One of the most difficult steps in calculating sample size estimates is determining the smallest scientifically meaningful effect size.

Here’s the logic:

The power of every significance test is based on four things: the alpha level, the size of the effect, the amount of variation in the data, and the sample size.

You will measure the effect size in question differently, depending on which statistical test you’re performing. It could be a mean difference, a difference in proportions, a correlation, regression slope, odds ratio, etc.

When you’re planning a study and estimating the sample size needed for (more…)

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 something. For example, it could measure agreement with a statement, such as 1=Strongly Disagree to 5=Strongly Agree. It can be a 1 to 5 scale, 0 to 10, etc. (more…)

You may have heard that using SPSS syntax is more efficient, gives you more control, and ultimately saves you time and frustration. It’s all true.

….And yet you probably use SPSS because you don’t *want* to code. You like the menus.

I get it.

I like the menus, too, and I use them all the time.

But I use syntax just as often.

At some point, if you want to do serious data analysis, you *have* to start using syntax. (more…)

Okay, maybe these SPSS shortcuts won’t make your *whole* life easier, but it will help your work life, at least the SPSS part of it.

When I consult with researchers, a common part of that is going through their analysis together. Sometimes I notice that they’re using some shortcut in SPSS that I had not known about.

Or sometimes they could be saving themselves some headaches.

So I thought I’d share three buttons you may not have noticed before that will make your data analysis more efficient.

(more…)

One of the difficult decisions in mixed modeling is deciding which factors are fixed and which are random. And as difficult as it is, it’s also very important. Correctly specifying the fixed and random factors of the model is vital to obtain accurate analyses.

Now, you may be thinking of the fixed and random effects in the model, rather than the factors themselves, as fixed or random. If so, remember that each term in the model (factor, covariate, interaction or other multiplicative term) has an effect. We’ll come back to how the model measures the effects for fixed and random factors.

Sadly, the definitions in many texts don’t help much with decisions to specify factors as fixed or random. Textbook examples are often artificial and hard to apply to the real, messy data you’re working with.

Here’s the real kicker. The same factor can often be fixed or random, depending on the researcher’s objective. (more…)

*Updated 12/20/2021*

Despite its popularity, interpreting regression coefficients of any but the simplest models is sometimes, well….difficult.

So let’s interpret the coefficients in a model with two predictors: a continuous and a categorical variable. The example here is a linear regression model. But this works the same way for interpreting coefficients from any regression model without interactions.

A linear regression model with two predictor variables results in the following equation:

Y_{i} = B_{0} + B_{1}*X_{1i} + B_{2}*X_{2i} + e_{i}.

The variables in the model are:

- Y, the response variable;
- X
_{1}, the first predictor variable;
- X
_{2}, the second predictor variable; and
- e, the residual error, which is an unmeasured variable.

The parameters in the model are:

- B
_{0}, the Y-intercept;
- B
_{1}, the first regression coefficient; and
- B
_{2}, 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 (X_{1}) and whether the plant is located in partial or full sun (X_{2}).

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*X_{1} + 11*X_{2}

### Interpreting the Intercept

B_{0}, the Y-intercept, can be interpreted as the value you would predict for Y if both X_{1} = 0 and X_{2} = 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 X_{1} and X_{2} can be 0, and if the data set actually included values for X_{1} and X_{2} 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 X_{2} sometimes is 0, but if X_{1}, our bacteria level, never comes close to 0, then our intercept has no real interpretation.

### Interpreting Coefficients of Continuous Predictor Variables

Since X_{1} is a continuous variable, B_{1} represents the difference in the predicted value of Y for each one-unit difference in X_{1}, if X_{2} remains constant.

This means that if X_{1} differed by one unit (and X_{2} did not differ) Y will differ by B_{1} units, on average.

In our example, shrubs with a 5000/ml bacteria count would, on average, be 2.3 cm taller than those with a 4000/ml bacteria count. They 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 measurement unit for bacteria count is 1000 per ml of soil, 1000 bacteria represent one unit of X_{1}).

### Interpreting Coefficients of Categorical Predictor Variables

Similarly, B_{2} is interpreted as the difference in the predicted value in Y for each one-unit difference in X_{2} if X_{1} remains constant. However, since X_{2} is a categorical variable coded as 0 or 1, a one unit difference represents switching from one category to the other.

B_{2} is then the average difference in Y between the category for which X_{2} = 0 (the reference group) and the category for which X_{2} = 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. It would if it were the only predictor variable in the model. Or if the predictors were independent of each other.

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 means that adding or removing variables from the model will change the coefficients. This is not a problem, as long as you understand why and interpret accordingly.

### Interpreting Other Specific Coefficients

I’ve given you the basics here. But interpretation gets a bit trickier for more complicated models, for example, when the model contains quadratic or interaction terms. There are also ways to rescale predictor variables to make interpretation easier.

So here is some more reading about interpreting specific types of coefficients for different types of models: