformation.  So if there are not major advantages to running a linear model, you are usually better off with a more sophisticated and more accurate Poisson model, or one of its derivatives.

If you’d like to learn more about the different models available for Count data, you can download a recording of the webinar: Poisson and Negative Binomial Regression for Count Data. It’s free.

It seems every editor and her brother these days wants to see standardized effect size statistics reported in journal articles.

For ANOVAs, two of the most popular are Eta-squared and partial Eta-squared.  In one way ANOVAs, they come out the same, but in more complicated models, their values, and their meanings differ.

SPSS only reports partial Eta-squared, and in earlier versions of the software it was (unfortunately) labeled Eta-squared.  More recent versions have fixed the label, but still don’t offer Eta-squared as an option.

Luckily Eta-squared is very simple to calculate yourself based on the sums of squares in your ANOVA table. I’ve written another blog post with all the formulas. You can check it out here.

But if you’re still wondering about the details of the differences between partial Eta-squared and Eta-squared and which one you ought to be using, I recommend reading this article:

Levine, T.R. & Hullett, C.R. (2002). Eta Squared, Partial Eta Squared and the Misreporting of Effect Size in Communication Research.  Human Communication Research, 28, 612-625.

Centering predictor variables is one of those simple but extremely useful practices that is easily overlooked.

It’s almost too simple.

Centering simply means subtracting a constant from every value of a variable.  What it does is redefine the 0 point for that predictor to be whatever value you subtracted.  It shifts the scale over, but retains the units.

The effect is that the slope between that predictor and the response variable doesn’t change at all.  But the interpretation of the intercept does.

The intercept is just the mean of the response when all predictors = 0.  So when 0 is out of the range of data, that value  is meaningless.  But when you center X so that a value within the dataset becomes 0, the intercept becomes the mean of Y at the value you centered on.

What’s the point?  Who cares about interpreting the intercept?

It’s true.  In many models, you’re not really interested in the intercept.  In those models, there isn’t really a point, so don’t worry about it.

But, and there’s always a but, in many models interpreting the intercept becomes really, really important.  So whether and where you center becomes important too.

A few examples include models with a dummy-coded predictor, models with a polynomial (curvature) term, and random slope models.

Let’s look more closely at one of these examples.

In models with a dummy-coded predictor, the intercept is the mean of Y for the reference category—the category numbered 0.  If there’s also a continuous predictor in the model, X2, that intercept is the mean of Y for the reference category only when X2=0.

If 0 is a meaningful value for X2 and within the data set, then there’s no reason to center.  But if neither is true, centering will help you interpret the intercept.

For example, let’s say you’re doing a study on language development in infants.  X1, the dummy-coded categorical predictor, is whether the child is bilingual (X1=1) or monolingual (X1=0).  X2 is the age in months when the child spoke their first word, and Y is the number of words in their vocabulary for their primary language at 24 months.

If we don’t center X2, the intercept in this model will be the mean number of words in the vocabulary of monolingual children who uttered their first word at birth (X2=0).

And since infants never speak at birth, it’s meaningless.

A better approach is to center age at some value that is actually in the range of the data. One option, often a good one, is to use the mean age of first spoken word of all children in the data set.

This would make the intercept the mean number of words in the vocabulary of monolingual children for those children who uttered their first word at the mean age that all children uttered their first word.

One problem is that the mean age at which infants utter their first word may differ from one sample to another. This means you’re not always evaluating that mean that the exact same age.  It’s not comparable across samples.

So another option is to choose a meaningful value of age that is within the values in the data set. One example may be at 12 months.

Under this option the interpretation of the intercept is the mean number of words in the vocabulary of monolingual children for those children who uttered their first word at 12 months.

The exact value you center on doesn’t matter as long it’s meaningful, holds the same meaning across samples,  and within the range of data.  You may find that choosing the lowest value or the highest value of age is the best option. It’s up to you to decide the age at which it’s most meaningful to interpret the intercept.

If you want more information on using and interpreting regression coefficients for interactions, dummy variables, and centered predictors, get the recording from my webinar: Interpreting Regression Coefficients: A Walk Through Output. It’s free.

As someone who focuses on data analysis, I don’t often have to calculate probabilities, but I always love it when I do.  It brings me back to the basics I learned in stats classes.

A client asked what is the probability of choosing exactly 3 outliers in a sample of 30, if 3.5% of the population is outliers.

I got to pull out my old theoretical stats book to make sure I still remembered the pdf of a binomial distribution (I almost remembered it).

I calculated it out, but wanted to double-check that it was correct, and found this fabulous website for calculating binomial probabilities:

StatTrek

(I have to admit, the geek in me loves the name as well).

So if you ever need to calculate a binomial, or other probability, check it out.

I recently received this great question:

Question:

Hi Karen,  ive purchased a lot of your material and read a lot of your pdf documents w.r.t. regression and interaction terms.  Its, now, my general understanding that interaction for two or more categorical variables is best done with effects coding, and interactions  cont v. categorical variables is usually handled via dummy coding.  Further, i may mess this up a little but hopefully you’ll get my point and more importantly my question, i understand that

1)  given a fitted line Y = b0 + b1 x1 + b2 x2 + b3 x1*x2, the interpretation for b3 is the diff of the effect of x1 on Y, when x2 changes one unit, if x1 and x2 are cont.  ( also interpretation can be reversed in terms of x1 and x2).

2) given a fitted line Y = b0 + b1 x1 + b2 x2 + b3 x1*x2, the interpretation for b3 is it affects Y by changing the slope (whereas b2 affects the intercept), if x1 is cont. and x2 is cat 0,1,  namely dummy coded.  Thus, b1 is not a main effect for x1.

But what was not clear to me is what happens when x1 and x2 are effect coded cat variables, let’s say (-1,1) and the fitted line is Y = b0 + b1 x1 + b2 x2 + b3 x1*x2, the interpretation for b3  is  ????  this was only hinted at in the outlines .. I’d like to know how to interpret the b3 in this case, and I’d like to know why in this case of interaction, b1 is still considered a main effect for say x1?    Thanks for any clarifications!

Any help would be greatly appreciated,

J

Answer:

Hi J,

Great question.  First, yes, everything you state in points 1 and 2 are correct. I would add, though, that the effect-coded coefficients are not all that easy to interpret by themselves.  They do give you p-values for true main effects, and that’s useful, but you usually need to interpret using multiple comparison tests of the means.

To answer your questions:

It all comes down to what is a 0 point in dummy and effect coding.

In dummy coding, we’ve set the value of 0 to one of the categories, so:

1. the coefficients reflect actual cell mean differences, and have meaningful interpretations as such

2. when there is an interaction, the value of b1, eg., is the effect of X1 when X2 = 0.  Since X2=0 for one category of X2, b1 is not a main effect (an overall effect of X1 across all values of X2). It’s a marginal effect–an effect of X1 at a single value of X2.

In effect coding, we’ve set the value of 0 to being in between the two categories, so:

1. the coefficients are differences between cell means and grand means, and do not have particularly meaningful interpretations.  So you can use the p-value to tell that there is a significant interaction, but the only way to interpret that interaction in a meaningful way is to actually look at the means.  That’s why we usually just look at F statistics and means in ANOVA, which uses effect coding.  The coefficients themselves aren’t very interpretable.

2. when there is an interaction, the value of b1, eg., is the effect of X1 when X2 = 0.  Since X2 = 1 at the mean of the two categories of X2, b1 is a main effect.  It’s the effect of X1 at the mean value of X2.

Below, I’ve inserted an example.  I’ve displayed the means of each group and the coefficients when the variables are dummy and effect coded.

In dummy coding, the intercept is one of the four means–when both Poverty and Gender=0.  The various coefficients are differences from that mean.  Grab a calculator and see if you can figure out how to get each mean.

In effect coding, the intercept is the grand mean.  That point right in the middle of the graph.  The coefficients are differences from that point to the points marked with X on the graph.  Once again, see if you can figure out what each coefficient is telling you.

If you want more information on using and interpreting parameter estimates in regression using SPSS, get the recording from my webinar: Interpreting Regression Coefficients: A Walk Through Output.  It’s free.

You’ve probably noticed it looks a little different around here.

The look of the site has changed, clearly.  It’s a lot less blue.

But more importantly, we’ve made a big effort to simplify the website and make it easier to get around.  As we’ve grown a lot over the past 3+ years, little bits were added here and there, and it was starting to get a bit jumbled.

So we’ve streamlined.  Here are some highlights of the website changes.

Website Updates

1. We’ve combined the web site and the blog.  The tips, articles, and resources on our blog are some of our most popular pages, but a link to “blog” doesn’t always capture that.  So we’ve made those articles front and center.

2. We’ve made a lot of our free offerings easier to get.  You used to have to sign up separately for each webinar recording.  Now you can sign up just once, and get access to all our recordings.

3. We’ve just plain cleaned up the navigation.  Drop-down menus, a sidebar with popular pages and functions, and fewer duplicate pages should make everything much easier to find.  We’re still working on the Workshop section, but hope to have that updated soon.

Service Updates

We’re also making some changes to our consulting services that will help us grow and still be able to give you great service and continue to offer new workshops, webinars, and resources:

1. We’re simplifying how we administer consulting.  We used to ask for a deposit, then do monthly billing.  Now we’re just keeping it simple.  You can pre-purchase one consultation or four (at a discount).  Just pick what you need.  Not sure what you need?  Start small, and you can always add on.  If your organization requires an exception, just contact us.

2. I am offering consulting only.  I realize when a lot of people say consulting, they mean doing projects as a contractor.  I have offered both services, and always made a distinction between projects and consulting–guiding you through analyses you’re doing yourself by helping you choose appropriate analyses, explaining how to conduct them, and helping with interpretation.

Although I love getting my hands dirty with data, my true love, (and I honestly believe my true gift and the best way to serve you) is through understanding where you’re stuck, asking the right questions, and getting you unstuck by explaining what you’re not understanding.

There are a lot of good data analysts, but not a lot of good statistical consultants.  So I am going to limit myself to what I do best, and personally offer only consulting services.  Learn more about my consulting services here.

3. BUT, I also realize there is great need for many researchers to hand off their data analysis.  So I have partnered with two expert freelance data analysts so you’re not left without the data analysis project services.

So with pride, I introduce Dr. Jane Yank and Dr. Maike Rahn, who will be available to run your analyses for you.  Jane and Maike have different backgrounds in their areas of research, statistical expertise, and the statistical software they use.  This will greatly expand the types of analyses and software we’re able to assist with.  Learn more about Jane and Maike and how to work with them on our Statistical Project Services page.

“Because mixed models are more complex and more flexible than the general linear model, the potential for confusion and errors is higher.”

- Hamer & Simpson (2005)

Linear Mixed Models, as implemented in SAS’s Proc Mixed, SPSS Mixed, R’s LMER, and Stata’s xtmixed, are an extension of the general linear model.  They use more sophisticated techniques for estimation of parameters (means, variances, regression coefficients, and standard errors), and as the quotation says, are much more flexible.

Here’s one example of the flexibility of mixed models, and its resulting potential for confusion and error.

In repeated measures and longitudinal studies, the observations are clustered within a subject.  That means the observations, and their residuals, are not independent.  They’re correlated.  There are two ways to deal with this correlation.

The Marginal Model

One is to alter the covariance structure of the residuals.  What this means is that instead of assuming that all observations are independent, as you do in a linear model, you assume the residuals from a single subject are related.  Their covariances are non-zero.  So you have to estimate the covariances among all the residuals from a single subject.

This approach is called a Marginal or Population Averaged approach.  It’s not truly a mixed model, although you can use Mixed procedures to run them.  You get these models in SAS Proc Mixed and SPSS Mixed by using a repeated statement instead of a random statement.

The Mixed Model

The other way to deal with non-independence of a subject’s residuals is to leave the residuals alone, but actually alter the model by controlling for subject.  When you control for subject as a factor in the model, you literally redefine what a residual is.  Instead of being the distance between a data point and the average for everyone, it’s the distance between a data point and the mean for that subject.

You could, theoretically, include Subject as a fixed factor, but that usually uses up most of the degrees of freedom.  If instead, you treat Subject as a random factor, you are still controlling for Subject, you’re still able to redefine the residuals and deal with non-independence, while using up only a few degrees of freedom.

Because the model now contains both fixed and random effects, it is now officially a Mixed Model.  You get these models in SAS Proc Mixed and SPSS Mixed by using a random statement.

Putting them together

Most of the time, controlling for Subject is enough to deal with all the non-independence of the residuals for each subject.

But every once in a while it’s not. If there is extra non-independence (or even non-constant variance) among the residuals, you can still estimate those non-zero covariances by adding a Repeated statement.

It’s fine to include a Repeated statement right along with a Random statement, and is sometimes necessary to have a good fitting model.  The repeated statement still controls the covariance structure of the residuals for a single subject.  It’s just that now those residuals have been redefined as the distance between each point and the subject’s mean.  In the Marginal model, they’re not.  They still represent the distance between each point and the overall mean.

If you want to learn more about mixed models, check out the recording of my Random Intercept and Random Slope Models webinar.  These two models are the basic building blocks of all mixed models.

Get it all here.  It’s free.

Has this ever happened to you?

You need to implement a statistical technique you haven’t encountered before.  Maybe someone told you you need SEM or linear mixed models or logistic regression, for example.

Luckily, you’ve found a pretty good book on the topic and are cruising along learning it, when it refers to some mathematical concept that you don’t understand.  Like the rank of a matrix.  Or a logarithm.

Maybe you never took matrix algebra.  Maybe you know you understood this many years ago.  But after 10 (or many more) years of not using linear algebra on a regular basis, and you just don’t remember exactly what a rank means.

Sure you can do a web search, but it’s still a chore to find a good source that explains what the heck it means.

I just came across this fabulous tutorial website: Khan Academy.

It has video tutorials on many topics in math.

I was particularly pleased to find it because I recently wrote about how complicated issues in statistics, like a Covariance Matrix, are not so intimidating if you understand some linear algebra and can remember what covariance means.

Khan Academy has tutorials on both linear algebra (a lot of them) and basic probability and statistics.  So even if the concept you don’t quite remember is a statistical one–like covariance–you can get those as well.

In this webinar we’re doing something a little different – rather than give you an overivew of a topic, we will interpret together the regression coefficients table from a real data set.

This data set is from the dissertation of a client I worked with a few years ago.  She has graciously allowed us to use her data as an example.  It’s a great example because it includes many of the coefficients that can be really tricky

• a categorical dummy-coded variable
• an interaction between the categorical and a continuous variable
• centering

Date: October 19, 2011

Time: 12pm Eastern Time (11am Central, 10am Mountain, 9am Pacific)

Where: Anywhere you have a fast internet connection

Length of Program: An Hour

Cost: Always FREE

Space is limited.

Sign up for our next webinar here.