When I was in graduate school, stat professors would say “ANOVA is just a special case of linear regression.” But they never explained why.
And I couldn’t figure it out.
The model notation is different.
The output looks different.
The vocabulary is different.
The focus of what we’re testing is completely different. How can they be the same model?
Have you ever heard that “2 tall parents will have shorter children”?
This phenomenon, known as regression to the mean, has been used to explain everything from patterns in hereditary stature (as Galton first did in 1886) to why movie sequels or sophomore albums so often flop.
So just what is regression to the mean (RTM)? (more…)
When you put a continuous predictor into a linear regression model, you assume it has a constant relationship with the dependent variable along the predictor’s range. But how can you be certain? What is the best way to measure this?
And most important, what should you do if it clearly isn’t the case?
Let’s explore a few options for capturing a non-linear relationship between X and Y within a linear regression (yes, really). (more…)
In a simple linear regression model, how the constant (a.k.a., intercept) is interpreted depends upon the type of predictor (independent) variable.
If the predictor is categorical and dummy-coded, the constant is the mean value of the outcome variable for the reference category only. If the predictor variable is continuous, the constant equals the predicted value of the outcome variable when the predictor variable equals zero.
When your predictor variable X is categorical, the results are logical. Let’s look at an example. (more…)
A normally distributed variable can have values without limits in both directions on the number line. While most variables have practical limitations, most of the time, this assumption of infinite tails is quite reasonable as there is no real boundary.
Air temperature is an example of a variable that can extend far from its mean in either direction.
But for other variables, there is a practical beginning or ending point. Age is left-bounded. It starts at zero.
The number of wins that a baseball team can have in a season is bounded on the upper end by the number of games played in a season.
The temperature of water as a liquid is bound on the low end at zero degrees Celsius and on the high end at 100 degrees Celsius.
There are two types of bounded data that have direct implications for how to work with them in analysis: censored and truncated data. Understanding the difference is a critical first step when working with these variables.
Censored data have unknown values beyond a bound on either end of the number line or both. It can exist by design. When the data is observed and reported at the boundary, the researcher has made the decision to restrict the range of the scale.
An example of a lower censoring boundary is the recording of pollutants in our water. The researcher may not care about (or instruments may not be able to detect) the level of pollutants if it falls below a certain threshold (e.g., .005 parts per million). In this case, any pollutant level below .005 ppm is reported as “<.005 ppm.”
An upper censor could be placed on temperature in a science experiment. Once the temperature goes above x degrees the scientist doesn’t care. So s/he measures it as “>x”.
Data can be censored on both ends as well. Income could be reported as “<$20,000” if the actual is below $20,000 and reported as “ >$200,000” if above that level.
There are potential censored data not created by design. Test scores or college admission tests are examples of censored data not created by design, but by the actual bounds. A student cannot score above 100% correct no matter how much better they know the topic than other students. These are bounded by actual results.
Truncation occurs when values beyond a boundary are either excluded when gathered or excluded when analyzed. For example, if someone conducting a survey asks you if you make more than $100,000, and you answer “yes” and the surveyor says “thanks but no thanks”, then you’ve been truncated.
Or if a number of arrests is measured from police records, then everyone with 0 arrests will, by definition, be excluded from the sample.
Excluding cases from a data set at a preset boundary has the same effect. Creating models on middle income values would involve truncating income above and below specific amounts.
So to summarize, data are censored when we have partial information about the value of a variable—we know it is beyond some boundary, but not how far above or below it.
In contrast, data are truncated when the data set does not include observations in the analysis that are beyond a boundary value. Having a value beyond the boundary eliminates that individual from being in the analysis.
In truncation, it’s not just the variable of interest that we don’t have full data on. It’s all the data from that case.
Jeff Meyer is a statistical consultant with The Analysis Factor, a stats mentor for Statistically Speaking membership, and a workshop instructor. Read more about Jeff here.
Outliers are one of those realities of data analysis that no one can avoid.
Those pesky extreme values cause biased parameter estimates, non-normality in otherwise beautifully normal variables, and inflated variances.
Everyone agrees that outliers cause trouble with parametric analyses. But not everyone agrees that they’re always a problem, or what to do about them even if they are.
Sometimes a nonparametric or robust alternative is available — and sometimes not.
There are a number of approaches in statistical analysis for dealing with outliers and the problems they create. It’s common for committee members or Reviewer #2 to have very strong opinions that there is one and only one good approach.
Two approaches that I’ve commonly seen are: 1) delete outliers from the sample, or 2) winsorize them (i.e., replace the outlier value with one that is less extreme).
The problem with both of these “solutions” is that they also cause problems — biased parameter estimates and underweighted or eliminated valid values. (more…)