This page shows an example simple regression analysis with footnotes explaining the output. The analysis uses a data file about scores obtained by elementary schools, predicting api00 from enroll using the following SAS commands.
proc reg data="c:sasregelemapi2"; model api00 = enroll; run;
The output of this command is shown below, followed by explanations of the output.
Dependent Variable: api00 api 2000 Analysis of Variance Sum of Mean Sourcea DFb Squaresc Squared F Valuee Pr > Fe Model 1 817326 817326 44.83 <.0001 Error 398 7256346 18232 Corrected Total 399 8073672 Root MSEf 135.02601 R-Squarei 0.1012 Dependent Meang 647.62250 Adj R-Sqj 0.0990 Coeff Varh 20.84949 Parameter Estimates Parameter Standard Variablek Labell DFm Estimaten Erroro t Valuep Pr > |t|p Intercept Intercept 1 744.25141 15.93308 46.71 <.0001 enroll number of students 1 -0.19987 0.02985 -6.70 <.0001
a. This is the source of variance, Model, Residual, and Total. The Total variance is partitioned into the variance which can be explained by the indendent variables (Model) and the variance which is not explained by the independent variables. Note that the Sums of Squares for the Model and Residual add up to the Total Variance, reflecting the fact that the Total Variance is partitioned into Model and Residual variance.
b. These are the degrees of freedom associated with the sources of variance. The total variance has N-1 degrees of freedom. In this case, there were N=400 observations, so the DF for total is 399. The model degrees of freedom corresponds to the number of predictors minus 1 (K-1). You may think this would be 1-1 (since there was 1 independent variable in the model statement, enroll). But, the intercept is automatically included in the model (unless you explicitly omit the intercept). Including the intercept, there are 2 predictors, so the model has 2-1=1 degree of freedom. The Residual degrees of freedom is the DF total minus the DF model, 399 – 1 is 398.
c. These are the Sum of
Squares associated with the three sources of variance, Total, Model & Residual. These can be computed in many ways. Conceptually, these formulas
can be expressed as:
SSTotal. The total variability around the mean. S(Y – Ybar)2.
SSResidual. The sum of squared errors in prediction. S(Y – Ypredicted)2.
SSModel. The improvement in prediction by using the predicted value of Y over just using the mean of Y. Hence, this would be the squared differences between the predicted value of Y and the mean of Y, S(Ypredicted – Ybar)2. Another way to think of this is the SSModel is SSTotal – SSResidual. Note that the SSTotal = SSModel + SSResidual. Note that SSModel / SSTotal is equal to .10, the value of R-Square. This is because R-Square is the proportion of the variance explained by the independent variables, hence can be computed by SSModel / SSTotal.
d. These are the Mean Squares, the Sum of Squares divided by their respective DF. For the Model, 817326.293 / 1 is equal to 817326.293. For the Residual, 7256345.7 / 398 equals 18232.0244. These are computed so you can compute the F ratio, dividing the Mean Square Model by the Mean Square Residual to test the significance of the predictor(s) in the model.
e. The F Value is the Mean Square Model (817326.293) divided by the Mean Square Residual (18232.0244), yielding F=44.83. The p value associated with this F value is very small (0.0000). These values are used to answer the question "Do the independent variables reliably predict the dependent variable?". The p value is compared to your alpha level (typically 0.05) and, if smaller, you can conclude "Yes, the independent variables reliably predict the dependent variable". You could say that the variable enroll can be used to reliably predict api00 (the dependent variable). If the p value were greater than 0.05, you would say that the independent variable does not show a significant relationship with the dependent variable, or that the independent variable does not reliably predict the dependent variable.
h. Coeff Var. The coefficient of variation for the residuals is defined to be the root mean square error divided by the mean of the dependent variable. It is an indicator of how well the model fits the data. It is useful in comparing different models since it is unitless.
i. R-Square is the proportion of variance in the dependent variable (api00) which can be predicted from the independent variable (enroll). This value indicates that 10% of the variance in api00 can be predicted from the variable enroll.
R-square. As predictors are added to the model, each predictor will explain some of
the variance in the dependent variable simply due to chance. One could continue to
add predictors to the model which would continue to improve the ability of the predictors
to explain the dependent variable, although some of this increase in R-square would be
simply due to chance variation in that particular sample. The adjusted R-square
attempts to yield a more honest value to estimate the R-squared for the
population. The value of R-square was .10, while the value of Adjusted
R-square was .099. Adjusted R-squared is computed using the formula 1 – (
(1-R-sq)(N-1 / N – k – 1) ). From this formula, you can see that when the number of
observations is small and the number of predictors is large, there will be a much greater
difference between R-square and adjusted R-square (because the ratio of (N-1 / N – k – 1)
will be much less than 1. By contrast, when the number of observations is very large
compared to the number of predictors, the value of R-square and adjusted R-square will be
much closer because the ratio
of (N-1)/(N-k-1) will approach 1.
k. This column shows the predictor variables below it (enroll). The last variable (_cons) represents the constant, also referred to in textbooks as the Y intercept, the height of the regression line when it crosses the Y axis.
Ypredicted = b0 + b1*x1
The column of estimates (coefficients or parameter estimates, from here on labeled coefficients) provides the values for b0 and b1 for this equation. Expressed in terms of the variables used in this example, the regression equation is
api00Predicted = 744.25 – .20*enroll
Thise estimate tells you about the relationship
between the independent variable and the dependent variable. This estimate indicates
the amount of increase in api00 that would be predicted by a 1 unit increase in the
predictor. Note: If an independent variable is not significant, the
coefficient is not significantly different from 0, which should be taken into account
when interpreting the coefficient. (See the columns with the t value and p value
about testing whether the coefficients are significant).
enroll – The coefficient (parameter estimate) is -.20. So, for every unit increase in enroll, a -.20 unit decrease in api00 is predicted.
o. These are the standard errors associated with the coefficients. The standard error is used for testing whether the parameter is significantly different from 0 by dividing the parameter estimate by the standard error to obtain a t value (see the column with t values and p values). The standard errors can also be used to form a confidence interval for the parameter, as shown in the last 2 columns of this table.
p. These columns
provide the t value and 2 tailed p value used in testing the null hypothesis that the
coefficient/parameter is 0. If you use a 2 tailed test, then you would compare each
p value to your preselected value of alpha. Coefficients having p values less than
alpha are significant. For example, if you chose alpha to be 0.05, coefficients
having a p value of 0.05 or less would be statistically significant (i.e. you can reject
the null hypothesis and say that the coefficient is significantly different from 0).
If you use a 1 tailed test (i.e., you predict that the parameter will go in a
particular direction), then you can divide the p value by 2 before comparing it to your
preselected alpha level. With a 2 tailed test and alpha of 0.05, you can reject the
null hypothesis that the coefficient for enroll is equal to 0. The coefficient of
-.20 is significantly different from 0. Using a 2 tailed test and alpha of
0.01, the p value of 0.000 is smaller than 0.01 and the coefficient for enroll would still
be significant at the 0.01 level.
The constant (_cons) is significantly different from 0 at the 0.05 alpha level. However, having a significant intercept is seldom interesting.