Lecture 18
R^{2}
and Tests of Significance
The Relationship
Between r and b
Correlation and regression are similar. Regression concerns the prediction of y
from x. Correlation concerns the association of x and y. Note that with
correlation, it does not matter which variable is x and which is y. With
regression, however, we are more concerned with x as a predictor and y as an
outcome (sometimes called "criterion" variable). Thus, with
regression analysis, we need to determine which variable is the independent variable
and which variable is the dependent variable. The x variable is the
hypothesized cause, and y is the hypothesized consequence. Of course, just as
with correlation, a predictive relationship does not mean that x causes y. We
need a well controlled study to determine that; and if we are not able to rule
out alternative explanations for the relationship (e.g., a third variable), we
cannot assume a causal relationship. However, regression analysis does require
that one variable is specified as a predictor of another.
We can compute b from the
correlation coefficient with a little help from our friend, standard deviation:
and we could go back the other way:
R^{2}
There is another way in which regression and correlation are related. In the
previous lecture, I discussed the variation that is explained and unexplained
by the regression equation. We can think of this as similar to the overlapping
Venn diagrams for correlation. The amount of variation explained by the
regression line in regression analysis is equal to the amount of shared
variation between the X and Y variables in correlation.
So that means we can create a ratio
of the amount of variance explained (sum or squares regression, or SSR)
relative to the overall variation of the y variable (sum of squares total, or
SST) which will give us rsquare.
In regression analysis, r^{2}
is usually printed as R^{2} because later we will add more predictors
(independent variables) and a capital letter is used to indicate we are dealing
with something bigger. For now, r^{2 }is identical to R^{2}.
Ftest of R^{2}
Just as with correlation, we may
want to see if the value of r is significantly different from zero, so we could
use the same ttest we used before. Usually, in practice, researchers (and
computers) tend to use the Ftest for testing the significance of rsquare
rather than the ttest for correlations when conducting regression analysis.
Because we know that the ttest and the Ftest are related, we are dealing with
equivalent tests.
In order to create an Ftest, we
need to have a ratio of variances. Any idea what those variances might be?? You
guessed it (at least I hoped you guessed it). To create an Ftest we will use these
same variances, the variance explained (SSR), the variance unexplained, and the
sum of squares total (SST). We first need to obtain the mean square before the
F. To do that, we just need to know what the d.f. will be for dividing the Sum
of Squares values (and our instructor always gives us that).
Source of Variance 
Sum of Squares Abbreviation 
SS Formula 
Interpretation 
Mean Square 
Ftest 
Regression 
SSR 

Variation in the y variable explained by the regression equation. Deviation of the predicted scores from the mean. Improvement in predicting y from just using the mean of y. 


Residuals (or error) 
SSE 

Variation in the y variable not explained by the regression equation. Deviation of y scores from the predicted y scores. Error variation. 


Total 
SST 

Total variation in y. Deviation of y scores from the mean of y. 
not used 

Does this look familiar at all?? It
should. It looks exactly like the ANOVA table we created!! Oh no! Not again!!
Yes, again. Regression and ANOVA are the same.
Imagine a predictor X that is
dichotomous and a criterion variable (dependent) that is continuous. In this
case, ANOVA and regression analysis are equal. Technically speaking, the
equivalence depends on a couple of detailshow you decide to code the
dichotomous predictor and if the two groups are of equal size. But the bottom
line is that ANOVA is a special case of regression analysis, because under
certain conditions, they are equal. Regression analysis is more general,
however, because it is not limited to independent variables that are
dichotomous.
ttest of b
Sometimes we wish to test the
slope, b, in the regression equation for significance. Although b and r are not
exactly equal, their tests are equivalent. If we have test r, we don't really
need to test b. However, a little later on, we will want to test them
separately. You should arrive at the same ttest value for both.
When we test b for significance, we
are testing the null hypothesis that in the population, the slope is 0. The
population slope is represented by the Greek letter for b, b (or
"beta"). So, our statistical hypotheses are this.
Below are the equations for testing
b for significance. As usual, we need an estimate of the standard error of our
statistic b to get to the ttest.
In the above equations, s_{b}
is the standard error of b, representing the estimate of the variability of the
slope from sample to sample. MSE is the mean square error we obtained above
(Daniel uses a different notatio,,for the MSE sometimes). Notice that x's are
involved in the computation of the standard error of b. That is because we use
x's to get an estimate of b, the slope. d.f. for this test is 1 and the usual t
table is used.
Standardized b
The slope, b, is interpreted as the number of units (or points on the scale) of
increase in y as a result of 1 unit of increase in x. The slope then depends on
what scales we are using for x and y. If we measure fat intake with the
percentage of daily recommended intake rather than the grams of fat, we would
wind up with a different slope, because 1 percentage change is not the same as
one gram change. Similarly, if we used some different scale of measurement for
cholesterol the slope would be different. Notice that in these situations, the
actual variables being measured (fat intake and cholesterol) are identical
either way, it is just that they are represented by a different scale. Think
about measuring height in feet and inches. They are both measures of height and
they are equivalent measures, but they use different scales. Although using
different scales of measurement will affect the slope value, the relationship
between x and y will be the same. The correlation will be the same regardless
of which scaling you use. This creates a problem, because the slope value is
not that informative sometimes. We need a standardized measure of the slope,
just like the standardized measure of association, correlation.
I will call the standardized version
of b, "B". Some texts and the output in SPSS called the standardized
version "beta." Beta is also used as the population version of the
unstandardized slope. So, this all gets quite confusing. Here is the formula to
find the standardized version of the slope, B (and the one for getting back):
My guess is that this formula looks
awfully familiar to you. Yes, it's true: the standardized slope is equal to the
correlation coefficient. (note that this only applies when there is one
predictor, which I will return to later).
To b or not to b
SPSS will print two slopes, one standardized and one unstandardized. Often,
they are referred to as regression "coefficients" on the printout.
Researchers have to make a choice about which they should report or examine.
The significance test of the slope applies to both the standardized and the
unstandardized, and indicates that there is a relationship between them.
The standardized B can be
interpreted like a correlation coefficient. It's possible values range from
1.0 to +1.0, with zero indicating no relationship. The advantage to the
standardized B is that it is scale free. A slope of .6 between two variables is
the same as .6 between two completely different variables. The disadvantage is
that, under some circumstances, it does not provide very good information about
particular predicted values. For instance, in our fat intake example, we might
want to know exactly how much the cholesterol will be reduced if we reduce the
fat intake each day by 30 grams. For this type of question, the standardized B
is of no assistance. However, in many other situations, the measures used do
not have very meaningful scaling (e.g., a Likerttype questionnaire), and the
unstandardized coefficient does not provide very useful information. This is
why most social scientists tend to report the standardized coefficients.
Researchers in biology and physics will often report unstandardized
coefficients, because they are interested in the particular scale values.
Health researchers may use either, depending on the context and the specific
researcher's preferences.
A Comment on the
Intercept
The intercept in regression analysis is not used very often. It represents the
value of y when x equals zero, and it does not have a standardized version. It
is tested for significance on the printouts, but usually the test does not make
much sense. The significance test of the intercept coefficient is just the test
of whether it is significantly different from zero. Most of the time this is
not very meaningful and researchers usually do not even report it in articles.
There are some conditions under which the intercept might be useful, but this
usually applies to situations in which the values of x have a meaningful zero
point (e.g., x is a ratio scale). In these cases, it might be useful to know
what value of y is predicted when x is zero.