linear regression

SPSS GLM: Choosing Fixed Factors and Covariates

The beauty of the Univariate GLM procedure in SPSS is that it is so flexible. You can use it to analyze regressions, ANOVAs, ANCOVAs with all sorts of interactions, dummy coding, etc.

The down side of this flexibility is it is often confusing what to put where and what it all means.

So here’s a quick breakdown.

The dependent variable I hope is pretty straightforward. Put in your continuous dependent variable.

Fixed Factors are categorical independent variables. It does not matter if the variable is [Read more…] about SPSS GLM: Choosing Fixed Factors and Covariates

Interpreting Regression Coefficients

by Karen Grace-Martin 30 Comments

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:

Y, the response variable;
X₁, the first predictor variable;
X₂, the second predictor variable; and
e, the residual error, which is an unmeasured variable.

The parameters in the model are:

B₀, the Y-intercept;
B₁, the first regression coefficient; and
B₂, the second regression coefficient.

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

Interpreting the Intercept

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.

Interpreting Coefficients of Continuous Predictor Variables

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).

Interpreting Coefficients of Categorical Predictor Variables

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.

Interpreting Coefficients when Predictor Variables are Correlated

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.

Centering for Multicollinearity Between Main effects and Quadratic terms

by Karen Grace-Martin 7 Comments

One of the most common causes of multicollinearity is when predictor variables are multiplied to create an interaction term or a quadratic or higher order terms (X squared, X cubed, etc.).

Why does this happen? When all the X values are positive, higher values produce high products and lower values produce low products. So the product variable is highly correlated with the component variable. I will do a very simple example to clarify. (Actually, if they are all on a negative scale, the same thing would happen, but the correlation would be negative).

In a small sample, say you have the following values of a predictor variable X, sorted in ascending order:

2, 4, 4, 5, 6, 7, 7, 8, 8, 8

It is clear to you that the relationship between X and Y is not linear, but curved, so you add a quadratic term, X squared (X2), to the model. The values of X squared are:

4, 16, 16, 25, 49, 49, 64, 64, 64

The correlation between X and X2 is .987–almost perfect.

To remedy this, you simply center X at its mean. The mean of X is 5.9. So to center X, I simply create a new variable XCen=X-5.9.

These are the values of XCen:

-3.90, -1.90, -1.90, -.90, .10, 1.10, 1.10, 2.10, 2.10, 2.10

Now, the values of XCen squared are:

15.21, 3.61, 3.61, .81, .01, 1.21, 1.21, 4.41, 4.41, 4.41

The correlation between XCen and XCen2 is -.54–still not 0, but much more managable. Definitely low enough to not cause severe multicollinearity. This works because the low end of the scale now has large absolute values, so its square becomes large.

The scatterplot between XCen and XCen2 is:

Plot of Centered X vs. Centered X squared

If the values of X had been less skewed, this would be a perfectly balanced parabola, and the correlation would be 0.

Tonight is my free teletraining on Multicollinearity, where we will talk more about it. Register to join me tonight or to get the recording after the call.

Regression Through the Origin

by Karen Grace-Martin 3 Comments

I just wanted to follow up on my last post about Regression without Intercepts.

Regression through the Origin means that you purposely drop the intercept from the model. When X=0, Y must = 0.

The thing to be careful about in choosing any regression model is that it fit the data well. Pretty much the only time that a regression through the origin will fit better than a model with an intercept is if the point X=0, Y=0 is required by the data.

Yes, leaving out the intercept will increase your df by 1, since you’re not estimating one parameter. But unless your sample size is really, really small, it won’t matter. So it really has no advantages.

Regression models without intercepts

by Karen Grace-Martin 8 Comments

A recent question on the Talkstats forum asked about dropping the intercept in a linear regression model since it makes the predictor’s coefficient stronger and more significant. Dropping the intercept in a regression model forces the regression line to go through the origin–the y intercept must be 0.

The problem with dropping the intercept is if the slope is steeper just because you’re forcing the line through the origin, not because it fits the data better. If the intercept really should be something else, you’re creating that steepness artificially. A more significant model isn’t better if it’s inaccurate.

Can Likert Scale Data ever be Continuous?

by Karen Grace-Martin 56 Comments

A very common question is whether it is legitimate to use Likert scale data in parametric statistical procedures that require interval data, such as Linear Regression, ANOVA, and Factor Analysis. A typical Likert scale item has 5 to 11 points that indicate the degree of agreement with a statement, such as 1=Strongly Agree to 5=Strongly Disagree. It can be a 1 to 5 scale, 0 to 10, etc.

The issue is that despite being made up of numbers, a Likert scale item is in fact a set of ordered categories.

One camp maintains that as ordered categories, the intervals between the scale values are not equal. Any mean, correlation, or other numerical operation applied to them is invalid. Only nonparametic statistics should be used on Likert scale data (i.e. Jamieson, 2004).

The other camp maintains that while technically the Likert scale item is ordered, using it in parametric tests is valid in some situations. For example, Lubke & Muthen (2004) found that it is possible to find true parameter values in factor analysis with Likert scale data, if assumptions about skewness, number of categories, etc., were met. Likewise, Glass et al. (1972) found that F tests in ANOVA could return accurate p-values on Likert items under certain conditions.

Meanwhile, the debate rages on.

What is a researcher with integrity supposed to do? In the absence of a definitive answer, these are my recommendations:

Understand the difference between a Likert type item and a Likert Scale. A true Likert scale, as Likert defined it, is made up of many items, which all measure the same attitude. But many people use the term Likert Scale to refer to a single item. Confusion about what a Likert Scale is, no doubt, has contributed to the debate.
Proceed with caution. Research the consequences of using your procedure on Likert scale data from your study design. The fact that everyone uses it is not sufficient justification. There are some circumstances and procedures for which it is more egregious than others.
At the very least, insist that the item have at least 5 points (7 is better), that the underlying concept be continuous, and that there be some indication that the intervals between points are approximately equal. Make sure the other assumptions (normality & equal variance of residuals, etc.) be met.
When you can, run the nonparametric equivalent to your test. If you get the same results, you can be confident about your conclusions.
If you do choose to use Likert data in a parametric procedure, make sure you have strong results before making claims. Use a more stringent alpha level, like .01 or even .005, instead of .05. If you have p-values of .001 or .45, it’s pretty clear what the result is, even if parameter estimates are slightly biased. It’s when p-values are close to .05 that the effect of bending assumptions is unclear.
Consider the consequences of reporting inaccurate results. Will anyone ever read your paper? Will your research be published? Will it be used to shape public policy or affect practices? The answers to these questions can inform the seriousness of potential problems.

References:
Carifio, J. & Perla, R. (2007). Ten Common Misunderstandings, Misconceptions, Persistent Myths and Urban Legends about Likert Scales and Likert Response Formats and their Antidotes. Journal of Social Sciences, 2, 106-116. http://thescipub.com/PDF/jssp.2007.106.116.pdf

Glass, Peckham, and Sanders (1972). Consequences of failure to meet assumptions underlying the analyses of variance and covariance, Review of Educational Research, 42, 237-288.

Jamieson, S. (2004). Likert scales: how to (ab)use them. Medical Education, 38, 1212-1218.

Lubke, Gitta H.; Muthen, Bengt O. (2004). Applying Multigroup Confirmatory Factor Models for Continuous Outcomes to Likert Scale Data Complicates Meaningful Group Comparisons. Structural Equation Modeling, 11, 514-534.