The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
One of the most useful and intuitive statistics we have in linear regression is the Coefficient of Determination: R²
It tells you how well the model predicts the outcome and has some nice properties. [Read more…] about The Difference Between R-squared and Adjusted R-squared
Odds ratios have a unique part to play in describing the effects of logistic regression models. But that doesn’t mean they’re easy to communicate to an audience who is likely to misinterpret them. So writing up your odds ratios has to be done with care. [Read more…] about Guidelines for writing up three types of odds ratios
Updated 11/22/2021
Probability and odds measure the same thing: the likelihood or propensity or possibility of a specific outcome.
People use the terms odds and probability interchangeably in casual usage, but that is unfortunate. It just creates confusion because they are not equivalent.
They measure the same thing on different scales. Imagine how confusing it would be if people used degrees Celsius and degrees Fahrenheit interchangeably. “It’s going to be 35 degrees today” could really make you dress the wrong way.
In measuring the likelihood of any outcome, we need to know [Read more…] about Logistic Regression Analysis: Understanding Odds and Probability
One issue that affects how to interpret regression coefficients is the scale of the variables. In linear regression, the scaling of both the response variable Y, and the relevant predictor X, are both important.
In regression models like logistic regression, where the response variable is categorical, and therefore doesn’t have a numerical scale, this only applies to predictor variables, X.
This can be an issue of measurement units–miles vs. kilometers. Or it can be an issue of simply how big “one unit” is. For example, whether one unit of annual income is measured in dollars, thousands of dollars, or millions of dollars.
The good news is you can easily change the scale of variables to make it easier to interpret their regression coefficients. This works as well for functions of regression coefficients, like odds ratios and rate ratios.
All you have to do is create a new variable in your data set (don’t overwrite the individual one in case you make a mistake). This new variable is simply the old one multiplied or divided by some constant. The constant is often a factor of 10, but it doesn’t have to be. Then use the new variable in your model instead of the original one.
Since regression coefficients and odds ratios tell you the effect of a one unit change in the predictor, you should multiply them so that a one unit change in the predictor makes sense.
Here is a really simple example that I use in one of my workshops.
Y= first semester college GPA
X1= high school GPA
X2=SAT score
High School and first semester College GPA are both measured on a scale from 0 to 4. If you’re not familiar with this scaling, a 0 means you failed a class. An A (usually the top possible score) is a 4, a B is a 3, a C is a 2, and a D is a 1. So for a grade point average, a one point difference is very big.
If you’re an admissions counselor looking at high school transcripts, there is a big difference between a 3.7 GPA and a 2.7 GPA.
SAT score is on an entirely different scale. It’s a normed scale, so that the minimum is 200, the maximum is 800, and the mean is 500. Scores are in units of 10. You literally cannot receive a score of 622. You can get only 620 or 630.
So a one-point difference is not only tiny, it’s meaningless. Even a 10 point difference in SAT scores is pretty small. But 50 points is meaningful, and 100 points is large.
If you leave both predictors on the original scale in a regression that predicts first semester GPA, you get the following results:
Let’s interpret those coefficients.
The coefficient for high school GPA here is .20. This says that for each one-unit difference in GPA, we expect, on average, a .2 higher first semester GPA. While a one-unit change in GPA is huge, that’s reasonably meaningful.
The coefficient for SAT math scores is .002. That looks tiny. It says that for each one-unit difference in SAT math score, we expect, on average, a .002 higher first semester GPA. But a one-unit difference in SAT score is too small to interpret. It’s too small to be meaningful.
So we can change the scaling of our SAT score predictor to be in 10-point differences. Or in 100 point differences. Chose the scale for which one unit is meaningful.
In a linear model, you can simply multiply the coefficient by 10 to reflect a 10-point difference. That’s a coefficient of .02. So for each 10 point difference in math SAT score we expect, on average, a .02 higher first semester GPA.
Or we could multiply the coefficient by 50 to reflect a 50-point difference. That’s a coefficient of .10. So for each 50 point difference in math SAT score we expect, on average, a .1 higher first semester GPA.
Multiplying the coefficient is easier than rescaling the original variable if you only have one or two of these and you’re using linear regression.
It doesn’t work once you’ve done any sort of back-transformation in generalized linear models. So you can’t just multiply the odds ratio or the incidence rate ratio by 10 or 50. Both of these are created by exponentiating the regression coefficient. Because of the order of operations in algebra, You have to first multiply the coefficient by the constant, and then re-expontiate.
Likewise, if you are using this predictor in more than one linear regression model, it’s much simpler to rescale the variable in the first place. Simply divide that SAT score by 10 or 50 and the coefficient will .02 or .10, respectively.
Updated 12/2/2021
Updated 9/20/2021
Imputation as an approach to missing data has been around for decades.
You probably learned about mean imputation in methods classes, only to be told to never do it for a variety of very good reasons. Mean imputation, in which each missing value is replaced, or imputed, with the mean of observed values of that variable, is not the only type of imputation, however.
Better, although still problematic, imputation methods have two qualities. They use other variables in the data set to predict the missing value, and they contain a random component.
Using other variables preserves the relationships among variables in the imputations. It feels like cheating, but it isn’t. It ensures that any estimates of relationships using the imputed variable are not too low. Sure, underestimates are conservatively biased, but they’re still biased.
The random component is important so that all missing values of a single variable are not exactly equal. Why is that important? If all imputed values are equal, standard errors for statistics using that variable will be artificially low.
There are a few different ways to meet these criteria. One example would be to use a regression equation to predict missing values, then add a random error term.
Although this approach solves many of the problems inherent in mean imputation, one problem remains. Because the imputed value is an estimate–a predicted value–there is uncertainty about its true value. Every statistic has uncertainty, measured by its standard error. Statistics computed using imputed data have even more uncertainty than their standard errors measure.
Your statistical package cannot distinguish between an imputed value and a real value.
Since the standard errors of statistics based on imputed values are too small, corresponding p-values are also too small. P-values that are reported as smaller than they actually are? Those lead to Type I errors.
Multiple imputation solves this problem by incorporating the uncertainty inherent in imputation. It has four steps:
Remarkably, m, the number of sufficient imputations, can be only 5 to 10 imputations, although it depends on the percentage of data that are missing. A good multiple imputation model results in unbiased parameter estimates and a full sample size.
Doing multiple imputation well, however, is not always quick or easy. First, it requires that the missing data be missing at random. Second, it requires a very good imputation model. Creating a good imputation model requires knowing your data very well and having variables that will predict missing values.
Effect size statistics are extremely important for interpreting statistical results. The emphasis on reporting them has been a great development over the past decade. [Read more…] about Odds Ratio: Standardized or Unstandardized Effect Size?
