Karen Grace-Martin

What It Really Means to Take an Interaction Out of a Model

When you’re model building, a key decision is which interaction terms to include.

As a general rule, the default in regression is to leave them out. Add interactions only with a solid reason. It would seem like data fishing to simply add in all possible interactions.

And yet, that’s a common practice in most ANOVA models: put in all possible interactions and only take them out if there’s a solid reason. Even many software procedures default to creating interactions among categorical predictors.

How Big of a Sample Size do you need for Factor Analysis?

by Karen Grace-Martin Leave a Comment

Most of the time when we plan a sample size for a data set, it’s based on obtaining reasonable statistical power for a key analysis of that data set. These power calculations figure out how big a sample you need so that a certain width of a confidence interval or p-value will coincide with a scientifically meaningful effect size.

But that’s not the only issue in sample size, and not every statistical analysis uses p-values.

Effect Size Statistics: How to Calculate the Odds Ratio from a Chi-Square Cross-tabulation Table

by Karen Grace-Martin Leave a Comment

Lest you believe that odds ratios are merely the domain of logistic regression, I’m here to tell you it’s not true.

One of the simplest ways to calculate an odds ratio is from a cross tabulation table.

We usually analyze these tables with a categorical statistical test. There are a few options, depending on the sample size and the design, but common ones are Chi-Square test of independence or homogeneity, or a Fisher’s exact test.

Three Designs that Look Like Repeated Measures, But Aren’t

by Karen Grace-Martin 1 Comment

Repeated measures is one of those terms in statistics that sounds like it could apply to many situations. In fact, it describes only one specific situation.

A repeated measures design is one where each subject is measured repeatedly over time, space, or condition on the dependent variable.

These repeated measurements on the same subject are not independent of each other. They’re clustered. They are more correlated to each other than they are to responses from other subjects. Even if both subjects are in the same condition.

That non-independent clustering needs to be accounted for in the analysis. It’s the only way to get accurate standard errors, p-values, and confidence intervals.

This gets confusing because this clustering can look very different in different studies.

Additionally, a few study designs look like they should be called repeated measures but, for the purpose of analysis, are not.

So, to reiterate, we have repeated measures if we have three important elements:

Multiple clustered measurements of the dependent variable
that are more correlated to each other than to others’ measurements because they’re measured on the same subject
repeatedly over time, space, or condition.

A great way to figure out if your study requires a repeated measures analysis (via mixed or marginal model) is to define these three elements. Here are a few examples:

– measuring the graduation rate in 50 high schools in the four years before and four years after an intervention.

Multiple clustered measurements of the DV: 8 measures of pass rate per school
Subject: school
Repeated over: time

– measuring the time it takes second language learners to read a sentence under 4 different grammar structures.

Multiple correlated measurements of the DV: 4 measures of reading time of each sentence per person
Subject: Person learning a language who reads the sentences
Repeated over: grammar structure condition

Studies that aren’t “repeated measures” even though something is measured repeatedly:

1. A single subject measured over time

A common design is to measure a single “subject” for many time points. An example is a data set with one company’s valuation measured each year for 10 years.

This design is missing element #2. The responses from our single subject cannot be more correlated to each other than to other subjects’ responses because there are no other subjects.

This is a time series. In a time series, there is serial autocorrelation — similarities in responses that are closer in time. Luckily, there are time series models that account for serial autocorrelation without accounting for clustered correlation.

In contrast, a design with 30 companies, each measured over 10 years, is repeated measures. You have 300 data points clustered into 30 sets of 10 (30 companies with 10 non-independent measurements). Each company’s data may also have autocorrelation, and good repeated measures models can account for that too.

2. A new random sample of subjects at each time point

A second common design is a cross-sectional longitudinal design. For example, you measure the annual revenue of 30 companies for 10 years. But each year you randomly selected a new sample of 30 companies. That’s not repeated measures. With new subjects at each time point, we’re missing element #2.

When you cannot match a subject’s measurement in year 1 to the same subject’s measurement in year 2, that is not repeated measures. If each measurement in year 1 could be equally similar or different to any measurement in year 2, you have no clustered non-independence. A model that assumes independence would work here.

3. A predictor variable is measured repeatedly over time for each subject, but the dependent variable is measured once

I don’t know of a name for this design, but I see studies like this every so often. For example, a study where 30 companies’ revenues over 10 years were used to predict whether or not they had an IPO in year 11.

This is missing element #1. The repeated measurements have to be on the dependent variable. Since you have only a single measurement of the dependent variable, that’s not repeated measures, even though there are 10 measures of the predictor.

Conclusion

Once again, statistical vocabulary is inconsistent and less intuitive than we’d like. A good practice is to always be precise in describing your design, your variables, and your study. Define the terms you’re using rather than assuming your audience understands.

Three ‘Rules’ of Statistical Analysis from Your Statistics Class to Unlearn

by Karen Grace-Martin Leave a Comment

When you are taking statistics classes, there is a lot going on. You’re learning concepts, vocabulary, and some really crazy notation. And probably a software package on top of that.

In other words, you’re learning a lot of hard stuff all at once.

Good statistics professors and textbook authors know that learning comes in stages. Trying to teach the nuances of good applied statistical analysis to students who are struggling to understand basic concepts results in no learning at all.

And yet students need to practice what they’re learning so it sticks. So they teach you simple rules of application. Those simple rules work just fine for students in a stats class working on sparkling clean textbook data.

But they are over-simplified for you, the data analyst, working with real, messy data.

Here are three rules of data analysis practice that you may have learned in classes that you need to unlearn. They are not always wrong. They simply don’t allow for the nuance involved in real statistical analysis.

The Practices to Unlearn:

1. To check statistical assumptions, run a test. Decide whether the assumption is met by the significance of that test.

Every statistical test and model has assumptions. They’re very important. And they’re not always easy to verify.

For many assumptions, there are tests whose sole job is to test whether the assumption of another test is being met. Examples include the Levene’s test for constant variance and Kolmogorov-Smirnov test, often used for normality. These tests are tools to help you decide if your model assumptions are being met.

But they’re not definitive.

When you’re checking assumptions, there are a lot of contextual issues you need to consider: the sample size, the robustness of the test you’re running, the consequences of not meeting assumptions, and more.

What to do instead:

Use these test results as one of many pieces of information that you’ll use together to decide whether an assumption is violated.

2. Delete outliers that are 3 or more standard deviations from the mean.

This is an egregious one.

Yes, it makes the data look pretty. Yes, there are some situations in which it’s appropriate to delete outliers (like when you have evidence that it’s an error). Yes, outliers can wreak havoc on your parameter estimates.

But don’t make it a habit. Don’t follow a rule blindly.

Deleting outliers because they’re outliers (or using techniques like Winsorizing) is a great way to introduce bias into your results or to miss the most interesting part of your data set.

What to do instead:

When you find an outlier, investigate it. Try to figure out if it’s an error. See if you can figure out where it came from.

3. Check Normality of Dependent Variables before running a linear model

In a t-test, yes, there is an assumption that Y, the dependent variable, is normally distributed within each group. In other words, given the group as defined by X, Y follows a normal distribution.

ANOVA has a similar assumption: given the group as defined by X, Y follows a normal distribution.

In linear regression (and ANCOVA), where we have continuous variables, this same assumption holds. But it’s a little more nuanced since X is not necessarily categorical. At any specific value of X, Y has a normal distribution. (And yes, this is equivalent to saying the errors have a normal distribution).

But here’s the thing: the distribution of Y as a whole doesn’t have to be normal.

In fact, if X has a big effect, the distribution of Y, across all values of X, will often be skewed or bimodal or just a big old mess. This happens even if the distribution of Y, at each value of X, is perfectly normal.

What to do instead:

Because normality depends on which Xs are in a model, check assumptions after you’ve chosen predictors.

Conclusion:

The best rule in statistical analysis: always stop and think about your particular data analysis situation.

If you don’t understand or don’t have the experience to evaluate your situation, discuss it with someone who does. Investigate it. This is how you’ll learn.

Five Ways to Analyze Ordinal Variables (Some Better than Others)

by Karen Grace-Martin Leave a Comment

There are not a lot of statistical methods designed just for ordinal variables.

But that doesn’t mean that you’re stuck with few options. There are more than you’d think.

Some are better than others, but it depends on the situation and research questions.

Here are five options when your dependent variable is ordinal.
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