Interpreting the Intercept in a regression model isn’t always as straightforward as it looks.
Here’s the definition: the intercept (often labeled the constant) is the expected value of Y when all X=0. But that definition isn’t always helpful. So what does it really mean?
Regression with One Predictor X
Start with a very simple regression equation, with one predictor, X.
If X sometimes equals 0, the intercept is simply the expected value of Y at that value. In other words, it’s the mean of Y at one value of X. That’s meaningful.
If X never equals 0, then the intercept has no intrinsic meaning. You literally can’t interpret it. That’s actually fine, though. You still need that intercept to give you unbiased estimates of the slope and to calculate accurate predicted values. So while the intercept has a purpose, it’s not meaningful.
Both these scenarios are common in real data. (more…)
Have you ever had this happen? You run a regression model. It can be any kind—linear, logistic, multilevel, etc. In the ANOVA table, the effect of interest has a very low p-value. In the regression table, it doesn’t. Or vice-versa.
How can the same effect have two different p-values? In this article, let’s explore when this happens and what it means.
What the statistics in each table measures
The ANOVA table is a table of F tests. It may not be called the ANOVA table on your output, but it always includes a set of F tests. Some software procedures only give one F test for the model as a whole, but most will break it down into a series of F tests, one for each predictor variable or term in your model.
The regression coefficients table is a table of t tests. It includes each regression coefficient, along with its standard error, and usually a t test (some generalized linear models will have Wald or z tests instead, but they have the same role here).
Both tables often list out each predictor variable, along with a p-value for that variable’s conditional effect on Y.
There are two situations in which the p-values will match. Both must be true.
- The F test has one df. This happens in two situations. Either the predictor, X, is numerical or it’s categorical and binary (only two groups).
- The predictor is not involved with any interactions with a variable that is not centered at is mean.
If both of those are true, not only will the p-value match, but the t-statistic in the regression coefficients table will be the positive or negative square root of the F statistic.
An Example ANOVA Table with Matching and Unmatching Regression Coefficients
Here’s an example of an ANOVA table from a linear regression. In this example, there are four treatment groups, two genders, and age in years (measured continuously and centered at its mean). The response variable, Y, is a satisfaction score with a training. The four groups represented four learning strategies the adult learners were trained to use.
Let’s compare this to the regression coefficients table.
If you compare p-values across the two tables, you can see that Gender and Age have the same p-values, but Group doesn’t.
Gender and Age meet both conditions. Both have 1 df in the F table. Gender because it’s binary (two categories) and Age because it’s numerical). There are no interactions.
Group doesn’t match because it has 3 df in the F test. The F test is testing the null hypothesis that there is no difference among the four means. The t-tests in the regression coefficients table are testing three specific contrasts. Each one compares one group mean to the group 4 mean. For example, the group=1 coefficient tests whether the difference between the mean group 1 satisfaction score differs only from the group 4 score. It’s a different null hypothesis than the F test.
This would be the case whether or not there were interactions in the model that contain Group. Any time you have more that one df in the F test (you can see group has 3), you’ll get as many p-values in the regression coefficients as you have df in the F table. The p-values can’t match because there are more of them in the regression coefficients table.
Gender, which is also categorical, does have the same p-value in both tables. It has 1 df in the F test, which tests the null hypothesis that the two gender means have no variance (they’re the same). Gender is involved in an interaction, so the only reason the hypothesis test, and therefore the p-value, is the same is because the variable it interacts with, Age, is centered.
In conclusion, most of the time, it’s fine if the results don’t match. It’s because the two tables are reporting results of different hypothesis tests, based on what’s in your model.
One of the important issues with missing data is the missing data mechanism. You may have heard of these: Missing Completely at Random (MCAR), Missing at Random (MAR), and Missing Not at Random (MNAR).
The mechanism is important because it affects how much the missing data bias your results. This has a big impact on what is a reasonable approach to dealing with the missing data. So you have to take it into account in choosing an approach.
The concepts of these mechanisms can be a bit abstract.
And to top it off, two of these mechanisms have really confusing names: Missing Completely at Random and Missing at Random.
Missing Completely at Random (MCAR)
Missing Completely at Random is pretty straightforward. What it means is what is (more…)
Some repeated measures designs make it quite challenging to specify within-subjects factors. Especially difficult is when the design contains two “levels” of repeat, but your interest is in testing just one.
Let’s look at a great example of what this looks like and how to deal with it in this question from a reader :
The Design:
I want to do a GLM (repeated measures ANOVA) with the valence of some actions of my test-subjects (valence = desirability of actions) as a within-subject factor. My subjects have to rate a number of actions/behaviours in a pre-set list of 20 actions from ‘very likely to do’ to ‘will never do this’ on a scale from 1 to 7, and some of these actions are desirable (e.g. help a blind man crossing the street) and therefore have a positive valence (in psychology) and some others are non-desirable (e.g. play loud music at night) and therefore have negative valence in psychology.
My question is how I can use valence as a within-subjects factor in GLM. Is there a way to tell SPSS some actions have positive valence and others have negative valence ? I assume assigning labels to the actions will not do it, as SPSS does not make analyses based on labels …
Please help. Thank you.
(more…)
When is it important to use adjusted R-squared instead of R-squared?
R², the the Coefficient of Determination, is one of the most useful and intuitive statistics we have in linear regression.
It tells you how well the model predicts the outcome and has some nice properties. But it also has one big drawback.
(more…)
In most regression models, there is one response variable and one or more predictors. From the model’s point of view, it doesn’t matter if those predictors are there to predict, to moderate, to explain, or to control. All that matters is that they’re all Xs, on the right side of the equation.
(more…)
