The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
No, a degree of freedom is not “one foot out the door”!
Definitions are rarely very good at explaining the meaning of something. At least not in statistics. Degrees of freedom: “the number of independent values or quantities which can be assigned to a statistical distribution”.
This is no exception.
The last, and sometimes hardest, step for running any statistical model is writing up results.
As with most other steps, this one is a bit more complicated for structural equation models than it is for simpler models like linear regression.
Any good statistical report includes enough information that someone else could replicate your results with your data.
[Read more…] about Member Training: Reporting Structural Equation Modeling Results
Linear regression with a continuous predictor is set up to measure the constant relationship between that predictor and a continuous outcome.
This relationship is measured in the expected change in the outcome for each one-unit change in the predictor.
One big assumption in this kind of model, though, is that this rate of change is the same for every value of the predictor. It’s an assumption we need to question, though, because it’s not a good approach for a lot of relationships.
Segmented regression allows you to generate different slopes and/or intercepts for different segments of values of the continuous predictor. This can provide you with a wealth of information that a non-segmented regression cannot.
In this webinar, we will cover [Read more…] about Member Training: Segmented Regression
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 [Read more…] about Should You Always Center a Predictor on the Mean?
Oh so many years ago I had my first insight into just how ridiculously confusing all the statistical terminology can be for novices.
I was TAing a two-semester applied statistics class for graduate students in biology. It started with basic hypothesis testing and went on through to multiple regression.
It was a cross-listed class, meaning there were a handful of courageous (or masochistic) undergrads in the class, and they were having trouble keeping [Read more…] about Confusing Statistical Terms #2: Alpha and Beta
Linear regression is one of the most popular statistical techniques.
Despite its popularity, interpretation of the regression coefficients of any but the simplest models is sometimes, well….difficult.
So let’s interpret the coefficients of a continuous and a categorical variable. Although the example here is a linear regression model, the approach works for interpreting coefficients from any regression model without interactions, including logistic and proportional hazards models.
A linear regression model with two predictor variables can be expressed with the following equation:
Y = B0 + B1*X1 + B2*X2 + e.
The variables in the model are:
The parameters in the model are:
One example would be a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2).
Height is measured in cm, bacteria is measured in thousand per ml of soil, and type of sun = 0 if the plant is in partial sun and type of sun = 1 if the plant is in full sun.
Let’s say it turned out that the regression equation was estimated as follows:
Y = 42 + 2.3*X1 + 11*X2
B0, the Y-intercept, can be interpreted as the value you would predict for Y if both X1 = 0 and X2 = 0.
We would expect an average height of 42 cm for shrubs in partial sun with no bacteria in the soil. However, this is only a meaningful interpretation if it is reasonable that both X1 and X2 can be 0, and if the data set actually included values for X1 and X2 that were near 0.
If neither of these conditions are true, then B0 really has no meaningful interpretation. It just anchors the regression line in the right place. In our case, it is easy to see that X2 sometimes is 0, but if X1, our bacteria level, never comes close to 0, then our intercept has no real interpretation.
Since X1 is a continuous variable, B1 represents the difference in the predicted value of Y for each one-unit difference in X1, if X2 remains constant.
This means that if X1 differed by one unit (and X2 did not differ) Y will differ by B1 units, on average.
In our example, shrubs with a 5000 bacteria count would, on average, be 2.3 cm taller than those with a 4000/ml bacteria count, which likewise would be about 2.3 cm taller than those with 3000/ml bacteria, as long as they were in the same type of sun.
(Don’t forget that since the bacteria count was measured in 1000 per ml of soil, 1000 bacteria represent one unit of X1).
Similarly, B2 is interpreted as the difference in the predicted value in Y for each one-unit difference in X2 if X1 remains constant. However, since X2 is a categorical variable coded as 0 or 1, a one unit difference represents switching from one category to the other.
B2 is then the average difference in Y between the category for which X2 = 0 (the reference group) and the category for which X2 = 1 (the comparison group).
So compared to shrubs that were in partial sun, we would expect shrubs in full sun to be 11 cm taller, on average, at the same level of soil bacteria.
Don’t forget that each coefficient is influenced by the other variables in a regression model. Because predictor variables are nearly always associated, two or more variables may explain some of the same variation in Y.
Therefore, each coefficient does not measure the total effect on Y of its corresponding variable, as it would if it were the only variable in the model.
Rather, each coefficient represents the additional effect of adding that variable to the model, if the effects of all other variables in the model are already accounted for. (This is called Type 3 regression coefficients and is the usual way to calculate them. However, not all software uses Type 3 coefficients, so make sure you check your software manual so you know what you’re getting).
This means that each coefficient will change when other variables are added to or deleted from the model.
For a discussion of how to interpret the coefficients of models with interaction terms, see Interpreting Interactions in Regression.
